LIGHT CALCULATIONS AND MEASUREMENTS

PHILIPS TECHNICAL LIBRARY

LIGHT CALCULATIONS AND

MEASUREMENTS

An introduction to the system of quantities and units in light-technology

and to photometry

H.A.E. KEITZ

SECOND REVISED EDITION

MACMILLAN

© N.V. Philips' Gloeilampenfabrieken, Eindhoven (The Netherlands), 1971

Softcover reprint of the hardcover 2nd edition 1971

All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission.

Published by MACMILLAN AND CO LTD

London and Basingstoke Associated companies in New York, Toronto, Melbourne, Dublin, Johannesburg and Madras

ISBN 978-1-349-00014-2 ISBN 978-1-349-00012-8 (eBook) DOI 10.1007/978-1-349-00012-8

PHILIPS

Trademarks of N.V. Philips' Gloeilampenfabrieken

No representation or warranty is given that the matter treated in this book is free from patent rights; nothing herein should be interpreted as granting, by implication or other­

wise, a licence under any patent rights

FOREWORD

Since the first edition of this book appeared 14 years ago, it has been widely used in lighting practice and as a textbook, and appears to have received an enthusiastic response from users. It was therefore possible to leave the form and content of the book basically unchanged in this second edition.

A new section on Dourgnon and Fleury's quasi-central projection has been added to the chapter on the representation of light distributions. A section on the determination of the reflection properties of road surfaces in the chapter on reflection, absorption and transmission has completely been revised. Finally, a new section on the special measures to be taken when measuring the pro­perties of gas-discharge lamps, with or without fittings, appears in the chapter on the measurement of luminous flux. The absorption of colour filters has also been dealt with in somewhat greater detail than in the first edition.

Measuring techniques have become much more mechanized and automated of recent years; electronics, and in particular digital techniques, are widely used in this connection. Such methods, which are also widely used for lighting measurements, belong to the field of electronics rather than that of lighting proper. They have therefore not been discussed in detail in this book, which is mainly concerned with basic photometric methods.

The author has made grateful use of the remarks and suggestions received from many readers since the appearance of the first edition. The whole text and the illustrations have also been carefully revised, and it is hoped that this second edition will also be of service to its readers in the practical and educa­tional fields.

Thanks are due to all who have helped in the production of this book, and in particular to Prof. Dr. H. W. Bodmann of the University of Karlsruhe (until recently attached to the Philips' Lighting Laboratory in Aachen) who read the entire revised second edition through, and to whom many improvements in the text are due.

H. A. E. Keitz

PART I

Chapter I

I-1. 2. 3. 4. 5. 6.

Chapter II

11-1. 2. 3.

CONTENTS

LIGHT CALCULATIONS

Introduction

The nature of light . . . . Light as a wave-phenomenon Polarisation. . . . . . . . . Photometry; the photometric system of Lambert The development of lighting engineering Subjects dealt with in this book .

Solid Angle

Solid angle, steradian . . . Some special solid angles . Significance of the conception ruinating engineering

"solid angle" in illu-

4. Small solid angles 5. Table of solid angles

Chapter III

III-I. 2. 3.

4. 5. 6.

Chapter IV

IV-I. 2. 3.

4.

Luminous Flux, Luminous Intensity, Quantity of Light

Luminous flux; luminous intensity . . . . . . . Units of luminous intensity and luminous flux Formulae giving the relationship between luminous intensity and luminous flux . . . . . . Horizontal, spherical luminous intensity. . . Luminous efficiency . . . . . . . . . . . . Quantity of light, lumen-second, lumen-hour

Light Distribution, Rousseau and Zonal Lumi-nous Flux Diagrams

Light distribution, the Rousseau diagram . . . Construction of the Rousseau diagram . . Derivation of the Rousseau diagram with the aid of the infinitesimal calculus . . . . . . . . Determination of the efficiency of a lighting fitting from the Rousseau diagram . . . . . . . . . .

Page

I 3 7 8 9

16

18 19

21 21 24

25 28

32 33 35 35

39 45

45

47

VIII CONTENTS

5. The zonal luminous flux diagram 48 6. The area enclosed within the luminous intensity

curve is not a measure of the luminous flux emitted 50 7. Average luminous intensity calculated from the

Rousseau diagram 51 8. Russel angles. 51 9. The long-base Rousseau diagram for narrow beams

of light 53 10. Detailed example: determination of a 1 000-lm curve

for a symmetrical lighting fitting . 54 11. Classification of lighting fittings according to their

light distibution 58

Chapter V !If ethods of Representing Light Dt:stribution

V-1. Luminous intensity table; polar and rectangular light distribution diagrams 60

2. The isocandela diagram . 60 3. Spherical co-ordinates . 62 4. Sinusoidal projection 62 5. Transformation of spherical co-ordinates 64 6. The azimuthal projection 67 7. Comparison of the sinusoidal and the azimuthal

projections . . . . . . 70 8. Quasi central projection . • 71 9. The application of preferred numbers in isocandela

diagrams . . . . 7 4 10. Examples of isocandela diagrams of a fitting ·. . .76

Chapter VI

VI-1. 2. 3. 4. 5.

Chapter VII

VII-1. 2. 3.

Illumination

Illumination; foot-candle; lux The inverse square law . . . Illumination with oblique incidence . Horizontal and vertical illumination Other units of illumination . , . .

Illumination Calculations and Diagrams

Introduction . . . . . . . . . . The lumen method; coefficient of The point-by-point method . . .

utilization

78 79 81 85 86

88 88 90

CONTENTS

4. Illumination diagrams . . . . . . . . 5. The rectangular illumination diagram. 6. The solid of illumination. 7. 8. 9.

10.

Chapter VIII

The plane isolux diagram . . . . . . The polar isolux diagram . . . . . . Determination of the luminous flux from the plane isolux diagram . . . . . . . . . . . . . . The isolux diagram in quasi central projection

Luminance and Luminous Emittance

IX

92 93 95 95 96

99 100

VIII-I. Luminance; the stilb, candelas per sq. in. . 104 2. Lambert's law . . . . . . . . . . . . . 105 3. The luminous flux of uniform diffusers . . 106 4. Luminous flux of a uniformly diffuse cylinder. 108 5. Luminous emittance . . . . . . . . . 111 6. Emittance and luminance of uniformly diffusing

surfaces . . . . . . . . . . . . . . . . . . . . 112 7. Units of luminance, based ~m the emittance of

uniform diffusers . . 112

Chapter IX Non-Faint Sources

IX-1. Luminous intensity of non-point sources . . . . . 115 2. The inverse square law in another form . . . . . 120 3. Universal formula for the illumination produced by

uniformly diffuse circular light sources . . . . . . 121 4. Alternative derivations of the formula E = :rcL sin2 8 124 5. Luminance of light beams. . . . . . . . . . . . 128 6. Comparison between the illumination values ob­

tained from equations E = Lw and E = :rcL sin2 8 128 7. Some special applications of the formula

E = :rcL sin2 8 = Ija2 •••••••••••••• 130 8. Illumination produced by a uniformly diffuse cir­

cular light source in planes parallel to the source . 131 9. Illumination produced by linear light sources . . . 136

10. Comparison of illumination values of linear light sources as obtained by exact calculation with those obtained by means of the inverse square law . . . 140

11. Other kinds of non-point source . . . . . . . . . 143 12. The significance of the foregoing considerations as

applied to practical forms of light source . . . . . 143

X CONTENTS

Chapter X Reflection, Absorption, Transmission

Reflection, absorption, transmission. . . . Regular reflection . . . . . . . : . . . . Reflection factors of non-conductive materials Reflection factor of conductive materials Diffuse reflection . . . .

X-1. 2. 3. 4. 5. 6. 7. 8. 9.

10.

146 147 149

0 ·152 152

Uniform diffuse reflection . . . . . . . . 15 7 Luminance factor . . . . . . . . . . . . 158 The luminance factor as a constant of the material 162 Gloss . . . . . . . . . . . 167 Reflection from road surfaces 168

II. Transmission . . 174 176 178 180 183 187 188

12. Density . . . . . 13. Diffusion factor . . 14. Mvltiple reflection 15. Absorption . . . . 16. Absorption of coloured filters 17. Colour of reflected light . .

Properties of Optical Systems

Introduction . . . . . . . . . . . . . 190

Chapter XI

XI-I. 2. Relationship between luminance and refractive index;

Abbe's law . . . . . . . . . . . . . . . . . . . 191 3. Luminance of images formed by lenses and mirrors,

and of the lenses and mirrors themselves . . . . . 196 4. Optical systems as light sources; the exit pupil of

optical systems . . . . . . . . . . . . . . . . . 200 5. Determination of the exit pupil of an optical system;

entrance pupil . . . . . . . . . . . . . . . . 203 6. Object at the focus of a lens; angle of divergence . 205 7. The exit pupil of lens systems; vignetting . . . . 207 8. The luminous flux of optical systems. Aperture of

lens and mirror systems . . . . . . · 212 9.

10. 11. 12. 13. 14.

Some remarks on mirror systems . Drum lenses . . . Cylindrical mirrors . . . . . . . Facetted mirrors . . . . . . . . Optical systems for the projection Diascopic projection. . . . . . .

of images

214 214 216 ·217 218 219

CONTENTS XI

15. Slide projection. . . . . 221 16. Standard-film projection . 225 17. Sub-standard film projection. 227 18. The luminous flux emitted by projection systems 228 19. Episcopic projection. . . . . . . . . . . . . . 230

Chapter XII

XII-I. 2. 3. 4. 5. 6.

PART II

7.

8. 9.

Chapter XIII

XIII-I.

2. 3. 4.

Chapter XIV

XIV-I. 2. 3.

The Photometric Measuring-units System

Introduction . . . . The Luminance criterion . . . Luminance of coloured light . The relative luminous efficiency The summation law ....

of radiation

Definitions of photometric quantities and units

233 234 236 239 242

based on v,\ .................. 243 Vision at high and low luminance levels; Purkinje effect . . . . . . . . . . . . . . . . . . . . 248 Equivalent luminance . . . . . . . . . . . . . . 251 Minimum perceptible luminance difference and sensitivity to luminance difference . . . . . . . . 254

MEASUREMENT OF LIGHT

General C on.sider at ions

Introduction. Principles of visual and physical photo-metry . . . . . . . . . . . . . . . . 263 Standard light sources . . . . . 264 Simultaneous and substitution methods . 268 Some practical hints . . . . . . . . . 269

Visual Photometry and Photometers

Principle of the visual photometer . . . . . . . 273 Forms of photometer field . . . . . . . . . 276 Methods used in photometers for obtaining the variation of luminance . . . . . . . . . . . 282

4. Photometer bench. Some examples of visual photo-meters ..................... 286

XII CONTENTS

5. Homochromatic and heterochromatic photometry . 291 6. Choice of observer . . . . . . . . . . . . . . . 292

Chapter XV Physical Photometers and Photometry

XV-I. 2. 3. 4. 5. 6.

Principles of physical photometers and photometry 295 Photo-emissive cells . . . . . . 296 Photo-voltaic cells . . . . . . 302 Bolometers and thermo-couples 313 Photographic photometry . . . 314 Physical photometers for heterochromatic photometry 315

Chapter XVI Measttrement of Luminous Intensity

XVI-I. Principle of luminous intensity measurement . 321 2. The photometer bench . . . . . . . . . . . 322 3. Apparatus for measurement of light distribution 324 4. Measurements on projectors . . . 329 5. Photometer with Maxwellian view . . . . . . 332

Chapter XVII Measurement of Luminous Flux, Quantity of Light and Luminous Emittance

XVII-I. The Ulbricht sphere photometer . . . . . . . . 335 2. Illumination of the sphere window in the "ideal"

integrating photometer . . . . . . . . . . . . . 336 3. Illumination of the window of a non-ideal integrating

photometer . . . . . . . . . . . . . . . . . . . 338 4. Measures to be taken with non-ideal integrating

photometers to approximate to the ideal sphere 340 5. The measurement of the luminous flux of fluorescent

lamps and the appropriate fittings . . . . . . . . 351 6. Determination of luminous flu~ from the light

distribution or from an isolux diagram . . 355 7. Measurement of quantity of light . . 355 8. Measurement of luminous emittance . 357

Chapter XVIII Jfeasttrement of Illumination

XVIII-I. 2. 3.

Introduction . . . . . . . . . . . . . . . . . 359 Measurement by means of laboratory photometers 360 Visual illumination photometers . . . . . . . . . 36 0

CONTENTS

4. Physical illumination photometers 5. Calibration of illumination photometers .

XIII

362 366

Chapter XIX AI easurement of Luminance

XIX-I. 2.

Direct visual measurement of luminance Visual and physical luminance measurements tained from measurement of illumination . . .

.. 367 ob-

372 3. Measurement of luminance distribution . . . . 374 4. Determination of the size of the exit pupil of lenses 380

Chapter XX ]I,J easurement of Reflection, Transmission and Absorption

XX-I. Measurement of reflection factor 381 2. Measurement of transmission factor. 388 3. Measurement of absorption . . . . 390 4. Measurement of luminance factors and glos~ 391

APPENDIX International co-operation in Illuminating Engineering 403

Table I Table II

- Table of solid angles . . . . . . . . . 40 7 - Values of cos31X cos3 p for a number of values of the

angles IX and p . . . . . • . • . . . . . . . . 408 Table III - Values of tanp /cos IX for a number of values of the

angles IX and P . . . . . . . . . . . . . . . . . 410 Table IV - Units of illumination and their mutual conversion

factors • . . . . . . . . . . . . . . . . . . . 4 12 Table V - The units of luminance and their mutual conversion

factors . • . . . . • . . . . . . • . . . . . . 4113 Table VI - International relative luminous efficiency of radiation

for photopic vision . • . . . . . . . • . . . . . 414 Table VII - International relative luminous efficiency of radiation

for scotopic vision . . . . . . . . . . • . . . . 415

Light distribution and zonal luminous flux diagrams for a number oflighting fittings· • 417

Index 426

ex ~ y 8 £

7J >. 1-' v 7T

p

a

T

'I' 'P w J e E <P 1

b c d I g

h i l n r r, R t u v E H I K>. L L M Q s T V>.

V>,'

TABLE OF MOST IMPORTANT SYMBOLS USED IN THE TEXT

alpha beta gamma delta epsilon eta lambda mu nu pi rho sigma

tau phi psi omega delta theta s1gma phi

Plane angle, absorption factor Plane angle, luminance factor Plane angle Plane angle Emission factor Efficiency of fittings, coefficient of ultilization Wavelength Micro Frequency 3.1415 .... Reflection factor Diffusion factor, transmissive exponent of diffusing

media Transmission factor Plane angle Plane angle Solid angle Small part. of a quantity Half-aperture angle of light beams Sum of a number of quantities Luminous flux Transmissive exponent of transparent media Geographical latitude Velocity of light Distance Focal distance Optical limit distance of photometry, gloss number

of Harrison Height Angle of incidence Geometrical longitude, length Refractive index Angle of refraction Radius Time Object distance Image distance Illumination, energy (power) Luminous emittance Luminous intensity Luminous efficiency of radiation at wavelength >. Luminance Equivalent luminance Mechanical equivalent of light, linear magnification Quantity of light Area, density Periodic time International relative luminous efficiency of radiation

for photopic vision at wavelength >. International relative luminous efficiency of radiation

for scotopic vision at wavelength >.

LIGHT CALCULATIONS AND

MEASUREMENTS

CHAPTER I

INTRODUCTION

I-1. The nature of light The natural phenomenon which we know as light governs to a large extent all our activities; it is therefore not surprising that ever since the earliest times man has endeavoured to produce artificial light. Sources of such light have in the course of time been evolved, from the primitive wood-fire by way of oil lamps, candles, paraffin and gas lamps in numerous varieties, to the present-day electrical sources of light, the electric filament and gas discharge lamps. Naturally enough, the efforts of scientists have from a very early date been directed towards finding an answer to the question: what is light? The ancient Greeks held several theories which for a long time amounted to the assumption that the eyes emitted radiations which located the objects around us. In this, they overlooked the fact that the sun is the pri­mary source of light. When, amongst others, A r i s t o t l e demonstrated the fallacy of this idea the learned men of the day came to the conclusion that light consisted of a current of very small particles emitted by incandescent substances; the particles, on entering the eyes, were sup­posed to be responsible for the sensation of light. Of those theories which were developed in more recent times those of Christiaan Huygens (1629-1695) and Isaac Newton (1642-1727) are the most well-known. Their views on the nature of light are to a large extent compatible with present-day conceptions. In 1678 H u y g e n s put forward the opinion (published in 1690) that light is a wave phenomenon (the wave theory); according to this theory light consists of vibrations in a hypothetical medium, the light-ether or, in short, the ether. Such vibrations would be propagated in straight lines. Newton had quite a different conception of the nature of light; he held the theory that light sources emit particles which, on entering the eye, produce the sensation of light, and this was actually an ex­tension of the theory propounded by the ancient Greeks. Newton's theory is known as the corpuscular theory.

2 INTRODUCTION rr Both H u y g e n s and N e w t o n had their supporters for a great many years. One of the followers of the wave theory was F r e s n e 1 (1788-1827) who made certain elaborations on it. Many phenomena could be explained in terms of both of these theories, but in the long run it proved to be impossible to explain certain newly discovered characteristics of light with the aid of the corpuscular theory, whereas these could be fitted in with H u y gens' wave theory which thus gradually gained supremacy. None the less, it was not until the middle of the 19th century that the wave theory came to be universally accepted. At the same time there were some very valid objections to the properties of the ether as postulated by the wave theory and on which that theory was based. These objections were overcome by the electromagnetic theory of light formulated by Max we 11 (1831-1879): according to this theory light is compounded of electrical and magnetic vibrations of the same kind as electromagnetic waves such as are produced . by an oscillating electric current, e.g. in a spark discharge. In the course of time it has been found that a large number of radiations are of the same nature as light, amongst these being x-rays and radio waves. All these consist of electromagnetic waves, amongst which light waves differ from the others in that, when they enter the eye, they constitute a stimulus which conveys to the brain the sensation of "light". This difference, from the physical point of view, will be discussed in the next section. In its turn the electromagnetic theory, too, failed to supply an ex­planation of certain light characteristics later discovered; for example it did not explain the photo-electric effect, a subject which we shall mention in connection with photo-electric cells. A new principle was next introduced to the science of physics, namely the quantum theory, as developed by P 1 a n c k. This theory states that all radiations consist in the emission of energy, not progressively as in the case of electromagnetic waves, but in certain very small discrete quantities at a time. These quantities are called light quanta, or photons. Although the photons are emitted at irregular intervals, there is still this much regularity that a constant-burning light source emits the same average number of photons in the same interval of time. So much evidence has. been brought forward in support of the photon that it is now a generally accepted convention. It may be noted here that the quantum theory really re-introduces Newton's corpuscular theory, albeit in a modified form.

I-2] LIGHT AS A WAVE-PHENOMENON 3

What then, of the wave-like character which light is assumed to possess and which, with one or two exceptions, has been the means of explaining all the properties of light? The answer to this was given in 1924 by d e B r o g 1 i e and again in 1925 by S c h r 6 d i n g e r in his wave mechanics; agreement was thus reached on the dual character of radi­ation. The modern conception of light may be expressed as follows. As far as materials are concerned (photo-electric cell, the eye, photo­graphic plate) light - and radiation in general - behaves as though it were composed of quanta, but results, i.e. the average number of quanta which reach a surface per unit of time can in every case be computed accurately by means of the wave theory. We may therefore regard light (and radiation in general) as a wave­motion propagated in straight lines and conveying energy with it. Light and all radiations are therefore energy and, when absorbed, this energy can be converted to other forms of energy such as heat or electrical energy. We shall refer to this again later. For so far as may be necessary in the discussions in this book to refer to the physical character of light, we shall regard light as a wave­phenomenon.

I-2. Light as a wave-phenomenon Before proceeding to a discussion of light as a wave-phenomenon, let us consider wave-motion in general. Fig. I shows a line AB. Assume that every point on this line from A to B is set in motion successively, i.e. that they move rapidly up and down. Since each point commences to vibrate later than the preceding one, a wave-like motion is set up, such as can be observed when a stone is thrown into the water. The wave-motion is thus propagated along the line AB and this is called the direction of propagation of the wave. If we now draw in the deflections exhibited by the points on AB at a given moment and join these points by a line, we obtain the wave-shaped curve shown in Fig. I. The direction of vibration of this curve is perpendicular to the direction of propagation, and waves of this kind are called transverse. When the direction of vibration coincides with the direction of propagation, the wave is said to be longitudinal. If the curve, of the kind depicted in Fig. I, is sinusoidal, we speak of a harmonic vibration or wave. It is now necessary to define some dimensions and conventions relating to wave motion.

4 INTRODUCTION ll

The greatest deflection from the position of rest is known as the amplitude of the vibration (a in Fig. I). Of those points which, from the point of view of distance and mobile conditions, are situated similarly with respect to AB, it is said that they are in phase (e.g. points P and Q, Fig. 1).

1- --A

v ~ /' ~ Av \ Qv \ 1/ \ v \

I ~ v ~ I 1\ v 1\~ I I ~ .,.1 ,..1

Fig. I. Transverse harmonic vibration. AB = direction of propagation. a = amplitude. .\ = wavelength. The points

P & Q, R & S are in phase.

The distance between two successive points having the same phase is termed the wavelength (A.); this is accordingly also the distance between two successive peaks in the wave (e.g. R and S). The time during which a point describes a complete oscillation, that is, the time taken by it to travel from the position of rest first in the one direction and then back in the other direction, through the point of rest, to return finally to the starting condition, i:s the periodic time (T). From Fig. I it will be seen that one wavelength is just completed during the periodic time, for, when the wave has travelled the distance A from R to S, the point R, in completing one vibration, has again acquired the same phase. Hence the speed of propagation, o:r the velocity (c) of the wave-motion is found to be

A c = T' (I-1)

By the frequency (v) of the vibration is meant the number of vibrations per second. If the periodic time of one vibration be denoted by T, the frequency is

, = -y· (1-2)

1-2] LIGHT AS A WAVE-PHENOMENON

From equations (I -1) and (I -2) it follows that

c =A.. v.

5

(1-3)

EVf~ry wave-motion can thus be characterised by the dimensions wave­length, velocity and frequency, and the relationship between these quan­tities is expressed by equation (1-3). In electromagnetic waves (and therefore also in light) we are concerned with an electrical field, the strength of which varies with the time, the value differing from one point to another. The electrical field strength is perpendicular to the direction of propagation, and electromagnetic waves are therefore transverse waves. The instantaneous values of the field strength along a ray of light may be represented by lines (vectors) in the manner shown in Fig. I. The electrical field strength is associated with a magnetic field which varies simultaneously with the electrical field; the magnetic field strength is also perpendicular to the direction of propagation and is at right angles to the electrical field. If we represent both of the field strengths by vectors, an electromagnetic wave-motion may be illustrated in the manner shown in Fig. 3; here, the wavelength is equal to the distance PQ. The waves are sinusoidal waves and it may accordingly be said that the "vibration" of the electrical and magnetic field strengths is harmonic. Of the characteristic dimensions v, A and c, the velocity for all electro­magnetic waves in a vacuum is the same, viz. 2.99792 x 1010 cmfsec, or, rounded off, 2.998 X 1010 cmfsec, which is almost 300.000 kmfsec. In all other media, (air, glass etc) the velocity is lower, but, whereas the velocity in vacuum is the same at all wavelengths, it is different at every wavelength in other media. In air the velocity is only slightly lower than in vacuum (only 3 per 10.000 for light in air at a pressure of 760 mm, and oo C), and the differences for the various wavelengths are only small. When the value of the velocity in equation (I-3) is varied, the question arises which of the two quantities in the second term (A and v) will vary with it. It is found that the frequency of a given radiation is a constant, i.e. it is independent of the medium through which the radiation is passing. From (I-3) it follows that the wavelength of a radiation varies proportionately with the velocity of propagation. It is customary to designate the different kinds of radiation by their wavelength, although, in view of the fact that the wavelength is de­pendent on the medium, it would be more logical to indicate the fre-

6 INTRODUCTION [I

quency. We shall follow the established practice and name the wave­length, the values being understood to be in respect of vacuum. These values differ only slightly from those in air. Radio waves are usually specified in metres. However, for the range of wavelengths of interest in lighting engineering, this unit is much too large to yield convenient values. Centimetres and millimetres are also too large for the purpose and various smaller units are employed, viz:

the micron (1 p.m = 10-3 mm) the millimicron or nanometre (1 mp. or lnm = 10 - 6 mm) the Angstrom unit (LA = t0-7 mrr1 = 0.1 m,u)

The symbols used throughout this book for the miGron and the millimicron will be p.m and mp. respectively.

It is generally assumed that the human eye is capable of perceiving radiations of wavelengths between 0.40 and 0.70p.m 400-700 m,u or 4000-7000 A). The range of_ wavelengths to which the eye is sensitive actually extends from 313 m,u to 1050 m,u, but it is only the range from 400 to 700 m,u that is of general interest in lighting technology. The difference between wavelengths in the visible range is seen by the human eye as colour. Radiation at 0.4 .urn is perceived as violet, while that in the range above 0.6 .urn is seen as red, with the colours visible in the rainbow lying between these two. The sequence of colours in the visible spectrum is violet, blue, green, yellow and red, with the inter­mediate colours like turquoise, yellow-green and orange between them.

Fig. 2 gives the electromagnetic spectrum and shows the various wave­length ranges with the names by which they are known.

cosm;c rod1otion

wsrble lrghl

ll-rays X-rays UV-rad Infrared rod. Radrowaves

;; EHFSHFUHFVHF SW MW LW 1 T ! : 1- ~ ~ t 1 I T I • I

;o ··· 10 '. 10 ' 10·' 10 ' TO 1 10· 5 TO· IL ·l 10··' 10·' I TO' 10 -' 10' 10" 10·· 10' 10 7 mm Imp 10 TOO lj.J 10 100 lmm 10 100 lm 10 100 TKm /0

Fig. 2. The electromagnetic spectrum.

I-3] POLARISATION 7

The wavelength zones which border on the range of light w;~ves are the ultra-violet (extending from 0.2.um to 0.4pm approx.) and the infra­red or heat rays (from 0 7 f..lm to 100 .um approx.). The characteristics of ultra-violet radiations are of interest mainly in chemistry and biology (tanning and reddening· of the skin, therapeutic action e.g. in rickets, germicidal properties). Infra-red radiations are known for the pronounced sensation of heat which they produce; for this reason they are often referred to as heat rays, although this is not strictly correct, seeing that visible. radiations (and ultra-violet) also produce a sensation of heat. In such cases the energy of the light wave is converted to heat energy. Now, the light emitted by temperature or incandescent radiators such as the sun or the electric filament lamp is always accompanied by a large amount of infra-red radiations; the light from such sources gives a pronounced sensation of heat which is sometimes pleasant, but some­times a distinct source of discomfort, for which reason cold light is often asked for. From the foregoing it will be seen tha:t there is really no such thing as "cold" light in the absolute sense·; when we speak of such light we therefore merely mean light that is accompanied by little or no invisible radiation such as infta-red *). Radiations consisting of light of only one single wavelength are known as monochromatic, and monochromatic kinds of light of different wavelengths are distinguished from each other by the colour sensation which they produce (spectral colottrs). Combinations of different wave­lengths, with which we are almost always concerned in practical work, also produce colour sensations; these are to be regarded as mixed colours of spectral colours. Certain combinations will give an impression of white light. Colour perception, however, falls outside the scope of this book and will not form part of our discussion 1)**).

I-3. Polarisation It is stated in the previous section that light can be regarded as an electromagnetic, transverse, wave-motion. From the aspect of the plane~ in which the el~ctrical and magnetic field strengths "vibrate", various possibilities exist. For example the "vibration" of the electrical field

*) The meaning of "cold" light intended above should not be confused with another meaning. often attributed to this term, viz. light of a certain colour such as blue which produces an unpleasant "cold" sensation. **) The numerals refer to the bibliography at the end of each chapter.

8 INTRODUCTION

E

Fig. 3. Electromagnetic wave. The electrical field strength E "vibrates" in the plane W, the magnetic field strength H in

the plane V, perpendicular to W.

[I

strength may lie in one plane only (with the magnetic field strength in a plane perpendicular to it), this being the in~tance depicted in Fig. '3. That plane which is at right angles to the electrical field (V, Fig. 3) is then known as the plane of polarisation and we say that the light polarised is in this plane. Possibly there may be no definite direction of polarisation, in which case the electrical "vibration" takes place in all directions without any preference for one or the oth.;:r; we then have natural, or non-polarised, light. The light from sources such as the sun, electric incandescent lamps and gas discharge lamps may be regarded as non-polarised light.

1-4. Photometry; the photometric system of Lambert Until the 18th century the study of light was limited almost entirely to ·geometrical optics which deals with the behaviour of rays in lenses, prisms etc. Little or nothing had thus far been done in the quantitative measurement of radiation. The first to announce a more or less successful C~;ttempt to measure light (photometry) was the Frenchman B o u g u e r ( 1729), who dealt with only some of the quantities and conceptions which we now employ in photometry and illuminating engineering. There is very little evidence of any mathematical treatment of the problems, or satisfactory defi­nitions of the conceptions in Bouguer's work.

1-5] THE DEVELOPMENT OF LIGHTING ENGINEERING 9

In 1760 L a m b e r t *) published his "Photometria sive de mensura et gradibus luminus, colorum et umbrae", i.e. "Photometry, or the measure­ment and classification of light, colour and shadow" 2).

In this, L a m b e r t developed a system of conceptions (photometric system), the principle of which is still in use unchanged today. Mathematically he established a large number of relationships between the different concepts and, although many of these were found. to be of little practical interest, it is surprising to note when reading his book that so many of his formulae have been adopted in publications on light and photometry of the last decade or two. Here one senses the genius of this founder of photometry who built up his system unaided. The practical methods of photometry described in Lambert's work were primitive in the extreme and there is no record of any photometer in the form in which it is known today. It was not until the second half of the 19th century, when lighting technology came to be developed, that justice was done to the work of Lambert. We shall have something to say about this development in the next section.

I-5. The develo~ment of lighting engineenng The term lighting engineering is understood to be the technique which embraces everything relating to the production and application of light •*). For that branch of lighting technique which deals with the actual production of light we have no specific term; we might refer to it as the technique of light production. The branch of the technique that relates to the applications of light, i.e. the illumination, falls under the heading of illuminating engineering. The technique of light production covers the development and manufac­ture of the primary sources of Jight, that is, the equipment which converts the energy supplied to it into visible radiation. Illuminating engineering includes not only the design and execution

*) J. H. L am b e r t, born at Mulhausen in Alsace in 1728; died 1777 in Berlin. Ht: ~as self-educated, a .fact that pr?bably accounts for the originality of his wntings. J:Ie was versed m many subJects and also wrote works on philosophy, mathematics, heat, sound and astronomy. • *)The term "technique" is taken to cover the entire equipment and methods employed in the execution of one of the arts.

10 INTRODUCTION [I

of lighting installations, but also the development of lighting fittings, which refers to the equipment in which the primary light source is contained and which serves to throw the light from the source in those directions where it is required by the lighting engineer. Or again, the lighting fitting may be such as will mask the light partially or wholly in directions where it would otherwise be found a hindrance (glare), alternatively the fitting may fulfil a decorative function, or it may merely protect the lamp. When we speak of the development of lighting technique we should first make it clear that, until the latter half of the 19th century, de­velopments related almost entirely to techniques in the production of light. By the second half of the 18th century there had been little question of any development, for the firebrand or torch, the candle and oil lamp were until then the ordinary sources of light which, in the technical sense, had not risen above the level of the primitive wood fire. Such light sources were too weak for the execution of more than the simplest domestic activities and therefore served mainly to maintain domestic and social life after sunset. There was little demand to extend the working day with the aid of increased or improved artificial lighting. The second half of the 18th century marked, particularly in England, the commencement of the industrial era which came about mainly as a result of the increased demand for merchandise, especially in the European colonies and America. In order to meet this demand production had to be increased constantly and the machine made its entry into the factories (spinning machines and looms, driven by a steam engine). The daily hours of work, too, had to be increased to keep pace with the demand, and the need for better sources of light arose. It is therefore only natural that at this time numerous improvements to existing light sources were introduced, amongst which we may mention the cylindrical lamp chimney (Quinque t, 1765) and the centre-draught oil burner which took the place of the solid wick (A r g and, 1786). At the beginning of the 19th century the technique of light production was much improved by the introduction of the coal gas jet (batwing burners). By the middle of the 19th century the replacement by paraffin of the oil used in oil lamps marked another important step forward. The greatest impetus to the production of artificial light was given by E d i s o n in 1879 when he succeeded in making a serviceable electric filament lamp suitable for manufacturing as an industrial product. Up to that time only the carbon arc was known as an electrical sourcP

1-5] THE DEVELOPMENT OF LIGHTING ENGINEERING 11

of light, but this was a powerful source of light and therefore unsuitable for the small rooms of private residences or offices. It was possible to manufacture Edison's lamp in units of relatively small power and these accordingly promoted the use of electric light in almost all lighting installations. The whole impact of the development of the electric lamp would have teen lost, however, if it were not for the fact that at the same period a development in electrical technology, namely the invention of the dynamo and suitable means of distribution of electric current made it possible to generate electrical energy on a large scale and to supply it ~~~~~~. . Edison's electric lamp contained a carbon filament which was heated to incandescence, thus making it a source of light, by passing electric current through it. To prevent the carbon filament from being burnt it was mounted in a glass bulb from which the air was exhausted. The carbon filament evaporates rather quickly, which is why it cannot be taken to very high temperatures. This means that the useful output of these lamps was relatively low. It was possible to iinprove this somewhat by metallising the carbon filament, but no basic increase in the useful output was achieved thereby. Such an increase was successfully brought about only when the carbon filament was replaced by metal filaments. Higher temperatures could be .used first with osmium ( 1902) and later with tantalum ( 1905), 'which therefore gave more light for less power. Even better results were obtained with tungsten (tungsten filament lamps, 1906). To this day tungsten is the material from which electric lamp filaments are made. Originally these lamps had a straight filament and evacuated bulb (vacuum lamps). Later still, in 1913, Langmuir introduced lamps filled with an inert gas. The gas filling reduces the rate of evaporation of the tungsten, b"yt also lowers the temperature of the filament because it dissipates the heat more quickly. Langmuir remedied this drawback by coiling the filament. Since the heat dissipation is proportional to the length of the incandescent body and depends only slightly on its diameter, the losses due to the gas were more than compensated by the coiling of the filament. Coiled coil lamps were first made in 1934. The purpose of this arrange­ment is to shorten the incandescent body even further and to reduce the losses through the gas even more. Development of incandescent lamps has been continued over the past few years and is still in progress. The life of an incandescent lamp, and its

12 INTRODUCTION [I

efficiency, are determined to a considerable degree by the evaporation of tungsten from the filaments. The evaporated tungsten is deposited on the envelope of the lamp in the form of a grey or black coating, gradually reducing the light output during the life of the lamp. If this evaporation process could be reduced, therefore, the lamp would last much longer and produce a greater useful output. It has been found that the effects of evaporation can be gr~atly reduced by the addition of a small quantity of iodine to the gas filling. As the tungsten evaporates, it combines with the iodine to form tungsten iodide at temperatures up to some 800 °C. At temperatures above 2000 °C, however, the tungsten iodide decomposes again to form iodine and metallic tungsten. The latter process, therefore, can take place only at the filament itself or in its immediate vicinity. The temperature of the glass bulb of the ordinary incandescent lamp remains below that which favours the combination of tungsten and iodine to tungsten iodide. If, therefore, the temperature of the bulb could not be raised, there would be no point in introducing iodine into it. It is, however, possible in quite a simple manner to attain the desired increase in the temperature of the bulb- it only has to be made smaller. Nevertheless, the temperatures of up to 800 oc required at the envelope do generally mean that the latter must no longer be made of glass but of quartz. An increase in the efficiency of about 25% has been obtained for the same life. Recently, other halogens* and mixtures of different halogens have been used in these lamps, too, which is why they are referred to as halogen lamps. This development has, in the initial stages, been directed towards special-purpose lamps. While on the subject of electrical light sources we should also mention gas discharge lamps, the development of which has taken place during the last 30 years 3). In these lamps, e.g. mercury vapour and sodium lamps, a pilot discharge in an auxiliary gas renders the vapour conductive by splitting it into ions and electrons. When current flows, processes occurring between the electrons and atoms of the vapour result in the emission of light. The colour of the light depends upon the type of vapour or gas and also upon the gas pressure. In these lamps the light is not produced as a result of incandescence due to high temperatures as in filament lamps; gas discharge lamps are therefore not temperature radiators.

*) The halogens are the clements chlonne, bromine, iodine and fluorine.

1-5] THE DEVELOPMENT OF LIGHTING ENGINEERING 13

Whereas temperature radiators emit all the visible wavelengths (as well as infra-red and some ultra-violet), thus producing a contznuous spectrum, gas discharge lamps radiate only those wavelengths which are character­istic of the vapour or gas in the lamp (line spectrum). The latest development of the gas discharge lamp embodies an entirely new principle in the form of the tubular fluorescent lamp 4). Here, the inside of the tube is coated with a powder which has the property of converting radiation of the shorter wavelengths into radiation of longer wavelenghts. By carefully blending these fluorescent substances it is possible to modify the colour of the light emitted and in this way lamps can be manufactured to give light of the same colour as that of a temperature radiator, or the colour of daylight. In these lamps, too, the spectrum is continuous. The main advantage of gas discharge la~ps as compared with filament lamps is their higher efficiency; on the other hand the fact that a current­limiting device is essential is a drawback. Improvements effected in the efficiency of light sources during the course of time are illustrated in graphical form in Fig. 4. In this diagraJ;n the efficiency of the light source is plotted vertically in lmfW, against time on the horizontal axis. The unit lm/W is explained in section 111-5; at this point it is sufficient to say that it is a measure of the amount of light emitted per second by the source, per unit of power in watts. In the case of combustion or flame sources of light such as the candle or paraffin lamp, the energy is supplied by the heat of combustion. It will be seen that the efficiency of the candle and paraffin lamp, which is 0.1 to 0.3 lmfW, is improved upon by the carbon filament lamp to the extent of a factor of 10. T4e introduction of the tungsten lamp yielded further improvement by a factor of 5 and the gas discharge lamps increase efficiency by another 2 to 10 times. A new method of producing light, known as electroluminescence, has recently been developed. Basically, an electroluminescence plate is a capacitor with one of its electrodes made of a translucent material and with a certain luminescent material as the dielectric. If an alternating electric field is applied to this device, light will be produced in the luminescent layer. Because the efficiency of these "elu plates" is very low, they are not suitable for lighting purposes. For the moment, their use is restricted to warning notices, house numbers, radio receiver scales, instrument dials, etc. No discussion of techniques of light production would be complete

14 INTRODUCTION [I

lm(w

~ $~ ~ -fO ~ m

1--l' ~·

,fi9 9( ~ t1W _.. ---- -- ~'""-"" ·-

6 J e~ v nl 4 --~

~c / ~

~ben fi discha)ge lamps :amen/ /on 1ps rungs/en :unps

I

ao ,..,

a2 I~ f--

at lb 50 11310 IS"' ISO &i/O /9 JO 19 !0 1!>50 '.160 '' '70

Physical and technological problems year Oplica-physiological problems

Psycha/ogicol problems

Fig. 4. Progressive increase in the efficiency of artificial light sources a) Wax candle b) Paraffin lamp c) Edison carbon filament lamp d) Carbon filament lamp e) Lamp with metallised carbon filament f) Tungsten filament lamp (straight filament, vacuum) g-) (single spiral, gas-filled) h) ( .. .. vacuum) i) .. .. .. (coiled coil,. gas-filled) j) Mercury vapour lamp k) Sodium lamp l) Fluorescent lamp m) Blended light lamp (filament and mercury vapour light) n) Electroluminescence

1-5] THE DEVELOPMENT OF LIGHTING ENGINEERING 15

without mention of the. important improvement in gas lighting in­troduced by Au e r von We 1 s bach in 1886, in the shape of the incandescent gas light. V o n W e 1 s b a c h made use of the peculiar radiating properties of the oxides of certain metals in the rare-earth group, chiefly cerium and thorium. When heated to incandescence these oxides radiate strongly those wavelengths which lie within the visible spectrum, with relatively little infra-red. The "incandescent mantle" may consequently be heated to a high temperature to give radiations in the zone of wavelengths to which the eye is particularly sensitive. The radiations of such oxides are accordingly said to be selective. This improvement in gas lighting arrived at a time when the invention of the electric lamp made it look as though electric lighting would take the world by storm and rapidly replace gas. Owing to v o n W e 1 s­b a c h's invention, however, the need for electric light was not by any means so strongly felt; gas lighting therefore held its ground for many years and is still used. Acetylene lighting may be mentioned as being the third great step forwarq in techniques of light production. For isolated buildings it was par­ticularly useful, but nowadays the carbide-lamp is employed almost exclusively for signal and portable lamps. Development of the E d i s o n lamp and systems for the distribution of electricity since 1880 made it comparatively easy to meet the demand for improved lighting, especially in industry, in not too costly a manner. With the increases in illumination levels - which should not be over­estimated, however - came the realisation that the new light sources could not be used in the same way as the old flame sources, that is, free to radiate light in all directions. Glare was a new factor to be reckoned with and, this being harmful to the eyes, lamps were fitted with shades; these constituted virtually the first lighting fittings as such and in this way illuminating engineering came into being. The next step was to make use of reflecting materials to direct the light towards those points where it was required, i.e. towards the working plane. At that time the associated problems were purely physical and technolo­gical, development being aimed mainly at higher and higher illumination levels. Later (ca. 1905) it was seen that this in itself was not enough, but that effective lighting was more an optico-physiological problem. Subsequently (ca. 1930) it was realised that vision also involves a psychological factor and that illuminating engineering would have to take this into account.

16 INTRODUCTION [I

So far we have only touched upon artificial light and its sources in lighting engineering, but it must not be overlooked that this subject also includes natural daylighting. Particularly within the last few decades numerous methods have been evolved for computing indoor daylighting from the geometrical proportioning of windows and other boundaries of the natural light from the sky. A discussion of this would take us beyond the scope of this work, however 5).

1-6. Subjects dealt with in this book The first part of the book is devoted to a review of the system of di­mensions and nomenclature which we have so far referred to as the photometric system. A number of the relationships existing between these quantities will be developed, these being essential for a thorough understanding of the photometric system, as well as for the various lighting computations which the lighting engineer will usually encounter. The reader may possibly say: all this is sufficiently clear as applied to white light (e.g. from an incandescent lamp), but what happens in the case of coloured light such as that produced by the sodium lamp? To this we reply that in many books the answer to this question is given at the start, but that in the author's opinion a discussion of the subject is more simply followed if the reader is conversant with the photometric system. Preference is therefore given to a postponement of an explanation of the significance of lighting nomenclature as ap­plicable to coloured light until the last chapter of the first part of the book. Until then the reader is asked to assume that the discussions refer to white light, bearing in mind that in chapter XII we shall ex­plain how the various considerations and definitions may be extended to cover coloured light. The second part of the book deals with methods of photometry and makes frequent use of the theory contained in the first part. The author considered that it would be useful to precede the section on lighting computations by a chapter covering the mathematical conception of solid angle which plays such an important part in illu­minating engineering. In this it is assumed that the significance of the term is not always sufficiently understood amongst those who may wish to know something more about lighting technology.

I-6J SUBJECTS DEALT WITH IN THIS BOOK 17

REFERENCES 1) See for instance: P. ]. Bouma: "Physical Aspects of Colour". Philips Tech­

nical Library, Eindhoven 1948 J. Bergmans, "Seeing Colours", Philips Technical Library, Eindhoven, 1960.

2 ) In 1892 an abridged German translation of Lambert's "Photometria" was published in the series "0 s twa 1 d's Klassiker der exakten Wissen­schaften" (nos 31, 32 and 33)

3) J. Funke and P. ]. 0 ran j e: "Gas Discharge Lamps". Philips Technical Library, Eindhoven 1951

4) W. W. Elenbaas, "Fluorescem Lamps and Lighting", Philips Technical Library Eindhoven, 2nd edition, 1963. '

") Some publications on daylightmg: "The Lighting of Buildings", Post-war Building Studies no. 12. H. M. Stationery Office, London 1944 W. A. A 11 en, Trans. I.E.S. London, 11, 1946, 205-218. "The Basis of Day­lighting Calculations" A. F. D u f ton: "Protractors for the Computation of Daylight Factors" Building Research Technical Paper no. 28. H. M. Stationery Office, London 1946 P. ]. W a 1 dram, ]. Jun. Inst. Eng. 54, 1943, 27. "Daylight Illumination in Factories and Workshops" Prof. Dr. Ing. W. Arndt: "Praktische Lichttechnik". Berlin 1938 In Dutch: R. Swier s t r a: "Licht en Zicht" ("Light and Vision"), Tome II: "Bezonning en Beschaduwing" ("Sun-lighting and Shadow") Haarlem 1954 J. W. T. Walsh, "The science of day-light", London, 1961.

CHAPTER II

SOLID ANGLE

Il-l. Solid angle, steradian In plane geometry, that part c f a plane area lying between two lines meeting at a point (the apex) is called an angle. The size of an angle can be indicated in two different ways, viz: a) the whole plane area may be divided (by lines passing through the

apex of the angle) into 360 equal parts (degrees), the angle being then expressed in degrees.

b) with the apex of the angle as centre a circle of any radius may be drawn. Use is then made of the fact that the arc of the circle enclosed by the two lines forming the angle is proportional to the angle. The length of the arc, expressed in terms of the radius is employed as a measure of the angle. The unit is the angle subtended by an arc of the same length as the radius of the circle, this unit being known as the radian: 1 radian= 57°17'44.8". A plane angle is therefore arc length ---::-,------ radians.

radius Since the circumference of a circle is 2n times the radius, an angle of 360° contains 2n radians, an angle of 180° contains :rc rad. and

a right angle ~ rad.

In the geometry of solids (stereometry) the term solid angle is employed by analogy with the idea of angle in plane geometry; instead of the two lines enclosing a plane angle, however, there will be a conical surface, and the space enclosed within this surface is the solid angle, usually denoted by the Greek letter w. Such conical surfaces can assume quite arbitrary and irregular forms; in practice they will often consist of a regular conical surface or the sides of a pyramid and here we have the means of computing their solid angles. The size of a solid angle is expressed in a similar way to the measurement of a plane angle in radians. To do this we imagine a sphere of any radius r from the apex of the solid angle (see Fig. 5); that part of the spheriCJ.l surhce which is enclosed by the conical boundary surface of the solid

11-2] SOME SPECIAL SOLID ANGLES 19

angle is then proportional to the solid angle. When the size of the portion of the spherical surface is equal to r2, we say that the associated solid angle is I steradian (abbrev. sterad). If the subtended part of the spherical surface is not equal to r 2, but may be denoted by 5, then

5 w=-. r2 (II-I )

The surface of a sphere of radius r is 4nr2;

the solid angle enclosed by the whole sphere is thus

Fig. 5. The solid angle w subtends at the surface of the sphere an areaS. When the radius of the sphere is denoted by Y and the subtended area is ,s, the solid angle is

4nr2 -- = 4n sterad. r2

equal to 1 steradian Hence the half sphere contains 2n sterad. Example:

A solid angle w subtends an area of 12 sq. ft. at the surface of a sphere of radius 3ft. From equation (11-1) it will be found that this solid angle is

s 12 w = ;z = 3i = 1.33 sterad .

II- 2. Some special solid angles Fig. 6 shows the cross-section of a sphere with centre M and radius r.

Fig. 6. The value of the conical solid angle of which the half-apex angle is ex rna y be expressed in

terms of ex.

( w = 211 (1- cos ex) =

= 4" sin1 i)

A solid angle w, enclosed by a right cone of which the half-apex angle is ex, subtends a circular area whose cross-section is AB. The solid angle may now be expressed in terms of ex.

5 According to formula (II-I) w = -. r2

The area subtended is a circular spherical segment; hence 5 = 2nrh, where his the height of the spherical segment (i.e. h = CD). We may now write:

5 2nr . CD 2n w = - = = -(MD- MC) = r2 r2 r

2n = - (r- r cos ex) = 2n {1 -cos ex). r

20 SOLID ANGLE [II

A conical solid angle may thus be expressed by

w = 2n (I -cos ex). (11-2)

For logarithmic applications this may be put in the form:

(II-2a)

Example: A conical solid angle w has a half-apex angle IX = 6.5". The solid angle then contains

• 2 IX • 2 6.5o w = 47T sm 2 = 47T sm 2 = 0.0404 sterad.

In the above we have defined the idea of "solid angle" as the space enclosed by a conical surface. Space can also be enclosed by hyo conical surfaces having a common apex; the space thus enclosed is also called a solid angle and the definition given in section II-I should be extended

G

Fig. 7. When a solid angle w is enclosed by 2 coaxial conical surfaces having half-apex angles

of IX1 and IX2:

w = 21r (cos IX1 -cos 1X2)

accordingly. Let us take the case of a solid angle w as depicted in Fig. 7, enclosed by two right cones having a common axis and half-apex angles cx1 and cx2; the solid angle can then be expressed in terms of cx1 and cx2•

Hence the solid angle encloses an area equal to the round surface of a disc of the sphere. Equation (II -I) once more tells us that w · Sfr2 , in which S is now the spherical surface of the disc ABDC; this is equal to 2nrh; where h = EF. Now, EF = MF- ME= r cos cx1 - r cos cx2 =

= r (cos cx1 - cos cx2);

therefore

S 2nr X r (cos cx1 - cos cx2) 2 ( ) w = - = 2 = n cos cx1 - cos cx2 . r2 r

w = 2n (cos cx1 - cos cx2)

or, for logarithmic treatment:

(II-3)

(II-3a)

11-4] SMALL SOLID ANGLES 21

If oc1 = oo, that is cos oc1 = 1, we have a solid angle bounded by a single conical surface, and equations (11-3) and (II-3a) revert to (II-2) and (II-2a). The problem dealt with at the commencement of this section is thus a special instance of the case discussed above. Equation (11-3) can also be derived by regarding the solid angle MABDC as the difference between solid angles MAGB and MCGD. It then follows that in order to apply equations (11-3) and (ll-3a) it is not necessary for the conical surfaces to be coaxial. However, the circle CD must not cut circle AB, or lie completely outside it.

Example:

A solid angle w is enclosed by conical surfaces having half-apex angles of 28° and 12° (ex1 = 12°, ex2 = 28°). This solid angle would contain

w = 47T sin ex 1 ; ex 2 sin ex2 2 ex 1 = 47T sin 20° sin 8° = 0.0598 sterad.

11-3. Significance of the conception "solid angle" in illuminating engineering

What are the applications of the idea "solid angle"? Anticipating somewhat the uses to which this conception will later be put, we may mention the following. When a certain surface appears at a given distance from the eye, that surface is seen because a beam of light rays reaches the eye. Now, if the surface is sufficiently far away from the eye, we may regard the eye as a point; the beam of light then constitutes a solid angle, the size of which will depend on the size of the surface and the distance from the eye. We say that this surface subtends a certain solid angle. Similarly a beam of light emitted by a light source which may be re­garded as a point source occupies a solid angle. Another example is to be found in a beam of light brought to a focus by a lens. It will thus be seen that the idea "solid angle" figures very largely in lighting technology: it will accordingly be encountered frequently in our computations.

II -4. Small solid angles Let us now take the case of a fiat circular surface of radius a and at a di~tance d from the eye with the line joining the centre of the eye to the

22 SOLID ANGLE [II

surface perpendicular to the latter. What is now the size of the solid angle w subtended by this surface? Let AB be the cross-section of

Fig. B. The circular sur­face AB, as seen from a point M subtending a solid angle of which the half-apex angle is ex. When ex is small, the solid angle can be ob-

7Tex2 tained by means of ---;J,2

instead of from the quotient of the spherical surfaceAB divided by the square of the radius MA

of the sphere

the surface (Fig. 8) and let M be the place occupied by the eye; then AC = CB = a, and MC =d. The angle AMC is denoted by IX.

In order to compute the solid angle we take M as the centre of a sphere, the circumference of the surface being a small circle on this; in the cross-section the sphere is then a circle having M as centre and passing through A and B. The solid angle to be computed is conical and equation (ll-2a) therefore applies:

• IX w = 4n s1n2 -2"

We now calculate the value that would be obtained for the solid angle with respect to the flat surface instead of to the spherical surface, and for the perpendicular distance (d) from the eyes to the flat surface instead of the radius of the sphere. Then

nAC2 na2

w = MC2 = d2 (11-4)

in place of which we may write, since ajd = tan IX:

w = n tan2 IX. (ll-4a)

What would be the difference if we employed equations (11-2) and (ll-2a) or (11-4) and (ll-4a)? For angles IX which are so small that the angle, sine and tangent are interchangeable, the results show no difference, as will be seen from the following: Equation (ll-2a) reads:

• IX w- 4nsm2 -- 2"

For small angles we may put this in the form

w = 4n (~r = niX2• (11-5)

11-4] SMALL SOLID ANGLES 23

Equation (II-4a) for small angles becomes

(II-6)

Using (II-5) and (II-6) we therefore obtain the same result for w with angles IX for which IX = sin IX = tan IX.

With larger angles IX there is a difference in the results; the variations as percentages are given for a number of instances in Table A. Apart from the angle IX, this table includes the relationship between the diameter AB" ( = 2a) of the circular surface to the distance d from the eyes to that surface. The last column contains the amount of error as a percentage, involved when equations (II--4) and (II-4a) are em­ployed instead of (II-2) and (II-2a).

TABLE A For the solid angle {half-apex angle a:, column 1) within which a circular disc of diameter 2a is seen from a distance d, column 2 gives the ration 2a : d ap­propriate to each value ohx. Columns 3 and 4 show the relative solid angles in steradians computed from formulae (II-2) and (II-4a) respectively. Column 5 indicates the difference as a percentage between the values in columns 3 and 4.

rx I 2a: d I w 1 = 47T sin2 i I w2 = 1T tan2 rx I w 2 : w1 X 100%

-----~--------~--------------c--------------T---

1 : 7.2 1 : 5. 7 1: 4.7 I : 4.1 1 : 3.5 1 : 3.2 1 : 2.8

0.004872 X 11

0.007611 X 11

0.01096 X 11

0.01491 X 11

0.01946 X 11

0.02462 X 11

0.03038 X 11

0.004890 X 11

0.007654 X 11

0.01105 X 1r

0.01507 X 11

0.01'975 X 11

0.02509 X 11

0.03109 X 11

0.4% 0.6% 0.8% 1.1% 1.5% 1.9% 2.3%

From the table it will be seen that with a ratio of 2a : d of about I : 4 the error attendant on the use of the equation

is only about I%-

na2 w = - = n tan2 IX d2

In practice, therefore, there will be very many cases where the solid angle can be computed by dividing the flat surface (5v) by the square of the distance from the eye (d); thus

51) w = d2'

24 SOLID ANGLE [II

II-S. Table of solid angles In Table I (see p. 407) will be found the values of conical solid angles with half-apex angles of from oo to 180° in steps of S0 , together with the difference in solid angle with respect to the angle ex which is smaller by S0 •

From this table it will be noted that up to goo the increase in w becomes more and more pronounced, but that from that point up to 180° it decreases. Annular solid angles for the same arc length are thus much greater at goo than at 180° and it is important to bear this in mind, as will appear later.

CHAPTER III

LUMINOUS FLUX, LUMINOUS INTENSITY, QUANTITY OF LIGHT

III-I. Luminous flux; luminous intensity Light sources emit energy in the form of electromagnetic waves which spread out in all directions. The amount of energy radiated during a unit of time (the power) may be expressed in physical units as ergs per second, or in watts. Apart from the total energy, we may also consider the amount of energy passing through a certain part of a plane, or within a certain solid angle, and express this in the same units. Generally speaking, only a part of the energy entering the eye produces an impression of light, viz. that part of which the wavelengths lie between about 0.4 and 0.7 ~tm. As we shall see later (Chapter XII), the sensation of light induced in the eyes by a certain amount of energy in the form of electromagnetic waves is not the same at all wavelengths. Because of this fact the energy emitted by a light source is not expressed in watts or ergs/sec, even for so far as it lies between 0.4 and 0. 7 ~tm; instead, the energy is evaluated in terms of the sensation of light produced in the eyes (Cf. definition of· photometry: the measurement of radiation evaluated in accordance with the visual impression). Radiated energy thus evaluated on the basis of the impression of light which it induces in the eyes is termed. the luminous flux (symbol f/J). The unit of luminous :flux is the lumen, the definition of which follows later.

The quantity lumindus flux is frequently compared with analogous conceptions employed in other branches of physical science. It is the most conveniently comparable with a flow of liquid or electrical current. The first instance would refer to the quantity of water passing a certain point in a pipe within a unit of time; the second would represent the quantity of electric current (coulombs) flowing at a given point in a conductor within the unit of time.

If we regard a light source as a point from which light is emitted, we can imagine a sphere round the source, with the source as centre; the whole of the luminous flux will then pass through this sphere. Now, if measurements be taken of the luminous :flux passing through areas of a certain size at different points on the surface of the sphere, it will be found that the :flux differs more or less between one point and another.

26 LUMINOUS FLUX, LUMINOUS INTENSITY, Q.UANTITY OF LIGHT [III

The radiated luminous flux is in effect not uniformly distributed in space, but varies with the direction. When the size of the sphere is increased or decreased and the size of the small area from which the measurement is taken is altered in pro­portion, we find exactly the same values of luminous flux, seeing that light is propagated in straight lines. We thus measure the luminous flux radiated in different solid angles of equal size. The manner in which luminous flux from a light source is distributed in space is, of course, of considerable importance to the illuminating engineer, since a knowledge of this enables him to direct the light in. the most effective and economical manner towards the objects .which he wishes to illuminate. It is necessary, therefore, to measure values of luminous flux in the different directions and to express these in a certain unit. We might decide to measure the flux radiated in a certain solid angle (e.g. with an apex-angle of 10°), but this would not be very prac­tical, if for no other reason than that the energy in such a solid angle might be unevenly distributed. The solution to this problem can be solved along the lines of the de­finition of velocity; in mechanics the velocity of a moving body is the quotient of the distance travelled, divided by the time. For example, the distance covered in I minute is measured and this, divided by the time (in this case 1 minute) gives the velocity in, say metres per minute. It is quite possible, however, that the speed of the moving body is not constant during that particular minute. In that case we would have measured the average speed in that minute, but we should know nothing about any variation that may have occurred in the speed during that time. In order to arrive at this we must reduce the period of the mea­surement to so small a value that the speed in that space of time can be regarded as uniform. Accordingly, we approximate the time to zero, so that it becomes in­finitely short, and express this mathematically in such a way the velocity (v) is the quotient of the distance travelled (s) divided by the time (t) with respect to t -> 0. As a formula this is

. s v = hm -.

t-o t

Needless to say, practical measurements of velocity based on measure­ments of distance and time have to be effected within a finite period, but this should be so short that the relevant velocity can be regarded as constant.

111-1] LUMINOUS FLUX; LUMINOUS INTENSITY 27

To return to the problem of the distribution of luminous flux in space, we require to know what luminous flux is radiated within a certain

solid angle and, by analogy with the idea of velocity, . , we (distance) tzme

now employ the quotient of luminous flux divided by the solid angle (~). In the same way that the time is made to approximate to zero for the velocity, we now approximate the solid angle to zero. , . . . . . . . luminous flux fh1s s1mllarly prov1des a hm1tmg value of the quotient .

soltd angle . with the solid angle approaching zero; this is known as the luminous intensity (symbol I): formula:

if> I= lim

w-*0 W

Formerly, the name candle-power was generally given to this ratio. In the consideration that has led to this definition the light source is assumed to be a point source, i.e. infinitely small. All practical forms of light source are naturally of finite dimensions, hut in the following we shall nevertheless consider practical light sources to be point sources, assuming thereby that all the light emitted emanates from one single point at the source. In Chapter IX it will be shown that in most cases this assumption is permissible, as it involves no serious error. The conditions under which the source may be considered to be a point source are also dealt with in that chapter. In the same way that measurement of velocity in terms of distance covered and time must be effected in a finite period of time, in the measure­ment of luminous intensity the luminous flux must be measured within a finite solid angle which should be so small that the distribution of the luminous flux in that angle can be regarded as uniform. In place of the unit of time we employ the unit of solid angle, i.e. the steradian. It is possible to express the luminous intensity in the unit lumens per steradian, but this term is never employed. Formerly, the unit was known as the candle. The standardised unit at present is the candela, to which reference will be made in the next section where the units of luminous intensity are defined. When the luminous intensity, that is, the luminous flux per steradian, is constant within a certain solid angle, the luminous flux in that

28 LUMINOUS FLUX, LUMINOUS INTENSI~Y, QUANTITY OF LIGHT LIII

solid angle is obtained by multiplying the luminous intensity by the solid angle, v1z.

C/J = w. I. (III-I)

The average luminous intensity in a solid angle in which a luminous flux C/J is radiated is the quotient of luminous flux divided by the solid angle; thus

(III-2)

Vsing the symbols employed in the infinitesimal calculus we express the luminous intensity in accordance with what has been said above

as the differential ~ :. Hence

In general, therefore

<P = JI.dw.

111-2. Units of luminous intensity and luminous flux

The unit of luminous intensity

(111-3)

(III-4)

In the preceding section it was stated that the definition of the lumen as the unit of luminous flux would be given later. The reason for this is that the unit of luminous flux is derived from the unit of luminous intensity, it being therefore necessary to define the latter first. There have been numerous units of luminous intensity during the course of time. They have all had this much in common that they represented the luminous intensity of certain light sources, defined as accurately as then possible. It is a requirement of such light sources that they must not vary with the time and that it must be possible to reproduce the source at any time and place. Moreover, the luminous intensity must not vary when the source is in use, that is, when measurements are taking place. In those days when the first endeavours were being made to measure light (the middle of the 18th century) the wax candle of all the sources in use at that time came closest to meeting the conditions of constant intensity and reproduceability; quite naturally, therefore, the luminous intensity (in the horizontal direction) of a wax candle was taken to be the unit of luminous intensity. This unit thus became known as the "candle".

III-2] UNITS OF LUMINOUS INTENSITY AND LUMINOUS FLUX 29

Needless to say, the luminous intensity of a flickering candle, which was not invariably of the same composition and of which the flame was dependent on the condition of the wick, was not particularly constant, and it was not long before a more constant source of light was sought. Of the many standard sources devised and used since that time there arc two which have remained in use for a considerable period. These are the Hefner candle (HK) employed in Germany and certain other European countries, and the International candle (ic) which was favoured in the English-speaking countries and France. As there were objections to both the Hefner a~d the international candle, these were replaced as standards on 1st January 1948 by a new unit, originally called the New Candle, but now known as the Can­dela (cd). The Hefner candle represents the luminous intensity radiated horizontally by the Hefner lamp, designed by von Hefner A 1 ten e c k in 1884 and consisting of a kind of oil lamp which, however, burned amyl acetate instead of oil. When the flame height is properly adjusted, all Hefner lamps will, under specified conditions of humidity, carbon dioxide content and atmospheric pressure, yield an intensity of I Hefner candle. For other humidity and pressure conditions the correct intensity is calculated with the aid of a correction. formula. Notwithstanding all this, the Hefner lamp has serious disadvantages; not only is the flame size difficult to control, but the fact that the in­tensity of the flame is not more than one candle results in difficulties in measurement. There are also objections to the yellow colour of the flame. In 1909 the United Stq.tes, Great Britain and France decided to standard­ise the unit of luminous intensity and this gave rise to the International candle, established with the aid of the intensity of a number of carbon filament electric lamps. A certain number of international candles was attributed to the luminous intensity of these lamps in the horizontal direction, and from these lamps, which served as primary standard, secondary standards were prepared. The primary standards were maintained in the State Laboratories in the United States, Great Britain and France and were very seldom used, so as to limit as far as possible the blackening of the bulb which evaporation of the carbon filament produces. At the same time, however slight the utilisation and consequent blackening, there must ultimately be a point where the blackening results in a~ appreciable deterioration

30 LUMINOUS FLUX, LUMINOUS INTENSITY, Q.UANTITY OF LIGHT [III

of the luminous intensity, with a change in the value of the unit of intensity. This was duly foreseen and another method of standardising the unit has been adopted. It was agreed internationally that the new unit should be known as the Candela (abbrev. cd). This represents one sixtieth of the luminous intensity of 1 sq. em of the surface of the black body at the temperature of solidifying platinum, radiated perpendicular to that surface 1).

The black body or full radiator is a body that absorbs all radiations falling upon it. The radiation characteristics of such bodies are accurately known and the radiations at all wavelengths and temperatures can be very precisely calculated by means of a formula*). No existing freely radiating materials will actually meet such absorption requirements, but an artificial black body can be made in the following manner. A

------th I I I I

Fig. 9. Appara tus employed for the primary standard of

the Candela. B = thorium oxide tube (black body), K = thorium oxide

crucible, Pt = platinum

hollow body of any material, in the wall of which there is an aperture that is small compared with the diameter of the body, will absorb all the light entering through the aperture. If the inside of the body be rendered incandescent, the radiations emitted through the aperture will be those of the black body, and such equipment can be employed for the purposes of measurement. For the practical realization of the primary standard to be used for calibrating secondary standards in candelas the Bureau of Standards at Washington have designed an apparatus shown schematically in Fig. 9. A crucible K is filled with pure platinum; into this is placed a tube of thorium oxide (B) containing powdered thorium oxide. To ensure effective thermal insulation the whole is placed in a larger container, also filled with thorium oxide. The platinum is heated by generating H.F. eddy currents in the metal. When the metal has been heated to a temperature above the melting

point and then allowed to cool, the temperature, and with it the radiation, remains constant during the time required by the platinum to solidify; this time is utilised for taking the measurements. The new unit was originally to have been introduced in all countries on 1st January 1940, but, as the various government laboratories had

*) This calculation is dealt with further in Section XII-6.

111-21 UNITS OF LUMINOUS INTENSITY AND LUMINOUS FLUX 31

not all completed the necessary arrangements for the introduction and also because of hostilities, the introduction was postponed until 1st January, I948. In concluding this section we would like to add a few words on the relationship between the Hefner and international candles and the candela. Originally, it was generally accepted on the basis of comparative tests that

Hefner candle = 0.9 intern. candles, or Intern. candle = I.II Hefner candles.

In the course of time, however, more accurate measurements have shown that, owing to various circumstances, these conversions are only valid for light sources having an incandescent temperature equal to that of carbon filaments, that is to say, for lamps of the same kind as the primary standards of the international candle. Lamps operating at higher temperatures yielded other ratios, these being l.I4 for tungsten filament vacuum lamps and I.I65 for gas-filled tungsten lamps. This fact obviously served to increase the need for a new unit. The candela lies in value between the Hefner and international candles; it is slightly smaller than the latter, the differ­ence for vacuum lamps being 0.5% and for Pas-filled lamps 1.5 to 2%.

The unit of luminous flux

The unit of luminous flux is obtained by means of equation (III-I): ifJ = w. I, I= I (c or cd) and w = I (sterad.). Then ifJ = I lumen, which may be defined in the following manner: the lumen (abbrev. lm) is the luminous flux radiated by a light source of uniform luminous intensity, at unit intensity in 1 steradian. (Fig.lO). If we express the luminous intensity in Hefner candles, the luminous flux will be in Hefner lumens (Him): if we employ the international

Fig. 10. Illustrating the definition of the lumen. If the light source L has a uniform himinous in­tensity of 1 cd within the solid angle depicted, the luminous flux in this solid angle will be 1 lm

candle the result will be in international lumens (ilm). The unit thus derived from the candela is known merely as the lumen. For a number of years in some countries the dekalumen ( = 10 lm) was used as well as the lumen, but as it has been agreed internationally that preference should be given to units differing from the basic unit only

32 LUMINOUS FLUX, LUMINOUS INTENSITY, QUANTITY OF LIGHT [Ill

by 103 , 106 etc (i.e. micro, milli, kilo, mega etc), the dekalumen is now no longer employed.

111-3. Formulae giving the relationship between luminous intensity and luminous flux

Let us now illustrate the application of formula (111-1), viz.

W = w. I.

Example: A light source has a luminous intensity of 280 cd within a conical solid angle with a half-apex angle of 5°. From equation (111-l). then

. ex <I>= w. I= 41T sm2 2 X I= 0.0239 X 280 lm = 6.7 lm.

In section 11-1 we have seen that space with respect to a point contains 4n steradians. A light source having a luminous intensity in all directions of 1 cd (i.e.a. uniform intensity of 1 cd) thus gives in accordance with equation (111-1) a total luminous flux of 4n lumens. Let I denote the uniform luminous intensity; the luminous flux is then 4ni lm. Hence we may write:

(/> = 4ni.

Example: A light source having a luminous intensity of 25 cd in all directions gives a total luminous flux of

<I> = 41TI = 41T X 25 lm = 314 lm.

In place of the uniform intensity we may also take the average intensity in all directions (I 0 ), in which case we put:

Example:

(/> (/> = 4ni0 or I 0 = -.

4n (111-5)

An incandescent lamp gives a luminous flux of 1400 lumens. 1'he average luminous intensity of this lamp wilr be

<I> 1400 I 0 =- = -- 1:::::! Ill cd.

41T 41T

Before it became standard practice to qualify a light source by its luminous flux, the average luminous intensity was known as the mean spherical candle-power (M.S.C.P.), expressed in spherical candles. The

TII-4] HORIZONTAL, SPHERICAL LUMINOUS INTENSITY 33

conceptions of mean lower (I 0 ) and mean upper (I 0 ) hemispherical candle-power were also employed, these being, as the terms imply, the mean luminous intensities of the radiation in the lower and upper hemispheres respectively. If we denote the luminous flux emitted in the lower and upper hemis­pheres by t/J 0 and t/J0 respectively, it may be said that, as each hemisphere contains 2:n: steradians,

111-4. Horizontal, spherical luminous intensity In many instances it will interest us to know the luminous intensity of a source in a horizontal direction. Take for example projector lamps, the light from which reaches the lens mainly in a horizontal direction. The mean horizontal luminous intensity (Ih) may also be important, e.g. in the case of the old straight filament vacuum lamps, which give very little light downwards, but emit most of it horizontally. Before quick methods of measuring luminous flux were known, the mean hori­zontal luminous intensity of such lamps was measured by rotating the lamp rapidly about its axis. Owing to the uniform arrangement of the filament of these lamps, the light emission could be sufficiently accurately indicated by the average horizontal luminous intensity, in accordance with which the lamps were typified. This system resulted in lamps of, say, 10, 16 and 25 candle power. All this was changed when the coiled filament was introduced. The greatest intensity of lamps having this type of filament lies in a down­ward direction; furthermore, the horizontal luminous intensity is subject to considerable variation, since it depends on the extent to which the filament sags between the pigtails. The greater the number of coil sections screened by the other sections, the lower the horizontal in­tensity. In the meantime simple methods of effecting measurements of luminous flux had been evolved and these lamps were accordingly photometered spherically. It would have been possible to rate the lamps in spherical candles, but this would yield lower values for spirallized filaments than the "candles" of straight filament lamps of the same luminous flux. To overcome this difficulty, lamps with spiral filaments were rated for some time in accordance with the nominal candle-power *), defined

*) This term was used only for a time and is merely of historical interest.

34 LUMINOUS FLUX, LUMINOUS INTENSITY, QUANTITY OF LIGHT [III

as the horizontal luminous intensity which a straight filament lamp would yield for the same luminous flux, (and hence also spherical lu­minous intensity), as the lamp with spiral filament. As we shall show later, the theoretical ratio of the spherical to the horizontal luminous intensity of straight filament lamps should be n/4, and practical values do not in fact deviate much from this. For spiral filament lamps the nominal candle-power was thus found by multi­plying the spherical candle-power by 4/n. This was obviously not very satisfactory, and a system of standardisation of lamp wattage was resorted to, the rated power being shown on the bulb. This has its disadvantages, too, seeing that for the same amount of power, the luminous flux of lamps of different manufacture may vary considerably and may be increased by technical improvements in the luminous efficiency *). The next step, adopted in various countries, was to standardise the luminous flux of the lamps, the unit Dekalumen being introduced for this purpose. Consequently the luminous flux, as far as the numerical value was concerned, was almost identical with the horizontal and nominal luminous intensity previously employed.

n _As already stated, the equation I 0 = 4 I,. is valid for straight filament lamps. Now

where lP is expressed in lumens. Putting this into Dekalumens, we have

In the case of coil-filament lamps I,. can be substituted £or the nominal luminous intensity in this equation, making it possible to tell the customer that instead of the lamp which he formerly purchased to give n candle­power, he could ask for a lamp of n Dim and receive one that would give practically the same luminous flux. The rating of electric lamps according to their luminous flux was adopted by a limited number of countries only and in these it was not found possible to accustom users to the idea of the Dekalumen.

*) See section III-5.

III-6] Q.UANTITY OF LIGHT, LUMEN-SECOND, LUMEN-HOUR 35

In recent years, therefore, electric lamp manufacturers have reverted to the standardisation of wattages, the lamps now being branded with the wattage as well as the luminous flux in lumens. In the case of gas discharge lamps it is impossible to standardise the luminous flux, seeing that the ballasts are manufactured to suit a certain lamp wattage; the wattage of a given lamp type must therefore always be the same, so that the same ballast can be used each time. Any technical improvement resulting in an increase in the luminous efficiency will accordingly represent an improvement in the luminous flux for the same wattage.

Ill-5. Luminous efficiency Electrical light sources consume a certain amount of electrical power per second (watts) and yield a certain luminous flux. The efficiency of a light source is higher according as the output of light per watt, i.e. the quotient of luminous flux divided by the power consumed, is greater. This quotient is known as the luminous efficiency of the light source and it is expressed in lmfW.

Example: 1. The luminous efficiency of a 100 W-filament lamp giving a luminous

flux of 1425 lm is 1425 LOO = 14.25 lmfW.

2. The efficiency of a 40 W fluorescent lamp (lamp only) giving a luminous flux of "2800"lm is

2800 40= 70 lmjW.

Suppose that the losses in the ballast amount to 10 W and that these are included in the calculation; the total consumption of lamp and ballast will then be 50 W and the efficiency is

2800 = 56 l /W 50 m ·

III-6. Quantity of light, lumen-second, lumen-hour In section III-I we have seen that luminous flux is the quantity of light radiated per second. When the luminous flux of a source is known, the quantity of light (symbol Q) emitted by the source within a certain time is found by multiplying the flux by the time. Thus

Q = cp. t. (III-6)

36 LUMINOUS FLUX, LUMINOUS INTENSITY, QUANTITY OF LIGHT [III

If the luminous flux is not constant with time we must write: Q = 1:<1>. At, or preferably Q = j<~>. dt. (III-6a)

When the time in equation (111-6) is in seconds, Q is in lumen-seconds (lmsec); if it is in hours, Q is expressed in lumen-hours (lmh).

The idea of light quantity is of practical importance in two special con­nections, viz: a) for flash bulbs (e.g. Philips Photoflux); b) when an appraisal of the total quantity of light emitted by a source

during the working period is required. a) The density *) of a photographic plate or film is' in the first instance dependent on the quantity of light reaching the emulsion. When the

:3'.5

1.0

!J.. .lr 70: I v ""' i\. 7

I I ~

:3'.0

2.5

2.0

1.5

) "" v ............ ~ 1--

0.5

10 20 :30 40 50 -millisec

Fig. ll. Curve of the luminous flux v. time (flash curve) of the Philips PF 60 Photoflux flash bulb. The area included between the curve and the horizontal axis is a measure of the quantity

of light emitted

plate or film is exposed to the light from a flash bulb for the whole of the available flash, the density is proportional to the quantity of light

*) See section X-12 for further details.

III-6] QUANTITY OF LIGHT, LUMEN-SECOND, LUMEN-HOUR 37

emitted by the bulb. The manner in which the luminous flux waxes and wanes during the flash does not affect the density. Fig. 11 shows the complete flash curve of Philips Photoflux flash bulb type PF 60 plotted with the time in milliseconds horizontally and the luminous flux (i.e. the quantity of light per second) vertically, this latter being the light emitted at every moment during combustion of the wire in the flash bulb. It is seen that after ignition has taken place the luminous flux increases up to a maximum value and then drops to the point where the bulb finally extinguishes. The quantity of light is obtained by multiplying the luminous flux at every moment by the (infinitely) short time dt during which light is emitted, and subsequently totalling these values; it is thereby assumed that the luminous flux is constant for this infinitely short period. This can also be expressed graphically by determining the size of the rectangles formed by the infinitely short times (abscissa) and the luminous flux values relating to those times (ordinate). In Fig. 11 a number of these rectangles are shown, obviously for finite periods. When the periods are made smaller and smaller until infinitely small, the sum of all the small rectangles will be equal to the area enclosed between the curve and the horizontal ordinate.

Expressed mathematically, this area is J q, . dt and is, in accordance with equation (III-6a), equal to the quantity of light Q.

It will be seen from the curve that a maximum of 3 X 106 lm is emitted. The average quantity of light given by the PF 60 is 62.000 lmsec, this being emitted in 52 milliseconds. Using equation (111-6) we can now find the average luminous flux emitted in this 52 msec: from (111-6) it follows that (/) = Q jt, hence

62 000 (/Jm = 0_052 = 1 200 000 lm = 1.2 million lm.

A comparison of such values of luminous flux with those of ordinary 100 W gas-filled electric lamps for 220-230 V which yield about 1400 lm, will give some idea of the powerful source of light which the Photoflux flash bulb places in the hands of the photographer and explain how it is that instantaneous exposures can be made with this very small light source. b) The luminous flux of all known sources of artificial light decrea­ses during the life of the source. The amount of the depreci-

38 LUMINOUS FLUX, LUMINOUS INTENSITY, Q.UANTITY OF LIGHT [III

¢(225v) /500 lm --f r--

/000

500

0

a

r-- r-e-r--

I

0 500 - h /OCX)

Fig. 12. Luminous flux as a function of time, for two filament lamps of the same

rating (100 W) a) coiled coil lamp b) single coil lamp

ation differs between lamps of different make and type. In order to make a com­parison between two lamps of the same rating it is necessary to consider not only the initial value of the luminous flux, but also the total quantity of light produced by the lamps during their life. These factors can be obtained from a graph showing the luminous flux as a function of the number of hours of burning. As an example Fig. 12 shows the lines repres­enting the average depreciation of the luminous flux of a 220-230 V Philips 100 W coiled coil "Biarlita" lamp and also of the earlier 100 W single coil "Arga" lamp for the same voltage. The areas representing the quantities of light of the two lamp types for an average life of 1000 hours on 225 V is found to be

Coiled coil Single coil

1.37 X 1 ()6 lmh 1.14 X 106 ,

Whereas the initial luminous flux of the coiled coil averages 14% more than that of the single coil type, the quantity of light is 20% higher. The coiled coil lamp therefore not only has the advantage of the higher initial flux, but the decline in output during its life is less than in the single coil lamp.

REFERENCES 1) G. He II e r, Ph. T. R. *) 5, 1940, 1-5. "The new luminous Standard"

W. de Groot, Ph. T. R. 10, 1948, 150-153. "The New Candle" J. W. T. Walsh, W. Barnett, R. G. Berry andJ. S. Preston, Proc. Inst. Electr. Eng. JOB, 1961, 173-181. "Units and standards oflight maintained at the National Physical Laboratory, 1915-1960."

*) Ph. T. R. = Philips Technical R<view.

CHAPTER IV

LIGHT DISTRIBUTION, ROUSSEAU AND ZONAL LUMINOUS FLUX DIAGRAMS

IV -1. Light distribution, the Rousseau diagram In order to be able to evaluate the luminous characteristics of light sources it is necessary to know, amongst other things, the radiated luminous flux in the various directions within the space around the source, i.e. the distribution of the luminous intensity, termed the light distribution of the source. It must also be possible to express this numeri­cally or graphically. Once the luminous intensity in a number of directions in a plane passing through the source has been established, the light distribution in that plane can be represented graphically by plotting the intensity values as radius vectors in the direction of measurement and subsequently joining the extremities of these vectors to form a curve. The curve thus obtained is the light distribution curve to which the designation polar may be added, since the values represented are plotted from a single point known as the pole. If the light distribution of a source is axially or rotationally symmetrical, the whole distribution may be repre­sented by plotting it for one half of a plane through the axis of symmetry. When the distribution is for all practical purposes rotationally symmetrical, the average light distribution around the axis is determined and plotted as the average light distribution diagram. Fig. 13 is an example of a light distri­bution diagram for an axially symme­trical light source. The latter should be regarded as being suspended with the axis of symmetry vertical. The (average) luminous intensities are determined for a number of angles around the axis of symmetry (e.g. every

Fig. f3. Polar light distri­bution diagram of a light

source

40 LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM [IV

10 degrees) and these are plotted from the pole in the diagram, in the direction of measurement. A smooth curve is drawn through the ends of the radius vectors as shown in Fig. 13. As a general rule the direction vertically downwards is taken to be oo. In cases where the light distribution is not axially or nearly axially symmetrical it may be represented by curves plotted for different planes. If we plot the luminous intensities from a single point in the direction of- measurement we can imagine a curved surface passing through the ends of the radius vectors, and the solid thus obtained is called a solid of light distribution. In the following chapter we shall show how this solid may be represented as a plane figure. When the light distribution is axially symmetrical, the solid of light distribution is a solid of revolution whose axis of rotation coincides with the axis of symmetry. It is usual to convert and plot the distribution of lamps and lighting fittings for a luminous flux of 1000 lumens. For lighting fittings this luminous flux is taken to be that of the bare lamp. This conversion to a fixed value of luminous flux has the advantage over the use of the actual luminous intensity of the lighting fitting that the curves can be employed for lamps of different luminous flux. Accordingly it is not necessary to plot separate curves for lamps of different rated voltages, and variations in the luminous flux due to modifications in lamps do not make the curves invalid. Moreover, the curves can be used for lamps of different power, provided that the light distribution of the fitting is not thereby affected. Such is the case with opal glass globes or vitreous enamelled and lacquered reflectors in which the lampholder is adjustable so that the filament or gas discharge of the different lamp types can be made to assume a constant position. When a 1000-lm curve is available, the values on the curve can be multiplied in every case by the number of times 1000 lm at which the lamp is rated, in order to find the actual luminous intensity in a given case. From the foregoing it would seem obvious that the light distribution curves for 1000 lm are derived by measuring the actual intensity of a lighting fitting with a calibrated lamp (i.e. of known luminous flux) placed in it. The values in respect of I 000 lm for the lamp could then be found simply by dividing the actual luminous intensities as obtained, by the number of times 1000 lm by which the reference lamp is the greater.

IV-1] LIGHT DISTRIBUTION, THE ROUSSEAU DIAGRAM 41

There are, however, various practical objections to such a procedure in that it would entail the use of numerous calibrated measuring instruments (and lamps), whilst it would also be necessary carefully to maintain correct calibration (i.e. of intensity meter, voltmeter andfor ammeters, and lamps). Preference is usually given to another method which, although apparently, more complicated, is more reliable and, since it involves no calibration fulfils its purpose just as quickly as the seemingly shorter method suggested above. The method is as follows. The luminous intensity of the naked lamp is measured with suitable apparatus*) in different directions on a relative scale; the fitting with the same lamp in it and operated at the same current or voltage, is then measured on the same scale; the rest is purely a matter of computation and plotting curves. In this method it should be noted that all calibration, even of the meter used for checking the lamp voltage or current, is quite unnecessary; the only condition is that all the instruments, as well as the luminous flux of the lamp, shall remain constant. Having thus obtained the light distribution of lamp and fitting according to the same relative scale, we convert these to values in respect of 1000 lm and, to do this, we must connect the average light distribution curve with the luminous flux of the lamp. Obviously, this 1s established by means of equation (III-1) which states that

t:P = w. I.

In section III-3 it was shown that when I is constant throughout the whole of the solid angle 4n, ([>is equal to 4ni; in practical forms of light source, however, I is not constant; it varies between one direction and another and ([> can be obtained from the luminous intensity only by dividing the entire solid angle 4n into a large number of solid angles Llw, small enough to justify the assumption that the luminous intensity within them is constant, or that the mean intensity can be sufficiently accurately approximated to without difficulty. The total luminous flux can then be represented by the sum of all the values for these small solid angles, in which case

t:P = EiiJw. Llw,

*) See section XVI-3.

42 LIGHT DISTRIBUTION AND THE ROUSSEAU Dl'I\GRAM [IV

where lt:lw denotes the average luminous intensity in each of the solid angles L1 w concerned. From the measurements we know the average relative luminous intensity for all the directions making the same angle with the axis. As solid angles L1 w we now consider disc-shaped solid angles of so small a height

that the mean intensity within them can be taken as being equal to the intensity towards the centre of the arc subtended by the solid angle. Let us now turn to Fig. 14a which

~---t---15'UQ----dl-----i---l shows a polar light distribution curve L drawn on an arbitrary scale. The cross-section of a hemis­phere of radius r is shown with the

-+---....6-----' pole 0 as centre, with also one of L the small solid angles L1 w corres-

Fig. 14'l. Polar light distribution curve from which the Rousseau

diagram Fig. 14b is derived

ponding to a height CD = h. The solid angle subtends a disc on the surface of the sphere, corresponding

with the arc AB. The luminous intensity in this solid angle varies from IA. to IB, and we take as average intensity It:lw the intensity in the direction of the point mid-way between A and B. The luminous flux in the disc-shaped solid angle L1 w is then

Lll/> = lt:lw. Llw. (IV-1) The solid angle is determined as in section Il-l, by dividing the area subtended at the surface of the sphere by the square of the radius. Now, the area Lla of the disc is

Lla = 2nrh hence

Lla 2nrh 2nh L1w=-=--=-

r2 r 2 r

For (IV-I) we may therefore write:

2nh Lll/> = ft:lw­

r

and for the total luminous flmc of the source:

2nh (/> = 1: lt:lw-

r

IV-1] LIGHT DISTRIBUTION, THE ROUSSEAU DIAGRAM

. 2n. Smce - 1s constant, we may also put r

2n cp = - E I Jw . h.

r

43

(IV-2)

If we now represent I Jw and h as the sides of a rectangle, the area of that rectangle will represent the product of IJw and h. Proceeding thus for each Llw and combining the areas so obtained, we then have the means of determining the luminous flux graphically from the I Jw values in the luminous intensity difl,gram. Now let us look at Fig. 14b. This shows IJw projected horizontally and h vertically. The area EFGH is therefore equal to IJw. hand represents the luminous flux in the solid angle Ll w divided by 2nfr. Conversely, the luminous flux Ll(f> is obtained from EFGH by multiplying this by 2nfr. The same procedure can be followed for the whole of the solid angl~ 4n, in which case all the h's placed together will give us KL, or 2r. Clearly, then, instead of constructing rectangles such as EFGH, we may join the ends of the lines representing IJw by means of a smootr curve. In this way we obtain as sum of all the areas IJw. h the figure KLMN, whose area may be denoted by 5. This area can be measured with a planimeter, but we can also measure IJw at regular intervals and multiply the average of all the values of I.dw by the base*) KL = 2r of the figure KLMN. The construction of diagrams such as that shown in Fig. 14b was origi­nated in 1882 by the Belgian engineer Rousseau, after whom these diagrams are named 1).

How can we now use the Rousseau diagram to convert luminous intensity curves known on a purely relative basis, to a certain luminous flux of the source? Let us assume that r = 12 em, so that the base of the Rousseau diagram KL = 24 em. The scale on which I is represented is as yet unknown and we assume tentatively that 1 em = a cd; I is then represented

I by- em.

a

*) Although the line LK is usually drawn vertically with the luminous intensity shown horizontally, we still refer to LK as the base of the diagram.

44 LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM

According to (IV -2):

2n (/) = - E ILJw . h,

r

where E IiJw. h is represented by the area 5. So, if 1 em = a cd,

E IiJw. h =a. 5

[IV

since, as each horizontal centimetre of KLMN represents a cd, the area 5, measured in sq. em, must be multiplied by a in order to yield E ILJw. h. We may therefore write:

2n (/)=-a. 5,

r

from which it follows that

(/)r a=-.

2n5

When (/) = 1000 lm we may write

1000 >< r a=----

2nS

When r = 12 em the value of a is

1000 X 12 1910 a= 2n5 =s·

(IV--3)

In this case, then, the luminous intensities for 1000 lm are found by . . . . 1910

mulbplymg the relative values by the conversiOn factor s· Conversely we can compute the area represented by 1000 lm, assuming a luminous intensity scale and radius r. If a= 1 cdjcm, then with r = 12 em, 5 = 1910 sq. em and 1000 lumen will be represented by an area of 1910 sq. em. If a= 20 cdjcm, then 5 = 1910: 20 = 95.5 sq. em and 1000 lm will be represented by an area of 95.5 sq. em. The conversion factor 1910/5 is valid for all values of the luminous intensity obtained from the same arbitrary scale, and therefore also for that of the lighting fitting. This operation having been carried out, the light distribution curves of lamp and fitting can be drawn.

IV-3] DERIVATION OF THE ROUSSEAU DIAGRAM 45

IV -2. Construction of the Rousseau diagram Fig. 14b shows the Rousseau diagram as derived from the light dis­tribution curve; it is also possible to construct the Rousseau diagram independently. Let ex. denote the angle AOD in Fig. 14a; then

OD = rcos ex..

In Fig. 14b, therefore we also have

O'H = rcoscx..

Taking L as the origin of the Rousseau diagram, we find for the angle oc along the base LK

LH = O'L-O'H = = r- r cos oc = r (I- cos oc).

In Fig. 15 a Rousseau diagram is depicted in which the lines associated with the various angles are shown.

.aor= t-

19~

181-t-

171-1-

15f--

15~ f--

14~ 13 1-

f--l.dl= Il l-

/0~ 9~ -8--7 --5 ::::;

~5= "'4= 03_ ~ t-s: .d 1-' t-I f-- '-'-

A

Fig. 15. Rousseau diagram

= sd'

~

= 14d --/30 -~,;;d = no• ~

1-Jod f--

f--1-

9<f 1-1-1-

eif ~ 1-f--

70° ~ f--

60° r= ~

5if ~

40° ~ 30 1-

f--1-'--

61191

These lines lie at a distance r (I - cos IX) from the abscissa. The base of the diagram is 7.5 em (3 in.) long. In order to ensure sufficient accuracy when calculating the area of the Rousseau diagram it is desirable to employ a diagram with a longer base, and for practical purposes a base of, say, 24 em (about I 0") is convenient. With a base of 24 em and a scale of 20 cd = I em, an area of 95.5 sq. em will correspond to 1000 lm luminous flux at the source (see previous section) .

IV - 3. Derivation of the Rousseau diagram with the aid of the in-finitesimal calculus With the aid of the infinitesimal calculus, the relationship between light distribution and luminous flux is expressed in the following manner. Fig. l6a shows the light distribution curve of a light source within the half cross section of a sphere of radius r. In the solid angle d w between angles ot and ot+dot the luminous flux emitte:l is equal to

d <l> = lot. dw.

46 LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM

Now d -~ = da = 21T1'2 sinadiX 2 . d

- r• r• = 1rSin1X IX

and d~ = 21rl1X sin ada. The luminous flux between IX1 and IX2 is thus found to be

Fig. 16. Derivation of the , Rousseau diagram from the light distribution by

means of the infinitesimal calculus

IX•

C!>IX IX = 21rJ I IX sin IX diX " .

[IV

(IV-4)

(IV-5)

and the total luminous flux between IX = 0° and 180° = 1r rad., as 1T

C1> = 21r J I IX sin IX diX. (IV-6) 0

This integration is carried out graphically in the Rousseau diagram in the following manner (fig. 16b): The surface element dS in the Rousseau diagram corresponding to dOt is equal to

dS = /IX r sin IX diX. The area between the luminous intensity lines IX1 and a 2 is

a. sa,, Ots = r f I IX sin IX diX

Ott

and the whole area enclosed within the curve is 1T

(IV-7)

S = r J I a sin IX diX. (IV -8) 0

Equations (IV-7) and (IV-8) differ only in the factor, from equations (IV-5) and (IV-6) for the luminous flux. The luminous flux is there­fore obtained from the area enclosed within the curve in the Rousseau diagram by multiplying this by 21rjr.

IV-4] EFFICIENCY OF A FITTING FROl\1 THE ROUSSEAU DIAGRAM 47

IV -4. Determination of the efficiency of a lighting fitting from the Rousseau diagram

The Rousseau diagram also gives us the means· of ascertaining the efficiency of a lighting fitting, by which is meant the ratio of the total luminous flux emitted by the fitting to that of the bare lamp. Efficiency is usually denoted by the Greek letter 'YJ and is expressed as a percentage. It was showp. in section IV-I that the luminous flux values of the lamp and fitting can both be represented by areas in the Rousseau diagram; division of the one area by the other therefore gives us the efficiency. It is a condition, however, that it must be possible to plot an average light distribution curve for the fitting, which can usually be done only in the case of axially symmetrical fittings. The efficiency of a lighting fitting can also be determined by direct measurement with a luminous-flux meter, this being a method of verifying the result obtained from the Rousseau diagram. This direct method of efficiency measurement also provides a simple means of ascertaining the light distribution of symmetrical fittings, since, when the efficiency is known from the direct measurement, mea­surement of the relative light distribution of the lighting fitting is all that is needed to determine the light distribution for I 000 lm. The luminous intensity values thus determined on an arbitrary scale are set out in the Rousseau diagram and the area S thus obtained is measured with the planimeter. In order to make the conversion for 1000 lm of the bare lamp, the scale ( 1 em = a cd) is found for the luminous intensity with r = 12 em, as

'YJ 1910 a= 100 X S'

. 1910 for, instead of the 1000 lm to w1th the factor S applies, we now

have _!}_ x 1000 lm, and the relative luminous intensity values 100

obtained for the fitting must also be multiplied by this factor a. As a rule the graphical method of obtaining the efficiency of a fitting, with a check by means of direct measurement will be preferred to the system just described. For asymmetric fittings the method is not employed, since determination of the average light distribution is as a rule too cumbersome.

48 LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM [IV

If the light distribution from a fitting is uniform in every plane passing through one axis of the fitting, the luminous flux can easily be found by a Moerman method 2) for which the light distribution in only two planes need be known. This method is particularly useful for fluorescent lamp fittings. Here, the light distribution curves are generally uniform in the planes through the longitudinal axis, i.e. in every plane the luminous intensity at every angle p to the axis bears the same proportion to the luminous intensity vertical to this axis. If the light distribution in the vertical plane containing the longitudinal axis of the fitting and that in the plane perpendicular to the longitudinal axis are known, the light distribution in all other planes can readily be found. If, now, the mean luminous intensity in the plane perpendicular to the axis is calculated and it is found that a proportion a of the mean luminous intensity is directed vertically downwards, it then becomes necessary merely to multiply all luminous intensities in the vertical plane through the axis by a to find the mean light distribution. This can then be plotted on the Rousseau diagram in order to find the luminous flux or the efficiency of the fitting. In practice this is made simpler by transferring the light distribution in the vertical plane through the axis to the Rousseau diagram, measuring with a planimeter and subsequently multiplying by a.

IV -5. The zonal luminous flux diagram

It is often necessary to know what part of the luminous flux radiated by a lighting fitting occurs within certain solid angles, i.e. in certain (/) (%) zones and, in the case of

100 symmetrical fittings, the "" -

80 ....

" " 1--1-, ""' 60 I' , , v

40 1--1/ ,

-~ v "

BCfl ~

_ ... ooo 20 40° 60 00 KXP f2(J" HCI' 160 I

Fig. 17. Zonal luminous flux diagram - - - - - - - lamp, fitting

zonal luminous flux diagram is plotted for this purpose (Fig. 17). In this diagram the half-apex angles of the conical solid angles which have the axis of the fitting as axis are plotted horizon­tally and the luminous flux radiated within those solid angles vertically.

IV-5] THE ZONAL LUMINOUS FLUX DIAGRAM 49

Usually the luminous flux is expressed as a percentage of the flux of the bare lamp, so that the over-all efficiency of the fitting may be read off at 180°. From the example shown in Fig. 1 7 it will be seen that the efficiency is 74%, also that between oo and goo (i.e. in the lower hemisphere), 33% is radiated, and between goo and 180° (upper hemi­sphere) 41%. The ratio of the flux radiated down­wards (direct) to that radiated upwards (indirect) is thus as 1 : 1.25.

A Between 0° and 10° the emission is 4%, between oo and 20° 11%, hence between 10° and 20° it is 7% of the luminous flux of the lamp. The luminous flux values of the zonal luminous flux diagram can be obtained from the Rousseau diagram; the area between the oo and 10°-lines cor­responds to the flux radiated between oo and 10°, and the planimeter reading, divided by the area corresponding to 1000 lm, gives the percentage of luminous flux emitted between 0° and 10°. Re­peating this process for 0°-20° and 0°-30° etc we obtain the values needed for the zonal flux diagram. This is not a quick method, however; another method will be found simpler but not quite so ac­

Fig. 18. The lumi­nous flux of a light source can be de­termined from the polar distribution curve by measure­ment of the length of the lines AB = lex sin ex for ex = 5°, 15°, 25° etc. The sum of these lengths, multiplied by 1.095 R;j l.l gives

the luminous flux

curate, although sufficiently accurate for ordinary practical purposes. Let us turn to Fig. 18. This shows the polar light distribution curve of a light source. We now compute the luminous flux between the angles 1X1 and IX2, assuming (and this is the source of the error) that the average luminous intensity I ex within the particular solid angle is equal to the intensity radiated in the direction of the bisector of the angle 1X2 - IX1,

i.e. at the angle (II-3a))

IX + IX 1 2 2• The solid angle between IX1 and 1X2 is (Equation

4 . IX2 - IXl • IXl + IX2 w = nsm 2 sm 2 .

In this solid angle the luminous flux is therefore

• IX2 - IX1 • ot1 + ot2 lf> ex,ex, = 4nl ex sm 2 sm 2 .

50 LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM [IV

Assuming a constant difference between ot1 and ot2 of, say, 10°, then

ot2 - otl so d otl + ot2 ---,-----= an =ot

2 2 '

so we may write:

<PIX, IX• = 4n sin so I IX sin ot = 1.09S I IX sin ot ~ I. I I IX sin ot.

In Fig. 18 IIX sin ot is denoted by the line AB, and the value of IIX sin ot

can be measured directly from the light distribution curve with a rule. For the calculation of </> in lumens it is of course necessary to take into account the scale. If I em represents a candelas, the length of IIX sin ot

as measured must be multiplied by a as well as by the factor 1.09S. The luminous flux between oo and I oo is founft by measuring I IX sin -ot

for ot = S0 , and that between 10° and 20° by taking the measurement at ot = IS0 and so on. The luminous flux between oo and ot1 is found by adding up the values of ot lying between those limits, and the over-all efficiency of a lighting fitting can be obtained by totalling the values of ot for all the angles between 0 and I80°, as expressed by

</> = 1.09S E IIX sin ot. (IV-9)

IV -6. The area enclosed within the luminous intensity curve is not a measure of the luminous flux emitted

This is demonstrated by Fig. 19a which shows the luminous intensity curves for two light sources of the same luminous flux. Fig. 19b is the Rousseau diagram of the two curves; the areas enclosed are of equal size, whereas the area for the curve of source A in Fig. 19a is much larger than that of source B. The reason for this is that, as we have seen in section 11-5

I" '\

I \s I

'J A

!/ .... .. g

o ~

0

(Table I), the solid angles within the same plane angle ( ot2 - ot1) increase as ot ap­proaches 90°. This is not ex-pressed in the luminous in-

Fig. 19. The area enclosed by the 7rf' polar light distribution curve is not a

60 measure of the luminous flux.

0 Whereas the area within the full-line 0 curve in Fig. 19a is considerably 0 smaller than that of the dotted-line 50

40 :30 oo

curve. the Rousseau dia~ram in Fi~ . 0 19b shows that the lummous flux lS

the same in each case

IV-8] RUSSEL ANGLES 51

tensity diagram, which contains only plane angles, but it is inherent in the Rousseau diagram in which, as ex approaches 90°, the parts of the base corresponding to the same plane angle (cx2 - a 1) increase progressively.

IV-7. Average luminous intensity calculated from the Rousseau diagram Equation (IV-2) tells us that the luminous flux of a source may be represented by

2n t/J = - J: I IJw • h. r

The spherical luminous intensity I 0 is derived by means of equation (III-5a) from the luminous flux:

tP Io=-.

4n

Substitution of the expression in (IV-2) for tP in equation (III-5a) gives

2n - J: I11w. h r

Io=--4-n--

We have seen that 1: I 11w. h is represented by area S in the Rousseau diagram, and that 2r is the base of this diagram; from the Rousseau diagram, therefore, I 0 is found to be

s Io = Tr· (IV-10)

In other words the spherical luminous intensity of a light source is found by dividing the area of the Rousseau diagram by its base. It will be clear that, in order to find the average luminous intensity in solid angles smaller than 4n we must divide the area of the diagram corresponding to the luminous flux in the solid angle by that part of the base which also corresponds to the solid angle. Hence the lower and upper hemispherical luminous intensities (I 0 and I 0 )

are evaluated by dividing respectively the areas between oo and goo, and goo and 180° by the half base r of the Rousseau diagram.

IV-8. Russel angles In the absence of a planimeter for measuring the area of a Rousseau

52 LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM [IV

diagram, an ordinary rule can be used to determine the area fairly accurately (as pointed out in section IV-1), by measuring the distances representing lAw at regular distances. The average of these measurements is then multiplied by the length of the base of the diagram. This can be facilitated by. drawing blocks at each side of the Rousseau diagram in the manner shown in Fig. 15. Here, the diagram is thus divided into 20 parts, each with a centre line. The rule is laid along these centre lines and the value of I Aw in em or inches is measured at the centre of the block, this being taken as the average luminous intensity in that block. The average of the measured values of lAw represents the average for the whole Rousseau diagram and this, multiplied by the base, gives the area enclosed by the curve. Comparing this method with the system for ascertaining the spherical luminous intensity from the Rousseau diagram mentioned in section IV-7, we see that the average for the measurement of IiJw is at the same time the spherical intensity. The centre of each block in the diagram corresponds to a certain direction of measurement, which can be calculated as follows. In the diagram in Fig. -15:

AB = r (1 -cos oc).

At the centre of the lowest block:

AB = 0.05r.

From 0.05 r = r (I -cos oc) it follows that cos oc = 0.95. At the centre of the second block:

AB = 0.15 r;

hence cos oc = 0.85. The value of oc is thus calculated for the centre of each block and the results are tabulated as in Table B. Instead of measuring the intensities at every 5 or 10 degrees and plotting these in the form of a Rousseau diagram to obtain an area corresponding to the luminous flux, which may be measured by the method just described, we may also measure the luminous intensity in those directions which correspond to the centres of the blocks, i.e. at angles as given in Table B. The average of these 20 luminous intensity values then gives us 10 (see section IV-7) and multiplication by 4n yields the luminous flux.

IV -9] LONG-BASE ROUSSEAU DIAGRAM FOR NARROW BEAMS OF LIGHT 53

In English publications this is referred to as the method of R u s s e 1, in Germany that of B 1 o c h and the angles in Table B are known as Russel, or Russel-Bloch angles.

1 2 3 4 5 6 7 8 9

10

TABLE B Russel-Bloch angles

18°12' ll 31°47' 12 4e25' 13 49°28' 14 56°38' 15 63°15' 16 69°31' 17 75°31' 18 81°22' 19 87° 8' 20

9~52'

98°38' 104°29' ll0°29' ll6°45' 123°22' 130°32' 138°35' 148°13' 161°48'

This method appears at first sight to be very attractive, but it does suffer from the disadvantage that the angles are not round values. They can of course be inscribed on the protractor on the test instrument, but it is difficult to plot the luminous intensity in the form of a polar diagram with proportional angular divisions. The Russel method is not suitable for narrow-angle fittings or for fittings with a freakish form of the light distribution curve unless the number of measuring directions is considerably increased.

IV-9. The long-base Rousseau diagram for narrow beams of light When the light distribution is of the kind produced by a projector; for instance, - in which case the luminous flux is radiated wholly or for the greater part within a very narrow solid angle, there is an objection to ascertaining the luminous flux of the beam from a Rousseau diagram having a base of 24 em (about 10"). The greater part of the area corresponding to the luminous flux would thus be compressed into a long narrow space which would be very difficult or impossible to measure at all accurately with a planimeter. In such an attenuated diagram the thickness of the lines themselves would affect the result and, for this reason, the diagram in such cases is extended vertically so as to provide sufficient height. In effect this consists in enlarging the base of the Rousseau diagram, to the extent, say, of r = 24, l 00 or 1000 em ( 10", 40" or 400").

54 LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM

When r = 24 em, a = 3820/S (Eq. IV-3) r = 100 em, a= 15920/S, and r = 1000 em, a= 159200/S.

[IV

With so large a base it is not practicable to construct the complete diagram (i.e. with the whole base of 2r), nor is this even necessary. It is sufficient to show that part which corresponds to the solid angle containing the luminous flux to be determined. If desired, the luminous flux can be split, one part being determined from a diagram in which r = 1000 em for instance, and another with r = 100 em. The division and bases of the diagrams should be such that the form of the area will lend itself to accurate measurement with the planimeter.

IV -10. Detailed example: determination of a 1000-lm curve for a symmetrical lighting fitting

Let us now work out in detail the light distribution curve and zonal luminous flux curve of a lighting fitting. This fitting contains a 300 W incandescent lamp and is, for practical purposes, axially symmetrical. Measurements taken from the lamp and fitting give the average values of luminous intensity listed in Table C on an arbitrary scale.

TABLE C

Relative luminous intensity values of the lamp and fitting of Figs. 20 and 21

I Lamp I Fitting II I Lamp I Fitting

oo 11.9 13.3 100° 9.2 8.3 100 11.75 12.75 ll0° 9.75 ll.O 20° 11.55 ll.35 120° 10.25 12.25 30° 11.2 9.2 130° 10.8 12.95 40° 10.9 7.55 140° ll.25 12.75 50° 10.85 6.7 150° ll.5 11.35 60° 10.25 6.1 160° 10.05 8.3 70° 10.1 5.7 170° 6.1 5.25 80° 9.7 5.55 180° 0.2 4.75 90° 8.1 6.1

The results with respect to the lamp are plotted as a Rousseau diagram with a base of 24 em, and a curve is drawn through the points as plotted.

IV-10] DETERMINATION OF A 1000-LM CURVE FOR A FITT~G 55

The resultant Rousseau diagram is depicted in Fig. 20, reduced in size by a factor of 3. The area enclosed by the curve, and as measured with the planimeter, was found to be 243 sq. em. To obtain the luminous intensity in cd from the readings, that is, the intensity scale (1 em= a cd), the values in Table C are multi­plied by 1910j243 = 7.86, and the results are as given in Table D. These data now enable us to plot the required light distri­bution curves for lamp and fitting, and these are shown in Fig. ·21. It now remains only to ascertain the efficiency and plot the zonal luminous flux curve. The efficiency can be determined in accordance with the method described in section IV - 5 for the

15 20 /SG

1"'>. 18

1- I

I i/ r--1- /15

v I /10 14 I

- --1- ?'

~ / / /2

/ I -r- -90

\:- KJ \ I BO

I A I 8 .. ?0

~

\ \

1- 1- ,__ \

50 4 1'- 40

' 1- 1- - 0 2 1- I I-'- 0

0 24 68KJ/214

Fig. 20. Rousseau diagram with curves for an mcandescent lamp (dotted line) and for a lighting fitting with the same

lamp (full line)

zonal diagram, by measurement of I or. sin oc. The values of I oc sin oc between

TABLED

Luminous intensity values in cd of the lamp and fitting of Figs. 20 and 21, referred to a lamp luminous flux of 1000 lm

I Lamp I Fitting II Lamp Fitting

oo 93.5 104.5 100° 72 65 10° 92 100 110° 76.5 86.5 20° 90.5 89 120° 80.5 98 30° 88 72.5 130° 85 101.5 40° 85.5 59 140° 88.5 100 50° 85 52.5 150° 90 89 60° S0.5 48 160° 79 65 70° 79.5 45 170° 48 41 80° 76 43.5 180° 1.5 34.5 90° 63.5 48

56 LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM [IV

0 and 180° when totalled and multiplied by 4n sin 5° (= 1.095) give the luminous flux of the fitting, and from this the efficiency is computed. The totalling of so many values each of which is subject to a certain amount of error in measurement necessarily yields a less accurate result than that obtained by planimeter measuremen.t of the area in the Rousseau diagram. To ascertain the efficiency of the lighting fitting we therefore utilise the Rousseau diagram to check the result obtained from Irx sin ex. For a luminous flux of 1000 lm we obtain from equation (IV-9) the value which should be found for the sum of all the measured lines Irx sin ex. Equation (IV-9) states that

(/) = 1.095 E Irx sin ex. For (/) = 1000 lm,

. 1000 E Irx sm ex = 1.095 = 913.

Now, if 1 em= 1 cd, E Irx sin ex con•esponds to 913 em. In the original light distribution curve from which I rx sin ex was measured, 100 cd represented 6.25 em, so that 1 em = 100/6.25 = 16 cd. Hence .E Irx sin ex would be 913/16 = 57.1 em= 571 mm. Irx sin ex as measured from the luminous intensity curves yields the values shown in Table E. It will be seen that the value of E I rx sin ex for 1000 lm is not the same as the calculated value of 571 mm and this is due to inaccuracies in plotting and measurement. The measured value of 569 is taken as basis for the calculation of the percentages. "'' The efficiency is also obtained .from the Rousseau diagram. The readings in respect of the lighting fitting are shown plotted in Fig. 20, and planimetry of the area within the curve gives us 208.5 sq. em; the

efficiency is thus 2~:~5 X 100% = 86%, which is the same value as

that obtained by measuring Irx sin ex. Instead of employing the readings we could also plot as Rousseau diagram the luminous intensity of the fitting, calculated for 1000 lm, and divide the area thus obtained by the area corresponding to 1000 lm. Let us now determine the areas in the Rousseau diagram by measuring the luminous intensities in 20 blocks of equal height; lines are included at the side of the diagram in Fig. 20 to represent the centres of the 20 blocks, and the result of this measurement is reproduced in Table F.

IV-10] DETERMINATION OF A 1000-LM CURVE FOR A FITTING 57

TABLE E Actual measured values of I a. sin a in the case of Fig. 21, together with totals for the construction of the zonal luminous flux diagram. In the original instance

100 cd corresponded to 6.25 em.

ZONE

0°- 10° 10°- 20° 20°- 30° 30°- 40° 40°- 50° 50°- 60° 60°- 70° 70°- 80° 80°- 90° 90°-100°

100°-110° 110°-120° 120°-130° 130°-140° 140°-150° 150°-160° 160°-170° 170°-180°

cd

so 15° 25° 35° 45° 55° 65° 75° 85° 95°

105° 115° 125° 135° 145° 155° 165° 175°

Lamp

I a. sin a.' L'l a; sin a.' % of (mm) (mm) 1000 lm

5 14.5 23 30.5 37 42 45 47 46 41.25 45 44.5 42 38 32.5 23 11.5

1.25

5 19.5 42.5 73

110 152 197 244 290 331.25 376.25 420.75 462.75 500.75 533.25 556.25 567.75 569

50

40

1 3.5 7.5

13 19.5 26.5 34.5 43 51 58 66 74 81 88 93.5 97.5 99.5

100

"

Lighting fitting

I a. sin a;' L'l a; sin a; I % of (mm) (mm) 1000 lm

5.5 5.5 1 15.5 21 3.5 21 42 7.5 23 65 11.5 24 89 15.5 25 114 20 25.75 139.75 24.5 26 165.75 29 27.25 193 34 34.5 227.5 40 46.5 274 48 52.75 326.75 57.5 51 377.75 66.5 44.75 422.5 74 34.5 457 80 20.5 477.5 84

9 486.5 85.5 2 488,5 86

-,

, "

>0 0 20 °o 40 80 120 w Fig. 21. Light distribution curve and zonal luminous flux

diagram derived from the Rousseau diagram Fig. 20

58 LIGHT DISTRIBUTION AND THE ROUSSEAU DIAGRAM [IV

TABLE F

Relative luminous intensity values measured at the centre of each of the 20 bands in the Rousseau diagram Fig. 20, in order to ascertain the areas enclosed by the

Rousseau curves.

I Lamp Fitting II I Lamp I Fitting

1 11.6 11.6 11 8.2 6.4 2 11.1 8.9 12 9.0 7.8 3 10.85 7.5 13 9.75 9.6 4 10.8 6.8 14 9.8 11.05 5 10.45 6.3 15 10.1 11.9 6 10.15 6.0 16 10.4 12.6 7 10.1 8.7 17 10.8 12.95 8 10.0 5.6 18 11.2 12.85 9 9.8 5.6 19 11.55 12.0

10 8.6 5.8 20 10.9 10.8

In this way we find the areas to be 10.25 em X 24 em= 246 sq. em and 8.89 em X 24 em= 213.5 sq. em, both of which are slightly higher than the results obtained with the planimeter. The efficiency as obtained in this manner is 86.5%.

IV -II. Classification of lighting fittings according to their light distribution

There are many different kinds of lighting· fittings, of which the light distribution differs very considerably between the one kind and another. It is therefore not very surprising that many endeavours have been made to classify these. We shall not discuss all the suggestions put forward to this end, but will confine ourselves to the proposal of the C.I.E., made in 1935. This classification is based on the ratio of upper to lower hemispherical flux of the fitting; this ratio is important to the lighting engineer in that it largely determines the system on which a lighting installation may be based. Let us imagine a lighting fitting suspended with its axis vertical above the horizontal surface to be illuminated (working plane). If the fitting radiates the whole of its luminous flux in the lower hemisphere, the flux will fall directly upon the working plane; this method of lighting is termed direct lighting and the fitting is a direct-lighting fitting. If the fitting emits the whole of the flux in the upper hemisphere, the working plane receives only light reflected from the ceiling and walls,

IV -11] CLASSIFICATION OF FITTINGS ACCORDING TO LIGHT DISTRIBUTION 59

and none direct from the fitting. This is called indirect lighting and the kind of fitting which produces it is an indirect-lighting fitting. Between these two extremes the working plane may be lighted in part directly and in part indirectly and, according as the mode of radiation is either more direct or more indirect, we speak of semi-direct and semi-indirect lighting and lighting fittings. When the distribution of the luminous flux between the upper and lower hemispheres is roughly half and half, the lighting is known as general diffuse. Limits have been laid down for these different kinds of lighting, the C.I.E. classification being as follows:

Percentage of luminous flux

Lower hemisphere I Upper hemisphere

DIRECT 100-90 0-10 SEMI-DIRECT 90-60 10-40 GENERAL DIFFUSE 60-40 40-60 SEMI-INDIRECT 40-10 60-90 INDIRECT 10-0 90-100

The illustrations in the Appendix (A to U) (p. 417-4 25) are examples of different kinds of lighting fittings with their light distribution, classified in accordance with the above.

Since much less of the luminous flux radiated upwards reaches the working plane (i.e. with lower efficiency) than of that emitted direct towards the working plane, the direct luminous flux of general diffuse lighting fittings and in many cases also of semi-indirect fittings (i.e. radiated straight towards the working plane) will be greater than the flux derived from the indirect or upward radiations. The terms employed in the classification, of "general diffuse" and "semi-indirect", therefore, convey nothing of the ratio of the direct to the indirect luminous flux in the working plane, or of the proportion in which the flux reaching the working plane is direct or indirect; they refer only to the distribution of the flux leaving the lighting fitting. In this respect the two terms are rather misleading.

REFERENCES 1) In addition to the Rousseau method there are a number of other graphical

methods for the determination of luminous flux which are dealt with fully in the book by E. L. J. M a t t h e w s : ''Etude critique des procedes graphiques ou analytiques pour la determination du flux lumineux des sources de lumiere". Ghent 1929

2) J. J. B. Moerman, Elektrotechniek, 34, 1956, 407-412, "Bepaling van het licht­stroomverdelingsdiagram van armaturen" (Determination of the luminous flux distribution diagram of lighting fittings).

CHAPTER V

METHODS OF REPRESENTING LIGHT DISTRIBUTION

V-I. Luminous intensity table; polar and rectangular light distribution diagrams

In section IV -I it was stated that, if the distribution of the luminous intensity of a light source is known, it can be reproduced either numeric­ally or graphically. The numerical method is very simple, entailing only a table specifying the different directions and associated intensities; this is obviously a very convenient method when the data are required for computations, as it is then unnecessary to read the required values from graphs. If a table compiled from measurements of the light distribution is available, the curve is therefore superfluous, but, should it be necessary to interpolate from the values given, a curve will be necessary. On the other hand, a table is seldom used to characterise a light source as it does not present a clear picture of the light distribution. The following methods of graphical representation are employed: I) the polar diagram, 2) the rectangular diagram (cartesian co-ordinates), 3) the isocandela diagram.

Polar and rectangular diagrams have already been discussed in Chapter IV, so that on the subject of these we need only be brief. The polar diagram is the more generally used of the two, but in some cases the rectangular is preferred, for example when it is necessary to plot the curves of narrow beams of light. If the light distribution to be plotted covers a wide range of values, it is advisable to employ logarithmic graph paper. Both kinds of diagram can be plotted for symmetrical as well as asymmetric lighting fittings. Fully symmetrical sources can be characterised by showing the distribution in only one half of the plane of symmetry. The distri­bution of asymmetric sources is plotted in several planes all of which usually pass through a single line. In the case of bilateral symmetry it is sufficient to show the curves for the half-plane only. For pendant fittings this reference line is generally taken vertically through the fitting, but, as will be seen later, there are instances in which it is better to place the line elsewhere.

V -2. The Isocandela diagram

When the light source is imagined as being at the centre of a sphere,

V-2] THE ISOCANDELA DIAGRAM 61 the luminous intensities can be shown at the points of intersection of the radius vectors for the different directions of measurement with the surface of the sphere. Points of equal luminous intensity on the surface of the sphere can be found by interpolation, and these points are joined together to form lines; such lines are known as isocandela lines and the diagram thus obtained is called an isocandela diagram. The isocandela diagram for the 3-dimensional spherical surface has now to be represented in a flat plane. This is not a new problem, as it is substantially the same as that which confronts the cartographer with his maps. E u 1 e r has already demonstrated that it is not possible to depict spherical objects in the flat in such a way that both the area and the form are faithfully reproduced; stated more simply this means that a spherical projection system can be accurate (or proportionate) either from the area or from the angular point of view. Numerous systems of projection are in use in which either the one or the other of these two methods is adopted. For representing light distribution, a spherical projection system is employed which is proportional in area since the spherical surfaces to be represented correspond to solid angles. In order to determine the luminous flux within a certain solid angle we can measure the area corresponding to this solid angle from the plane diagram (by planimetry), and multiply this area (converted to steradians) by the luminous intensity (or average lum. intensity) in that solid angle. Two of the systems employed in cartography, both of which are area-proportional, have been adopted in illuminating engi­neering, namely the sinusoidal

Meridian Pole

projection of Sanson-Flamsteed Zero meridian

and Lambert's azimuthal Pole

projection. For some years now a projection described in a publication by Dourgnon and Fleury I) has been used to represent isocandela

Fig. 21. Fixing the bearing of a point P on a sphere by means of spherical co-ordinates. The latitude b is. measured along the zero meridian or along the P-meridian, and the longitude Z along the

equator or the parallel

curves. This projection, which we shall call "quasi central projection", is not area or angle-proportional. It does, however, have other advantages about which we shall learn later.

62 METHODS OF REPRESENTING LIGHT DISTRIBUTION [V

z Fig. 23. Develop­ment of the sinus­oidal projection. The surface of a hemi­sphere (a) is divided into plane and in-

XI-+++-Ir++++-lr+++-HH-t-+1X finitely narro'Y seg­ments (b). When

V -3. Spherical co-ordinates

Before discussing these systems of spherical projection let us see how the relative position of a point on a spherical surface is determined. We use for this purpose the method and terminology employed in geography. In geography the position of a point on the earth's surface is fixed by means of two co-ordinates, the latitude and the longitude (see Fig. 22). Of these, the latitude b is the distance from the

these segments are closed up horizon­tally, whilst main­taining the original width of each one, the spherical surface assumes the form of an onion-shaped fi -gure in which the meridians are sinus­

z oidal (c)

c z

equator, this being measured as an angle in degrees or arc-length, along the meridian (the great circle passing through the poles) on which the point lies. The longitude l is the distance of the point from a given meridian (the zero meridian); it is an angular measure along the small circle or parallel (parallel to the equator) on which the point lies. Latitude can of course also be measured along the zero meridian, and is then the distance from the equator to the parallel on which the point lies. Longitude can also be measured along the equator, and is then the distance from the zero meridian to the meridian on which the point lies.

V -4. Sinusoidal projection We imagine the surface of the sphere, with its parallels, cut into sections by a number of meridian planes to form semi-circular wedges (like segments of an orange) (Fig. 23a). The spherical surfaces of these segments are then laid out fiat (Fig. 23b) ;

V-4] SINUSOIDAL PROJECTION 63

these surfaces- as is the whole surface of the sphet:e- are double curved and cannot therefore be developed as plane surfaces, but we assume that the segments are so narrow that they may be regarded as curved in one direction only, and that the included portions of the parallels are similarly narrow enough to be considered to be straight. The strips representing the segments are thus divided by the straight parallel lines into figures bounded by two parallel straight lines and two curved lines. If we now take the spacing of the parallels to be sufficiently small, these curved lines may be regarded as straight and the strip can then be said to consist of equilateral trapezoids. The parallel sides, that is, the parts of the parallels in these trapezoids are now displaced horizontally until they meet, so that all the parts of each parallel form a straight line of the same length as the original parallel. This results in distortion of the trapezoids, but the distortion does not alter the area, seeing that the height and the parallel sides remain the same. The figure thus obtained, which is reproduced for a hemisphere in Fig. 23c, will then have the same area as that of the strips of which it is .composed, i.e. of the hemispherical surface from which the strips were obtained 1). z If the radius of the sphere be denoted by R, the area of the onion-shaped figure as shown in Fig. 23c will be 2nR2• Let the radius of any of the parallels be denoted by rand the arc of the Xr------7rl'-v-'-t...-=-=:::..::::...--Jx portion of the parallel by L1l; then, since L1l is assumed to be so small that the angle, sine and tangent are interchangeable, the length of this part of the parallel will be r X L1l. If the latitude of the parallel (in the geographical sense) is b; then

r = R cos b

z Fig. 24. Derivation of the plane co-ordi­nates x andy of a point P from the latitude

and longitude

and the length of the section of the parallel will be

R cos b L1l.

The joint height of the segments and therefore also of the ultimate

64 METHODS OF REPRESENTING LIGHT DISTRIBUTION [V

figure (ZZ in Fig. 23c) is equal to half the circumference of the sphere, i.e. nR. The length of the half-equator (XX) is also nR and the length of the half-parallel is nR cos b. In the projection the distance OA from the parallel to the equator (X-axis) (Fig. 24) 1s

b OA = 90 X !nR

with b in degrees, or

b X R

with b in radians. On the equator, the distance OB from the meridian to the polar axis (Z­axis) is

l OB = 90 X !nR

with l in degrees, or

l X R

with l in radians. The location of a point on the sphere being fixed by the spherical co­ordinates longitude l (rad) and latitude b (rad) the point is characterised in the projection by the plane co-ordinates:

x = AP = OB cos b = R . l cos b, and

y = OA = R. b. (V-1)

V -5. Transformation of spherical co-ordinates In Fig. 23a the Z-axis is the polar axis of the meridians and parallels. Obviously, any diameter of the sphere can be taken as polar axis and the associated meridian and parallels drawn, and the latitude and longitude of a point on the sphere can be indicated in accordance with any such system. Among all the possible polar axes apart from the Z-axis shown in the diagram, however, two are of interest. these being the X- and Y-axes in Fig. 23a. The two systems of meridians and parallels can be drawn in the sinusoidal projection as in Fig. 23c and the co-ordinates of a point, indicated as latitude and longitude in the X- or Y-system can be converted to co-ordinates in the Z-system; by means of this conversion of co-ordinates the meridians and parallels in the projection of Fig. 23c can be calculated.

V-5] TRANSFORMATION OF SPHERICAL CO-ORDINATES 65

Fig. 25 shows the projection of a number of meridians and parallels drawn in the three systems. This conversion will now be illustrated with reference to a particular example; in the other cases only the formulae will be given. Fig. 26 shows a hemisphere represented by its sinusoidal projection. The meridian of the point P relative to the X-system is given. The latitude of P can be represented by the curve P A ( = bz) and the longitude by the straight line A Y ( = l z). The meridian of P relating to the Z-axis is also included, the

z

z Fig. _25i" Sinusoidal representation of a hemisphere, showing a number of me­ridians and parallels corresponding to

the X-, Y- and Z-systems - - - - - - X-system ---- Y-system -- Z-system

latitude of P in this system being given by the curve PB ( = bz) and the longitude by the line BY ( = lz). Required: to calculate bz and lz from bz and lz. The figure PBX represents a right-angled spherical triangle with the right angle at B. The sides are:

z PB = bz, XB = 90°-lz, XP = 90°-bz.

Further, the angle PXB is equal to the longitude of P in the

X X-system: lz. ~~--+..~.--4~----------~X

z Fig. 26. Derivation of formulae for the conversion of the co-ordinates in one system (e.g. the X-system) to those of

another (e.g. Z-system)

Employing two formulae used in spherical trigonometry, it is a simple matter to compute bz and l. from bz and lz. In the spherical triangle PBX:

sin bz = sin lz sin (90° - b.,) = =sin lz cos bz;

further tan (90° - lz) = = cos lz tan (90°- bz),

66 METHODS OF REPRESENTING LIGHT DISTRIBUTION [V

from which it follows that

cot z. = cos ltz cot b~.

The formulae for any transposition can be similarly derived, viz:

X--+ Y: sin b11 = cos l., cos b., (V-2a) ~~=~~~~ ~~

Y --+ Z: sin b. = sin l 11 cos b11

tan z. = cos l 11 cot b11

Z --+ X: sin b., = sin z. cos b. cot ltz = cos z. cot b.

Y --+ X: sin b., = cos l 11 cos b11

tan l., = sin l 11 cot b11

Z--+ Y: sin b11 =cos z. cos b. cot l11 = sin z. cot b.

X --+ Z: sin b. = sin l., cos b., cot l. = cos l., cot b.,

(V-2c) (V-2d)

(V-2e) (V-2/)

(V-2g) (V-2h)

(V-2i) (V-2j)

(V-2k) (V-2l)

To draw the projections of meridians and parallels in the X- and Y­systems, the co-ordinates of a number of the relevant points can be referred to co-ordinates in the Z-system, after which equations (V-1) can be used to compute the plane co-ordinates for plotting the points on the circles. There is a much simpler method, however, which will now be demon­strated by means of an example. Let us draw the meridian in the X­system, having a longitude l.,. For calculation purposes we select those points on the meridian whose positions are determined by the latitude b. in the Z-system. l., and b. in respect of the point are then known. We can compute z. from equation (V-2/):

cot l., = cos z. cot b •.

From this it follows that

cot l., cos l, = -- = cot l~ tan b •.

cot b,

Having thus obtained z., we find from equation (V-1) the plane co-

V-6] THE AZIMUTHAL PROJECTION 67

ordinates of the points required for drawing in the projection of the meridian desired. In practice it amounts to this, that lz may be laid out on the straight horizontal lines representing the parallels, since in Fig. 26, RQ is the projection of a quarter-parallel and PQ is proportional to lz. The equations employed for computing the projections of the meridians and parallels are as follows:

lmeridians: X-system

parallels:

l meridians: Y-system

parallels:

cos lz = cot l., tan bz

sin b., sin lz = --

cos b%

sin lz = cot !11 tan bz

sin b11 coslz = --cos bz

V-6. The azimuthal projection

(V-3a)

(V-3b)

(V-3c)

(V-3d)

In this system the surface of a hemisphere is reproduced within a circle. The meridians are drawn as they appear when the sphere is regarded in the direction of the polar axis, thus as diameters of the outermost circle which is in this case the equator. The parallels are represented as circles, concentric with the outermost circle. The parallels are drawn in such a way that the areas enclosed by them are equal to the areas bounded by the parallels on the· sphere; this gives the projection of the meridians and parallels, with the Y-axis as polar axis. Let us denote the radius of the sphere by R, the radius of the outermost circle in the projection by r, the latitude of the circle (i.e. the distance in degrees of the parallel from the equator or outermost circle in the figure) by b11 , and the radius of the projected parallel by r1; the area S of the spherical disc bounded by the parallel is then (see Fig. 27)

S = 2nRh = 2nR2 (I- sin b,.).

In the projection, the area enclosed by this parallel is (see Fig. 28)

Putting now S' = S, then

nr12 = 2nR2 (1- sin b11)

68 METHODS OF REPRESENTING LIGHT DISTRIBUTION (V

from which it follows that

r1 = Rv2 (1 -sin b11) = 2R sin (45°-=- }b11 ). (V-4)

y

Fig. '17. Fig. 28.

Fig. 27-28: Derivation of the plane co-ordinates of a point on a sphere from the spherical co-ordinates in the azimuthal projection

At the equator, b11 = 0° and r1 = r, so that we may write:

r = Rv'2. Solving for R, we obtain

r R= -v2·

Substitution of this value for R in (V-4) yields:

r1 = rVz. sin (45° -}b11).

The plane co-ordinates of a point P (l 11 , b11) on the sphere are thus

x = r1 cos l 11 = rVz. sin (45° -!bti) cos 111 ,

(V-5)

and (V-6)

Taking successive values of 0°, 10°, 20° etc for b11 in equation (V-5). we can now compile table G.

V-6] THE AZIMUTHAL PROJECTION 69

Fig. 29 depicts the azimuthal representation of a hemisphere with a number of parallels and meridians (full lines). As stated above, the axis of these is the Y -axis (see Fig. 23a). In the same way as in sinusoidal

TABLE G

Hadii of the parallels of latitude b in the azimuthal projection in terms of the radius of the projected equator. Llr1 is the increment of r1 for increases of 10° in b.

b I rl I Llr1

oo r 100 r X 0.909 r X 0.091 20° r x 0.811 r X 0.098 30° r X 0.707 r X 0.104 40° r X 0.598 r X 0.109 50° r X 0.484 r X 0.114 60° r X 0.366 r X 0.118 70° r X 0.246 r X 0.120 80° r x 0.123 r X 0.123 90° 0 r X 0.123

projection, it is again possible to draw the projection of the meridians and parallels in the other (X- and Z-systems). In this case the latter must be converted to the Y -system. For this purpose equations (V-2) can be em­ployed and the peculiarity is then observed that the Z­system projection can be obtained from the X-system by rotating the diagram through 90°. It is thus only necessary to compute the plane co-ordinates of the points in the X-system, the Z-system co-ordinates being then found by substituting y for x and vice versa. For the conversion from the X-system to the Y -system equations (V-2a and b) are

Fig. 29. Azimuthal representation of a hemis­phere with a number of meridians and

parallels

70

used, viz:

METHODS OF REPRESENTlNG LIGHT DISTRIBUTION

sin b11 = cos l., cos b.,, and

tan l 11 = sin l., cot b.,.

[V

In equations (V-6) b11 and l 11 must now be expressed in terms of l., and b.,; this gives for the plane co-ordinates of the points in the X-system:

x=r sin b.,

VI +cos l., cos b.,

sin l., cos b., y=r---;====:==== VI +cos l., cos b.,·

and

(V-7)

A number of parallels and meridians of the Z-system are shown by the dotted lines in Fig. 29. As the azimuthal method of representation is also area-proportional, we are once more in a position to ascertain luminous flux values by planimetering the area between 2 isocandela lines and, after conversion to the corresponding solid angle, to multiply it by the average luminous intensity in the measured area. The solid angle can be computed from the measured area in the following manner. Let S denote this area; then the ratio of this to the area of the hemisphere is as S : :n:r2• The hemisphere contains 2:n: steradians; hence S corresponds to

s 25 - X 2:n: sterad. = - sterad. :n:r2 r2

V-7. Comparison of the sinusoidal and the azimuthal projections When a comparison is made between the sinusoidal and azimuthal projections it is found that the latter has certain advantages to offer: the diagram as obtained by Lambert's method shows much less distortion and, moreover, in the azimuthal projection, the amount of distortion for the same distance from the centre of the figure is the same in all directions, whereas the nature of the distortion in the sinus­oidal system, for the same angular distance, depends on the particular position with respect to the centre of the figure. The relatively slight distortion in Lambert's projection is due to the fact that the scale on which one degree is plotted radially differs only slightly over the different sections of the diagram. Within the circle of radius 30° about the axis (i.e. between b = 60° and b = 90°)

V-8] Q.UASI CENTRAL PROJECTION 71

the scale is practically constant (Cf table G). It is also an important advantage that interpolation from the diagram is accordingly much simplified. The various meridians and parallels in the Z-system in Fig. 29 do not intersect each other precisely at right angles, but the discrepancies at 90° are not so great as in the sinusoidal projection (Fig. 25). In the sinusoidal projection it is not possible to see whether a given light distribution is rotationally symmetrical, as is possible in the Lambert diagram. This is not important for practical purposes, seeing that symmetrical light distributions are seldom represented by means of isocandela diagrams.

V -8. Quasi central projection

1 his method of plotting a light distribution on one plane was published by Dourgnon and Fleury under the title "Universal Diagram". It is our opinion, however, that the method is better described by the name "quasi central projection". A point P on a sphere (Fig. 30) can be centrally projected on a horizontal

Fig. 30. Point P on : sphere is centrally projected on to the plane H in point P' (coordinates on the Z-system).

72 METHODS OF REPRESENTING LIGHT DISTRIBUTION [V

plane li with the centre M of the sphere as the centre of projection. Point P' is centrally projected as the point of intersection of the produced straight line MP and plane H. The position of projection P' can be established with the aid of the same coordinates used to find the position of point P on the sphere, i.e. the geographic longitude and latitude. All points or lines on the sphere can be projected on to plane H in this way. If, with a number of points P', the luminous intensities of the appropriate points P on the sphere were to be written, isocandela curves could be determined by interpolation in plane H. The great disadvantage of this metod is that projections P' move further and further away from the projection of pole Z as the latitude decreases, i.e. the closer points P are to the equator. The points on the equator would be projected further and further away from Z' ad infinitum. According to Dourgnon and Fleury it is better with this projection not to use the latitude b but the distance b' from the pole which is meas­ured from the pole and is the complement of the latitude b. Distance Z'P' is equal to MZ' tan b' and this expression becomes in­finitely great where b' = goo. In order to remedy this drawback, Dourgnon and Fleury do not locate point P' on plane H at its natural position obtained by central projection but they use a system of coordinates in which the angles are shown proportionally (Fig. 31). They take the longitude l 11 and the distance b'y of theY system from the pole as Cartesian coordinates. This has the advantage that the important lines parallel to the road axis correspond in the system of coordinates to lines parallel to one another and to the Y -axis. It must now be asked how the systems of meridians and parallels are represented on this chart if XX, YY and ZZ are taken as polar axes. The simplest case is provided by theY-system, since here the meridians and parallels give a system of straight lines intersecting one another at right angles (Fig. 31). A point P on the sphere with the geographic coordinates b' 11 and l 11 corresponds to a point P' on the chart with Cartesian coordinates b' 11 and j 11 •

With polar axis ZZ the meridians and parallels on the chart are repre­sented by curves, which can be calculated with the aid of spherical trigonometrical formulae (Fig. 32). The X-system gives a projection of meridians and parallels in curves as shown in Fig. 33. This curve network can be derived from the foregoing by drawing the grid in Fig. 32 symmetrically and then effecting a goo shift parallel to the axis of l 11 •

V-8] Q.UASI CENTRAL PROJECTION 73

0

Fig. 31. Plotting point P' on a diagram in which the angular magnitudes of b'y und /y are entered proportionally.

y

0 30° 45° 60° 75° ly

F1g. 32. Plotting the meridians and parallels of the Z-system on the diagram with b'y and /y in Fig. 31.

74 METHODS OF REPRESENTING LIGHT DISTRIBUTION [V

Fig. 33 .. Plotting the meridians and parallels of the X-system on the diagram with bjy and 7& in Fig. 31.

As we shall see in VII .1 0, this system has considerable advantages in the calculation and plotting of illumination diagrams. The diagram with b' y and ly shown in perspective in Figs. 31, 32 and 33 is, of course, transferred to the drawing plane for practi~al use (Fig. 34). The isocandela curves can then be plotted on such a diagram.

V-9. The application of preferred numbers in isocandela diagrams Isocandela diagrams can be constructed, say, for 100, 50, 25, 10, 5, 2.5, 1 cd and so on, but it is much better so to arrange the isocandela lines that the luminous intensity of each successive line is greater than that of the preceding line by a constant factor. To this end the use of preferred numbers is recommended and we shall now devote a little space to this subject 3). Preferred numbers are the terms of certain geometrical progressions, every nth term of which is a whole power of 10; n may be 5, 10, 20 or 40. When we taken = 5, we have the so-called R5 series of which the ratio (i.e. the ratio of one term to the next) is ~TO ~ 1.6. In other words,

V-9] PREFERRED NUMBERS IN ISOCANDELA DIAGRAMS 75

25

Fig. 34. lsocandela magram of the lighting fitting of Fig. 35, on Dourgnon and Fleury's quasi.central projection.

the terms increase in steps of about 60%; the terms between I and IO in this series become 1.6-2.5-4--6.3. If the steps of the R5 series are too large for a given purpose, the RIO series (n = 10) may be used, the ratio then being -\o/ 10 R; 1.25 and the increment about 25%. The terms from 1 to 10 are then I.25-I.6-2-2.5-3.I5-4-5-6.3-8. The R I 0 series can be derived from the R5 series by interpolating one term between each pair in the latter series. Interpolation between the terms of the R I 0 series gives the R20 series, in which n = 20 and the ratio

20-is ylO R; 1.12 (increment approx. I2%). Similarly an R40 series can

40-be set up with n = 40, ratio viO R; 1.06 (increment approx. 6%). For the luminous intensities in isocandela diagrams we shall in general employ the R5 series. The lines are then for 1-1.6-2.5-4--6.3-10-16-25-40-63-1 00-160_:_250-400-630-1 000 cd. According to our requirements we shall utilise a part of these or extend the series upwards or downwards.

76 METHODS OF REPRESENTING LIGHT DISTRIBUTION [V

When an isocandela diagram is plotted from luminous intensity values according to a preferred number system, the concentration of the isocandela lines yields more information than with intensities of, say,

Fig. 35a.

Fig. ;351b.

10, 20, 50 and 100, since the intensities of the successive lines then bear a constant ratio to each other.

• V -10. Examples of iso­candela diagrams of a fitting

Figs. 35a and b show the light distribution of a highly asymmetric street lightingunitin bothsinus­oidal and azimuthal pro­jections; the values of the luminous intensities for the isocandela lines are based on the R5 series of numbers. If the fitting is suspended at a height of I 0 m, overhanging the roadway at a distance of I m from the curb of a street I 0 metres in width, the light between the meridians + 42° and -5! 0

as shown, will fall on the

Fig. 35. lsocandela diagram of an asymmetric lighting fitting (a) sinusoidal pro­jection; (b) azimuthal pro-

jection

roadway. In these figures only the meridians and parallels of the Y­system are shown. We have here an instance, then, in which instead of the projection on the Z-system, the Y-system as computed from the Z-system is preferred. Both figures are isocandela diagrams from

V-10] EXAMPLES OF ISOCANDELA DIAGRAMS 77

the point of view of the horizontal line through the lighting unit, parallel to the axis of the roadway. The meridians are cross-sections of planes passing through this line. These planes are accordingly re­presented by straight lines in the azimuthal projection, and curved lines in the sinusoidal system. The azimuthal representation is clearer than the other. Fig. 34 shows the same light distribution on Dourgnon and Fleury's quasi central projection. Because the perspective diagram in Fig. 31 represents only an eighth of the surface of the sphere and because a quarter of the sphere ought to be shown with this asymmetric fitting, the diagram of Fig. 31 has been duplicated here. The shape of the iso­candela curves in the upper hemisphere cannot be shown here on the same diagram. This is a disadvantage of this method, but one which is not generally obtrusive, because the light radiated in the upper hemisphere does not contribute to the lighting of the street. There is another advan­tage to this method of depicting a light distribution. If the fitting is moved through, say, 10° in a plane perpendicular to the axis of the street, the new isocandela diagram is obtained in a simple way by shifting the locus of the curve in the diagram horizontally through 10°. If light then falls on the surface to be illuminated from what was the upper hemisphere in the initial position, the abovementioned disad­vantage becomes evident.

REFERENCES

1) This derivation of the sinusoidal projection is taken from a publication by R. Swier s t r a, Euclides 12, 1935, 56 (in Dutch)

2) J. Dourgnon and D. Fleury, Lux 28, 1Y60, 53-68, ,Diagramme univenel pour-la representation des repartitions lumineuses des sources dissymetriques."

3) I.S.A. Bulletin 11, "Preferred numbers". Dec. 1935

CHAPTER VI

ILLUMINATION

VI-1. Illumination; foot-candle; lux Obviously, the amount of luminous flux falling on an object emitting no light of its own is very important from the aspect of observation of that object. We wish to know, therefore, how strongly an object or a surface is illuminated and the natural course is to express the intensity of the illumination, or illumination (symbol E) in terms of the luminotts flux per unit area of the surface thus illuminated. Denoting the luminous flux by f/J and the area by 5, we may say that

Hence

f/J E=s-·

f/J =E. S.

(VI-I)

(VI-la)

The luminous flux falling on a surface is thus ascertained by multiplying the illumination by the area. Generally speaking, the illumination on a surface will differ between one point and another, so that division of the incident luminous flux by the total area of the surface in accordance with equation (VI-I) yields the average illumination of that surface. If we now consider a part of the surface only, it may be said that the smaller the area, the less the variation in the illumination in that area and, if this process be continued to produce an area that is in fact only a point, the illumination of that point will be uniform. We may then refer to the actual illumination of that point in a plane. It must be remembered, however, that in speaking of such a point we really mean a very small area, so that, when referring to the illumination of a point, it is necessary to state the plane in which that point lies.

In terms of the infinitesimal calculus, illumination can be expressed as

E = dtl> dS

the incident luminous flux being obtained from <P = jE. dS.

VI-2] THE INVERSE SQ,UARE LAW 79

If the luminous flux is expressed in lumens and the area in square feet in equation (VI-I), the illumination is obtained in foot-candles (fc). Hence, 1 fc is the illumination produced by 1 lumen falling on a surface 1 sq. ft in area. The foot-candle unit is employed only in English-speaking countries; the internationally standardized unit of illumination is the lux (abbrev.lx). One lux is the illumination of an area of 1 sq. m. produced by a luminous flux of 1lm. In equation (VI-I), with S =1m2 and if>= 11m, E = I lux. Since I foot is 0.305 m,

Examples:

1 fc = (o.~o5r lx = 10.76lx ~ 10 lx, and

(0.305)2 llx = -- fc = 0.0929 fc ~ 0.1 fc.

. I

1. If the incident luminous flux on a surface of 4.5 sq. ft is 400 lm, the average illumination on that surface is

tf> 400 E = S = 4.5 = 89 fc = 89 X 10.76lx = 958 lx.

2. An area of 12 m 2 receives an average illumination of 150 lx. In ac­cordance with equation (VI-la) the incident luminous flux must then be

tf> = E X 5 = 12 X 150 lm = 1800 lm.

VI-2. The inverse square law In Fig. 36, L represents a light source, the luminous intensity of which, in all directions within a solid angle w, is denoted by I. The luminous flux if> in this solid angle is then if> = w . I. If we now draw a sphere with L as centre and with radius r, the spherical surface corresponding to w will receive the luminous flux if>. The illumination of this surface is as­certained by dividing the incident lu­minous flux by the area; this area is S = wr2 and the illumination is therefore

L

if> wi I E=-=-=-.

S wr2 r 2 {VI-2) Fig. 36. Derivation of the

inverse square law

80 ILLUMINATION [VI

Hence the illumination on the spherical surface is equal to the luminous intensity in the direction of that surface, divided by the square of the radius. · Now, if we imagine the surface diminished in size to a mere point, that point can be regarded as representing a plane surface perpendicular to the line passing between the point and the light source. We may therefore state that the illumination at a point in a plane per­pendicular to the line joining the point and the source is equal to the luminous intensity of the source in the direction of the point, divided by the square of the distance between point and source. If we denote this distance by d, equation (VI-2) becomes

I E = d2 • (VI-3)

In equation (VI-3), when

E = I fc and I= I cd, d = I foot, or

when E = I lux and I= I cd, d = I metre.

The lux and the' foot-candle may thus also be defined as the illumination produced on a surface 1 metre or 1 foot from a light source of1 cd luminous intensity, measured in a plane perpendicular to the direction of radiation .

.Formula (VI-3) can be expressed in words as follows. The illumination at a point in a plane is proportional to the luminous intensity, and inversely proportional to the square of the distance from the light source. For this reason equation (VI-3) is generally known as the inverse square law. In our consider~tion of this law, as well as that of luminous intensity and luminous flux, the light source has each time been assumed to be a point source, although in practice all light sources have finite dimen­sions, these being sometimes even quite large. The extent to which the assumption of a point source is justifiable in practical applications of the inverse square law is discussed in Chapter IX (Non-point sources).

Example: A light source is suspended 10 feet above a horizontal surface; the lu­minous intensity of the source vertically downwards is 420 cd.

The illumination of the plane, at a point vertically below the source is I 420

E = (j2 = W fc = 4.2 fc.

VI-3] ILLUMINATION WITH OBLIQ.UE INCIDENCE 81

VI-3. Illumination with oblique incidence Equation (VI-3) applies only when the plane of which the illumination is to be determined is perpendicular to the line connecting the light source and point, i.e. with normal incidence of the light at that point in the plane. Let us now ascertain the illumination when the plane lies at an angle -90°-oc to the ray of light, in other words, when the light is incident at an angle oc to the normal to the plane. In Fig. 37 L is a light source with a luminous intensity of I cd in the direction of the point P at which we wish to know the illumination E. The point P lies in a plane V and the line LP is at an angle oc to the normal to· the plane. The illumination to be computed is that of this plane. It is once more necessary to imagine that we wish to know the illumination of a very small area in the plane V, instead of that of the actual point P. If we provide · a plane V', perpendicular to LP at P, the illumination E' at P in that plane will be

E' = !__ d2

The luminous flux thus reaches a small

\ d

Fig. 37. When light from a source L falls at a point P on a plane V at an angle ex to the normal at P, the illumination of V at P is cos ex times the illumination at P nerpendicular

to LP.

( E = ~2 cos ex)

area of V' which may be represented in cross-section by the line CD. Now, if the same luminous flux falls on the plane V, the area concerned may be represented in cross-section by AB; then

CD AB=--.

cos oc

Since the dimensions of the illuminated areas of V and V' are the same in the direction perpendicular to the plane of the drawing, the relevant

I . area of V will be -- times as large as that of V'. The illumination

COS IX

82 ILLUMINATION [VI

of V is therefore cos ot times that of V' and we may accordingly write:

E = E' cos ot, or

I E = d2 cos IX. (VI-4)

Example: A light source L is 12 feet from a point P in a plane V. The normal to V lies at an angle rx = 25° to the line LP. The luminous intensity of L in the direction of Pis 200 cd. The illumination on plane Vat the point P is then

I 200 o E = (ji" cos rx = W cos 25 = 1.27 fc.

In practice it is often necessary to determine the illumination at different

t

Fig. 38. If a light source L is suspended at a height h above a plane H and the light falls on a point P on H at an angle rx, the luminous intensity of Lin the direction of P being denoted by I, the illumination of H at P is

I E = JiB cos sa;

points in a plane lighted from above by a source at a known height above the plane. This height, which we will denote by h, is thus the perpendicular distance of the source L from the plane H (see Fig. 38 . Formula (VI-4) gives the illumination of the plane H at the point P, viz:

Now

hence

I E = d2 cos IX.

h d=--,

COS IX

I I E = -;-(---::-----,.)-::- X cos ot = hs cos 3ot.

CO: IX 2

I E =- cos 3ot h2 (VI-5)

If we now apply equation (VI-5) for various points in respect of which ot is constant, we at once obtain the illumination values for circles of radius AP = h tan ot. If the luminous intensity for all such points of the same angle otis constant, we thus obtain in one operation the illumi­nation value for the whole of the circle whose radius is h tan ot. Equation (VI-5) is therefore very useful for ascertaining the illumination of

VI-3] ILLUMINATION WITH OBLIQ.UE INCIDENCE 83

surfaces lighted by a source the distribution of which is wholly or nearly symmetrical around the axis.

Examples: I. A light source L is suspended 15ft above a point A in a horizontal

plane. A point Plying in the plane is so positioned that L ALP= 35°. If the luminous intensity of L in the direction of P is 275 cd, the illumination on the horizontal plane at point P will be

E I a 27 5 a 35o 0 6 f = Jii' COS IX = 151 COS = . 7 C.

2. If the point Pin the first example is 8 feet from point A, it is neces­sary first to calculate the angle IX:

tan IX = ~: = 185 = 0.533; hence IX 1'1::1 30°.

The illumination of the plane at Pis then, taking I = 275 cd, I 275

E = - cos3 IX = - cos3 30° = 0.8 fc. h1 151

When illumination calculations are to be effected for elongated areas, another formula will in many cases be found more useful, especially when the light distri­bution is asymmetric. A good example of this is a roadway with lighting standards. In Fig. 39, L represents a light source at a mounting height LA = h above the plane H. H may thus be a roadway bounded by the parallel lines QQ' and RR'. The illumina­tion at a point P on the

Fig. 39. The illumination of a plane H at a point P is I

E = }il cos1 IX cos3 {3

plane H can be calculated from equation (VI-5):

I E = hz coslly.

The point P lies in the plane LBP which makes an angle at with the normal to plane H and the line LP makes an angle P with LB. LB lies in a plane through L which is at right angles to H. The illumination at P can be computed when at and P are known.

LA In tJ. LAP, cos y = LP

84 ILLUMINATION

LB In 6 LAB, LA = LB cos oc, and in 6 LBP, LP = --.

cos{J

LA LBcosoc Hence cos y = LP = LB = cos oc cos {J.

cos fJ

Substituting this for cosy in the equation for E, we have

I E = };2 cos 3oc cos 3/J.

The distance AB is found to be

AB = h tan oc

and tan fJ BP = LB tan{J =h--.

cos (X

[VI

(VI-6)

For calculations of illumination values by means of equation (VI-6), Table II on p. 408/9 provides the values of cos3 oc cos3 fJ for a number of angles oc and f3.

tan{J Table Ilion p. 410/1 I gives the values of h tan oc and h -- for a number of angles oc and {J. cos oc If the plane H represents a roadway, as suggested, the plane LAB will be taken perpendicular to the direction of the road. BP is then parallel to the centre-line of the roadway. In this way we calculate the illumination along different lines, all of which are parallel to the road axis. It is essential for the purposes of the calculation that the luminous intensity at the various angles fJ in planes at various angles oc to the vertical LA shall be known, in other words the light distribution in these planes must be ascertained. All these planes pass through the line TT' lying parallel to the roadway and passing through the light source L. There is another advantage in this method of selecting the planes of measurement, for the same family of light distribution curves can be utilised for different positions of the lighting fitting. Suppose the unit to be canted so from the position to which the angles oc refer; this will mean that, in order to compute the illumination for the line for which oc = S0 , we must take the values for the plane where oc = oo from the set of light distribution curves relating to the original position.

VI-4] HORIZONTAL AND VERTICAL ILLUMINATION 85

For the line whereby ot = 25°, the values of the plane where ot = 20° are taken, and so on. Usually, the calculation will be confined to those planes which are essential for determining the illumination of a road of a given width and with a certain mounting height.

Example:

Suppose the mounting height of the light source L in Fig. 39 to be 25 ft, with ot = 30°, {J = 40° and I = 540 cd. The illumination of the plane H at the point P is then

I 540 E = }jl cos3 ot cos3 {J = 25, cos3 30° cos3 40° = 0.25 fc.

VI-4. Horizontal and vertical illumination

The illumination at a point, i.e. that of a very small area, in a horizontal plane, is referred to as the horizontal illumination at, or of, that point (EH). If, on the other hand, the point occurs in a vertical plane, we speak of the vertical illumination of the point (Ev)· In this connection it must be borne in mind that only one horizontal plane, but an infinite number of vertical planes, can pass through. a given point. When stating and computing the vertical illumination of a point, the vertical plane in which the point lies must be denoted. In the case illustrated in Fig. 38 we calculate the illumination of the horizontal plane H, at the point P, that is, the horizontal illumination at P. We therefore put:

(VI-7)

If we regard Pas a point in the vertical plane V j_ AP, it is readily seen that the illumination of V at the point P, i.e. the vertical illumination at P, is

I Ev = hi cos 2ot sin ot.

It follows from (VI-7) and (VI-8) that

Ev sin ot - = -- = tan ot. EH cos ot

(VI-8)

86 ILLUMINATION [VI

In the example shown in Fig. 39 we have calculated the illumination at the point Pin the horizontal plane H, i.e. the horizontal illumination at P; hence we can state:

(VI-9)

For the vertical illumination at P in a plane perpendicular to BP (i.e. perpendicular to the road axis if we assume that H represents a road), we put:

I . Ev = h2 cos 21X sm fJ cos 2/J.

From (VI-9) and (VI-10) it then follows that

Ev tanfJ En cos IX

BP LA Now, tan fJ = LB and cos IX = LB.

hence

Ev tanfJ BP En cos IX LA

(VI-10)

As will be seen in Part II (Light Measurements) it is easier and more accurate- if the incidence of the light is very oblique- to measure the vertical instead of the horizontal illumination. When the vertical il­lumination Ev is measured at a point perpendicular to the road-axis, En can be determined by multiplying Ev by the quotient of the mounting height h divided by the distance BP from the measuring point to the vertical plane through the lighting fitting.

VI-5. Other units of illumination The foot-candle and the lux are not the only units of illumination in use. Low illumination levels are often expressed in millilux (mlx) ( = 0.001 lux). We also sometimes come across the phot, this being the illumination

VI-5] OTHER UNITS OF ILLUMINATION 87

produced by 1 lm on 1 sq. em, or by 1 cd or candle from a distance of 1 em. This unit fits into the C.G.S. system (1 phot = 10,000 lux). English publications on the subject of coast lighting mention the sea­mile candle, which is the illumination resulting from 1 cd at a distance of l sea-mile. As the sea-mile is 6080 feet

sea-mile candle= (60~0r fc = 2.70 X 10-8 fc

(= 2.9 X t0-7 lux).

In table IV (p. 412) the various units of illumination are listed together with the relevant conversion factors.

CHAPTER VII

ILLUMINATION CALCULATIONS AND DIAGRAMS

VII -1. Introduction Of the various factors which determine the quality of a lighting in­stallation as an aid to effective vision, the illumination is one of the most important. In the design of a lighting installation the lighting engineer will ascertain roughly the illumination level which the installation will have to produce and, in some cases the degree of non-uniformity that may be tolerated. He chooses a type of lighting fitting to suit the particular method of illumination and, on the basis of the required illumination level and permissible inequalities therein, he calculates the number of fittings required and the power of the lamp to be installed in each fitting. For this purpose details of the lighting fittings are required and these must be provided by the manufacturer. Calculations for complete lighting installations are outside the scope of this book, but are dealt with extensively in most manuals on illu­mination engineering 1). In the following sections, however, we shall describe briefly the more usual methods employed, with the emphasis on those calculations which the lighting engineer has to make from the distribution characteristics of the lighting fitting and which usually have to precede any calculations on the complete installation, for which they serve as basis. There is a choice of two methods of calculation for lighting installations. viz: 1) the lumen method,

2) the point-by-point method.

VII-2. The lumen method; coefficient of utilization Lighting installations usually comprise a number of lighting fittings. The engineer knows from experience based on measurement in practice that a given type of fitting will ensure a certain degree of uniformity in the illumination when given a certain height in relation to the spacing. Once the mounting height has been decided upon to suit local conditions, the most satisfactory spacing is thus determined and so also the number of fittings required. If it is also known what part of the luminous flux of each lamp with its appropriate fitting reaches the plane to be illuminated (the working plane), the luminous flux which will have to be supplied by each lamp can be calculated, and this in turn indicates the lamp rating.

Vll-21 THE LUMEN METHOD; COEFFICIENT OF UTILIZATION 89

Let the size of the working plane be denoted by 5 sq. ft (or sq. m) and the required average horizontal illumination by E fc (or lux); the luminous flux required in that area will then be E X 5 lm. If we now denote that portion of the light from the lamps which reaches the working plane by 'YJ, the total luminous flux of the lamps will have to be

Ex5 (/)tot = -- lm.

'YJ

With n fittings, the luminous flux per lamp will be

Ex5 (/Jia = -- lm.

nxn

n= 11 X {[Jza

(VII-IJ

where {[Jza is the total luminous flux of the lamps in every lighting fitting. The values of E, 5 and n can be determined at once by the lighting engineer, but the factor rJ is a different matter. This is the coefficient of utilization of the room and the lighting installation therein, this being mainly determined by the following:

a. The light distribution of the fittings. When all the light from a fitting is directed downwards (direct lighting), the degree to which the ceiling is capable of reflecting light hardly affects the illumination in the working plane at all. On the other hand, when the fitting radiates all the light upwards (indirect lighting), the condition of the ceiling is the all-important factor. If the fitting distributes its light in all directions, the ceiling and walls are important, but to a lesser degree than in indirect lighting.

b. The efficiency of the lighting fitting. c. The amount of incident light reflected by walls and ceiling, of which

a part reaches the working plane. Obviously a whitewashed ceiling will reflect more light downwards than a grey one, and light-coloured walls similarly contribute more towards the illumination in the working plane than dark ones.

d. The ratio of the over-all dimensions of the area to be illuminated to the height above this area at which the lamps are mounted. The influence of this on the illumination is included in the data relating to 'YJ under the name of room index and methods of computing this index may be found in any existing book on illuminating engineering.

90 ILLUMINATION CALCULATIONS AND DIAGRAMS [VII

Curves and tables are in existence from which approximate coefficient of utilization values may be read or interpolated in relation to all practical forms of lighting installation 2).

It is usual to make allowance in the coefficient of utilization for a certain depreciation in the efficiency of the lighting equipment during use, such being due to: 1. drop in the luminous flux of the lamps, 2. reduction in the efficiency of lighting fittings due to dust and dirt;

even with regular maintenance a certain average "dirty condition" must be taken into account.

3. deterioration in the reflection factors of walls and ceiling, to which the same remarks apply as in 2. regarding average working conditions.

The desired illumination must be produced as an average during the working period of the equipment and, owing to the three factors men­tioned above, this average is below the initial value for the new equipment. It is necessary, therefore, to plan an installation for a higher illumination level than the average output. A suitable correction is usually incor­porated in the tables or charts of illumination values. The factor by which the initial illumination level should be divided or, conversely, that by which the desired average should be multiplied to allow for depreciation in output during use is known as the depreciation factor. This may therefore be defined as the ratio of initial to average efficiency (or illumination) of the equipment during use; hence it is greater than unity. The method described above is based only on the luminous flux of the light sources as installed and is appropriately referred to as the lumen method.

VII-3. The point-by-point method The lumen method is adopted for calculations of lighting installations in cases where a given average illumination is envisaged in which some latitude is allowed in the uniformity of the lighting, or in which a certain, degree of uniformity is specified. Such uniformity is generally ensured by installing a number of light sources working in co-operation; the illumination at any given point in the working plane is then produced by several light sources in the vicinity of that point. In most cases the lumen method is suitable for interior lighting schemes; the lighting of streets, squares, in fact all outdoor lighting presents a rather different problem. As will be seen later (Ch. X), the illumination at the different points of the road, i.e. the distribution of the illumina-

VII-3] THE POINT-BY-POINT METHOD 91

tion, is our main concern in street lighting. In order to know something of this distribution we ascertain the illumination values at a number of points and if necessary interpolate the levels at intermediate points. The distribution of the illumination in any street lighting project will have to be derived from the light distribution of the fitting on the basis of a given mounting height and spacing, and in conjunction with the width of the street or roadway to be illuminated. Here again, thus, it is essential to have details of the distribution characteristics of the lighting fittings to be used. We can here, however, go a step further and say that the makers could be expected to provide the data in a form that would render the application of such data universal, that is, suitable for obtaining the distribution for any combination of mounting height, spacing and road width. Whether or not such data are determined by calculation from the light distribution or by direct measurement is not important. We shall discuss various forms in which such illumination data might be presented. Of the two methods of originating such data we shall concern ourselves here only with the calculation method. The direct method is described in Part II, Light Measurements (section XVIII-4). Since the illumination is computed for a number of points in the plane to be illuminated, the method under review is known as the point-by­point method. This system is not employed exclusively for exterior lighting; it is also used for interior illumination. In section VII-2 we have said that the lighting engineer usually knows from experience how much uniformity of illumination may be expected from a given project and that such experience is backed by practical measurement; this is only half the truth, however, for the expert can also calculate the degree of uniformity from the distribution of illumination in the working plane produced by one fitting. The illumination in this plane is two-fold; one part comes from the source itself (direct) and the other is reflected from walls and ceiling (indirect). The illumination due to the direct light can be computed from the characteristic distribution of the lighting fitting by the point-by-point method. Thus, also in the case of indoor lighting fittings the illuminating engineer must have data concerning the illumination at his disposal, especially in the case of direct lighting, and here too the data supplied should be universal in their application.

92 ILLUMINATION CALCULATIONS AND DIAGRAMS [VII

VII-4. Illumination diagrams Illumination diagrams are graphical representations of illumination patterns in a plane or along a line. A diagram of this kind will illustrate either the illumination distribution of a complete lighting installation, or local differences in the illumination produced by a single light source. In the first instance the number of light sources, the mounting height and the spacing are all specified, and also the condition, and therefore the influence, of walls and ceiling is similarly fixed. It is difficult and sometimes impossible to base a diagram of this kind on calculation, and recourse is usually taken to direct measurement of the illumination, this being, as it were, proof by results.

Fig. 40. Polar light distribution dia­gram of a lighting fitting suspended at a height h above a horizontal plane. The isolux diagrams in Figs. 42, 45 and 47 were constructed with the aid

of this Cl!rve

For estimating the effect of a lighting installation in advance, various forms of diagram are em­ployed, showing the illumination resulting from the use of par­ticular lighting fittings. These do not, of course, take into account the effect of the condition of the walls or ceiling. The diagrams are usually so arranged that they can be used in relation to any mounting height by interpolation or simple conversion. We shall now describe the more current types of illumination distri­bution diagram and the method of working them out from the light distribution of the fitting. There are other forms than those which we shall review, but the reader will have no difficulty in understanding these on encountering them in the literature, once the following has been studied. For the derivation of the diagrams relating to a single lighting fitting

let us assume an axially symmetrical and also a bilaterally symmetrical fitting. In the first type we take as starting point the light distribution of a fitting the axis of symmetry of which is vertical and which illuminates

VII-5] THE RECTANGULAR ILLUMINATION DIAGRAM 93

a horizontal surface; the situation is as shown in Fig. 40. The bilaterally symmetrical arrangement is depicted in Fig. 4la which is similar to that in Fig. 39. The light distribution of the fitting is given in planes through a line parallel to the illuminated plane (Fig .. 41 b). In principle, a distinction can be made between two types of diagram, viz: 1. those in which the illumination is plotted as a function of its location. 2. isolux diagrams in which

lines are drawn joining points of equal illumina­tion. These lines are called isolux curves. It is useful to employ the preferred number system for the values in isolux diagrams and the R5 series will generally be found the most suitable.

The illumination can of course be expressed in foot-candles or lux. In the following we shall use the foot-candle and the foot, but the values can be read as they stand in lux by substituting metres for feet. In the figures both notations are shown. The following are the most usual forms which the illu­mination diagrams assume: 1. rectangular diagram, 2. solid of illumination, 3. plane isolux diagram, 4. polar isolux diagram.

--.T-·-. L Ill --

It I It I

I I I I I I

I 1 I I I I

~ ~0

~ I I

cd

·-·-·-·r ;71

1/ I II I

I 1 I I I :

/ I I

0

Fig. 't-1. The light distribution curves in Fig. 4lb refer to planes making an angle Ot with the vertical line passing through the lighting fitting. The assumed position of the fitting with respect to the illuminated plane is seen in Fig.4la. The curves in Fig. 4lb serve as basis of calculation for the isolux

diagrams in Figs. 43, 44 and 46

VII-5. The rectangular illumination diagram In rectangular illumination diagrams the illumination is plotted with respect to various points on a line, the horizontal axis representing the distance of the points from a reference point and the vertical axis the illumination values. For the axially symmetrical fitting (Fig. 40) the illumination produced

94 ILLUMINATION CALCULATIONS AND DIAGRAMS [VII

Fe {lx)

250

200

150

100-

50

u

h=lft(m) cfi= 1000/m

can be wholly characterised by a single line (Fig. 42), but if the fitting is not axially symmetrical, more than one line is needed (Fig. 43). The diagram in Fig. 42 can be easily constructed with the aid of equation (VI-5):

I E =- cos 3:x. ft2

0 0.5 1.5 2 2.5 ft(m) The distance a from the axis, at which the computed illumination is produced at an angle oc, is found from: a = h tan oc. In a universal diagram h may be I foot or I 0 feet; when h = I foot,

~~-~ .. a- tan a.

Fig. 42. Illumination diagram for the lighting fitting shown in Fig. 40 ;_ the illumination is shown as a function of the distance from the vertical

through the fitting

E =I a. cos 3oc and

a= tan oc.

Fig. 42 is based on the light distribution illustrated in Fig. 40 for a height h = I foot and a luminous flux of the bare lamp used in the fitting, of 1000 lumens. For other mounting heights the illumination values are divided by h2, and the distances Illumination curves derived Fe (lx)

a arc multiplied by h.

from the ligh~ distribution 200 shown in Fig. 41b are depicted a.=O.b=O

/ a.=l5~b=h fan15"=2.68ff(mJ in Fig.43;here the illumination is plotted as produced along the lines of intersection QQ', SS' etc of the illuminated plane and the planes for which the light distribution curves are shown in Fig. 41b. The illumination values are calculated by means of equation (VII-6):

I E =-cos 3oc cos 3/J. ft2

100

50

..... , /

' ~ h=70ft{mJ </>=1000/m

/ ct.=30~ b= h fan .30° ="5.77ff (mJ a=45?b=hfan 45°=10ftrmJ

--.a

Fig. 43. Illumination diagram for the lighting fitting shown in Fig. 41. The illumination along the lines such as H.H.' ;md SS' in Fig. 41 a, is plotted as a function of the distance from the vertical plane

through the fitting

VII-7] THE PLANE ISOLUX DIAGRAM 95

The distances along the lines SS' etc. (see Fig. 4la) are obtained from tan ,8

a= h--, cos ex

the positions of these lines being in accordance with the distances b = h tan ex. The curves in Fig. 43 refer to a height h = I 0 feet and a luminous flux of the bare lamp of I 000 lm. Conversion for other mounting heights

is carried out by . h

distances by 1o·

I02 multiplying the illumination value by - and the h2

VII-6. The solid of illumination The illumination at any point in a plane can also be plotted along perpendiculars to that plane to form a solid figure; when the upper

~ I I

01'tm,J

qSI'ttn,.l -<7"tr;,l'tr~

Fig. 44. Solid of illumination for the lighting fitting shown in Fig. 41.

ends of the lines are joined, a curved surface is obtained and this three­dimensional figure gives a very good idea of the illumination in the plane concerned. Fig. 44 shows such a figure in relation to the lighting fitting illustrated in Fig. 41 and as derived from Fig. 43. This type of figure is not used for calculations on lighting installations, but merely provides a pictorial representation of the illumination levels. Plaster models are sometimes constructed in the same manner for demonstration purposes.

VII -7. The plane isolux diagram Plane isohtx diagrams consist of isolux curves drawn in the illuminated plane.

96 ILLUMINATION CALCULATIONS AND DIAGRAMS [VII

Clearly, the isolux curves for an axially symmetrical lighting fitting must be circles (Fig. 45b) .. In such case the isolux diagram is best con­

a

structed by derivation from the rectangular diagram, and the

H=lft(m) example given in Fig. 45a-b ~=IOOO!m needs no further explanation.

1,6

Isolux diagrams of fittings which are nol axially symmetrical can similarly be derived from the rectangular diagram, although it is more usual to compute or measure the illumination for a number of points, to record the values obtained in respect of those points and interpolate p0ints of equal illumination. Smooth curves are then drawn

10 joining up these final points in 76 the manner shown in the 25 40 example in Fig. 46 which refers 63 to the fitting in Fig. 41. 700 The same method can· be em-160

Fig. 45. Construction of the plane isolux diagram (b) for the axially symmetncal fitting shown in Fig. 40, from the illumination

ployed when the light distri-bution of the light source is given in planes cutting a vertical line curve (a) through the source; in this case

the points for which the illumination values are calculated lie on lines, all of which pass through the foot of the vertical line through the source.

VII-8. The polar isolux diagram

This is described in reference to Fig. 47a-b. In Fig. 47a, A denotes the

Fig. 46. Plane isolux diagram for the bilaterally symmetrical fitting in Fig. 41. Mounting height l ft

(m). Luminous flux of the lamp 1000 lm.

VII-8] THE POLAR ISOLUX DIAGRAM 97

position of the lighting fitting shown in Fig. 40. The plane of the drawing represents a vertical plane through the axis of the fitting. We now have to determine a number of points in this plane where a certain value of horizontal illumination occurs (say I fc). The manner in which this takes place will be des­cribed later. The points thus obtained are joined by a smooth curve which then represents the )-foot-candle line. Other lines are drawn for other illumination values. Concerning the point P on the I-fc line it may be said that the horizontal illumi­nation at this point is I fc; the location of this point P is determined by the vertical and horizontal distances AQ and PQ, of which AQ is the height of the source above the horizontal plane through P, i.e. the mounting height; PQ is the distance of P from the vertical through the source. Conversely, gi\·en a certain mounting height, say 5 feet, we might wish to know something about the illumi­nation in the plane 5 feet

5

Fig . .47. a) Polar isolux diagram for the axially symmetrical fitting in Fig. 40; b) Illumination curve of the same fitting, as

derived from the diagram in Fig. 47a

below the fitting. \\re then draw a horizontal line RS at 5 feet below A. This line intersects the isolux curves for 6.3, 4, 2.5 and I fc; hence at these points of intersection the horizontal illumination is equal to the values of the isolux curves. The distance from these points to the axis of the fitting can be read from the chart and from this it is possible to construct a rectangular illumination diagram. Thus, for a height It = 5 feet we obtain from Fig. 47a the curve shown in Fig. 47b. The polar isolux diagram is accordingly always a universal diagram, that is, it can be used for different mounting heights .. Although this applies in principle also to asymmetric lighting fittings, the use of the

98 ILLUMINATION CALCULATIONS AND DIAGRAMS [VII

diagram is straightforward only in the case of symmetrical ones and, in practice, such diagrams are used only for the last mentioned type of fitting. The method of calculation for a polar isolux diagram may be explained with reference to Fig. 48 in which A represents the light source. The

polar light distribution curve is that of Fig. 40. In a direction that makes an

A angle oc with the axis of symmetry we

Fig. 48. Illustrating the cal­culations for a polar isolux

diagram

can now calculate the distance r along the radius vector to a point P at which the fitting produces a certain horizontal illu­mination E fc. According to equation (IV-4) this illumination is

hence

I a. E =-cos oc, r2

1/ I a. . r = f E cos oc. (VII-2)

Using the light distribution curve of Fig. 40, let us now calculate the distance r almig the radius vector in respect of oc = 40°,

withE= 2.5, 4, 6.3 and 10 fc. From the figure we find that 141p = 198 cd. In equation (VII-2) therefore, I a.= 198 cd, cos oc =cos 40° = 0.766, and E is successively 2.5, 4, 6.3 and 10 fc. For E = 2.5 we then have

Similarly,

l/198 r = r- X 0.766 ft = 7.8 feet.

2.5

E = 4 fc ; r = 6.2 ft,

E = 6.3 fc ; r = 4.9 ft,

E = 10 fc ; r = 3.9 ft.

The values of r are plotted along the radius vectors. All the points of equal illumination on the various vectors are then connected, producing a diagram as shown in fig. 47a. Since we are interested only in the horizontal and vertical distances, the radius vectors themselves arc omitted from the final diagram, these being replaced by a system of horizontal and vertical lines.

VII-9] THE LUMINOUS FLUX FROM THE PLANE ISOLUX DIAGRAM 99

VII-9. Determination of the luminous ftux from the plane isolux diagram The plane isolux diagram provides a means of ascertaining the luminous flux radiated in the direction of a plane or part of a plane. According to (VI-la):

<P =EX S. If we measure the area between two isolux curves with a planimeter and multiply this by the average illumination in that" area, we obtain the luminous flux incident to that area between the two curves. In some cases it may be found difficult to determine the average illumination, but, generally speaking, this can be sufficiently accurately estimated for all ordinary purposes. If the isolux curves are more or less concentric it must be remembered that, assuming the distribution of the illumination to be linear from the centre outwards, the arithmetical mean of the values of the isolux curves will yield luminous flux values which will be slightly too high. The reason for this is that the area on the "high" side of the line representing the average is smaller than that on the "low" side; accordingly a somewhat lower value than the arithmetical mean should be employed. As an example let us turn once more to the isolux diagram in Fig. 46 and assume that we wish to know the luminous flux falling on the roadway depicted. We accordingly measure the areas between the isolux curves with the planimeter *), convert these to square feet and multiply the values by the appropriate (estimated) average illumination; in this way Table H is obtained.

Since the lighting fitting is bilaterally symmetrical and Fig. 46 only shows one half of the isolux diagram for the whole roadway, the total luminous flux intercepted by the road is 2 X 142 = 284 lm. As the diagram refers to a luminous flux of 1000 lm for the bare lamp, about 28.5% of the flux reaches the roadway. From the table it is seen that the light falling beyond the 2.5 fc line is very little indeed and that, although the figure does not go beyond 4 feet, we can safely say that the total flux on the roadway is equal to that as ascertained for the part shown (and even for a smaller section). It will be obvious from this example that when the foot-candle is replaced by the lux and the square foot by the square metre, the values remain the same. The fact that as an example for this method of determining the amount of *) For accurate planimetry the diagram should be drawn on a larger scale than that employed in the figure in order to save space.

100 ILLUMINATION CALCULATIONS AND DIAGRAMS [VII

TABLE H Calculation of luminous flux reaching the road surface in the isolux diagram

Fig. 46

Between isolux Area in Estimated illumination Luminous flux in curves sq.ft in fc lumens

160 fc 0.13 175 22.8 160 and 100 0.33 125 41.3 100 and 63 0.37 80 29.6

63 and 40 0.36 52 18.7 40 and 25 0.36 32 11.5 25 and 16 0.35 20 7.0 16 and 10 0.34 13 4.4 10 and 6.3 0.35 8.2 2.9

6.3 and 4 0.31 5.2 1.6 4 and 2.5 0.27 3.1 0.84 2.5 and 1.6 0.36 2.0 0.72 1.6 and 1 0.21 1.3 0.27 1 and 0.63 0.19 0.8 0.15 0.63 and 0.4 0.07 0.45 0.03

Total I 4.00 sq.ft I I - 142 lm

luminous flux the case of an illuminated roadway has been chosen, does not imply that it is of any great importance in street lighting calculations. The object of the example is merely to demonstrate the manner in which the luminous flux may be evaluated from the isolux chart.

Vll-10. The isolux diagram. in quasi central projection

In Section V-8 we learnt about the Dourgnon and Fleury quasi central projection method of representing the light distribution and pointed out that this method has particular advantages in the calculation and plotting of illumination diagrams. Fig. 49 shows the quasi central projection of the isocandela diagram of the fitting in Fig. 41. To calculate the illumination at point P' of this diagram, we must multi­ply the luminous intensity in the direction of P', i.e. 100 cd, by the third power of the cosine of the angle formed by the direction towards P' and the Z-axis, i.e. by cos3 b' z, and then divide it by the square of the suspen­sion height. If we standardise the height at 1 metre, we need then only multiply by cos3 b' z.

VII-10 THE ISOLUX DIAGRAM IN Q.UASI CENTRAL PROJECTION 101

Fig. 49. lsocandela diagram of Fig. 34. For the luminous intensity values their logarithms are also indicated.

Fig. 50. Diagrams with curves of log cos3 b' z with the same coordinates as m Fig. 49 (only half of the symmetrical diagram is represented).

This multiplication is a simple matter if we draw curves for cos3 b'~n a diagram with the same coordinates as in Fig. 49 (Fig. 50). If this is drawn on transparent paper and laid over the diagram in Fig. 49, the appro­priate value of the illumination can be calculated by multiplication at each point of intersection of the curves in Figs. 49 and 50.

102 ILLUMINATION CALCULATIONS AND DIAGRAMS [VII

_,..

F1g. 51. Isolux diagram in angular coordinates converted from Figs. 49 and 50.

Fig. 52. Isolux diagram in Cartesian coordinates converted from Fig. 51.

The cos3 b' z curves can now. be calculated for round numbers. If the cos3 b' z curves and the isocandela curves for preferred numbers are plotted, the products at the points of intersection once more provide preferred numbers, so that multiplication can be quickly done without any arithmetic with the aid of a table. If, the illuminations at the points of intersection of the curves are given on the transparent diagram in preferred numbers, the isolux curves required are found by joining the points with the same preferred numbers. The result of graphical calculation with the help of Figs. 49 and 50 is given in Fig. 51. This isolux diagram is plotted on angular coordinates. They can be converted into Cartesian coordinates by a simple process of calculation. The result of this conversion for Fig. 51 is given in Fig. 52.

VII-10 J THE ISOLUX DIAGRAM IN Q.UASI CENTRAL PROJECTION 103

REFERENCES 1) J o h. Jansen: "Beleuchtungstechnik", Philips Technical Library, Eind­

hoven 1954 J. W. Favie, "Lighting", Philips Technical Library, Eindhoven, 1962. W. B. Boast: "Illuminating Engineering", 2nd edition. New York 195:l "Illumination Design for Interiors", Published by ELMA Lighting Service Bureau, London 1948 H. H. Higbie: "Lighting Calculations", New York 1934

2) W. Harrison and E. A. Andersen, Trans. I.E.S. Am. 11, 1916, 67-91, "Illumination efficiencies as determined in an experimental room" and 15, 1920, 97-123, ~·coefficients of Utilization" H. Z ij 1, Ph. T. R. 7, 1942, 97-103. "Efficiencies of Lighting Installations". H. Z ij 1 : "Manual for the Illuminating Engineer on Large Size Perfect Dif­fusors". Philips Technical Library, Eindhoven, 1951 W. B. Boast: "Illuminating Engineering", 2nd edition p. 182 et. seq. New York 1953 "Illumination Design for Interiors" p. 81 et. seq. (see under 1 )

"I.E.S. Lighting Handbook", 4th edttion, section 9, New York 1966. 3) J. van H u 1 sen, Electrotechniek 19, 1941, 130-131. .,Een weinig bekende

methode voor het aangeven van verlichtingssterkten" ("A little-known Method for the Indication of Illumination Values")

4) j. t5ourgnon and D. Fleury, Lux 28, 1960, 53-68. "Diagrarnme umversel pour 1a representation des repartitions lumineuses des sources dissymetriques".

CHAPTER VIII

LUMINANCE AND LUMINOUS EMITTANCE

VII I -I. Luminance; the stilb, candelas per sq.in. When two light sources of the same luminous intensity, one of which has a larger area than the other, are regarded successively, the smaller appears to the eye to be brighter than the other. In the smaller of two such light sources the luminous intensity per unit area is higher than in the larger and we accordingly say that the luminance of the one is higher than that of the other. Luminance (symbol L) is defined as the luminous intensity radiated per unit area:

I L-­- s. (VIII-I)

\\"ith I in candelas and S in sq. em, I is expressed as cdjsq. em, or in stilbs (sb). Hence I stilb is I cdjsq. em. The stilb is the internationally standardized unit of luminance; in English-speaking countries the candela per sq. in. (cdjsq. in.), or the cdjsq.ft is employed. The luminance is also expressed in cdjm2. This unit is known under the name nit (symbol nt). The choice of unit usually de­pends on which of the two is the more convenient under given conditions In formulae this alternative is not always open, but may depend on the units employed for the other quantities. The term luminance was introduced by the C.I.E. in 1951 to take the place of brightness (Symbol JJ) which, used exclusively in older literature and to a certain extent in modern publications, is synonymous with luminance, although no longer the universal term. Generally speaking, the luminance of an area is not uniform, so use is made of the average luminance, this being the quotient of the total It.iminous intensity divided by the total area. In many cases this average luminance can be used in calculations without the need for knowing the distribution of the luminance throughout the area.

VIII-I] LUMINANCE; THE STILB, CANDELAS PER SQ.. IN. 105

An example of this will be found in the filaments of projector and cinema lamps which usually comprise a number of parallel coil sections. The luminance of the open spaces between the coils is zero, but for the purposes of calculations on optical systems the luminance of the area enclosing the entire filament is the important factor; this is the average luminance or, in short, the luminance of the filament.

Example: The luminous intensity of an area 8 mm in width and 10 nun in length is 800 cd. The (average) luminance is then

I 800 L = 5 = 8 X 10 cdfmm2 = 1000 sb (cdfsq.cm) = J07 cdfrn2,

VIII-2. Lambert's law In the same way that the luminous intensity of a light source is stated

s

Fig. 52. If the lu­minous intensity per­pendicular to a uni­formly diffusing sur­face S be denoted by I., the intensity I rx at an angle rx from the normal to that surface is I 0 cos rx (L a m b e r t's law)

with respect to the direction of radiation, it is also necessary to indicate the direction from which luminance is observed. When the direction of observation is perpendicular to the luminous surface, and provided that this surface is plane, the actual area must of course be taken when use is made of formula (VIII-I) (see Fig. 52). If the direction concerned is at an angle ex from the normal to the surface of the light source, the surface appears to ce smaller, to the extent of 5 cos ex; the luminance at angle ex is then

L = Irx a 5 cos ex (VIII-2)

and the luminous intensity is

I rx =--= Lrx . 5 . cos ex. (VIII-2a)

The more precise definition of luminance is thus the quotient of the lu­minous intensity divided by the apparent surface of the light source. Cnder this definition it is not necessary for the source to be a plane source; it may be of any cross-section. When the luminance of a source is the same in all directions, L is in­dependent of ex, the index may ce omitted and equation (VIII-2a) then reads:

I rx = L . 5 . cos ex,

106 LUMINANCE AND LUMINOUS EMITTANCE [VIII

where L. S = I 0 (the luminous intensity perpendicular to the plane, 1.e. direction of radiation 0°). Hence we may also write:

I= Io cos IX (VIII-3)

which is the mathematical expression of Lambert's law. This law states that if a surface has the same luminance in all directions, its luminous intensity in a given direction is equal to the luminous intensity perpendicular to the plane (Io), multiplied by the cosine of the angle between that direction and the normal to the plane.

L a m b e r t arrived at this law by observing the luminance of the sun's disc. This he found to be uniform over the whole surface, although the surface at the periphery is perpendicular to the centre part. From this uniformity of luminance at the. periphery and centre L a m b e r t concluded that the luminous intensity is proportional to the apparent area as given by the product of the actual area and the cosine of the angle of radiation. This law, which we nowadays have little difficulty in understanding, was not regarded as being quite so apparent in Lambert's day, as illustrated by the fact that, according to so sagacious a man as E u I e r, luminous intensity was held to be independent of the position of the radiating surface and, hence, that the sun as a sphere produces just as much illumination on the earth as it would do if the hemisphere which we see were flattened out.

There are no light sources which conform wholly to Lambert's law; such hypothetical sources do, however, provide a very usefu1 basis for theoretical arguments, and we shall make frequent use ot them in this book. Light sources of this kind are called uniformly diffuse sources or uniform diffusers. Practical light sources radiating approximately in accordance with Lambert's la~ are called diffuse radiating or, briefly, diffuse light sources.

VIII-3. The luminous flux of uniform diffusers We have seen in the previous section that when a surface radiates in accordance with Lambert's law and has, therefore, the same' luminance in all directions, the luminous intensity I a. at an angle IX from the normal, is given by equation (VIII-3):

I a.= 10 cos ot.

By means of this formula we can now construct the light distribution curve of a uniform diffuser, which is found to be a circle, touching the radiating surface (see Fig. 53a).

VIII-3] THE LUMINOUS FLUX OF UNIFORM DIFFUSERS 107

From the right-angled £::.01)1Xin Fig.53a it will be seen that IIX = Io cos ex.

.------,tad' 1-----1/Sd'

1-----11200

Fig. 53. Light distribution (a) and Rousseau diagram (b) of a uniformly

diffusing surface

If we plot the luminous intensity in a Rousseau diagram in order to be able to calculate the lu­minous flux from the light distri­bution curve (Fig. 53b), we find that the line joining the ex­tremities of the intensity lines is straight; this may be veri­fied in the following manner. Draw a line from the end of the line representing I IX to P, and let e be the angle between this line and the Y -axis; then:

IIX I 0 cosex I 0 tan€J=--=--=-.;-. r cos ex r cos ex r

Since I 0 fr is constant, the value of e is the same for all lines joining the ends of each of the I IX lines to P; in other words all these lines coin­cide, or, there is only one (straight) line that can be drawn through the ends of the I IX lines. The area enclosed by this line can be calculated as being that of the right-angled triangle of height r and base I 0 •

This area is

!ri0 •

In accordance with equation (IV-2), this area must be multiplied by 2nfr to give the value of the luminous flux. Hence

2n (/) = - X !ri0 = ni0 • r

(VIII-4)

In the case of uniform diffusers, the value of the radiated luminous flux is n times the luminous intensity perpendicular to the surface. Conversely

(VIII-4a)

The luminous flux ((/}IX) between the angles of emission 0 and ex corresponds to the area enclosed by the lines I 0 and I IX in the Rousseau diagram

108 LUMINANCE AND LUMINOUS EMITTANCE [VIII

This area is a trapezoid, the parallel sides of which are I 0 and I IX and whose height is equal to r (I -cos tX). Hence this area is

!r (I -cos tX) (I0 +I IX) = !r (I- cos tX) (I0 + I 0 cos tX) = = !r (I- cos tX) I 0 (I +cos tX) =!rio (I--- cos2tX) = !ri0 sin2tX.

In accordance with section IV -I we now obtain the luminous flux by multiplying by 2nfr the area as computed above:

2n 2 I . </J - - X 1ri sin tX- n sm2tX IX- r 2 o - o •

(VIII-5)

Equations (VIII-4) and (VIII-5) can also be derived from the integral of equation (IV -4) which states that

IX,

.P = 27T J I IX sin IX diX. IX,

In the case m point I IX = I. cos IX, so that IX,

.P = 27T J I 0 cos IX sin IX diX.

Solution gives us: IX,

tP = 1TI0 sin2 IX J·

7T If 1X1 = 0 and IX2 = 9 , we obtain .P = 7Tl0 , if IX 1 = 0 and IX2 = IX,

tP = 1Tl0 sin2 IX. -

Examples: l. A surface radiating in accordance with Lambert's law and having

a luminous intensity of I 0 = 60 cd in the direction of the normal, will have a luminous flux of

tP = 7TI0 = 7T X 60-lm = 188.5 lm. 2. If the luminous flux of such a surface is 445 lm, the luminous in­

tensity I 0 in the direction of the normal will be tP 445

I 0 = -=- cd = 141.5 cd. 7T 1T

VIII-4. Luminous flux of a uniformly diffuse cylinder The light distribution of a luminous cylinder, e.g. a wire, whose luminance is the same in all directions, can be expressed by means of a simple formula if we may assume that the diameter of the cylinder is very small in comparison with the length of the cylinder; here, the axis of symmetry coincides with the longitudinal axis of the cylinder. Fig. 54a depicts the light distribution of a cylinder of this kind. The luminous intensity perpendicular to the wire is denoted by I 0 • At angle

VIII-4] UNIFORMLY DIFFUSE CYLINDER 109

oc from the normal to the wire, or at an angle of 90° -oc from the wire itself, the luminous intensity is (VIII-3)

I ex= Io cos oc. (VIII-6) In this we are departing from the usual practice of indicating directions of radiation by the angles which these make with the axis of symmetry; instead, we state the angle from the normal to the uniform diffuser.

A glance at Fig. 54a at once r-=-----.;:g~ shows that the corresponding

light distribution curve is a 1--------',-------1+30° circle touching the axis of

symmetry at 0. The solid of

.tk----.-+t--JOa light distribution is toroidal ~ in form.

_ -:30_o_~A l--=-~---tL___-----l-30° !~x d~~:rm~:~ue~he 0/u~in~~: 1----::7""------J-600 ~ '-=-----'-90° plotted to give the Rousseau

Fig. 54. Light distribution (a) and Rousseau diagram (b) of a uniformly diffusing cylinder

diagram shown in Fig. 54h; for convenience a luminous in­tensity scale is used such that Io = r.

The Rousseau curve is thus found to be a semi-circle and this is easily proved in. the following manner. In the right-angled !:c. PAB:

PB2 = PA2 + AB2 •

Here, PA = r sin oc and AB = I ex = ! 0 cos oc; hence PB2 = r 2 sin2oc + + ! 0 2 cos2oc. Since r = ! 0 we may write:

PB2 . r 2 sin2oc + r 2 cos2oc = r 2 , and PB = r.

The line joining P to the end of each line representing lex is thus equal tor and the curve will be a semi-circle of radius r. The area of this semi­circle is i:nr2 and this, multiplied by 2nfr gives a luminous flux of

2n C/J =- X !nr2 = n2r.

r

Since we have said that ! 0 = r, we may also put: (/J = n2Jo

or (/J

Io= -. n2

(VIII-7)

(VIII-7a)

110 LUMINANCE AND LUMINOUS EMITTANCE [VIII

Provided that the distance at which 10 is used is great enough, this formula may also be employed for illumination calculations as applied to a number of luminous cylinders parallel to each other, but naturally only when such cylinders are sufficiently widely spaced to ensure that they will not mask any appreciable portion of each other's light. A practical example of this situation is found in a straight filament vacuum lamp in which a number of thin filament sections are mounted almost parallel to each other. The light distribution curve of such lamps conforms sufficiently well to equation (VIII-6). In this 10 is the horizontal luminous intensity 1h of the lamp, and the luminous flux of such lamps can be calculated quite accurately from

(/) = n21h.

The spherical luminous intensity is (/)

1o=-4n and the ratio of spherical to horizontal luminous intensity is therefore

(/)

1o 4n n --I,, cp 4"

n2

This is the proof of what has already been said in section III-4.

Equation (\'III-7): <P = TT2l 0 can also be obtained from the integral of equation (IV-5) which states that

a, <P = 2'" J I a sin IX da.

a, It must be remembered that for this formula a is measured from the axis of symmetry, whereas, in the case of the luminous cylinder, a is measured from the normal to the cylinder; equation (IV -5) must there­fore be adapted to this method of indication of the angle. To avoid confusion, let us refer to angle f3 in (IV -5} instead of angle a. Then

{3, <P = 2'" j I f3 sin f3 df3, (IV -5a)

{3, in which, in the present instance: I f3 = I o cos a, f3 = 90'~- a, with a positive in an upward direction and negative in the downward direction; thus sin f3 = cos a and df3 = da Equation (I\'-5a)· then becomes

~2 ~2

<P = 2'" JI. cos IX sin (90°- IX} drx = 2'" JI. cos2 IX drx. IX,

VIli-S] LUMINANCE EMITTANCE

For the total luminous flux we have to integrate from

1X 1 =- 90° =- i and IX2 = +90° = + i: then +::: 2

<P = 211 JI. cos2 IX' diX, 7T

2 the solution of which is: +::: 2

<P = 11!0 {IX + t sin 21X) / = 112! 0 •

7T

2 Examples:

111

l. A straight filament lamp of 25 cd (i.e. horizontal luminous intensity I 11 = 25 cd) gives an approximate luminous flux of

cp = 7T2J11 = 112 x 251m= 247 lm. 2. A similar lamp, having a measured luminous flux of 155 lm will

have a horizontal luminous intensity of <P 155 ~ ! 11 = 2 = - 2 cd = 15.7o cd.

7T 7T

VIII-5. Luminous emittance The luminous flux emitted or transmitted by, or incident to a surface per unit area is known as the luminous emittance or, briefly, the emittance of that surface (Symbol H).*) We have already seen that the luminous flux per unit area incident to a surface is termed the illumination; illumination, therefore, is another term for the emittance of an illuminated surface. Mathematically, luminous emittance can be defined along the same lines as illumination:

(/> H=s·

(H = ~;)

(VIII-8)

(VIII-Sa)

The considerations mentioned in section VI-I concerning average illumination as well as the illumination of a point in a plane apply equally to the more general conception of emittance and need not be repeated here. Emittance is evaluated by means of equation (VIII-8) in lmfm2, lmfcm2

or lmfsq. ft; it is only in the case of the emittance of an illuminated surface that these units have been given a specific name, viz. lux, phot, footcandle. *} The quantity luminous emittance was formerly called radiance or radiancy. In 1951 the C.l.E. recommended the discontinuance of the use of the term radiancy and to use in the future the term luminous emittance or emittance.

112 LUMINANCE AND LUMINOUS EMITTANCE [VIII

VIII-6. Emittance and luminance of uniformly diffusing surfaces A surface of area 5, radiating in accordance with Lambert's law with a luminance L will have a luminous intensity in the direction of the normal of I o = L . S candelas. According to equation (VIII-4) the luminous flux of that area would be

$ = nl 0 = nL . S. (/J

Hence s=nL, (/J

\\'here S is the luminous flux per unit of area, that is, the emittance H.

\Ve may therefore write: H=nL,

H L=-.

n

(VIII-9)

(VIII-9a)

This means that the luminous emittance of a uniformly diffusing surface is found from the luminance by multiplying it by n. Conversely, the luminance is the quotient of the emittance divided by n. When applying equations (VIII-9) and (VIII-9a) we have to take due care of the units employed. If L is expressed in sb (cdfcm2), H will be in lmfcm2 • L expressed in cdfm2 gives H in lmfm2, and, if expressed in cdfsq. ft, lmfsq. ft. .

Examples: l. A surface radiating in accordance with Lambert's law, with a

luminance of 16 sb will have an emittance of H = 1r X 16 = 50.2 1mjcm2 •

2. Should the emittance of such a surface be 700 lmfm2, the luminance 700 of that surface would be L = - = 223 cdjm2 = 0.0223 sb.

1T

VIII-7. Units ofluminance, based on the emittance of uniform diffusers From the above remarks it follows that the luminance of a uniform diffuser is fully determined once the emittance is known. A luminance unit can therefore be established as being the luminance of a surface which, radiating in accordance with Lambert's law, has a certain emittance. Units of luminance are derived from three emittance units, viz:

from 1 lmfm2:

from I lmfcm2 :

from 1 lmfsq. ft:

the. apostilb (asb), the lambert (L or la), the foot-lambert (ftla).

VIII-7] UNITS OF LUMINANCE 113

The apostilb, the lambert and the foot-lambert can thus be defined as the luminance of a surface radiating in accordance with Lambert's law and having respectively an emittance of I lmfm2, I lmfcm2 and I lmfsq. ft. What is the relationship between these and the stilb?

H According to equation (VIII-9a): L = -.

;'1;

Substituting su,ccessively I lmfm2, I lmfcm2 and I lmfsq. ft, we then have

I 1 asb = - cdfm2

;'1;

1 la 1

=- cdfcm2 ;'1;

1 1 ftla = - cdfsq. ft

;r

~-- sb = 3.183 X I0-5 sb, n X 104

1 =- sb = 0.3183 sb,

;r

3.282 ~~cc-: sb = 3.426 x I0-4 sb. ;r X 104

If L be expressed in these units and H in the corresponding units lmfm2 ,

lmfcm2 and lmfsq. ft, the luminance is numerically equal to the emittance so we write:

L=H (e.g. L in cdfsq. ft and H in foot-lamberts).

In Germany the apostilb is used as the unit of luminance, but the C.I.E. does not accept this and its use is therefore not recommended. The lambert is employed in the United States; the millilambert (mL or mla) is also used (1 L = 1000 mL). In Great Britain the foot-lambert is the more current unit, other expres­sions for the same unit being the equivalent or apparent foot-candle. For the origi_n of these terms see section X -6 (p. 15 7). On p. 413 the current luminance units are listed in Table V together with their conversion factors. In Fig. 228 on p. 412 the various units in order of their size are shown on a logarithmic scale; the different ratios are indicated alongside the arrows which show the relative positions of the units in the system.

Example: If a luminance of 25 mL is mentioned in a publication, this luminance is (see Table V or Fig. 228).

25 X 3.183 X I0-4 sb = 7.96 X I0-3 sb or 25 X 3.183 cd(m2 = 79.6 cd(m2 (or 25 X 10 asb = 250 asb).

114 LUMINANCE AND LUMINOUS EMITTANCE [VIII

The association of the luminance units discussed above with the luminance of a surface radiating in accordance with Lambert's law does not imply that only the luminance of surfaces radiating in this manner, or nearly enough in this manner, may be expressed in the units given. The luminance of any radiating surface in any direction can be ex­pressed in these units, proceeding from the following argument. If the luminance of a given radiating surface in a certain direction is known, say n lamberts, this means that the luminance of that surface in that direction is just the same as that of a surface radiating in accordance with Lambert's law and having a luminance of n lamberts, i. e. with an emittance of n lmfcm2•

CHAPTER IX

NON-POINT SOURCES

IX-I. Luminous intensity of non-point sources The formulae developed in the preceding chapters all refer to light sources which can be regarded as a point constituting the apex of a solid angle*). The derivations have each time been based on the cone-shaped propagation of the luminous radiation, i.e. on the inverse square law. The definition of luminous intensity is also founded on the conical propagation of luminous flux from a point. In Ch. III (section III-I) this is defined as the flux per unit solid angle, the light source being considered to be a point source. In practice, however, all light sources have finite dimensions. When dealing with sources of very small dimensions we can at once accept the approximation to a point source, but it is not quite so simple when the definition recalled above is applied to large sources such as search­lights or tubular fluorescent lamps with their appropriate lighting fittings. As always, the terms large and small are only relative conceptions and it is necessary to indicate the standard against which light sources can be considered large or small. In our case we have to apply these terms in reference to the distance from the source to the observer, or to the illuminated surface. Such relativity will often be intuitive; we have only to think of the sun or the stars, which we so readily accept as point sources, i.e. small sources, whereas they are, in fact, of very large dimensions compared with artificial light sources. In the case of large sources such as the fluorescent lamps already mentioned, it is difficult to look upon these as being point sources in the sense implied in the definition of luminous intensity given in Ch. III, viz. as representing the apex of a solid angle. The question then arises whether this definition is acceptable for such light sources.

*) We have thereby strictly assumed, and also treated, the light source as a mathematical point (which has no area), not as a physical point (a plane element having an infinitely small area). In our considerations we should of course justify this inaccuracy, but it will be seen upon closer investigation of light sources of finite dimensions that the discrepancy has no practical significance.

116 NON-POINT SOURCES [IX

The answer to this can be given in reference to Fig. 55, showing a light

L' Fig. 55. Demonstrating the fact that the inverse square law does not apply when the light source is large compared with its dist­ance from the illuminated point

source LL' which can be imagined as disc- or line-shaped. It is assumed that LL' radiates uniformly diffusely with a luminance L. Let us denote the area of the light source by S; the luminous intensity perpendicular to the surface of the source will then be according to the considerations given in Ch. VIII:

10 =LX S. Now, is it permissible to employ this

same 10 when applying the inverse square law to compute the illumi­nation at the point P in Fig. 55? A glance at the figure will show whether this is permissible or not. A surface element at L, of a size LIS, will have a luminous intensity at right angles to the source equal to: LII0 = L. LIS. But, in the direction of P, the luminous intensity is only LI/IX =Lifo COS at= L. LIS COS at.

Hence the illumination produced at P by LIS is

L1J IX L1/o (. d ) E p = -- cos APL = -- cos4 oc since LP=-- . LP2 d2 cos oc

That part of the illumination at P which is contributed by the surface element LIS at L is therefore really smaller by a factor of cos4 oc than the value that we should iQclude when applying the inverse square law, using ! 0 and d. The closer the elements of LL' approach A, the smaller this error becomes, and it follows, therefore, that the error in the total illumination at P will be smaller according as the angle at is reduced, that is as the distance d becomes greater in comparison with LL'. It will be seen, therefore, that for computations of illumination values for distances which are not great compared with the area of the light source, the conception of luminous intensity as defined in an earlier chapter, in conjunction with the inverse square law, no longer holds. In our further discussions it will be found that this is not such a very great drawback, but it is nevertheless necessary to determine just how far the inverse square law is valid in practice. It is also possible to demonstrate along other lines the peculiarity of the conception of a point source assumed in the inverse square law and the definition of luminous intensity as applied to large light sources. Using the inverse square law as based on a cone-shaped propagation

IX-1] LUMINOUS INTENSITY OF NON-POINT SOURCES 117

of luminous flux we can compute the illumination at a point, i.e. a surface element at a certain distance from the light source. Such a surface element should be regarded as the base of a cone with the source at its apex. Actually the "apex" of the "cone" may be a light source possibly several feet across, the "base" being infinitely small. The conception and the representation of a light source as being the apex of a solid angle for the purpose of defining luminous intensity, is thus forced. Fortunately, however, there is a way out of the difficulty, by taking into account the manner in which the luminous intensity is measured; this is never actually carried out by measuring the luminous flux and the solid angle within which it is radiated, and then dividing the one by the other according to the definition. All luminous intensity measurements are based on measurement of the illumination, the inverse square law then being applied to compute the luminous intensity, by multiplying the illumination by the square of the distance from the point of measurement to the source, i.e. I = Ed2.

We then employ the value of I thus obtained to compute E for other distances, e.g. £ 1 = Ijd12•

If the result is to be accurate we must write Ed2 = E 1d12, in other words the product of Ed2 must be constant; but, if some error is per­missible, Ed2 must be constant within certain limits. The point is, when is Ed2 constant? From Fig. 55 we have seen that the error entailed by the use of the inverse square law is smaller according as the angle ot is made smaller, i.e. as d is increased with respect to the light source LL'; this means that the product Ed2 will approximate more closely to a constant value when d is increased with respect to LL'. This is demonstrated in Fig. 56 which shows the product Ed2 plotted as a function of d (full line curve A) for a white diffuse lighting fitting (depicted in Fig. 40), containing a 200 W incandescent lamp. Ed2 is thus obtained from the illumination as measured on the axis of the fitting, and the distances. It is seen that when d R::i 2 m, the product Ed2 reaches the constant value 690; this is obtained with a ratio of lighting fitting diameter to d of about 1 : 5. Thus, when applying the inverse square law to this fitting we may use for I the value of Ed2

(expressed in candelas), as measured at d >2m, whilst, for the same distances, the inverse square law can .be employed to calculate the illumination values. If an error of at most 5% is permissible in a com­putation of the illumination, Fig. 56 shows that d may then be 2 0.85 m.

118 NON-POINT SOURCES [IX

The lighting fitting, of which the Ed2 (d) curve is reproduced in Fig. 56, consists of a reflector containing an incan­descent lamp whose filament is an inch or two above the plane through the rim of the reflector.

The light-emitting parts of the reflector and lamp are there­fore displaced in the direction of the axis

cd ~ 00,.---·

7

""' 5

62 5

600 0

@Q_ --

I I

I I I

--~

A

/ v -8 f-· --/ ./'" ··-

/ /

/

2 3--d 4m

Fig. 56. Curves showing the product Ed2 as a function of the distance d. The two curves A and B refer to distances measured from different points of

to the extent of some the light source

inches in respect of each other, and this gives rise to the question as to what part of the reflector should be taken for the measurement of d. The solution lies in a comparison of the dotted curve (B) in Fig. 5& with the full curve (A). Curve A refers to a distance d measured from a point 4 em from the plane of the reflector rim; for curve B, d was measured from the actual plane of the rim. The illumination measured at every point in curve A is thus multiplied by the square of a "d" which is 4 em larger than in curve B. At every distance, therefore, the values of Ed2 are lower in B than in A, as will be seen from the diagram. When d = 4 m the difference is about 2%, corresponding to a difference in d of I% (400 and 404 em). It can also be seen from the. diagram that, when d = 4 m, curve B con-· tinues to rise and Ed2 has not by then become constant, whereas curve A is horizontal from d = 2 m, and Ed2 is already contant from that distance. The greater the distance d, the closer will B approach A until, when d is infinitely great, the curves coincide and assume the same value of Ed2. If d were measured from a plane more than 4 em within the reflector, the calculated curve would lie above A and, after first yielding higher values than 690 for Ed2, would drop gradually to reach that value at a certain distance. The point from which d is measured should thus be the one that gives the shortest distance at which Ed2 becomes constant. As a rule the location of this point is not of much practical importance, but it is, nevertheless, useful to realise the effect of the particular choice. The fact that the practical importance is not so great may be demon­strated in reference to Fig. 56. If E be measured at d = 4 m from the

IX-1] LUMINOUS INTENSITY OF NON-POINT SOURCES 119

plane of the reflector rim, curve B will yield a luminous intensity of I= Ed2 = 675 cd. Using this intensity to compute the illumination at 2 m, we obtain a value that is I.5% too high (we should, of course, have taken the value 664 from curve B, for d = 2 m). When E is com­puted with d > 4 m, the resultant value is slightly too low, since with this distance a higher value of Ed2 should really be employed. At the very most the error will be 2%, that is, the difference between 675 and 690 cd; an error of this order of size can generally be accepted in lighting engineering. When smaller distances d are taken for tl)e measurement, the error is increased, and it is important, therefore, that this distance should not be too small. The tendency of Ed2 to approach a fixed value when the distance d is increased may now be utilised to formulate a definition of luminous intensity which will apply to all light sources, whatever _their dimen­sions. This must be expressed in a mathematical manner. We can do this by saying that, whem d is increased, it will approach infinity, and at the same time the product Ed2 acquires a limiting value. The definition would thus read: The luminous intensity of a light source is the limiting value of the product Ed2 with an infinitely high value of d; as a formula this will take the form

I= lim Ed2•

d-+ 00 (IX-I)

As shown in the above example (Fig. 56), the expression infinitely long may be interpreted as long compared with the light source. In practice it is often sufficient to accept a value which differs from the theoretical limit, depending on the degree of accuracy required. Our new definition of luminous intensity is of the greatest utility when applied to "large" light sources, generally known as non-point sources, as distinct from point sources (for which a better term might be quasi point sources). In equation (IX-I) we may write for E:

E = $fS

where (/J is the luminous flux incident on an area of size 5. Hence

d2 1 = lim (/J .5 . d-~ 00

120 NON-POINT SOURCES [IX

Now, ~ is a solid angle (w), of which the apex is at the light source, d2 so we may put:

I= lim~. (IX-la) d--'?-00 w

The luminous intensity of a light source can accordingly be further defined as the limit of the luminous flux per unit solid angle to which that quantity approximates with increasing distance from the source, the apex of the solid angle coinciding with an arbitrary point in the light source.

Equation (IX-la) can be more suitably written in the form of a dif­ferential, viz:

. ..dt/> . L1 tJ> dt;l> I= hm -- = hm-- = --.

d-+00 L1Sfd2 .d(l)--+0 Lfw dw (IX-lb)

Since the definition of luminance is bound up with the luminous in­tensity, this definition should also be revised to agree with equation (IX-1). Instead of (VIII-2) we therefore write:

I Ed2 La. = a. = lim . s cos Ot d--'?-00 s cos Ot

(IX-2)

For the remainder of this chapter we shall concern ourselves with the derivation of formulae for the illumination produced by non-point sources that will be valid for any distance fFom source to illuminated area. In this we shall confine ourselves to uniformly diffuse circular and linear light sources and, having derived the formulae, consider under which conditions the inverse square law can be used. We shall then also discuss in how far these considerations are of value when applied to sources which are not perfectly uniformly diffuse. To arrive at our formulae we shall make use of the inverse square law when the light source is very small compared with the distance and, to this end, non-point sources will be regarded as consisting of a large number of point sources.

IX-2. The inverse square law in another form If we are going to use the inverse square law E = I fd2 for diffuse non­point sources, we may substitute L . S for I, where L is the luminance and S the area of the light source; thus

s E =LX d2•

IX-3] UNIVERSAL FORMULA FOR THE ILLUMINATION 121

In this,

L 0

Sfd2 is the solid angle w under which the source is observed from a distance d, assuming d great com­pared with S; hence

Fig. 57. The deciding ele­ments in calculation of the illumination at the point P are the aperture of the diaphragm D, the distance of the diaphragm from P and the luminance of the light-emitting surface L be­hind it, but not the distance

from L to P

E = Lw. (IX-3) This, then, is the inverse square law in another form, which leads us to an im­portant conclusion. Let us consider Fig. 57. This shows a plane L radiating uniformly diffusely with a luminance L. At a distance from L there is an opaque screen D having in it an aperture S. Suppose that we wish to know the illumination at P, at a distance d from D. If equation (IX-3) is used for this purpose:

E = Lw ( = L X ; 2).

It will be seen that, apart from the luminance L, the illumination depends only on the aperture in D and the distance from D to P. The distance between L and D is quite immaterial, provided only that L is so large that, from the point of view of P, the aperture will be completely "filled" by the luminance of L (the aperture must be completely "flashed"). The location of the limiting element in the space is quite unimportant; the only factor that determines the illumination is the solid angle subtended by the flashed aperture. With diffuse sources the illumination is unaffected even by the form in depth of the light-emitting surface; hence the illumination at P from the three sources depicted in Fig. 58

L is the same in each case provided that the luminance is the same. It may be superfluous to add once more that the equation E = Lw can only be used when the inverse square law is ap- L'

plicable. The conditions under which this Fig. 58. The shape in the depth of the light source LL'

is permissable will be investigated in the does not affect the illu-next section. mination at the point P as

IX-3. Universal formula for the illuini-

long as the apparent size is the same

nation produced by uniformly diffuse circular light sources

For the construction of this formula we shall make use of the law of

122 NON-POINT SOURCES [IX

reciprocity, the derivation of which is as follows. Suppose the luminance of two diffuse sources S1 and S2 of any given shape (Fig. 59) to be L1

and La respectively. S1 then re­ceives a certain luminous flux from S2 and so also does Sa from S1•

We now compute the luminous flux received by one from the other, for which purpose we first

d

10409

Fig. 59. The law of reciprocity

determine the amount of luminous flux which: an element Lf5a of the surface Sa receives from a similar element Lf51 in S1.

Let us denote the distance between these elements by d, and the angles between the normals to the elements and d by oc1 and oca respectively. The luminous intensity of .151 in the direction of L15a is then

L 1 • Lf51 cos oc1;

the illumination of L15a will be

L1 • Lf51 • cos oc1 • cos oca da

and the luminous flux received by Lf52 from Lf51

tP _ L 1 • .151 • .152 • cos oc1 • cos oc2 .ds, _.. .ds. - d2 •

Similarly, the luminous flux received by .151 from Lf52 is found to be

L2 • .151 • Lf52 • cos oc1 • cos !Xz tP .ds •. - .ds, = d2 .

Division then gives

tP .ds,- .ds. L1 tP .ds.- .ds, = L2 ·

The two luminous flux values are thus proportional to the luminance values of the radiating surfaces. As the two sources S1 and S2 can be imagined as composed of a large number of surface elements of which each and every combination pos­sesses the property just mentioned, this rule applies to all surfaces; it can be expressed as

tPs,-+- s. EtP .ds,-+ .ds. Ll tPs.-+-s, = EtP.ds,-+.ds, = L2.

IX-3] UNIVERSAL FORMULA FOR THE ILLUMINATION 123

This, then, is the equation for the law of reciprocity, which states that if any two diffuse light sources illuminate each other, they deliver luminous fluxes to each other in the same proportion as their luminance (or emittance) values. When the luminance (or emittance) values are the same, the luminous flux values are also the same. Using this law of reciprocity we can now very easily derive a formula

A

B

for the illumination produced by uniformly diffuse circular light sources. In Fig. 60 AB represents a light source of this kind, emitting light with a luminance L. Let us now compute the illumination perpendicular to CP at the point P; L APC =e. The point P

80410 must once more be regarded as a surface element, the size of which we shall denote by L1 S. Now, AB emits just as much luminous flux towards L1S as L1S would emit towards AB if L1S were

Fig. 60. Illustrating the derivation of the for­mula E = 1rL sin1 8

of the same luminance, viz. L. The last mentioned luminous flux can be very easily computed, for the luminous flux in the solid angle of half-apex angle e, produced by a diffusely radiating surface L1S of luminance L is (VIII-5):

fP = nL . L1S . sinz e.

This is therefore also the luminous flux radiated to L1S by AB. The illumination of L1S is now obtained as the quotient of the incident luminous flux divided by the area L1S; thus

E = nL sin2 e.

60 . AC h f (X In Fig. , sm e = AP' so t at we may put or I -4):

AC2 E=nL -­AP2

(IX-4)

where n AC2 is the area of AB; hence n L . AC2 is the luminous intensity I of AB according to the definition given in section IX-I. If we now denote AP by a, we can write in place of (IX-4):

I E=-. az

This is another inverse square law, which differs from that relating

124 NON-POINT SOURCES [IX

to point sources in so far that the distance is not measured perpen­dicular to the light source, but along a line drawn to the periphery of the circular source. If the source is small compared with the distance, the difference between the perpendicular and oblique distances will also be small, and the former distance may be used in the formula, in which case E = Ifa2

becomes E = lfd2• The size of the error thereby introduced is dis­cussed in section IX-6.

IX-4. Alternative derivations of the formula E = nL sin2 e We shall now describe another way of deriving the formula: E = nL sin2€J. Our object in so doing is that this method makes use of certain very interesting introductory considerations. Let us consider a sphere of radius R (see Fig. 61). It will be assumed that the sphere transmits light without loss. In the surface of this sphere, at A, there is a very small light-emitting area L1S radiating according to Lambert's law, with luminance L. We will now calculate the illumination on the surface of the sphere at any given point, say P. As L1S is assumed to be quite small com­pared with AP, the inverse square law may be applied. The luminous intensity of L1S in the direction of the centre M of the sphere is L . L1S. If L MAP = oc., the lu­

Fig. 61. A point in a uni­formly diffuse light-emitting sphere uniformly illuminates

the wall of the sphere

minous intensity Ia. in the direction of P will be Ia. = L. L1S. cos oc.. Since AB = R cos oc. and AP = 2 AB, we may write AP = 2R cos oc.. The illumination E'a. at P, measured perpendicular to AP is therefore

, Ia. L.L1S.cosoc. L.L1S Ea.=--= -----AP2 4R2 cos2 oc. 4R2 cos oc..

Since AP makes an angle oc. with the normal at P (i.e. the radius PM), E' a. must be multiplied by cos oc. to give the illumination E' p at P on the surface of the sphere:

L . L1S . cos oc. L . L1S E'p= ----4R2 cos oc. - 4R2 . (IX-5)

Since the angle oc. no longer occurs in this equation, it follows that the

lX-4] ALTERNATIVE DERIVATIONS OF THE FORMULA E = 1rL sins 8 125

illumination produced by a uniformly diffuse light-emitting part of a sphere is the same at every point on the sphere. If we now introduce the emittance H to take the place of the luminance, so that L = Hfn, we may express equation (IX-5) in the following form:

H. LIS E'p = 4nR2 • (IX-6)

The product H . LIS represents the luminous flux emitted by the surface element LIS, and 4nR2 the surface of the sphere. Equation (IX-6) may therefore be expressed in words as follows: The illumination on a sphere produced by a uniformly diffuse light-emitting element of the surface of the sphere is equal to the quotient of the luminous flux emitted by that element, divided by the surface of the sphere. It may also be said that the luminous flux emitted is uniformly distributed over the whole of the sphere. If we now consider a part of the sphere, consisting of a number of small elements, formula (IX-6) will apply to each such element. The total illumination produced by the particular radiating portion of the sphere's surface is thus the sum of the illumination values produced by the

individual elements. If the area of the s radiating portion be denoted by S = ELlS,

80412

then H. S = EH. LIS, and

(IX-7)

What has been said above regarding the uniformity of distribution of the luminous flux of an element over the whole surface of a sphere thus applies equally to the luminous flux of a part of the sphere's surface of any given size.

Fig. 62. A sphere at the periphery of a circular, uni- Let us now look at Fig. 62. This again formly dfffuse light source represents a perfectly transparent sphere (A C) is uniformly illuminated

throughout of radius R. The upper segment ABC emits (M c A 11 is t e r's equilux uniformly diffuse light with an emittance H.

spheres) The area of that part of the sphere is S = 'b&R. BE and this area is bounded by the circle AEC (radius r). The illumination EABC of the wall of the sphere produced by the segment ABC according to formula (IX-7) is

126 NON-POINT SOURCES [IX

H . S H . 2nR . BE BE EABC = 4nR2 = . 4nR2 = H 2R" (IX-8)

Let us now compute the illumination at the wall of the sphere below the circle AEC, supposing the spherical segment ABC to be replaced by a circular light source of the same size and in the same location as the circle AEC. This light source is assumed to have the same emittance H as the segment ABC. We wiU. now calculate the luminous flux passing through the circle AEC from the segment ABC. This is the luminous flux f/> ADC received by the remainder of the sphere ADC and distributed uniformly over it. Let the area of this part be S'; then

(/>ADC = EABC. S',

where BE

EABC = H 2R and S' = 2nR. ED.

Hence BE

f/> ADC = H 2R . 2nR . ED = H . n . BE . ED.

In the right-angled .6ABD, AE2 = BE . ED, so that we may write for (/> ADC:

(/> ADC = H . 'J'l • AE2 = H . 'J'l1'2.

This, however, is also the luminous flux emitted by AEC, assuming this to be a uniformly diffuse light source of emittance H. We see, therefore, that the illumination of the part of the sphere ADC, when illuminated by the circle AEC, is the same as when illuminated by the portion ABC. This result confirms the statement made at the end of section IX-2, viz. that the determining factor for the illumination is the solid angle subtended by the flashed uperture which is placed in front of a diffuse radiating light source. It is also clear that a uniformly diffuse circular light source uniformly illuminates any spherical surface passing through its circular bourtdary. The illumination of such spherical surfaces is the ·same at every point. M c A 11 i s t e r, who published these considerations in 1911, referred to such spherical surfaces as equilux spheres l). Equation (IX-8) for EABC can be written in another form by intro­ducing the angle e (B = L ADB = L BAE).

IX-4] ALTERNATIVE DERIVATIONS OF THE FORMULA E = '"L sin1 8 127

AD a Then BE= AEtan 8 = rtan8, and 2R = BD = -- = --.

cos e cos e Since the illumination produced by the circular light source AEC is identical with that produced by the segment ABC, the index ABC can be omitted, and we may then write:

A

BE rtan 8 r E = H 2R = H --a- =H.-;; sin 8.

cos e Now, rfa = sin 8; hence

E = Hsin2 8 = H

AsH= nL,

or:

E = nL sin2 8

r2 E = nL -

a2

a2·

Fig. 63. Illustrating the deri­vation of the formula

Since nr2 is the area of the circular light, sourc~ AEC, L. nr2 is the luminous in­tensity of this disc, so that it is per­missible to write:

E = '"L sin2 9 by means of the infinitesimal

calculus

I E=-.

a2

The formula E ·= '"L sin2 9 can also be derived with the aid of the infinitesimal calculus, as follows. The uniformly diffuse circular light source AB (see Fig. 63) illuminates a point P on the normal to the centre of the circle. We now want to know the illumination at point P perpendicular to the normal, as produced by the spherical segment enclosed within two circles of radii rand r + dr. All points on this ring are therefore similarly oriented with respect to P (we shall waive the calculation for a surface element and integration through 360° around the axis CP). The area of this ring is

dS = '" { (r + dr) 2 - r 2} = '" (r2 + 2r dr + d2r- r 2) = 2~ dr.

Now dr = _;___ da. - 1- = . r da. , sm a. cos a. sm a. cos a.

so that the area can also be expressed as

dS= .2'"r2da. . stn a. cos a.

128 NON-POINT SOURCES

The apparent area of the ring as seen from P is therefore dS' = . 2..,.• dr~. cos r~. = 2":"• dr~.

Sln fl. COS fl. Sln fl. '

and the luminous intensity in the direction of P is -dl = !_"~ ,.. dr~..

sm r~.

[IX

This luminous intensity produces an illumination at P at a distance rjsin r~., the light being incident at an angle r~. to the normal at P; the illumination at P produced by the ring is thus

2"L r• dr~.

dE = sin r~. cos r~. = 2"L sin r~. cos r~. dr~.. ,.. sin1 r~.

The total illumination at P produced by the surface AB as enclosed within the angle 9 is then

9 E = J 21rL sin r~. cos r~. dr~.,

0

the solution of which is: E = 1rL sin• 9.

IX-5. Luminance of light beams In section IX-2 we have seen that when a diaphragm is placed in front of a light source the diaphragm must be regarded as a source of light, having a luminance equal to that of the actual light source. It is a condition, however, that, in the direction in which the luminous intensity of the diaphragm is measured or computed, the diaphragm shall be completely filled with light from the source (completely flashed). This leads us to tbe conclusion that the quantity luminance need not be limited to an actual light source, but that it can be determined and stated in respect of any point or plane in a beam of light. In the example in Fig. 57, seeing that a layer of air is tacitly assumed to exist between Land n; in which practically no light is absorbed, the luminance of D. is equal to that of L. Should a light-absorbing medium occur between L and D, it would be necessary in equations E = Lw and E = nL sin2 e not to employ the luminance at L, but that at D.

IX-6. Comparison between the illumination values obtained from equations E = Lw and E = nL sin11 e

In the foregoing we have shown that the formula E = nL sin2 8 is valid for all circular light sources and all values of the angle e, that is to say, it yields the exact value of E for all ratios of light source diameter to distance, whereas formula E = Lw may only be used when w is small, i.e. when the light source diameter is small compared with the distance.

IX-6] COMPARISON BETWEEN DIFFERENT ILLUMINATION VALUES 129

oo~o~~~--~~~o~~ffi~o~--a~aoo

In order to ascertain the extent of the error entailed when formula

E = Ifd2 = Lw is applied instead of the exact expression

E = nL sin2 e we shall now compute the results for a range of ratios of light source diameter to distance, using the two forms of equation. For E = lfd2 = L. w we may write E = nL tan2 e, so that we obtain for comparison E = nb sin2 e and

E = nL tan2 e. 1' 1•5 .3f The comparative results are illustrated

82544

1:2 1:20f.IO 1:5 1:4 1:3

Fig. 64.' Demonstrating the amount of the error when the illumination produced by a uniformly diffuse circular light source is calculated by means of the inverse square law instead of the exact formula

E =TTL sin2 8

in Fig. 64 for angles e of 0 to 20°; below the diagram a number of ratios of light source diameter 2r to the distance d are also given. In this figure the error as a percentage, involved when the inverse square law is used in place of the exact formula, is plotted vertically; this error is

tan2 e- sin2 e --:--::---=--- X 100% (= tan2 @ X 100%).

sin2 e Since tan e = rfd, we may compute the error as (rfd) 2 X 100%, or, if we denote the ratio 2r: d with q, as !q2 X 100%. It will be seen from the figure that a ratio of 1 : 7 entails an error of l% when the inverse square law is employed. A ratio of 1 : 5 shows an error of about 1%. Should an error of 5% be permissible, and this is usually the case in rough practical calculations, the use of the inverse square law can be extended to distances corresponding to rather more than twice the diameter of the light source. This comparison demonstrates once more the significance of non-point, or large light sources, viz. sources of which the dimensions are large (or preferably "not small") compared with the distance from the surface which they illuminate. The "largeness" is therefore relative and does not refer to the absolute dimensions of the source.

130 NON-POINT SOURCES [IX

The term point source *) is applied to a source whose dimensions are so small in relation to the distance from the illuminated surface that the product Ed2 approximates to its limiting value. "Large" light sources can be regarded as point sources in conjunction with large distances, whereas a "small" source may be looked upon as "large" when the resultant illumination is to be measured only a short distance from the source.

IX-7. Some special applications of the formula E = nL sin2 8 = Ifa2

a. When 8 = 90° as applied to equation IX-4, in which case sin 8 = 1, the formula reverts to E = nL. This happens when a point in a plane is enclosed by a light source of luminance L (see Fig. ~5). The illumination on the plane AB is at all points nL. For L we may put Hfn, so that E = H (E in lux, H in lmfm2, or E in fc and H in lmfsq. ft). It may be said that in this case A+----...I...J::l----:!:

the emittance in the plane AB is equal to that of the dome-shaped light source (illumination is, after all, a particular case of emittance). But it is not only in the plane AB that E, and hence also the emittance, are at all points the same and equal to the emittance of the dome; this applies in every horizontal plane above AB.

80414

Fig. 65. The illumi­nation produced by a uniformly diffuse dome­shaped light source is equal to the emittance

of this source

Accordingly it may be stated that, within the space between the light-emitting dome and the plane A B, the emittance of the light passing downwards through every horizonta~ plane is constant. Taking the luminance of the sky to be uniform, we can accordingly compute the illumination in any open space by means of the formula E = nL = H or, conversely, compute the luminance of the sky from the measured illumination. From the considerations mentioned in section IX-5 it follows that we need not ask ourselves how the sky should be represented as an actual light. source.

*) In lighting technology another meaning of point source is also recognised, namely a source that is perceived by the eye as a point. The image of such light sources as formed on the retina is so small that it occupies only one of the light­sensitive elements of the retina. A reduction in the size of the source, or an in­crease in the distance from which it is observed, does not therefore alter the ap­parent size of the (point) source. When the si?=e of the source is increased, or the distance reduced, the point as seen by the eye only becomes larger when the image on the retina covers more than one of the light-sensitive elements.

IX-7] SOME SPECIAL APPLICATIONS OF THE FORMULA E = nL sin1 8 131

b. Fig. 66 shows a spherical light source, the luminance L of which is uniform. Let the radius· of the sphere be r. Using the exact formula

p

we shall now compute the illumi­nation perpendicular to MP at a point P located a small distance d from M.

10415

Now, E = nL sin2 8 = nL d2 , m

Fig. 66. The illumination from a uniformly diffuse spherical light source may be computed by means of the inverse square law in respect of any distance, when the distance from the illuminated point is measured from

the centre of the sphere

which nr2 is the apparent area of the sphere as seen from infinity or, in more practical terms, from a point at a great distance from the sphere. nr2L therefore represents the luminous intensity of the sphere, in

accordance with the definition in section IX-1. Hence we may write: I

E=-d2'

the distance being measurea trom the centre of the sphere. With spherical light sources of uniform luminance, then, the inverse square law applies for all ratios of light source diameter to distance, when the distance is measured from the centre of the sphere.

IX-8. Illumination produced by a uniformly diffuse circular light source in planes parallel to the source

Before working out the illumination in such cases, let us first consider one or two further points in connection with M c A 11 is t e r's equilux spheres. In Fig. 67 AC represents a uniformly diffuse circular light source. We have already shown in section IX-4 that the illumination over the surface of all per­fectly transparent spheres coinciding with the circle AC is constant. The illumination

Vmax

is therefore computed for a plane tangent- Fig. 67. Demonstrating that the horizontal illumination on

ial to the sphere at P. an equilux sphere is equal to What is now the horizontal illumination that on the wall of the sphere at P? To obtain this we make use of the following characteristic of illumination (see Fig. 68 ).

132 NON-POINT SOURCES [IX

If we introduce a very small surface .15 in any beam of light and then rotate this surface so that the illumination on it is as great as possible (Emaz) (full line, Fig. 68), the illumination Ea. in any other position of the surface is equal to Ema., times the cosine of the angle (ex} through which the surface is rotated from the optimum position.

--~ ~-... _.. --

~s

Fig. 68. If at any point in a beam of light optimum illumination occurs in the plane shown by the elliptical full line, the illumination in a plane at angle a. to the optimum plane is

Thus Ea. = Ema,. COS ex. (IX-9)

The illumination Eex follows this rule up to the direction in which part of the light beam falls no longer on the surface under conside­ration but on the back of it. This theorem can be proved in the following way:

Ea. = Emaz COS a.

The luminous flux incident to the surface AS may be regarded as con­sisting of an infinitely large number of beams with an infinitely small solid angle, which can be looked upon as emanating from an equal number of point sources. If such a small beam, or a number of them with their axes parallel, be allowed to fall on a surface which is rotated about any line in its own plane, the illumination will follow a cosine curve. This can be seen from Fig. 69, in which the line LP represents a beam of light coming

J; L ---

- -- '::...'--'/

from a light source L and il­luminating the point P in the plane V 1. If we denote by E the illumination at P as measu­red perpendicular to LP, the illumination E1 of V1 at P will be E cos y1, where y1 is the angle between LP and the normal LA1 to V 1 .

If we now rotate the plane vl about the line QR through an angle b and denote the plane in the new position by V 2, the

Fig. 69. Illustrating the derivation of the illumination E 2 of V2 at P will formula Ea. = Emaz cos a. be: E cos Y2, where Y2 is the

angle between LP and the nor-mal LA2 to V 2•

IX-8]

Hence

UNIFORMLY DIFFUSE CIRCULAR LIGHT SOURCE

E cos y 2

E cos y1

cos 1'2 cos 'Yl

133

(IX-10)

We now draw lines A1B and A2B perpendicular to QR and denote the angles PLB, A1LB and A2LB by {3, cx1 and cx2; according to section VI-3 we may then write:

cos y1 = cos cx1 cos {3, and cos y2 = cos cx2 cos {3.

Substituting these expressions for cos y1 and cos y2 in equation (IX-I 0) we obtain:

cos cx2 cos {3 cos cx2

cos cx1 cos {3 cos cx1 (IX-II)

The angle of rotation is here <5 = cx2 - cx1 •

Suppose that cx1 = 0, in which case the plane is perpendicular to LB; then cos cx1 = 1 and <5 = cx2 - 0 = cx2• In this position the illumination at P is at its maximum when the plane is rotated about QR. Let us denote this value by Emaz; then equation (IX-11) becomes

E 2 cos cx2 -- = -- = cos <5 Emaz 1 '

or E = Emaz COS <5.

Fig. 70. Since the component parts of the illumination elements at a point describe cosine curves (C1 , C2 , C3} ac­cording to the position of the illuminated surfaces, the total illumination also

follows a cosine curve (C)

The variation in the illumination produced by the elementary beams of light accordingly fol­lows a cosine curve. For beams of different directions the po­sition at which E = Emax also differs, and the curves for beams in three different directions may be represented as the cosine curves C1, C2 and C3 in Fig. 70. The total illumination is then represented by adding together the cosines of the individual bf!ams. It can be shown mathe­matically that the resultant curve C is also a cosine curve;

134 NON-POINT SOURCES [IX

in accordance with equation (IX-9) the total illumination is thus E = Ema:x: COS a;,

It should also be pointed out that the cosine can become negative but illumination cannot. The validity of expression (IX-9) is therefore restricted to the angular range in which the rays of the beam fall only on the front of the surface under consideration. In the case of Fig. 70, the total illumination thus follows this formula only between the directions indicated by A and B. Let us now turn once more to Fig. 67. If we can determine the position of the plane for Ema:x: in respect of P, we can compute the horizontal illumination at P from the illumination at the wall of the sphere at P. Now, the beam which illuminates P is symmetrical with respect to the plane of the drawing and to a plane through BP and perpendicular to the plane of the drawing (arc AB = arc BC, and accordingly L APB = L BPC). The maximum illumination Ema:x: at P therefore occurs in the plane V ma:x: perpendicular to BP, so that BP is the normal to V ma:x:

at P. The normal to the surface of the sphere at P is the radius MP and the normal to the horizontal plane H at P is FP. Further, L MPB = L MBP = a: and L FPB = L MBP = a:, so that the horizontal plane at P makes the same angle with V ma:x: as the tangen­tial plane to the sphere. Hence the horizontal illumination at P is equal to that at the wall of the sphere, and the characteristic of M c A ll i s­t e r's equilux spheres is just as applicable to the horizontal illumination at all points on the surface of the sphere as to the illumination of the wall itself. This characteristic is of value in computing the illumination in planes parallel to uniformly diffuse circular light sources. In Fig. 7 i, AB represents a circular source of radius r and luminance L. We wish to calculate the illumination E p at the point P in a plane parallel to the source, at a distance CD = d from AB. The distance from P to CD is p. A sphere is imagined, passing through P and the periphery of AB, this being shown in Fig. 71 as a circle having M as centre and R as radius. This sphere is then an equilux sphere of which the illumination is E = nL sin2 8 (8 = L CFB).

f Fig. 71. Illustrating the deri­vation of the formula for calculating the illumination produced by a uniformly dif­fuse circular light source in planes parallel to the source

IX-8] UNIFORMLY DIFFUSE CIRCULAR LIGHT SOURCE 135

Since the horizontal illumination at P is equal to that on the wall of the sphere at P, it can also be said that

Ep = nL sin2 e.

To express sin2 8 in terms of d, r and p we draw the following lines in the diagram: BP, HGJJDP, so that BG = GP, MG = Rand BIJJCF.

Now sin2 8 = t (I- cos 28).

The angle 28 is obtained from !::::,. CMB:

CM cos28 =If·

In the similar triangles HGM and IBP:

HM : IP = GH : BI,

where r+P IP = p-r, GH = - 2 - and BI =d.

(IX-12)

(IX-13)

Inserting these in equation (IX-13) we find on solving for HM that

CM is obtained from

p2-y2 HM= -2d-.

P2 y2 J2 + p2 y2 CM = CH + HM = J.d + --- = 2 2d ----2~d~--

and the radius R can be calculated from !::::,. CMB:

It is now possible to compute cos 28 ( = c:), insert this value m

equation (IX-12) and then multiply by nL, which finally gives

E p = !nL 1 - _ _ . [ J2 + p2- y2 ] V (d2 + p2 _ r2)2 + 4d2r2

(IX-14)

136 NON-POINT SOURCES [IX

By means of equation (IX-14) the variation of the illumination in a plane parallel to the light source can be computed, and the results for a number of different ratios pjd and rjd are shown graphically in Fig. 72. The illumination is here given as a function of the ratio pjd; it is expressed as a percentage of the illumination in the centre below the light source.

E

fOOr--.~=r------.------.------.------. 0/o

60o~----,a~,,----,a~,2~--~a~.3~--~o~~~----~as -Pfd

Fig. 7~2. Illumination produced by a uniformly diffuse circular light source in planes parallel to the source, as a function of the ratio of distance (p) of the point from the axis of the light source, tu the height (d) of the source above the plane. The resultant illumination is illustrated for various ratius of the radius (r) to the height (d) of the source. The illumination with pjd = 0 is taken to be 100% for each ratio of r: d. The values for rjd = l/00 correspond to those as computed by means of the inverse

square law

The line for the ratio rfd = I joo represents the illumination produced by a point source, the light distribution of which is that of a plane light source radiating in accordance with Lambert's law.

IX-9. Illumination produced by linear light sources Among other forms of non-point sources, linear sources have become particularly interesting since the introduction of the tubular fluorescent lamp. Let us now see how the illumination produced by such sources can be computed. As the diameter of the tube is small compared with the length, such light sources may be regarded as linear, i.e. in .the computations it may be assumed that the light is concentrated along a line, viz. the axis of the tube. In Fig. 73, AB represents a linear light source suspended parallel to

IX-9] LINEAR LIGHT SOURCES 137

a plane H; we require to know the illumination at a point P in this plane. The radiation from AB is uniformly diffuse anu the solid of light distri­bution will accordingly be toroidal (see section VIII-4). Let the length

B

Fig. 73. Calculation of the illumination produced by a linear light source (AB) at any given point (P) in a plane

of AB be l and the luminous flux f/J; the luminous flux per unit length is then f/J1 = f/Jjl. The maximum luminous intensity 11 of a part of AB one unit in length will then be 11 = fPJn2 = f/Jjln 2•

A convenient formula for the illumination at P is obtained if this is expressed in terms of the height h of the source above H, the perpen­dicular distance PC = a of P from the source, and angles ex1 and ex2 •

These are the angles subtended by the parts of AB on each side of C, with respect to P (i.e. L L APC and CPB in Fig. 73). We shall compute E by calculating first the contribution dEa. of a (cylindrical) surface element, dl in length, towards the illumination at P, and then integrating dEa. between the angles ex1 and ex2• The element under consideration is shown at D in the figure. The distance DP is denoted by r and L QDP by fJ. In the length of the element, dl, an angle dex is subtended with respect to P. The luminous intensity of dl in the direction of P is

dla. = 1 1 dl cos ex,

(since DP is at an angle ex from the normal to AB at D). The contribution dEa. made by dl towards the total illumination at P in the plane H may be represented by

dla. I dE a. = -cos fJ = _~: dl cos oc cos {J,

y2 y2 (IX-15)

138 NON-POINT SOURCES [IX

in which r, dl and cos fJ can be expressed in terms of h, a and oc:

a r a h h r = --, dl = -- doc= ---doc, cos fJ =-=-cos oc.

cos oc cos oc cos2 oc r a

Substitution in equation (IX-J 5) then yields:

- h 2 dErx - I 1 -cos oc doc. a2 (IX-I6)

The total illumination at th~ point P is evaluated by integration between the limits oc1 and oc2, taking into account the fact that in Fig. 73 the angle oc2 is positive, so that oc1 must be negative. (For the perpendicular PC, angle oc = 0). This gives the equation:

h I I !oc, E = I 1 - - 2oc + sin 2oc a2 4 IXt

(IX-I7)

in place of which we may write:

h I [ . . ] E = I 1 a2 4 2 (oc2 - oc1) + sm 2oc2 - sm 2oc1 • (IX-I7a)

If h and a are in metres and I 1 in cdfm, E will be in lux; with h and a in em and ! 1 in cdfcm, a factor of I ()4 must be included to give E in lux. h and a in feet and I 1 in cdjft will give E in foot-candles, oc must be expressed in radians.

Example: Suppose that six 40 W tubular fluorescent lamps are mounted in a horizontal line. Let the luminous flux per lamp be 3000 lumen. We will assume that the lamps emit uniformly diffuse light. The length of each lamp is 4 feet and the space between them 2n, so that the luminous flux

per foot of tube, including the gaps is <P 1 = ~~~8° = 720 lmjft. The

luminous intensity per foot is then / 1 = 7~0 ~ 73 cdjft. These lamps '" are assumed to be suspended 12 feet above a horizontal surface.

The arrangement is shown in Fig. 73; P will b.e so located that PR = 10ft and AC = 15ft. In the figure, then, AB = 6 x 4If8 ft = 25ft (BC =10ft), h = 12 ft. In order to apply equation (IX-17a) we must first compute PC= a, and the angles lXI and ot2 :

a= PC= Vh2 + PR2 = Vl22 + 102 = 15.62 ft. BC -10

tan lXI = a = 15.62 = -0.64; OC1 = -0.57 rad.

AC 15 ta:1 OC2 =a= 15.62 = 0.96; 1X2 = 0.765 rad.

IX-9] LINEAR LIGHT SOURCES 139

All the values can now be filled in in equation (IX-17a) and this gives:

12 E = 73 X 15.621 X

x t { 2 (0. 765 + 0.57) + sin 2 X 0. 765- sin 2 (-0.57)} = 4.12 fc.

Let us now consider the universal formula (IX-17) in the special case where the point P at which the illumination is to be computed is located at a distance a from the light source that is small compared with the length l of the source. Here

n n oc1 = - 2 and oc2 = 2, so that

h n E = Iz. a2. 2"

Since ~ = cosy (see Fig. 73), this may be written as a

n 11 E = - . - . cos y. 2 a

For a surface element perpendicular to the shortest line between it and the light source, y = 0, and we then find that the illumination is inversely proportional to the distance from the source:

E = ~. !J = 1.57 !.J. 2 a a

(IX-18)

(/1 in cdfm and a in metres gives E direct in lux; if / 1 is in cdfcm and a in em, the result must be multiplied by 1()4 to give E in lux. 11 in cdjft and a in feet gives E in fc). Equation (IX-18) can also be derived in a more simple manner. The tubular light source of length l gives a luminous flux of

We now imagine a cylinder of radius a around the light source and concentric with it. If l is very long compared with a, it may be said that the whole of the luminous flux from the source will fall on the cylinder. The illumination of the cylinder wall is therefore found by dividing the luminous flux f/J of the source by the area 2nal of the cylinder, viz:

140 NON-POINT SOURCES [IX

E = _!!__ = n 2I 1 X l n !.J 2nal 2nal 2 a

which is identical with equation (IX-18). A diagram can be constructed for the practical application of equation

(IX-17), in which the value of the expression ! 120t + sin 2at 1:: is

plotted as a function of Ot. Values read from this diagram are then multiplied by I Na2 to give the value of the illumination. A further simplification can be effected in the application of this formula by calculating values of at2 - at1 which will represent the same con-

tribution towards the value of the expression ! 120t + sin 2at 1:: 2).

IX-10. Comparison of illumination values of linear light sources as obtained by exact calculation with those obtained by means of the inverse square law

In the same way as we have done for large circular light sources, we shall now ascertain the extent of the error involved when the illu­mination of linear light sources is computed with the aid of the inverse square law instead of accurately by means of equation (IX-17). When the inverse square law is employed, the luminous intensity is regarded as being concentrated at the centre of the line (in tubular lamps mid-way along the tube). This point is considered to be the point whence the light is emitted and it is therefore the point to which the inverse square law refers. Calculations will be made for fwo different sets of conditions, and these are now ex­plained in reference to Figs. 74 and 75.

Case 1 (see Fig. 74). A horizontal light source AB, the length of which is l, illuminates a horizontal sur­face H. The height of the source above H is denoted by h. The luminous intensity per unit length is I 1•

Fig. 74. Calculation of the illumination produced by a linear light source (AB} in its projection on a plane

parallel to it

We wish to compute the illumination at a point P, vertically below the centre of the source AB on the plane H. AB subtends an angle of 2at at P.

IX-10] LINEAR LIGHT SOURCES 141

In order to keep the argument general, calculations will not be made for specific dimensions of h and l, but for a range of ratios h : l. Applying equation (IX-17) to Fig. 74 we have

E = I 1 X : 2 X tl2oc + sin 2oc 1:: ; a= h and oc1 and oc2 are numerically equal, so that equation (IX-17) becomes

E = t X l (2oc + sin 2oc). (IX-19)

l Angle oc is found from tan oc = 2h. The factor l (2oc + sin 2oc) can now

be evaluated for various ratios of h to l. If AB be regarded as a point source at C, the formula for the inverse square law E = Ifd2 will contain I= I 1 X l and d = h, in which

0_--

Fig. 75 Calculation of the illumination produced by a light source AB at a point P in a plane parallel to the source, where

PQ = QR = CR = h

case I, . l I 1 l

E=J;2=;;·~t· (IX-20)

There is a third method of com­puting E, that is by means of equation (IX-18). The light source is here regarded as being infinitely long compared with the height h. Formula (IX-18):

n I 1 E = 2 . ;- then

. _ n I 1 form. E - 2 . },;

assumes the

(IX-21)

Equations (IX-19), (IX-20) and (IX-21) differ only in the factor by which I)h is to be multiplied to yield E, and these factors accordingly determine the difference in the results as obtained by these three methods of calculation. If we now work out these factors for a range of ratios h : l they can be plotted graphically as a function of hfl. With hfl on the horizontal axis, the vertical axis will show the illumination E divided by I 1fh. Figure 76 shows this for ratios hfl of from 0.1 to 10 (curves I), which means a

142 NON-POINT SOURCES [IX

variation in the height of the light source of from 1f10th of the length to 10 times its own lengt.h. From the figure it will be seen that the results are quite ac- E/f!. ceptable for technical purposes 10

as regards their accuracy when the inverse square law is used 5

for distances greater than twice the length of the source (with 2

"

hfl = 2 the error is about 4%). ..---,'( With distances less than roughly --n•

1'-. 'lj ....

' r-:: F- -c~

" .).;

one quarter of the light source length it can be said with suffi­

qa kcosacr c6s4f II

][

I 2a.+sin 2a /

--- -

""' " ..... cient accuracy that the illumi­nation is inversely proportional to the distance, in which case equation (IX-18) can be used. 0,1 18(.aa, .aa,+sL.aa.-sin~ Case 2 (see Fig. 75). 0, 0

,a Here the situation as regards light source and illuminated plane a is the same as in the above

-·

a ,Of instance, but we now wish to 0,1

know the illumination at the o,5

- -~

' '

" I\

10 G718J

point P of which the location in the plane H is determined by RQ = PQ =h.

Fig. 76. Illumination values for different ratios of h : l, relating to t~e conditions depicted in figs. 74 and 75. The values on the ordinate are not those of the

In order to apply formula.(IX-17) we determine oc1 and oc2 from

actual illumination, but of Eft (11 = luminous intensity per unit length of the

BD h-0.5l tan oc1 = DP = hy' 2 and

light source)

AD h + 0.5l tan oc2 = DP = hy' 2 a= DP = hy'2.

For (IX-17) we can therefore put:

E = I 1 X _h2 X ! [2oc +sin 2oc]ex• = I 1 X~ l2oc +sin 2oc]ex'. 2h ex, h 8 ex,

(IX-22)

If AB is regarded as a point source, equation (VI-5) can be used, viz.

IX-10] LINEAR LIGHT SOURCES 143

I E = - cos3 oc cos3 {J in which in the present instance h2 '

I = I 1 • z cos e, oc = <5, fJ = e, thus giving us

I . l I l E = - 1- cos3 <5 cos1 8 = __! • - cos3 <5 cos4 8.

h2 h h (IX-23)

Equations (IX-22) and (IX-23) thus again differ in the factor by which Izlh must be multiplied to give the value of E. These factors have also been calculated: for a series of ratios of h : l (between hfl = 0.1 and hfl = 10), and the results are depicted in Fig. 76 as a function of hfl (curves II). This diagram shows that the error involved in the approximating method of calculation is smaller than in the first instance. In general it may be said that the inverse square law yields values which approximate sufficiently closely for technical purposes to the exact values when the mounting height is more than twice the length of the source.

IX-II. Other kinds of non-point source Since the wa11s and ceiling of a room reflect light falling upon them, thus adding to the illumination in the working plane, it is also of interest to find formulae for the evaluation of this illumination as well as for the luminous flux radiated towards the working plane. For this purpose the reflecting surfaces are regarded as uniformly diffuse sources of light and, in practice, this is a fair approximation. Such sources of light are usually rectangular or square in shape; the plane in respect of .which the illumination is to be computed, when illuminated by a ceiling, is parallel to the plane of the light source, or, when i11umina­ted by the walls, perpendicular to the plane of the source. The method of arriving at the relevant formulae lies outside the scope of this book, however 3).

I X -12. The significance of the foregoing considerations as applied to practical forms of light source

a. The exact formula for the illumination produced by circular light sources (E = nL sin2 8) is not often used in practical work. This also applies to the form E = I Ja2•

The reason for this is that the luminance of practical forms of circular

144 NON-POINT SOURCES [IX

light source are hardly ever uniform, moreover their radiation is often not uniformly diffuse. A contributory reason is that the degree of error permissible in lighting calculations is usually so generous that for most distances the inverse square law yields sufficiently good results. It will be seen later (Chapter XI) that in optical systems (lenses etc.), the exact formula is used to give more accurate results. For calculations of illumination as produced by reflecting walls and ceilings, in which case the source of light is large compared with the distance, the inverse square law is not accurate enough, and the exact formula€ have to be employed. The shape of such reflecting surfaces is nearly always rectangular however, for which reason the exact for­mula for circular light sources is very rarely applicable in such cases as well. 1he significance of the formula and the considerations regarding its use, as far as the usual circular primary light sources are concerned, are centred mainly in the fact that the distance at which the inverse square law is applicable can be ascertained as an approximation, so that discretion must be exercised in the use of this law. Should the lighting engineer wish to calculate illumination values with an error of not more than 5% he will know that he can make use of the inverse square law when the distance is, roughly, 'not less than twice the overall dimensions of the light source. If data are needed in respect of shorter distances it will usually not be possible to calculate these, since the exact formula holds good only for perfectly uniform diffusers with uniform luminance; details can in such cases be obtained only by measurement. If we need to measure the luminous intensity of a light source and accordingly obtain this from the product Ed2, we must know roughly from what distance the measurement of the illumination must be carried out in order to remain within a certain percentage of the limiting value of Ed2•

If the permissible error is 1% we know that the distance at which the measurement is to be effected must be at least 5 times the diameter of the light source; for !% error the distance is lO times the diameter. As shown in section IX-1, when deciding upon the measuring distance we must take into account the particular point on the lighting fitting from which the measurement is to be made.

b. The practical point of view with regard to linear light sources is rather different from that adopted for circular sources. The emission of light from many present-day kinds of light source approximates

IX-12] PRACTICAL FORMS OF LIGHT SOURCE 145

so closely to uniformly diffuse radiation that it is worthwhile using the exact formula for short distances. On the other hand, when considering the lighting fittings in which such light sources are mounted, we cannot apply the exact formula and direct measurement must be resorted to. Here again, the value of the theoretical considerations lies more in the possibility of ascer­taining roughly the distance at which the inverse square law is still valid, with a view to the required accuracy of the results.

c. As already mentioned in a) above, the usefulness of the exact formulae for rectangular light sources lies in their application for calculating the illumination produced by walls and ceilings. In these cases the ratios of the dimensions of the light source to the distance are never such that the inverse square law will yield a sufficiently accurate result. The formulae can be used successfully for computing the coefficient of utilisation of lighting equipment in enclosed spaces.

REFERENCES 1) E. D. M c A IIi s t e r, Trans. I.E.S. Am. 9, 1911, 703-721. "The Law of

Conservation as applied to Illumination Calculations". See also: N. A. H a 1-b e r t s m a: "Der Lichtstrombegriff und seine Anwendungen", Berlin, 1921, p. 34 e.s.

2) H. Z ij I, Ph. T. R. 6, 1941, 147-152. "The Calculation of Lighting Instal­lations with Linear Sources of Light"

3) H. Z ij I : "Manual for the Illuminating Engineer on Large Size Perfect Dif­fusors". Philips Technical Library, Eindhoven, 1959

CHAPTER X

REFLECTION, ABSORPTION, TRANSMISSION

X-1. Reflection, absorption, transmission

When light passing from one homogeneous medium falls upon the interface between this and another medium, part of it is reflected and the rest passes into the second medium; the light entering the second medium is then partly or wholly absorbed. Any light that is not absorbed may pass through the medium, and this is referred to as transmitted light. These phenomena are accordingly known as reflection, absorption and transmission, respectively. The ratio of the reflected luminous flux (<Pe) to the incident flux is

. <Pe called the reflect~on factor (e); hence e = q;·

The ratio of the absorbed luminous flux (<Pa.) to the incident flux (<P) <P

is the absorption factor (oc); oc = ;.

The ratio of the transmitted luminous flux (<P-r) to the incident flux (<P) <P

is the transmission factor ( -r); -r = <PT

Since <PP, <Pa. and <P.T are together equal to <P:

For opaque substances, i.e. substances which do not transmit light, -r = 0; hence

e+oc= I.

Almost all substances which are opaque in the thicknesses in which they are normally used will transmit light when they occur in very thin layers; this can be observed in incandescent lamps which have a silver or aluminium mirror on the inside of the bulb; the filament is o.rten visible through this layer.

X-2] REGULAR REFLECTION 147

X-2. Regular reflection

A parallel beam of light falling on to a polished plane surface of glass or metal is reflected as a parallel beam. Before and after reflection the rays lie in the same plane through the normal at the point of incidence, and the angles made by the incident and reflected rays with the normal are equal (angle of incidence = angle of reflection). This kind of reflection conforms to the optical laws and is known as

A

specular or regular reflection. What is now the relationship be­tween the luminous intensity and luminance of the reflected light and the characteristics of the light before reflection? Fig. 77 shows a mirror M which gives

- ~- -- ~- -- -- ~ M' ==---~~~p~"""""";;.;------ regular reflection. A beam of light

1 l ~f -M from a source A of luminance L l ! /;1 falls on a point P (infinitely small I ,,!/ / area of M); the solid angle of the tV beam is w. A'

Fig. 77. The luminance of an image A', and of a point P on a mirror M, is equal to that of the reflected light source, multiplied by the reflection

factor of the mirror

In studying the reflection of the beam, P must be regarded as a small plane mirror lying in the plane M' which can be drawn tangentially to the convex mirror at P. The normal at P is PN. The light rays are

reflected symmetrically with respect to the normal, so that the reflected beam subtends the same solid angle as that of the incident beam. What, then, is the luminance as observed by the eye when looking into the reflected beam in the direction of P? The emittance at P (i.e. in this case the illumination) is H = Lw cos Cl.

If the observed luminance be denoted by L', then the emittance at P after reflection will be H' = L' w cos Cl.

Now, if we denote the reflection factor of the mirror by e, the luminous flux, and therefore also the emittance of the reflected light at P, will be equal to (! times that before reflection, i.e.

H' = (!H, or L' w cos Cl = (!Lw cos Cl.

Hence

L' = eL. (X-1)

148 REFLECTION, ABSORPTION, TRANSMISSION [X

This can be represented in the following manner. The flat mirror M' produces an image A' of A, and the luminance of the image is reduced by the mirror to a value QL. The plane element P may be regarded as a diaphragm which is flashed with the luminance of A'. The luminance of the image of the light source and that of the mirror are accordingly equal to the luminance before reflection has taken place, multiplied by the reflection factor of the mirror. This luminance is thus found to be independent of the emittance, i.e. the illumination, of the mirror. As this rule applies to every point or surface element of a specular surface, it is valid also for the whole surface, irrespective of the form of the mirror. With the aid of this law it is possible to compute the luminous intensity of curved mirrors; these can be regarded as light sources, the luminance of which is L' = eL. When the area S of the mirror is completely flashed with the reflected luminance of the light source, the luminous intensity is

I = SL' = SeL.

Exactly what is meant by mirrors and lenses being "flashed" is dealt with in chapter XI (section XI-6). Let us now see how the luminous intensity of a light source is changed by regular reflection. In Fig. 78 a point source A is depicted as being reflected by a plane mirror M. The optical laws tell us that the plane mirror forms an image A' of the light source A, whereby A' and A are symmetrically located with respect to the reflecting surface. What we wish to know is the luminous intensity of the image A' in the direction of A'R. The line A'R passes through the reflecting surface M at the point P, and the reflection in the direction of A'R thus takes place at the point P. Let I denote the luminous intensity of A in the direction of P; then the emittance H (the illumination) at P IS

I H = AP2 cos IX.

A

A'

Fig. 78. The luminous intensity of an image A' of a light source A as formed by a plane mirror M is equal to that of A, mul­tiplied by the reflection factor

of the mirror

If we denote the luminous intensity of A' in the direction of P by I',

X-3] REFLECTION FACTORS OF NON-CONDUCTIVE MATERIALS 149

the emittance at P will be I'

H' = A'P2 cos rx,

or, smce A'P = AP, I'

H' = AP2 cos rx.

As the emittance after reflection is: H' = eH, we may write: I' I

AP2 cos rx = 12 AP2 cos rx,

so that I'= ei. (X-2)

In the case of plane mirrors the luminous intensity of the image of a light source is equal to that of the source itself multiplied by the reflection factor of the reflecting surface. The most well known surfaces which produce regular reflection are those of polished metals, and glass with silvering on the back. Unsilvered glass will also produce specular reflection if the surface is quite smooth; so also will opal glass, glazed porcelain, vitreousenamel etc.

Examples: l. If in Fig. 77 L = 800 sb and p = 0.9:

L' = pL = 0.9 X 800 sb = 720 sb. If a curved mirror of circular cross-section 10 em in diameter, i.e. of which the apparent surface perpendicular to the boundary plane is ~ x· 102 em2 , is flashed with a luminance L' = 720 sb, the luminous intensity of this mirror (perpendicular to the boundary plane) will be

I = i X 102 X 720 cd = 56,500 cd.

2. If a light source of 250 cd luminous intensity is reflected by a plane mirror of which p = 0. 75, the luminous intensity of the image of the source will be

I' = pl = 0.75 x 250 cd = 187.5 cd.

X-3. Reflection factors of non-conductive materials

The reflection factor of a regularly reflecting surface is dependent on the angle· of incidence of the light. Now, there is a characteristic difference in this relationship as between electrical conductors and non-conductors.

150 REFLECTION, ABSORPTION, TRANSMISSION [X

In non-conductors which do not absorb light or of which the absorption is only slight*), the reflection factor at different angles of incidence is governed by the refractive index, formulated by F r e s-n e I as n,

[sin2 (i- r) tan2 (i- r)] (!; = ! sin2(T-=i--~) + tan2 (i + r) (X-3)

where e; is the refleGtion factor for an angle of incidence i, and r is the angle of refraction. The refractive indices determine the relation­ship between i and r according to Snell's law (see Fig. 79), viz.

Fig. 79. Snell's Jaw of refraction:

sin i sin r

sin i n'!

sin r n 1

(X-4)

where n1 and n2 are respectively the refractive indices of the media through which the light travels before and after refraction. If the first medium is a1r, n1 = I and equation (X-4) reads:

sin i . . . -- = n or sin t = n sin r. sin r '

In the process of reflection the light is polarised. The first term of (X-3) represents that part of the reflected light for which the electrical field is perpendicular to the plane of incidence, the second term to that part whereby this field is parallel to that plane. This polarisation can only be demonstrated by means of special apparatus such as the Nicol prism or polarising filter; the human eye is not capable of appraising the polarised condition of light, for which reason the two terms in equation (X-3) are made additive.

With perpendicular incidence equation (X-3) becomes

eo= (::~::r (X-5)

which, for n1 = I (air) and n2 = n, gives

eo=(:+ ~r (X-5a)

*) By "slight" is meant that the light absorbed in passing through a layer equal to the wavelength of the light is relatively only very little, e.g. at most 2 or 3 per cent.

X-3] REFLEXION FACTORS OF NON-CONDUCTIVE MATERIALS 151

Formula (X-5) can be derived from (X-3) by using the angles them­selves instead of the sines and tangents; for very small angles this is permissible.

When i = goo, i.e. with glancing incidence, e90 = I, in which case the reflection is complete. The curve representing the reflection factor of glass with a refractive

0,8

0,6

0,4

0,2

__ ,. 13"111 25°22' 34"511 40°5"31

6°39' d2B' 00'22 38oa9' '49' 41°

1 j_ I

1 I

/ l----'

index of 1.5 in air is shown in Fig. 80 (the fact that the refractive index differs according to the wavelength of the light is here disregarded). It is seen that, when i = 0°, glass gives about 4% reflection; when the incidence of light to a glass pane is perpendicular, 4% of the light is reflected from the front. Some of it is absorbed (very little, e.g. about 2% for a given thickness), so that about g4% falls on the rear surface, where 4% is again reflected, i.e. almost 4% of the incident light. In

0 0 300 600 .soo all, then, the glass reflects some 8%.

Fig. 80. The reflection factor P; as a fm1ction of the angle of incidence i on reflection of light from air to glass with n = 1.5. \\"hen light in the glass falls on the interface glass-air, p; is obtained by reversing

r and i in the chart

Up to an angle of about 60° there is hardly any increase in the reflection factor, but thereafter the factor rises quickly to unity at goo. When light is reflected in glass or water by a surface backing on to air, the values of i and r are reversed in equation

(X-3). In this case the curve in Fig. 80 will also give values of e if, instead of the angles i, the corresponding values of the angles r are plotted horizontally. This has been done in the figure, where the angle r

is shown at the top of the diagram. (! then attains unity at the critical angle, which is determined by

. sin goo stn r = ---- = -.

n n

If the angle of incidence exceeds the critical angle, the reflection factor remains unity, such being encountered in what is known in optics as the totally reflecting prism in which the angle of incidence at the hypotenuse face is 45° and thus greater than the critical angle.

152 REFLECTION, ABSORPTION, TRANSMISSION [X With non-conducting substances, therefore, the reflection factor must be stated in reference to the angle of incidence at which it is measured. If this is measured with light incident from all directions, i.e. with diffuse light, the reflection factor will be greater than with light incident at an angle of from 0 to 45°.

X-4. Reflection factor of conductive materials The reflection factor of electrically conductive materials, i.e. metals, too, is dependent on the angle of incidence. The region within which the reflec­tion factor is practically constant ranges from the perpendicular almost up to glancing incidence, in the neighbourhood of which direction it rises rapidly to unity when incidence is fully glancing, as in non-conductive materials.

For metals of sufficient thickness (e.g. > 1 Jlm) cc = l- p. Now the absorption factor is equal to the emission factor (<),that is, the emittance of an incandescent substance in a certain direction, divided by that of the black-body at the same true temperature and in the same direction. Black-body radiation is quite diffuse, but that of all metals is not wholly diffuse, and the extent of the departure from the diffuse is manifested in the glancing directions of the radiation. Whereas at radiating angles of 0° to nearly 90° the emission factor is fairly constant, it decreases as 90° is approached more closely and is zero at 90° (for, if p -+ I, l- p = cc = < -+ 0). This means that at 90° the absorption factor is also zero and the reflection factor accordingly unity.

X -5. Diffuse reflection Most surfaces found in practice do not give regular reflection; they spread or diffuse the light in all directions, and we accordingly speak of diffuse reflection. For the moment it will be sufficient merely to mention this fact; later we shall consider more closely the manner in which diffuse reflection occurs. Similarly, we speak of the diffuse reflection factor, by which is meant the ratio of the diffuse reflected luminous flux to the incident flux*). The ratio of regularly reflected luminous flux to the incident flux is known as the regul~r reflection factor. Whereas a surface producing regular reflection appears bright only in the direction of the (regular) reflected light, a diffuse reflecting sur­face reveals a certain luminance in all directions. The distribution of the light reflected by a diffuse reflecting surface

*) Older publications employ the term albedo, i.e. whiteness in the same sense as diffuse reflection factor. This word is still current in astronomy, but should not be employed in illuminating engineering.

X-5] DIFFUSE REFLECTION 153

element can be determined. Usually only one cross-section of the solid of light distribution (which is generally axially symmetrical only in the case of perpendicular incidence) is reproduced, i.e. the most character­istic, this being in the plane passing through the direction of incidence and the normal to the surface. When dealing with reflection and transmission the term light distribution

Fig. 81. Luminous intensity iridi­catrices. a and b are examples of spread reflection; c is the indicatrix of a uniformly diffusely reflecting

surface

Fig. 82. Luminance indicatrices derived from the luminous intensity

indicatrices in fig. 7a, b and c

curve is not used; in its place we use the term characteristic diffusion curve or indicatrix of diffusion. This can be plotted on either polar or cartesian co-ordinates. This curve can give the distribution of the luminous intensity of the reflected light, but we can also plot the luminance. In the latter case we will plot the relative luminance in respect to a certain illumination, prefer­ably per lux, or foot-candle. We accordingly determine the quotient of luminance divided by illumination for a certain direction of observation

154 REFLECTION, ABSORPTION, TRANSMISSION [X

and incidence. This quotient is called the luminance factor ({3); further reference will be made to this concept in section X-7. Two luminous intensity indicatrices are shown in Figs. 8la and b and these also indicate the direction in which the light is assumed to be incident. Figs. 82a and b depict the corresponding luminance indica trices. It is seen that the maximum luminance of the reflected light occurs in directions which make roughly the same angle with the normal to the surface as the incident light. The form of the diagram becomes less and less dependent on the direction of incidence according as the extent of the diff~sion is increased, and we can therefore visualise the case where the luminous intensity indicatrix is a circle, being entirely independent of the angle of incidence. We are then concerned with a uniform diffuser (Figs. 81c and 82c); the surface accordingly emits light in accordance with Lambert's law.

In his book "Photometria" mentioned earlier (see p. 8), Lambert formulated the law that diffuse reflecting surfaces (and also radiating surfaces such as the sun and incandescent solid bodies) have the same luminance in all directions. This is still sometimes referred to as Lam­b e r t's law. Lambert discovered it experimentally. Owing to the crudity of the apparatus with which he worked, he did not observe the departures from this law, particularly at glancing directions of observation. Furthermore, L a m b e r t worked with directions of incidence not far removed from the normal, for which this law certainly holds good in fair approximation. ·

The two extreme types of reflection are accordingly the regular and the uniformly diffuse, but neither occurs in perfect form with any known kind of surface. Even the most perfectly ground and polished mirror produces some, albeit a very small amount, of diffuse reflection in adqition to regular reflection. Between these two extreme, hypothetical kinds of reflection there are the more practical forms, and also combinations of more or less regular and diffuse reflection. If the luminous intensity indicatrix of diffusion differs to any great extent from a circle we use the term spread reflection; where a surface produces not only diffuse or spread reflection but also regular reflection, we refer to it as being mixed reflecting. The luminous intensity indicatrices of many surfaces are, for perpen­dicular incidence of the light, practically circular, the reflection being then almost uniformly diffuse. As the direction of incidence departs from the perpendicular, the form of the indicatrix differs more and more from the circular. With oblique incidence, therefore, all diffusely reflecting surfaces show spread reflection. This is in many cases of considerable importance. The high luminance observed in certain directions on

X-5] DIFFUSE REFLECTION 155

lighted roadways is due to the fact that the road surfaces which with perpendicular incidence give almost uniformly diffuse reflection, are strongly spread reflecting when the incidence is oblique. With· perpendicular incidence, surfaces of frosted opal glass, powders, matt paint etc all produce approximately uniformly diffuse reflection, but in every case spread reflection occurs with oblique incidence. Other surfaces, such as the matt side of frosted glass, give spread re­flection at all angles of incidence. All such surfaces giving diffuse reflection are known as matt.

Mixed reflection is produced by the smooth side of frosted glass, un­frosted opal glass, glazed porcelain, vitreous enamel, glossy paper etc. Diffuse reflection can arise in two different ways: 1. As a consequence of the fact that a surface is not smooth. Such

Fig. 83. Heflection from uneven surfaces

in metals with a matt surface.

surfaces may be regarded as consisting of a great number of small planes, all at different angles to each other, so that their normals all assume different positions. Now, if these small planes themselves reflect regularly, parallel rays falling on them are reflected in different directions and are thus dif­fused. This is illustrated schematically in Fig. 83. Such conditions are found

2. With inhomogeneous media at the surface of which diffuse reflection occurs, but which nevertheless transmit light to a certain .extent, so that some of the light penetrates the medium. That part of the light is then refracted and again reflected by the crystals or small homo­geneous areas within the medium, and some of it is thrown back; such light undergoes many changes in direction in its devious path through the medium and it is diffuse when it finally leaves the medium. This occurs with frosted opal glass, matt paint, powders and so on. In mixed reflection we are concerned with regular reflection at the surface, combined with diffuse reflection of the kind just described in 2) above. We shall now look a little more closely into the reflection characteristics of diffusely reflecting surfaces as a whole*).

*) Later, in section X-8, we shall revert to the manner in which diffuse reflection takes place at a surface-element.

156 REFLECTION, ABSORPTION, TRANSMISSION [X

The form of the reflection indicatrix of a matt surface depends on the structure of the surface, as well as on the angle of incidence of the light. Such surfaces exhibit small crests and troughs. With perpendicular incidence both the crests and troughs arc illuminated (Fig. 84a); .the former produce no masking effects and no shadows are formed that will reduce the luminance in any direction. The reflection indicatrix is thus axially symmetrical. When incidence is oblique, however, only those sides of the crests which

a b Fig. 84. With perpendicular incidence (a) both.· the crests and the troughs are il­luminated. With oblique incidence (b) only one side of the prominence is illuminated, and that side which faces the light source is seen to have a certain luminance. At the tops of the crests regular reflection takes place, so that the surface a lso appears light when regarded from the side remote

from the light source

face the light source are il­luminated, the far sides re­maining dark (Fig. 84b). This does not mean, of course, that the surface as observed against the light appears to be dark, for the tops of the crests, which are always more or less rounded, reflect the light fairly regularly (Fig. 84b). Such reflection can be quite con­siderable, since the reflection factor of non-conductors is high with glancing incidence. The form which the reflection indicatrix assumes depends

upon the depth of the troughs and the rounding of the crests. Deep troughs and sharp crests result in much light in the direction of the light source itself, whereas shallow troughs and flat crests give more light in the other direction. The more oblique the incidence of the light, the more these differences are accentuated and the lower the luminance of the reflected light in the direction of the normal to the surface. We can differentiate between micro- and macro-structures; the first refers to details that cannot be discerned with the naked eye, and the second to discernable irregularities. Surfaces exhibiting micro-structure without any macro-structure can be termed smooth; those with macro-structure are rough. Usually, rough surfaces also reveal micro-structure. In general, the troughs in smooth surfaces are shallow and the crests fairly flat, so that, when 1ight falls on them obliquely, the reflection in the direction away from the light source predominates. Only with glancing incidence is an appreciable quantity of light reflected in the direction of the source.

X-6] UNIFORM DIFFUSE REFLECTION 157

With rough surfaces the troughs may be so deep that the luminance as seen from the light source is greater than in the opposite direction,

1 even when the angle of + incidence is not so great.

This is illustrated in Fig. 85 which is taken from an article by

-60° J an sen 1).

Fig. 85. Luminance indicatrices of a rough surface which; with oblique incidence of the light reflects more light in the direction of the light source than

in the opposite direction. ---- perpendicular incidence - - - - - - angle of incidence 30° -·-·-·- angle of incidence 60°

Incidence in the direction of the arrows

From the foregoing it follows that it is pos­sible to influence the reflection characteristics of a material with mi­cro-structure by model­ling its surface.

X -6. Uniform diffuse reflection

This mode of reflection is mathematically defined and it may therefore be used as a basis for calculations. Perfectly uniformly diffuse reflection does not occur as such in practice, but many matt-surfaced materials approximate to the hypothetical conditions very closely, at all events when the direction of incidence is not too oblique. In many instances, therefore, we can base our computations with sufficient accuracy on the assumption of uniform diffuse reflection. As mentioned in the previous section, a surface that constitutes a uniformly diffuse reflector is to be regarded as a secondary light source radiating in accordance with Lambert's law*). The characteristics of such light sources have been dealt with in Chapter VIII, so that we can now employ the equations already derived. When a luminous flux f/> falls on a uniformly diffusely reflecting surface of reflection factor e, the reflected luminous flux is ecf>. According to equation (VIII-4a) the luminous intensity in the direction of the normal

to the plane is then ! 0 = ecf> and in the direction including an angle IX :n;

with the normal e. cJ>

Ia. =--cos IX. :n;

*) As applied to secondary light sources we speak of remission of the light, in contrast with emission from the primary sources.

158 REFLECTION, ABSORPTION, TRANSMISSION [X

If we compute these quantities per unit of area, the illumination E is substituted for <P and the luminance L for I. Then, per unit area, the reflected luminous flux will be eE, and the luminance, which is the same in all directions. is

L = eE_ n

(X-6)

If E is stated in lux, L will be in cdfm2• To obtain L in stilb (cdfcm2)

a factor of 10-4 must be added. E in fc gives L in cdjft2•

If E is expressed in lux and L in asb, n disappears from the equation, according to section VIII-6; hence L = eE. The same applies with L in ftla and E in fc. In the case of a surface for which e = 1, L = E, that is, the luminance in asb is numerically equal to the illumination in lux. Reflecting surfaces which are perfect diffusers, i.e. which reflect 100% of the light, are sometimes known as perfectly white, for which reason the asb was formerly known in Germany as Lux auf Weiss. An analogy with the linking of the luminance unit to the illumination unit through the concept of the 100% diffuse reflecting surface is to be found in the expression equivalent, or apparent footcandle for the foot­lambert.

X-7. Luminance factor

Let us now once more consider the concept luminance factor, the de­finition of which is given in section X-5 as the quotient of the luminance divided by the illumination for a certain direction of observation and a certain direction of incidence. This definition may be stated in the form of an equation as

L Po,I = E (X-7)

where the indices 0 and I indicate that {3 relates to the directions of observation 0 and incidence I. If we express L in cdfm2 and E in lux, {3 will be in (cdfm2)/lux; L in cdjft2 and E in fc give {3 in (cdfft2)/fc. Once f3o,I has been determined, the luminance in the direction 0 with the direction of incidence I can be obtained from the illumination:

L = {30 ,1 X E (X-7a)

in which {3 is a factor. The luminance factor fJ is also defined as the ratio between the luminance

X-7] LUMINANCE FACTOR 159

L of a sample for a given direction of observation and the luminance L of the perfect diffuser (Lw = canst.; Pw = 1) for the same illumination. Thus

p = LfLw.

Because L = Efn (Lw in cdfm2, E in lux) applies to the perfect diffuser, we can also write

P = 1t X LfE (cdfm2flx) The luminance factor thus defined, therefore, differs from the one first defined by the factor 7t. Furthermore, P = LfLw is an abstract number, while with P = LfE the units (e.g. cdfm2flx) must be given. The luminance factor is primarily used in the reflection from road sur­faces, and it is easier in this case to work with P = LfE (cdfm2flx). In the foregoing we have repeatedly spoken of one certain direction of incidence and one given direction of observation. It should be borne in mind, however, that the light falling on a point in a plane is always a pencil of rays with a finite solid angle, i.e. that it embraces a large number of different directions of incidence. Again, the acceptance surface of a photometer as used for the measurement of luminance, or the eye with which such luminance is observed, always intercepts a pencil of rays having a finite solid angle containing thus a great number of different directions of observation. Of these numerous directions of incidence and observation, one is taken as the direction of incidence, and one as the direction of observation, these being characterised by angular co-ordinates. It is obviously convenient to choose the axes of the solid angles of both the incident light and the light entering the eye or photometer for this purpose. In the hypothetical case of one single direction each for incidence and

L

Fig. 86. Principle of measurement of lu­minance factor. LL' light source. FF' acceptance surface of the photometer. 8 ac,ceptance angle of the photometer

observation these directions might be defined as the axes of infinitely small solid angles. In practice, however~ we in­variably employ various direc­tions of incidence, and various directions of observation, at the same time. There is a third factor to be taken into account, namely the size of the surface involved in the measurement. For dif­ferent points in this surface

160 REFLECTION, ABSORPTION, TRANSMISSION [X

the directions of observation differ and therefore also the luminance factors may differ. In principle the conditions under which the measurements arc taken are as shown in Fig. 8_6. The surface involved in the measurement is AB. LL' is the light source and FF' the acceptance surface of the photo­meter. By means of this arrangement we now wish to measure the luminance as observed by the eye in the "main direction" CE, of a surface element (point) C, illuminated in the "main direction" DC. Now, the illuminating directions vary, from L'A to LB, and the ob­servation directions from AF' to BF; in effect, then, we measure in this way the average of various luminance factors, and only if this average is equal to the luminance factor in respect of the main directions DC and CE will the result be correct. The fewer the rays deviating from the main directions that are included in the measurement, the more closely will the average {J approximate to the true {J relating to the main directions. In practice the main illuminating direction DC can be approximated by lighting the surface under measurement with a so-called parallel beam, this being obtained by placing a "point source" at the focus of a lens. It is true that the rays from the lens are not all parallel, but the axes of the solid angles of all the beams reaching the surface are parallel*). This method of illuminating an area may be represented diagrammatically in the manner shown in Fig. 87. Every point on the surface is thus illuminated by beams having the same solid angle and whose axes are pa­rallel. We shall call these solid angles the solid angles of illu­mination (wr)· The angle included by those rays which deviate most from the main direction during measurement of the luminance we shall call the acceptance angle ( 8). By reducing WI and e the true value of {J for the main directions DC and CE will be more nearly approximated.

Fig. 87. In a "parallel" beam of light the axes of the solid angles of illumi­nation w1 are parallel, and the solid

angles are equal

When the luminance indicatrices for a giveri surface are not too pointed,

*) This is referred to again in Chapter XI, (section Xl-6).

X-7] LUMINANCE FACTOR 161

i.e. when they include an area in which {3 is fairly constant, it will be noticed that if wr and f9 be reduced, a point occurs at which any further reduction yields no further change in the value of {3. In reducing wr and I or 8, therefore, we have a means of ascertaining whether the measuring arrangement used will yield values of {3 which can be used to compute the luminance as will be observed by the eye. It is not necessary, however, to reduce wr beyond the solid angles of illumination that occur in practice. If the luminance factor of a surface is constant for all directions within a certain acceptance angle, the luminance in that angle will be constant for a certain illumination, and the surface can be regarded as a light source radiating light of that luminance in the direction of observation. We can therefore apply the inverse square law to that surface as a light source, taking into account, of course, the question as to whether it may be regarded as a point source or a non-point source. When a surface, however, gives regular reflection, the inverse square law applies with the image as light source; in this case no fixed value of the luminance factor can be attributed to the surface. Hence, if the reflecting surface is a light source to which the inverse square law may be applied, i.e. if it can be regarded as the surface from which the rays originate, it will also have a luminance factor. If the inverse square law cannot be applied to the surface as a light source, i.e. the image behind the surface must be regarded as the origin of the rays, the luminance factor will have no fixed value. When the reflecting surface is a perfectly plane mirror then the position of the image is clearly defined; if, however, the surface is specular but gives a distorted image owing to irregularities in the surface, the point from which the rays originate is not fixed but may vary; here again, no fixed value of the luminance factor will be found. With mixed reflection, the reflecting surface being a true plane, the diffuse and specular reflections can be determined separately by making use of the difference between the points of origin of the diffuse and regular reflections. If the regular reflection is not completely regular, however, this cannot be done.

162 REFLECTION, ABSORPTION, TRANSMISSION [X

X-8. The luminance factor as a constant of the material*)

If we wish to regard the luminance factor as a constant of the material, it is necessary to recall the fact, already mentioned, that surfaces which reflect specularly have no luminance factor. No matter how far the values of Wz and e are reduced, no constant value of {J will be obtained, from which it may be deduced that a lu­minance factor can exist as a constant of the material only if reductions in wz andfor e will at a certain point yield a constant value of {J.

Naturally, there are limits, imposed by the technique, beyond which "'I and B cannot be reduced.

Now, between (hypothetical) uniformly diffusely reflecting surfaces and specular surfaces there are very many intermediate forms, and we shall now see for which of these there is, and for which there is not, a luminance factor. With uniformly diffuse reflecting surfaces the luminance is independent of the directions of incidence and observation; hence {J is independent Of (JJI and e. If the reflection factor be denoted by (!, we know from equation (X-6) that

eE L e L =- and {J =- =- (cdfm2)/lux

:n: E :n:

for all directions of illumination and observation. It will be clear that for surfaces whose reflective properties do not differ much from the uniformly diffuse, the values of Wz and fJ are not critical in a determination of {J. The position is different, however, in relation to surfaces giving a de­cidedly spread reflection and, when the incidence is very oblique, with surfaces giving almost diffuse reflection. In order to see what happens here let us consider the mechanism of reflection from a small area (i.e. the influence of the micro-structure). In their simplest form, the surfaces of materials consist of particles (granules) of various sizes. In general, the surfaces of these particles are curved, but they may also have flat facets with random orientation. Not all the tops of the particles will be on the same level; they exhibit miniature crests and troughs. Such surfaces are matt.

*) The term "constant of the material" is used here in another meaning from the usual one; the luminance factors are not constants of the material whatever its form, but only in relation to its actual surface.

X-8] LUMINANCE FACTOR 163

An example of this is the type of surface obtained by smoothing out the surface of a powder. The situation is entirely changed, however, if a binding medium is in­troduced between and over the grains. When light falls on a surface of this kind: I) it is reflected regularly from the surfaces of the grains and binder; 2) it penetrates both the grains and the binder; in the former repeated

refraction and reflection diffuse the light, part of which re-emerges from the surface.

This is what takes place in paints in which a resin or oil, which is in itself transparent and usuc~.lly nearly colourless, lies over and between the grains that give the paint its colour (the pigment). The surface of the binder will more or less follow the contours of the grains of pigment; the less it follows the outline of the grains of pigment, the more glossy it is, up to the point where the binder itself, with a perfectly smooth surface, will give regular reflection.

In the case of particles of the same order ot size as the waYclength ot the light, what is known as Rayleigh diffusion occurs instead of reflection and refraction. Each particle diffuses the light in all directions, and the laws of regular reflection no longer apply.

Of the light that emerges after penetration and diffusion in the particles it is clear that the points of origin lie in or on the particles, i.e. that

A

Fig. 8H. Diagram illustrating the mechanism of diffuse reflection: a) when the radii of the particles are of the

same order of size as the particles themselves·

b) when the ;adii are large compared with the particles

L L L A

B

164 REFLECTION, ABSORPTION, TRANSMISSION [X

the surface functions as a light source to which the inverse square !aw applies and for which a luminance factor can be established. In reference to the diagrams in Figs. 88a and b let us now consider in outline how reflection takes place at the surface under the different cir­cumstances. Fig. 88a shows a few particles adjacent to each other, these being drawn as spheres for the sake of simplicity. In practice they would not of course be true spheres nor exactly next to each other; the situation as depicted should be regarded as a model to which our considerations will apply. For our purpose, which is merely to explain qualitatively those facts already learned in practice, the situation represented in the figure is quite justifiable. Let us take the case of reflection from the centre sphere in Fig. 88a. Light falls on the sphere in the direction LM, M being the centre of the sphere. Owing to spherical aberration the spherical surface does not produce a sharply defined image of the light source. It may be said that the sphere forms an image of the light source on the line LM for every point of incidence. Since the exact positions of the images do not really affect our arguments, which are only qualitative, the com­posite image as a whole may- be denoted by L'. L' is thus the point from which a pencil of rays emanates, the size of this being determined by the amount of masking by the neighbouring particles. The average limiting rays in the cross-section are thus L' A and L'B, corresponding to the incident rays LP and LQ respectively. It is seen from the figure that the beam of rays at the side which is remote from the light source proper (the right hand side) runs further towards the horizontal than that facing the light source (left hand side). The angles of incidence with the normal to the sphere are greater on the right hand side than those on the left; consequently, the reflection factor of the surface of the sphere (and hence also the luminous intensity of the image) will be higher on the right hand side than on the other. These two facts together make it seem plausible that the luminance factor, with respect to the oblique directions of incidence shown in the figure will be higher in directions more remote from the light source than nearer to it. This also explains why the maximum luminance factor of many surfaces does not lie in the direction of specular reflection, but in directions that make greater angles with the normal. Let us now consider the effect of the solid angle of incidence (wi). If we vary wi, this means that the images L' are varied in size in pro­portion with wi. For a given luminance of the light source, the luminous intensities of the images will vary in proportion to the variation in wi

X-8] LUMINANCE FACTOR 165

(and hence with the illumination on the surface). The observed luminance of the surface accordingly also varies in proportion with wi, provided that the masking effect of the neighbouring particles does not alter the situation. Since the images in this case lie so close beneath the surface, the lu­minance of the surface remains proportional to the illumination, and the measured luminance factor will be constant provided that w I does not change too much. We will now consider the acceptance angle of the photometer. The light accepted by the photometer depends on the light distribution of the image L', as well as on the masking conditions. The light distri­bution curve will be very gradual and, except for glancing directions of observation, the masking conditions will not vary much with the direction of observation, so that the effect of the acceptance angle on the final result will only be small under the conditions shown in the figure. In practice, divergences from the idealised situation as outlined will have an equalizing effect on the light distribution, so we may say in general that, with surfaces consisting of particles whose radii of curvature ~re of the order of size of the dimensions of the particles themselves, the luminance factors are to a fairly large extent independent of wi and e. Let us now turn to the situation as depicted in Fig. 88b. Here the radii are large compared with the size of the particles. The images of the light source formed by the spherical surfaces lie relatively far below the surface; the greater the radii of curvature, the deeper the images, up to the point at which, with infinitely large radius (perfeCtly plane surface), the light source and its image will be sym­metrical with respect to the surface. The effect of variations in wro that is, in the size of the images, upon the masking of the reflected beam will be the more marked according as the beams become narrower and hence the radii of curvature greater compared with the size of the particles. From the form and direction of the beam AL'B it is seen that the spread in the reflected light is much less than in the case of the corresponding beam in Fig. 88a, and that the reflected light is then much more direct­ional. The indicatrix, therefore, is very much more pointed, which means that the effect of the size of the acceptance angle is much greater than that in Fig. 88a. The size of w r and 8 thus both have a pronounced effect on the measured

166 REFLECTION, ABSORPTION, TRANSMISSION [X

luminance factor and, when Wr and €J are reduced, the likelihood of finding a constant value of {3 becomes less according as the radii of curvature becomes greater compared with the particle size. Since the reflection factors of the convex surfaces of the particles as shown in Fig. 88b will not differ so much on average in the specular direction from those in a more horizontal direction, such surfaces will give a maximum luminance factor in or near the specular direction. This is in contrast with the surfaces shown in Fig. 88a, where the max­imum lies on the whole in more horizontal directions. This is borne out in practice. In paint coatings the particles as represented in Fig. 88a are covered with a clear layer which more or less follows the surfaces of the particles; in principle, such coatings assume the form shown in Fig. 88b and possess the reflection properties described. The light penetrating the outer layer is reflected in the manner shown in Fig. 88a, but the reflection factors of the particles are only low, seeing that the difference between the refraction indices of the outer layer and the particles of pigment is small. The diffuse reflection from the particles of pigment is thus of less significance than without the outer layer. The transmission of the outer layer is greater. This can be clearly de­monstrated by moistening a powder (e.g. glass powder) with water. The dry powder is white because it diffuses the. light in all directions, but, when the powder is wetted, the reflection factor is greatly reduced; the transmission, on the other hand, is considerably increased. The "whiteness", i.e. the relatively high luminance in all directions, almost entirely disappears. Recapitulating, then, we may say that luminance factors as constants of the material will be encountered only when the radii of curvature of the component parts of the surface are of the same order of size as these component parts themselves. Proceeding from the concept of uniformly diffuse, better: approximately uniformly diffuse, through spread to regular and mixed reflection, it is not possible to say just where the luminance factor ceases to be a constant of the material. In the doubtful intermediate zones, measurements with different solid angles of illumination and for photometer acceptance angles will have to prove the existence or otherwise of a constant value for {3.

X-9] GLOSS 167

X-9. Gloss A problem closely related to the considerations outlined in the previous s~ction is that of fixing a numerical scale for the effect generally referred to as gloss. Surfaces are described as glossy when, apart from some diffusion, they show more or less perfect specular reflection. Measurements of gloss are of considerable practical importance in the appraisal of paper and of paint finishes. To be effective, measurements of gloss must satisfy two requirements, viz. I) the results of measurement of two surfaces giving the same im­

,pression of gloss to a normal observer must be equal; 2) higher and lower values as the results of measurement must cor-

respond to impressions of higher and lower gloss. We have seen in the previous section that surfaces which are practically specular have no luminance factors as constants of the material. The relationship between the measured luminance and the illumination is wholly dependent on the measuring conditions, viz. the solid angle of the illumination and the acceptance angle of the photometer. The many investigators who have published definitions and results of mea­surement of gloss have all employed very widely divergent measuring conditions (usually a fairly large wr and 0), and it is therefore not surprising that their definitions and results show so little agreement. It would be going outside the scope of this book to examine this subject in greater detail. Readers wishing to know more about it are recommen­ded to read Harrison's work "Definition and measurement of gloss"2). This book contains an extensive discussion of almost all the literature on the subject of the definition and measurement of gloss. It finishes with a survey of exerything published on this problem up to 1945. The book gives a clear impression of the confusion which reigns in this field. Since I945 Harrison has carried out further extensive work on the subject of the appraisal and measurement of gloss. His investigations \vere concerned with samples of paper with various degrees of gloss and of different colours. On the one hand he evaluated the gloss of different kinds of paper by visual assessment, and on the other hand tried to find a system of photometric measurements which gives results that grade the papers in the same order as does the visual assessment. Harrison takes three measurements: the paper being illuminated at 45o he measures (I) the maximum intensity at or ncar the specular angle, 45°, (2) the intensity 5o nearer the normal, i.e. approx. 40°, (3)

168 REFLECTION, ABSORPTION, TRANSMISSION [X

the intensity normal to the surface. The results are combined in a rather complicated equation which yields a gloss number G. This gloss number gives a fairly good correlation with the visual assessment under mean viewing conditions but, according to Harrison's opinicn, the final solution of the problem is herewith not yet found, since the gloss num­ber thus established does not give an objective evaluation for all obser­vers and for all viewing conditions,

X- I 0. Reflection from road surfaces Amongst reflecting surfaces in general, roadways present a problem of their own; because of the importance of this in street lighting, there­fore, we shall devote a separate section to the subject. On a lighted roadway we see obstacles usually as silhouettes against a brighter background, the latter being the roadway itself. The luminance, or rather the distribution of luminance on the roadway is therefore the important factor. It is useful to be able to compute this distribution in respect of given lighting systems and roadways, and it is also im­portant for the designer of lighting fittings to be in a position to as­certain the particular light distribution that his fittings will have to provide in order to ensure the distribution of luminance on the roadway that will promote optimum seeing conditions along the road. Given the isolux diagram of the roadway, that is, the illumination at any point, we can compute the luminance at every point (with formula (X-7 a)), provided that we also know the luminance factor for each such point in respect of the associated directions of illumination and observation. To assist our considerations we depict the conditions relating to il­lumination and observation on a roadway in Fig. 89. A light source L is suspended above the kerb of the roadway AB-CD. 0 is the eye of the observer. In respect of a point P the directions of illumination and observation are determined by the angles rp and J, and If/ and r respectively. rp and If/ are meas­ured from the normal at P, J and r from a line passing through P parallel to the kerbs. The part of the road surface of

L

Fig. 89. Street lighting. AB-CD roadway, L = light source, 0 = observer's eye

X-10] REFLECTION FROM ROAD SURF ACES 169

importance to the direction of observation of the car driver is fairly far, e.g. 30 metres, in front of him. This means that only values of 'I' greater than 87.5° are of significance to our considerations. Measurements have shown that the angle 'If, provided that it is greater than 87.5°, has little effect on the luminance factor, and this, in turn,means that we can always use the value at 'I' = 89° for p. Angle r is rather small for points on that part of the road important as a background for the driver's observation, and the effect of angle r on p can be ignored between zero and these small values. If we introduce these two approximations (ignoring the effects of 'I' and ron p), the luminance becomes a function only of qJ and J. This simplifica­tion enables us to calculate the luminance of the road surface from the light distribution of the fittings and data on p for the values found for qJ and J, on the condition that the fittings can be regarded as point sources. This assumption means that P is constant for every part of a fitting, although qJ and J differ slightly because of the finite dimensions. Fortuna­tely it has been found that this is sufficiently accurate in the case of dry road surfaces. However, there are varying degrees of dampness for which the luminance distribution is equally important. As is known from experience, the reflecting properties of a roadway change appreciably when the surface becomes wet; they tend towards specular reflection and, if the rainfall is heavy enough to fill up all the irregularities in the surface and thus produce a more or less continuous layer of water, the reflection becomes almost completely regular and the luminance factor ceases to have a fixed value. With such regular reflection no other light reaches the eyes than that which is reflected in the vertical plane through the light source and the eye, so that, in very wet weather, narrow bright streaks appear on the roadway, the latter appearing almost completely dark outside the bright streaks. Between the dry and flooded conditions many different modes of reflec­tion occur, including those for which there is no fixed value of {J. B e r g m a n s 3) has shown that under conditions such as those that occur just after rainfall, that is, when most of the water has run off, all types of road surface yield a fixed value of {J for directions of illu­mination whereby q; < 80°, i.e. with lighting fittings of the cut-off type. This is important, because the wet concJitions persist, sometimes for a considerable time. The distribution of luminance in the wet condition

170 REFLECTION, ABSORPTION, TRANSMISSION [X

may therefore be satisfactorily characterised by the distribution in the condition described. We are giving h.ere a brief description of the method of de Boer, Onate and Oos trij ck4) as an example of a method of calculating the luminance of road surfaces. Let us refer to Fig. 90 which represents the plan of a road. The observer's

f.ll--.1

4---tF-4~<<~~ ·-Fig. 90. Diagram of the reflection properties of a road surface in which the curves represent projections of lamp positions for which point P has the same luminance factor for the observer above 0'. The plane of the drawing represents the road surface. The diagram can be used for distances between 0' and P greater

than 40 times the eye-level of the observer

eye 0 is 1.50 m above 0', and the observed point P is, say, 150 m from the observer. At some point on the road a light source is assumed to be at a height hand to have a luminous intensity I cd in the direction of P. The illu-

I mination at P is thus E = h2 cos3 rp. Under these conditions we now

measure the luminance L at P in the direction PO, which will enable us to compute the luminance factor from

L h2 {3=- =L---.

E I cos3 rp

We imagine the light source displaced .along the road at the same height, and measure in the above described manner the luminance factor at the point P in the direction PO for a large number of positions of the light source. At each position of the light source in which f3 is measured, we note the value of {3, and, from the values thus obtained we derive the locus by interpolation from the positions of the light source that give the same values f3 at P. In this way a diagram such as that shown in Fig. 90 is obtained. In order to render the diagram universal the mounting height of the light source is taken as unit of length (indicated in the diagram by the line h). The effect of the variation in "P upon the values of f3 is negligible at distances OP > 60 m.

X-10] REFLECTION FROM ROAD SURFACES 171

it is clear that the curves in Fig. 90 are symmetrical with respect to the line O'P provided that the structure of the road surface introduces no directional effects. The diagram in Fig. 90 does not give a spontaneous impression of the contribution by the various light sources in the system towards the luminance at P in the direction of the observer. A diagram can be derived from Fig. 90, however, that will do this. For positions of the light source which are far from the observed point the values of {3 will be high. The horizontal illumination at P produced in such positions, however, will be low; hence the contribution by these sources towards the total luminance of P will be small, in spite of the high values of {3.

h2 As we have seen above, we may write {3 = L . From this the

I I cos3 q; luminance can be computed as L = {3 cos3 q; h2• The value of {3 can be

read from the diagram Fig. 90. For each position of the light source in the diagram there is a corresponding value of q;. This leads to another diagram in which the curves are the loci of the light source positions yielding the same value of {3 cos3 q; at the point P in the direction PO. The new diagram is shown in Fig. 91, in which the unit of length is once more the mounting height h of the light source.

0.00'3

o' ~-

0.002 0.005

Fig. 91. Representation of the reflection properties of a road surface by means of an E.P. diagram. The observer is over point 0' and looking at point P on the road surface. The curves represent projections of lamp positions for which the product {3 cos3 qJ for point P as seen by the observer has the value indicated at each curve. The diagram can be used for distances O'P greater than 40 times

the eye-level of the observer

The authors call this diagram an E.P. diagram in which "E.P." stands for "equivalent positions", i.e. light source positions which are equivalent

172 REFLECTION, ABSORPTION, TRANSMISSlON [X

in so far as the value of fJ cos3 qJ at P is concerned. In order to compute the contributions towards the luminance at P we read the values of fJ cos3 qJ for different light source positions from a diagram of the kind shown in Fig. 91 and multiply these values by the luminous intensity I in the direction of P, dividing the result by h2•

For a lighting fitting that has the same luminous intensity in all direc­tions, Fig. 9 immediately gives the luminance contribution of the fittings in different positions. For any given lighting fitting, the light distribution is usually known and, if this be represented as an iso-candela diagram, projected on to the plan of the road, the combination of E.P. diagram and iso-candela diagram enables us to compute the luminance contributed by the fittings. Fig. 92 illustrates the manner in which the E.P. diagram is used to compute the luminance values for a number of selected points on a road surface, more than 60 m from the observer.

0.002 I 0.005

/ I / I I I ----1 I "' -I I I / -----O.o;

I --- ...... o ...... I 1/ ,--...... ..........,..:,9~ I 11 1 /~-- -.Q.o3 ............ _....L.-1-- -1- U.--!..Lfo4aos;£;;- "> _--:..... ·- -·-7- _L!._

0~ I I 1 ' \ -- .......... ,.. .,...,. \ \ \ \ \ ,,___ - ... ..- L3 ....... ' \ \ \ ' .- .............. .,.....-..---

...... _..... __ ........ -- oQf> -----\ l5\ ', ---- o ........ -\ \ ...... _..- .,.,. ....

' ' -- , .. "' o.o.92,--' h , --o'1o3' ' 1-----l "' ......

a

b

, ~00 640 • 400 2500 640 400\ \ I I rl'---, I --------, \ , .... --, \\ \

I I I ~.... J 1 p /' 150 ' \ , • J \.L.l I~ __ .., I o / ..,--roo---... ' \ , __ ..,. "'?-'~

\ I \ I • I / -"2-:::r- ' " \ \ I I Jo ·~\ / I I I'.,~ .. ' \ \ '~ 1,11 /VV '\ c I I I ' ) I \ ~ "

\ ' ' gg;• ® J 'Lilt-' I I \ \ --Ls \ \, L '.J I I 1 ~ I \ \ ,_.,... / h , ___ ..... ..-.... Fig. 92a. and b. Explaining the use of·the E.P. diagram.

a) To calculate the contributions of the light sources L 1 to L 5 towards the lu­minance of the point P, a diagram of the lighting system on transparent paper is placed over the E.P. diagram so that P coincides with point P in the E.P. diagram, with the direction PO' lying on the axis of symmetry of the E.P. diagr<).m. The values of fJ cos3 rp can then be read off and multiplied by the luminous intensities of the light sources in the direction of P to give the contribution of each source to the luminance of P.

b) !so-candela diagram of the light sources projected on the road surface, from which the luminous intensity values in the direction of P can bt> read

X-10] REFLECTION FROM ROAD SURFACES 173

A plan view of the road with its lighting system (lamps L1 to L5) is drawn on a sheet of transparent paper. The mounting height of the lighting fittings is used as unit of length, as in the E.P. diagram. Let P be one of the points in the plan, as seen by the observer. The trans­parent drawing is now placed over the E.P. diagram in such a way that P coincides with P in the diagram. The situation is then as shown in Fig. 92a. The values of {J cos3 cp for positions L1 to L5 of the light sources are now read off and tabulated (column II in Table 1), after which the luminous intensity in the direction of P is determined for each light source. The simplest way of doing this is to project an iso­candela diagram of one of the lighting fittings on to a plan of the roadway.

TABLE I

Lamp I {Jcos3 rp in P I I-+ p I Jt2.iJL

Ll 0.008 40cd 0.32 cd L2 0.0045 640cd 2.88 cd La 0.02 325cd 6.5 cd L, 0.015 205cd 3.08 cd L; 0.005 190cd 0.95cd

1: 112.AL = 13.73 cd

h=8m L = 0.214 cdfm2

This iso-candela diagram is in turn covered with the transparent plan of the road so that fittings L1 to L5 in the plan successively coincide with L in the iso-candela diagram. Care must be taken, that, if the light distribution is not .axially symmetrical, the iso-candela diagram is correctly positioned with respect to the roadway. In this way the luminous intensity in the direction of P can be ascertained for any given lighting fitting. Fig. 92b shows the iso-candela diagram for L3 . The third column of Table I gives the luminous intensity values, and we have now to mul­tiply the values in column II by those in column III. The result, entered in column IV, represents the contribution L1L of each lighting fitting to the luminance at the observed point P, multiplied by h2• Addition of the values in column IV, and division of this sum by h2 then gives the total luminance of the road surface at P, as seen by the observer at 0. This is repeated for each point, and, when this has been done for a sufficiently large number of points, iso-luminance curves can be plotted for an observer 0 on a road having a number of light sources Lt .... Ln.

174 REFLECTION, ABSORPTION, TRANSMISSION [X

This gives us the iso-luminance diagram in the plan of the road. For an evaluation of the distribution of luminance as seen by the observer, this plan should really be drawn in perspective, but, as the iso-luminance lines then fall very close to each other, the diagram becomes difficult to read. Conclusions can also easily be drawn from the plan view with regard to the uniformity, or points of highest and lowest luminance, etc. The method described and others known the literature 5) are much too complicated for general use by lighting engineers. It is doubtful whether a method of calculation can be found that is not too time-consuming and contains no simplifications that can adversely affect the required accuracy. In public lighting practice a satisfactory result can often be attained if research is restricted to the directions of observation of significance to road traffic 6). Here a number of simplifications become possible, especially in giving the average luminance of the road surface.

X-11. Transmission The same distinctions can be made for transmission, in accordance with the distribution of the transmitted light, as for reflection. Hence we speak of regular, diffuse, uniformly diffuse, spread and mixed transmission. As the corresponding terms as applied to reflection have already been defined, it is not necessary to give the relevant definitions with respect to transmission. Transmission can be characterised in the same way as reflection, by means of luminous-intensity and luminance indicatrices. It is also possible to determine luminance factors. With regard to the occurrence or otherwise of fixed values, the influence of the solid angle of the il­luminating beam of light and the acceptance angle of the photometer, similar consid&ations may be said to apply as those outlined in section X-5. We shall now give a few examples of materials displaying the different kinds of transmission:

a. Regular transmission takes place in clear glass and similar materials which are called transparent.

b. Uniformly diffuse transmission is closely approximated by perpen­dicular incidence of light on opal glass. Here again, deviations from uniformly diffuse transmission occur in the more oblique directions, up to glancing inCidence.

X-11] TRANSMISSION 175

With oblique incidence, opal glass also shows spread transmission. Owing to the deviation which opal glass shows from uniformly diffuse reflection and uniformly diffuse transmission, the luminance of an opal glass bulb such as that of some kinds of incandescent lamp is not the same over the whole of the apparent surface. Although the luminance of such bulbs ap­pears to the eye to be uniform, it is, at the periphery, only 50 to 60% of the value at the centre of the apparent surface. This also applies to both tubular opal lamps and tubular fluorescent lamps.

c. Spread transmission is producd by frosted glass. It is interesting to note that glass frosted on one side yields different luminance indi­catrices according to the side from which it is illuminated.

Fig. 93. Luminance indicatrices of glass frosted on one side, illuminated

at an angle of 45°; ---- Light incident to smooth

side - - - --- - Light incident to frosted

side

Fig. 93 depicts the luminance indicatrices of glass frosted on one side, with· light incident at 45o *). It will be seen that when the smooth side faces the light source the emergent rays are deflected towards the normal and that, conversely, with the frosted side illuminated, the emergent beam bends away from the normal. This is important from the point of view of lighting fitting design, since the illumination can be modified by frosting either the inside or the outside of the glass. The following may be noted in this connection. When light falls on the smooth side of the glass it is deflected towards the normal. If the other side were also smooth, refraction of the beam on emerging from the glass would restore it to its original direction (i.e. in Fig. 93, to 45°). If the side remote from the light source is frosted, that is, if it consists of facets all oriented differently, it is found that the principal direction of the transmitted beam is at an angle of 35° with the normal. For the principal direction of the transmitted light to show no variation when light falls on the frosted side, this direction would have to be at an angle of 28° (with n = 1.5). On balance, *) Measurements by the author.

176 REFLECTION, ABSORPTION, TRANSMISSION [X

therefore, the light is deflected less from the normal than when the reverse side is smooth. When the light is incident on the frosted side of the glass it is not deflected so much on average (in this case towards the normal) as when it enters from the smooth side. At the smooth reverse side it is deflected away from the normal, so that the principal direction of the transmitted rays includes a larger angle with the normal than if the glass were plain on both sides (according to Fig. 93 about 55°).

d. Mixed transmission. This occurs only rarely. With frosted glass it is very rare; the glass would have to be so lightly frosted that it would hardly be possible to speak of frosting at all. Some opal glasses give regular transmission of part of the light (opalin glass). When a clear incandescent lamp is placed behind an opalin glass, the filament can be seen through it; apart from a marked reduction in the luminance of the filament, a difference in the colour is to be seen, this being much redder than otherwise~ The reason for this is that in the process of diffusion by particles of the same order of size as the wave­length of light (Rayleigh diffusion), the shortwave rays are diffused more strongly than those of longer wavelengths. This phenomenon explains the blue colour of a cloudless sky the sunlight is diffused by the small particles suspended in the atmosphere and, since short-wave radiations are diffused to a greater extent than those of longer wavelength, the diffuse light is blue.

X-12. Density

In photography it is useful to know the transmission factor of developed negatives and diapositives, as a function of the exposure i.e. the quantity of light thrown onto the light-sensitive layer (sensitometry). This is also important in cases where plates or films are used for the measurement of light, e.g. in spectro-photometry or in measurements of luminance (photographic photometry; section XV-5). Instead of transmission factor the conception density is employed (symbol 5), the definition of density being the logarithm of the reciprocal of the transmission factor. We accordingly write:

S =-log T.

Suppose that the density is 3. Then T = I0-3 = 0.001. When it is remembered that the transmission factor is defined as the quotient of the total transmitted luminous flux divided by .the incident

X-12] DENSITY 177

luminous flux, it will be seen that the above definition of density does not correspond in most cases with photographic practice. It is only when prints are taken from negatives, the sensitive paper being in contact with the negative, that the total transmitted luminous flux is used. When an image is projected by means of a lens, (e.g. in enlarging a negative, in diapositive projection or in the examination of an X-ray film), only that light which falls on the projection lens, or on the lens of the eye is utilised, and light diffused beyond this is of no consequence. In such cases the total luminous flux falling on the image is of no con­sequence either; we are interested only in the luminous flux reaching the projection lens without the photographic image in front of it. The total incident luminous flux is important, however, in the case of contact prints. In accordance with the conception of density as actually employed in photography and as measvred by means of densitometers, this should be stated as the logarithm of the attenuation which the luminous flux of a given system of illumination fa!Ung on a "receiver" undergoes when a photographic plate or film is placed between the lighting system and the receiver. The receiver is understood to be a projection lens, the eye, or the sensitized paper used for making contact prints. if the attenuation is, say 100 times, the density will be equal to log 100 = 2. Owing to the diffusing properties of the silver granules in the photo­graphic material, one and the same object will yield different values according to the solid angles of the incident light and of the receiver as seen from the plate or film 7). The density measured, therefore, depends on the design of the densitometer employed. If we measure the density of a plate or film in accordance with our second definition, using a small acceptance angle, first with perpen­dicularly incident and then with diffuse incident light, we obtain a higher density with the former than with the latter. This is obvious when it is remembered that, owing to the diffusing properties of the silver granules, a certain amount of light is included in the measurement which originates from light falling obliquely on the plate and which, when the lighting system is measured without the plate, does not enter the receiver. C a 11 i e r ( 1909) found that the two densities measured thus differ by a factor which, in general, is a constant for any kind of plate or film. This is known as the Callier coefficient and the effect itself as the Callier effect.

178 REFLECTION, ABSORPTION, TRANSMISSION [X

Let us now demonstrate the practical importance of the Callier effect by means of a numerical example. We will suppose that we enlarge a plate or film, once with diffuse light and once with light from a condenser, which can be regarded as directional light. Let us also assume that two areas in the plate have densities of I and 2, respectively, when measured with diffuse light. The light pas­sing from the diffuse background of the plate to the projecting lens is thus attenuated down to 10% and I%, respectively, and the luminous flux values of the pencils of light in the two areas are thus in the ratio of I 0 : I. If the Callier coefficient of the plate is 1.2, the densities with directional light will be 1.2 and 2.4, corresponding to an attenuation of the beams to 7.6% and 0.575%. The ratio of the luminous flux of the beams is then 13.2 : I, i.e. the contrast between the two areas, which was I 0 : I in diffuse light is 13.2: I with light from the condenser. This explains why negatives enlarged in an enlarger with condenser, and thus receiving light from directions which differ but little, arc "harder", i.e. show much greater contrast than when enlarged with diffuse light. Apart from the method of illumination and the acceptance angle of the photometer, density values must be given in reference to the kind of light used for the measurement, since the diffusing properties of silve; granules are dependent on the wavelength of the incident light.

X -13. Diffusion factor

In the course of time numerous suggestions have been made for the characterisation, by means of one value, of the diffusing properties of diffusely reflecting or transmitting materials. For this purpose the obvious thing to do is to take the case where the incident light falls as a parallel beam perpendicular to the surface, since this yields a sym­metrical indicatrix.

Fig. 94. Explanation of the con­cept of diffusion factor according to H a I b c r t s m a. The surface of which curve A is the luminous intensity indicatrix, has a diffusion factor <;f

1,. a=--

1 ma.r

I max

X-13] DIFFUSION FACTOR 179

The total average illumination is thus found to be if> if>

Em= 5 (I + e + e2 + .... ) = 5 I_ e (X-9)

1 Suppose that e is, say, 0.75, so that -- = 4. The average illumi-

1-e nation owing to the multiple reflection is then 4 times as much as without

1 reflection at the boundary surfaces. If e = 0.2, i.e. --- = 1.25, the

1-e

increment is 25%. As a result of multiple reflection it is possible for the coefficient of utilisation to be more than 100%, which represents the curious situation in which more light falls on the working area than is emitted by the light source. However, this obviously conflicts with the law of the conservation of energy. It must be remembered that the light falls on the working surface, but is only partially absorbed; the energy is not dissipated at the working plane, but is in turn returned into space. Equation (X-9) can also be derived along other lines. The luminous flux absorbed by the boundary faces at any moment (if>abs) is equal to the

luminous flux if> emitted by the light source at that moment. If this were not so, the luminous flux in space W(}u]d increase steadily; hence

if> = if>abs"

The absorbed luminous flux is also equal to the total incident luminous flux if>; multiplied by the absorption factor oc (= 1- e), so that we may write

if> b =if>= if>. (1- e), a s l

from which it follows that

1 if>.= if>--.

• 1- e To·obtain the average illumination Em we divide the incident luminous flux by the illuminated area S, so that

if> 1 Em=-.--. s 1-e

If the boundary surfaces transmit light (transmission factor -r), it is possible to calculate what part of the luminous flux is transmitted. Suppose, for example, that in the case of a lighting fitting consisting

180 REFLECTION, ABSORPTION, TRANSMISSION [X

For a long time the diffusion factor propos<:;d by H a I bert s m a 8 )

has been in use. This may be defined in reference to Fig. 94. Let curve A represent the luminous intensity indicatrix of a surface element with perpendicular incidence of the light. Further, let I max denote the max­imum luminous intensity and C/J the luminous flux after diffusion. If this luminous flux were uniformly diffused, the indicatrix would be the circle B, for which the maximum luminous intensity is C/Jfn. According to Halbertsma the diffusion factor (a) is the ratio of maximum luminous intensity with uniform diffusion C/Jfn to the actual maximum luminous intensity (/max); hence

<P

n <P a=--=--.

!max nlmax (X-8)

Now, nlmax is the luminous flux of a uniform diffuser with 1 0 = [max (indicatrix is the circle in C Fig. 94). We can thus also define a as the ratio of the luminous flux actually diffused to that which the surface would radiate if it were a uniform diffuser having the same maximum luminous intensity. If the surface is actually a uniform diffuser, a = I. In 1939 the C.I.E. gave a different definition of diffusion factor, viz. the ratio of the average luminance at angles of 20° and 70° to that at 5° with perpendicular incidence.

X-14. Multiple reflection

In section VII-2 we have stated that the coefficient of utilisation (viz. the illumination) within a given space depends on the reflecting prop­erties of the walls and ceiling. If we assume that both these and the floor, i.e. all the boundary planes have the same reflection factor (!,

the effect of the reflection factor can be demonstrated by simple cal­culation. Suppose that a light source of luminous flux <Pis suspended in the room; this luminous flux will then produce an average illumination <PfS on the boundary surfaces, where S is the total area of these surfaces. Of this luminous flux a part e<P is reflected and distributed over the total space,

thus producing an average illumination of e;. Again, part of this

secondary incident flux f! 2<P is reflected, resulting in an illumination of e2<P . S' and so on, ad infinitum.

X-14] MULTIPLE REFLECTION 181

of a glass globe, e and o of the glass are known; we can ascertain the efficiency, assuming that the whole of the globe surrounds the light source, thus disregarding any cover plates for lamp entry-holes. Of the incident luminous flux f/>, a part of!> is transmitted, and another part ef!> is reflected; the reflected light again falls on the wall, causing o(!f!> to be transmitted and e2f/> to be reflected. Of this, oe2f/> is then transmitted and e3rP is reflected and the total transmitted light is equal to

o of!> (I + e + e2 + ea + .... ) = f/> --.

I-g

The efficiency of the lighting fitting is therefore

o 1J =I- e· (X-IO)

A globe made of clear glass, with o = 0.90 and e = 0.08 thus has an efficiency of

o 0.90 1J = -- = ---- R; 0.98, or 98%.

I- e I- 0.08

If the luminous flux were incident to a flat pane of the same glass, only 90% would be transmitted; the repeated reflections within the hollow globe increase this to 98%. Frosted glass gives various values of (!, ex and o according to whether the light falls on the plain or frosted side, and this explains why the efficiency of inside frosted lamps is higher than that of lamps with the same frosting on the outside of the bulb. Measurements of the reflection, absorption and transmission factors of some frosted glasses have been carried out by Pi rani and S c h 6 n born 9), who have come to the conclusion that when light falls on the frosted side (as with inside frosting)! the reflection factor is lower, the absorption factor also lower and, hence, the transmission factor higher than with incidence from the other side (as in outside frosting). Let us now take as an example one of the kinds of glass investigated by Pi rani and S c h 6 n born, in order to show that if lamp bulbs were to be made of such glass, the efficiency of the inside frosted bulb would be higher than that of the outside frosted one. They find that for a particular kind of glass with perpendicular incidence:

light on frosted side: e = 0.087, ex = 0.055, o = 0.865; light on plain side: e = 0.123, ex= 0.09, o = 0.78.

182 REFLECTION, ABSORPTION, TRANSMISSION [X

Calculation of the efficiency *) using these valu~s gives:

Inside frosted: 0.865

17 = I - 0.087 = 0·95•

outside frosted: 0.78

17 = I- O.I23 = 0·88·

The other samples of glass investigated by P i r a n i and S c h o n­b o r n yielded similar differences. The calculation thus explains the fact that the efficiency of inside frosted lamps is a few per cent higher than that of the outside frosted type.

P i r a n i and S c h 6 n b o r n suggest a very acceptable explanation of the fact that the absorption is higher when the light is incident on the plain side of the glass. In this case the likelihood of total reflection from a number of facets when the light emerges from the other (frosted) side, is much greater than when the light enters from the frosted side. If, after total reflection, the light leaves nevertheless at the frosted side, the greater absorption takes place because of the longer path that the light has to travel. The total and almost total reflection that occurs with smaller incidental angles when light passes from glass to air than when it enters the glass from the air is thus the reason for the higher reflection factor with incidence of the light on the smooth side.

In practice, of course, there is no such thing as a glass bulb consisting of a complete hollow globe in one piece. There is always an opening through which the lamp is inserted, and also some suspension device usually in the form of a cover plate, on which a lampholder is mounted with a bush and ring. If we wish to know the efficiencies of a number of different globes, t_he effect of reflection from the suspension device upon the efficiency must be eliminated as far as possible; this can be done by blackeqing the inside of the cover plate. The luminous flux of the lamp with suspension device, but without globe is measured first, then that of the complete lighting fitting, obviously with the lamp in the same position with regard to the suspension device. The ratio qf the one luminous flux value to the other is then a measure of the efficiency of the globe, or the glass efficiency.

*) For this calculation we should not actually use the values for perpendicular incidence since, owing to the diffusion, other angles of incidence occur as well, to which other values of p, ex and T relate. There is no objection, however, to the use of these values merely to show the tendency to a difference in efficiency.

X-15] ABSORPTION 183

X-15. Absorption We shall be looking iqto the way in which absorption is governed by the thickness of the absorbing layer (see Fig. 95). We assume that the absorption is a for a layer thickness of, say, d mm. a is the percentage of the penetrating luminous flux absorbed. The penetrating luminous flux is equal to the incident luminous flux minus the reflected luminous flux. 1 - a, therefore, is transmitted.

I I I I

d I d I d I .., .. •'• .. I I I I I I I

#; I I I #u I I I I I I I I I I I I I I I I I I I I I

Fig. 95. If the absorption in the layer thickness dis a, <1>11>11 = <Pt (1 - a)n will be transmitted through n layers of thickness d, <Pt being the penetrating luminous flux

If we add a layer of the same thickness to this layer, so that the total thickness of both layers becomes 2d, a luminous flux of 1 - a enters the second layer. Here again, a portion a is absorbed, i.e. a ( 1-a). In both layers together, then, a + a(l-a) is absorbed, so that part 1 - (a + a (1- a)) = 1-a- a + a2 ~ (1- a)2 emerges from the second layer. This luminous flux enters a (possible) subsequent layer of the same thickness (the total thickness thus becoming 3d mm), in which a part a (l- a)2 is absorbed. The total quantity, therefore, absorbed in the three layers is a + a(l -a) + a(l - a)2 and that transmitted is 1 -{a + a (l -a) + a (l - a)2} = (1 - a)3. The survey below shows the ratios on absorption.

absorbed transmitted

in lst layer a 1-a in 2nd layer a (1 -a) (l - a)2 in 3rd layer a (1 - a)2 (l - a)3 in nth layer a (1 - a)n-1 (l - a)n

If we designate the penetrating luminous flux by fiJi and the luminous flux transmitted through n layers of thickness d by f!Ju,n, we can write

f!Ju,n =fiJi (1- a)n

184 ·REFLECTION, ABSORPTION, TRANSMISSION [X

or, in logarithms, log cPu,n =log cPi + n log (1- a),

where a represents the part of the penetrating luminous flux absorbed in one layer. As can be seen, we are using the transmitted light here because this makes for simpler calculations than the absorbed light.* Once the absorption in a Ia yer of thickness d mm is known (e.g. a!), we can calculate the absorption (a2) in a layer of d2 mm in the following way. If we take the absorption in a layer 1 mm thick as a, with a thickness d1 :

cPu,d! = cPt (1 - a)di and at thickness d2 :

Now

and

We can therefore write

and

From the first equation we obtain for 1 - a 1-a = (1-a1) 1/di

If we insert this value for 1 - a in the second equation, we obtain 1- a2 = (1- aJ)di/d2 (X-11)

The absorption in the layer d2 mm thick is thus 1 - (1 - a1)d2fd1 (X-12)

Examples l. A layer I mm thick absorbs 10% (a = 0.1) of the penetrating light. A layer 5 mm thick will then transmit

C/Ju = C/11 (1-a)5 = cJ>1 (1-0.1)5 = 0.595 C/Ji

The absorption of 5 mm is therefore 40.5% of the penetrating luminous flux. 2. A pane of glass reflects 8% of the incident luminous flux. At a thickness of I mm, 20% (a = 0.2) of the penetrating light is absorbed. How much

*) To simplify we shall not take into acount the reflexion on the exit face of the absorbing medium

X-15] ABSORPTION 185

light will a 3 mm thick pane of this type of glass transmit? At a reflection of 8%, 92% enters the pane. Of this,

tP,. - tP1 (1-a)3 = tP1 (1~0.2)3 = 0.83 tP1 = 0.512 tP, will be transmitted through a thickness of 3 mm. Therefore,

0.512 X 0.92 = 0.471 = 47.1% of the incident luminous flux will be transmitted. 3. A 4 mm pane absorbs 15% of the penetrating luminous flux. How much light will a 6 mm pane of the same material absorb? According to equation (X-12), the absorption in the 6 mm layer is 1- (l-a1)d2fdt = 1- (1-0.15) &f4 = 1-0.85 &f4 = 1-0.784 = 0.216 = 21.6% of the penetrating luminous flux.

We shall now derive two equations for the absorption in diffuse and non­diffuse media, using infinitesimal calculus.

a. Absorption in a non-diffusing medium When a luminous flux fP0 is incident at right angles to the interface

Fig. 96. Derivation of the absorption for­

mula IIi,. = lfi; e-•d ....

of a non-diffusing medium, part of this flux is reflected and the rest enters the medium; in the latter, absorption takes place. When the light travels through a distance dx in the medium (see Fig. 96), that part which is absorbed is:

df/J = -a . f/J . dx (X-13) where ~ is a constant of the medium. The second term is given the negative sign to indicate that df/J is a reduction in the luminous flux. Equation (X-13) can also be written as:

df/J -=-a dx rp . . Integration gives the absorption for a finite distance of travel, viz.

f !rp = J- a . dx + C

lnf/J =-ax+ C.

The constant C can be computed for x 1 = 0, in which case C = In f/J; f/J is the luminous flux entering the medium, which we shall denote by f/J;. Hence:

In f/J = -ax + In f/J;

186 REFLECTION, ABSORPTION, T,RANSMISSION [X

from which it follows that:

If we imagine a medium of thickness d (e.g. a filter) and denote the luminous flux perpendicular to the exit face by f/J,.:

(X-14)

The coefficient a is known as the transmissive exponent. Formula (X-14) is valid only for perpendicular incidence and emergence of the light at the interfaces; for other directions the path travelled is not equal to d, but is greater.

b. Absorption in a diffusing medittm If the medium represented in Fig. 96 is diffuse, the luminous flux pas­sing through a layer dx in thickness will be partly absorbed. In contrast with transparent media, however, some of the light is diffused in all directions in the layer dx, i.e. also rearwards. This luminous flux is again in part absorbed by layers lying more towards the rear, and in part is diffused, the latter component being thus added to the light falling on the layer dx. This absorption and diffusion is repeated ad infinitum, so that ultimately a part of the light falling on the layer dx reaches the other side and another part does not. Let the last mentioned part be denoted by df/> = -af/>dx; we can then derive a formula for the light transmitted by a layer of finite thickness d, which will be analogous to equation (X-14), viz.

(X-15)

This equation is based on the assumption that the light entering all the layers dx is similarly diffused. If the incident light is diffuse, the distribution of the light entering the layer is the same for all layers dx from the moment of entering the diffusing medium, and equation (X-15) is then valid for any layer thickness. When light enters the diffusing medium from one direction only, the distribution of the transmitted light becomes uniform only after the light has penetrated to a certain depth; in the process of diffusion in the initial layers the light remains to a certain extent directional. For­mula (X-15) can be employed for directional light only when the medium is sufficient! y thick.

X-16] ABSORPTION OF COLOURED FILTERS 187

X-16 Absorption of coloured filters Coloured filters absorb differently in the different parts of the spectrum. Thus they alter the spectral composition of the light falling on the filter from a light source, if the light is not monochromatic, i.e. does not con­tain one wavelength only. This gives the visual impression of a change in the colour of the light. If a filter of a certain composition and thickness produces a certain colour in the light transmitt~d, the colour will change if the thickness of the filter is altered. This can be explained with reference to Fig. 97.

o.a

(' I

'To.&

o.s

11.4

Q;l

0,2

Ol

0.7

~m)

Fig. 97. Change in the relative spectral composition, and thus in the colour of light passing through a coloured filter, when the thickness of the filter is altered.

In Fig. 97, curve I shows the luminous flux transmitted through a filter of thickness d, i.e. the value of 1 -a in the previous considerations. If the thickness of the filter is doubled, the transmission at each wave­length can be calculated from (1 - a) 2• The result is shown in curve II in Fig. 97. The ratios of the intensities of the various wavelengths have now changed, e.g.

with 0.55 p.m, ( 1 - a) in curve I is 0.6 and (1 - a)2 in curve II is 0.36,

with 0.5 p.m,(l -a) in curve I is 0.22 an.d (1 - a)2 in curve II is 0.045.

188 REFLECTION, ABSORPTION, TRANSMISSION [X

Whereas the ratio for the wavelengths 0.55 and 0.5 pm in curve I is 0.6 : 0.22 = 2. 7, in curve II it has become 0.36 : 0.045 = 8. This change in the transmission ratio occurs at all wavelengths. Thus the colour of the light transmitted changes with the thickness of the filter.

X-17 Colour of reflected light

We have seen that with media which transmit light fairly well, a large part of the reflected diffuse luminous flux consists of light that has penetrated the granules and has ultimately been re-emitted after repeated reflection and refraction. If the medium is colourless, i.e. if light of all wavelengths is transmitted and absorbed to the same degree, the spectral composition (the colour) of the reflected light will be the same as that of the incident light. On the other hand, if the medium is coloured, that is, if more light of a certain wavelength is transmitted, or if less is absorbed than for other wavelengths, the spectral composition i.e. the colour of the transmitted light will differ from that of the incident light. For example, if transmission in the green is high, the diffuse light trans­mitted after repeated reflection and refraction of white light penetrating the medium will also be green. In such media there is relatively little reflection compared with transmission, so that, when there is not so much absorption, we have the curious fact that the colour of the diffusely reflected light is the same as that of the transmitted light. Thus the leaves of trees are green because the chlorophyll in them transmits more green than red or violet. A very different effect is produced by materials which absorb much light and transmit little, as do metals; the light is then absorbed on penetrating to a depth equal to only a part of the wavelength. At the same time, the penetration is so restricted that light entering is not attenuated, but is only reflected. In such cases, therefore, the colour of the reflected light is again determined wholly by the absorption factor, but in the sense that it is just those wavelengths for which ab­sorption is highest that are reflected best. An example of this is found in copper, which in very thin layers gives green transmitted light, but which when thicker gives a red reflection. The same thing occurs in some non-metallic substances such as methyl violet, which as a dilute solution absorbs green light and in the form of crystals also gives a green reflection.

X-17] COLOUR OF REFLECTED LIGHT 189

REFERENCES

1) J o h. J an sen, Ph. T. R 5, 1940, 125-130. "The Distribution of the Light :reflected by different Ceiling and Wall Materials"

2) V. G. N. H a r rison : "Definition and Measurement of Gloss". Published by the "Printing and Allied Trade Research Association", London 1945 V. G. N. Harrison and S. R C. Po u 1 t e r, Res. Sci. Appl. Ind. 7, 1954, 128-136. "Gloss Measurement of High-glass Papers"

3) J. B e r g m a n s : "Lichtreflectie door wegdekken" ("Light Reflection by Road Surfaces"). Thesis, Delft 1938; see also: J. Bergman s, Ph. T. R 3, 1938, 313-321. "The Brightness of Road Surfaces under Artificial Illumination"

4) J. B. d e B o e r, V. 0 ii a t e and A. 0 o s t r ij c k, Philips Res. Rep. 7, 1952, 54-76. "Practical Methods for Measuring and Calculating the Luminance of Road Surfaces" J. B. d e B o e r and A. 0 o s t r ij c k, Philips Res. Rep. 9, 1954, 209-2'24. "Reflection Properties of Dry and Wet Road Surfaces and a simple Method for their Measurement"

S) J. M. Waldram, Ill. Eng. 27, 1934, 305, "Road surface reflection characteristics and their influence on street lighting practice" A. Bloch, Tmns. I.E.S.. (London), 8, 1939, 113-128, "Light scattering by road surfaces. A theoretical treatment" A. J. Harris and A. W. Christie, Public Lighting 19, 1954, 553-569 "Relative importance of the variables controlling street lighting performance" H. R. Ruff and G. K. Lambert, Public Lighting 22, 1957, 177-190, "Relative importance of the variables controlling street lighting performance" J. B. de Boer, Monograph no. 4 of the I.E.S. London (1962), "The concept road surface luminance and its application to public, lighting"

6) J. Bergman s and W. L. Verve s t, Ph. T. R 5, 1940, 222-230. "A New Fitting for Road Lighting"

7) G. A. Bout r y: "Mesure de densites photographiques par la methode photo-electrique" Publ. scient. et techn. du minist. de l'air, Paris 1934 J. E. de G r a a f, Zts. f. Wiss. Photogr. 37, 1938, 147-159. "Zur Densito­metrie von Rontgenfilmen und ihrer Normung"

8) N. A. H a 1 bert s m a, E.T.Z. 39, 1918, 207-209. "Die Streuung (Diffusion) des Lichtes als Mittel zur Verringerung der Flachenhelle kiinstlicher Licht­quellen''

9) M. Pi rani and H. Schon born, Li. u. La. 15, 1926, 458-460, "LJber den Lichtverlust in mattierten Glasern"

CHAPTER XI

PROPERTIES OF OPTICAL SYSTEMS

XI -I. Introduction In optics, an optical system is understood to be a system of refracting andjor specular reflecting surfaces, i.e. lenses or mirrors, as used in combination with an object which can be reproduced by the optical system. Here, the object may be a primary or secondary light source. The paths travelled by light rays through optical systems are dealt with in geometrical optics, but this largely falls outside the scope of this book 1). Here we are interested in the characteristics of the light emerg­ing from optical systems, i.e. the luminous intensity, the luminous flux and its distribution, the luminance of the light and also the illu­mination produced by such systems. Nevertheless, in discussing these, we shall in many cases be unable to avoid taking into consideration the paths of the rays through the optical system. In the following, therefore, a certain knowledge of the behaviour of light rays in optical systems is essential and, as this work is not intended as a textbook on geometrical optics, we shall have to assume that the reader possesses the necessary knowledge. The luminous characteristics of optical systems will, in general, have to be considered in conjunction with the object, i.e. the light source. At the same time, certain characteristics can also be attributed to optical systems without an object which are important from the point of view of lighting calculations. In order to avoid ambiguity we shall speak of optical systems when we mean systems of refracting andjor reflecting surfaces combined with a light-emitting object, and of the optics where only the system of refracting or reflecting surfaces is concerned. In the following we shall show how the luminous characteristics of optical systems can be computed and, conversely, how such systems can be designed to meet given requirements as regards their luminous properties. The luminous characteristics of optical systems are governed by, apart from the geometrical-optical rules, Abbe's law, which we shall now explain.

XI-2] LUMINANCE AND REFRACTIVE INDEX 191

XI-2. RelationsJrlp between luminance and refractive' index; A b b e•s law

In the previous chapter we have seen that when light falls on the interface between two media, a part of it is reflected from the face and another part passes through it into the second medium; we have also seen what happens to the luminous flux; this is partly or even wholly absorbed. Let us now look into the question as to how the luminance and luminous intensity are modified when light passes from one medium into another. Obviously, there is no object in considering anything but a medium the absorption of which is low (clear glass etc). It is not possible a priori to say anything definite about what happens to the luminous intensity, since the refraction modifies the apparent area of the light source. It may be enlarged or reduced, dependent upon the shape of the refracting face, the ratios of one index of refraction to another and also upon the distance of the light source from the refracting face. As to the luminance, this quantity varies with the refraction, and the law to which this obeys can be ascertained; in doing this the reflection and absorption losses will for the moment be disregarded. In Fig. 98, L is a light-emitting surface radiating in a medium I, the index of refraction of which is n1• The light from L falls on the interface G of a medium II whose index of refraction is n2• At a point P on G we imagine a very small area of size LIS which alone is transparent; around LIS, therefore, no light is transmitted. NP is the normal to G at P. It is assumed that LIS is so small that all the normals at LIS are in the same direction, that is, parallel to NP. The luminance of L in the direction of P is denoted by L 1.

The rays are refracted at P and, after this refraction, the luminous flux incident at P again forms a beam. The aperture LIS is virtually a diaphragm situated within the luminous flux emitted by L. As shown in section IX-2, a diaphragm of this kind functions as a light source with respect to the space in front of it. The luminous intensity of the diaphragm is thus equal to the luminance of the light source that can be imagined behind it, multiplied by the area of the diaphragm. It is a condition, however, that the whole of the diaphragm be flashed. Since the solid angle after refraction ( w2) corresponds to that before refraction (w1), LIS as seen from w2 is completely flashed with the luminance produced by the radiation in medium II; this we shall denote by L 2•

192 PROPERTIES OF OPTICAL SYSTEMS [XI

Accordingly, LIS, for medium II, may be regarded as a light source emitting light in the solid angle w2 with a luminance L2•

L2 can be computed in the following manner. Fig. 99 shows a narrow beam of light originating from a part of the light source Lin Fig. 98. This beam is incident on the interface of medium II at P, which lies in the plane G' as drawn and representing the tangent plane at P to the interface G in Fig. 98: Hence the normal NP and all the normals to the small area LIS· are perpendicular to G'.

Fig. 98.

Figs. 98 and 99. Illustrating A b b e's law. The lumi­nance values in media I and II, having indices of re­fraction of n 1 amd n 2 respec­tively, are proportional to the squares of the indices of

refraction

Fig. 99.

For the purposes of our calculation we introduce in medium I a plane V1 at any point between P and L, and parallel to G'. The normal NP intersects V1 at M1.

The narrow beam of light from L is defined in the following manner. Two circles are drawn in V1 with M1 as centre; if points on these circles are now connected to P, two cones are produced, the half-apex angles of which are i 1 and i 2• From the area between the two circles we now select a part L' bounded by two radii of the concentric circles; these radii enclose an angle ex. The beam formed by connecting the periphery of L' to P is then that

XI-2] LUMINANCE AND REFRACTIVE INDEX 193

which we assume to be emitted by a part of L. L' is thus the equivalent light source for that part of L. L 2 is now determined by successively: a) calculating the luminous flux emitted by L' towards L1S with a

luminance L 1;

b) calculating the luminous flux emitted by L1S in the beam after refraction, with an unknown luminance L 2;

c) equating the luminous flux values as calculated in a) and b) and evaluating L 2 from the equation. The reflection at P and absorption in the media are thereby disregarded.

a) Calculation of the luminous flux emitted by L' towards L1S According to equation (IX-4) the illumination of G at P as produced by that part of V1 which is enclosed by the inner circle is: E 1 = nL1 sin2 i 1;

that produced by the part of V 1 which is enclosed by the outer circle is: E 2 = nL1 sin2 i 2•

The illumination of G1 at P, produced by that part of V1 which lies between the two circles, is therefore

Ep = E 2 - E1 = nL1 (sin2 i 2 - sin2 i 1}. (XI-I)

Since the circles are symmetrical with respect to the normal NP, each sector contributes in similar fashion to the illum~nation at P, and it may thus be said that the illumination at Pis proportional to the central angle of the sectors of the circles, i.e. also to the centre angle of the parts of the sectors lying between the two circles. The illumination E P in equation (XI -I) refers to the zone between the complete circles, that is, to the central angle 2n rad. For L', being the difference between two sectors of central angle IX rad, the illumination E L'-+ p at p is

IX IX E L' _ .. p = 2n n L1 (sin2 i 2 - sin2 i 1) = "2 L1 (sin2 i 2 - sin2 i 1}.

The luminous flux (/JL' ._..LIS incident to L1S from L' is then

</JL' -..LIS=~ L1S. L1 (sin2 i 2 - sin2 i 1). (XI-2)

b) Calculation of the luminous fhtx from L', emitted by L1S in medium II after refraction

To make this calculation we first see .what happens to the rays emanating from L' in the process of refraction, usi~g Snell's law.

194 PROPERTIES OF OPTICAL SYSTEMS [XI

Air the rays following th~ generators of the cone whose half-apex angle is i 1 (of the ihnex: circle), and which therefore make an angle i1 with the normal at P, <;~.re refracted at P in such a way that the angle from the normal changes to rl according to the equation:

Similarly, for the rays from· the outer circle on V1, we write:

--=-sin: r2 n1

These rays thus once more form conical surfaces, the half-apex angles of which are r1 and r2 respectively. The rays of light from the radii of the two circles bounding the area L' occur in two planes passing through the normal, and these rays remain in the same planes after refraction. In Fig. 99 a plane V2 is shown in medium II, parallel to V1 and G'. The conical surfaces o,n which lie the rays emanating from the two circles

. on vl intersect this plane to produce two circles; the rays from the lines M1A1B1 and M1J)1C1 thus pass through lines M2A2B2 and M2D2C2 res­pectively. The angle formed by these lines is again oc and the lines are parallel to the corresponding lines in plane V1• This is illustrated in the figure by two reference lines M1E 1 and M2E 2, both of which are horizontal in the figure. Angles E 1M1C1 and E 2M2C2 are then equal ({3). Now, i.f the luminance of L1S within the cone having a half-apex angle r1 be denoted by L 2 , L1S will be a light source, the luminous intensity of which in the direction of the normal is ! 0 = L1S. L2• According to equation (VIII-5) the luminous flux within this cone will be

f/>1 = nl0 sin2 r1 = n. L1S. L2 sin2 r1.

Within the cone with half-apex angle r2 the luminous flux is

f/>2 = n . L1S. L2 sin2 r2.

The flux if> .J.s m the solid angle between the two conical surfaces is therefore

if> .J.s = n . L1S . L2 (sin2 r 2 - sin2 r1).

Within these solid angles the luminous flux is uniformly distributed around the axis i.e. the normal PN, and a sector with a central angle

xt-2] LUMINANCE AND, REFRACTIVE .INDEX

of oc rad thus contains a part of the total luminous flu~ f/J within the cones 01:

equal to 2:n; x tP.

The luminous flux in the beam of light emitted by AS is equal to the flux emitted by L' to LIS and can also be denoted by f/JL'...,. .,5 :

f/JL' ...... As = ~ :n; • LIS . L 2 (sin2 r 2 - sin2 r 1) =

(XI-3)

c) Equating the luminous flux to and from LIS and evaluating L 2 from the equation

The expressions found for ·f/JL' ...... .::ls (XI-2) and (XI-3) a.re now equated to give the following equation

From this it follows that

According to Snell's law:

so that

L2 =

or

L 1 sin2 r2 - sin2 r1 -=

sin2 i 2 - sin2 i 1 •

n2 1

= n2 2

(XI-4)

Since this equation applies to every part of L, and because the beam of light from L can be regarded as consisting of a number of beams of the assumed form, formula (XI-4) is universally valid. Put into words, this formula means that the luminance of a light source is pro­portional to the square of the index of refraction o1 the medium in ·which it is measured, disregarding losses due to reflection and absorption.

This is known as Abbe's law.

196 PROPERTIES OF OPTICAL SYSTEMS [XI

XI-3. Luminance of images formed by lenses and mirrors, and of the lenses and mirrors themselves

The formula derived in the preceding section for the relationship between luminance and index of refraction implies that the luminance of the image of an object emitting light in air, formed by a lens or mirror in air, is the same as that of the object, disregarding reflection and absorption losses in the lens or mirror. We notice that the luminance of the image of an object produced in the air may not be interchanged with that of an image formed on a screen of, e.g. paper or ground glass. The luminance produced on the screen depends on the illuminatio~ on the screen and the reflective or transmissive properties of the screen itself. The illumination and thus the luminance of it are governed in this case by the aperture of the image forming system, as shown in XI-8. Concentration af the light from a source, say the filament of a projection lamp, by means of a lens or mirror does not therefore increase the luminance. Concentration refers only to the luminous flux which, in a lens, is "compressed" into a smaller solid angle. This is accompanied by an enlarged image of the light source; the luminance remains the same as before. Since light from sources in general is emitted in air and is returned to the air through other media, it may be said that luminance cannot be increased by optical means. Owing to the fact that when lenses and mirrors form images they themselves function as light sources for these images, the relationship between luminance and refractive index also means that the luminance of .lenses and

L

Fig.lOO.

mirrors is equal to that of the H H' I

p'

object of which they form an : :----._ P'

image*). P 9 i '~e Figs.lOO and 101. Demonstrating that the - •--·-T T--- -- --· luminance of a lens and that of the image formed by it (P') is equal to the luminance 1

of the subject P (disregarding reflection Fig. 101. and absorption losses in the lens)

*) This has already been shown in another manner as applied to mirrors in section X-2.

Xl-3] LUMINANCE OF IMAGES, LENSES AND MIRRORS 197

In relation to both lenses and mirrors the above statement can be proved in a different manner from that used in the previous section; as far as lenses are concerned, this is as follows. Fig. 100 shows an infinitely thin lens L producing an image P' of an object P. Now, if the object emits light with a luminance L, what we wish to know is the luminance L' of the image P'. We shall disregard reflection and absorption by the lens. The luminous flux emitted by P towards L is equal to the flux emitted by L towards P'. Now, since the rays in an image-forming system of this kind are reversible, we may say that the luminous flux which P' would emit towards L_- if the luminance were L' - is equal to the flux which P emits towards L. Let us first consider the case of image-forming in the paraxial zone (Fig. 100). This is the zone, close to the axis, in which the sine, the tangent

_and the arc of the angles between the rays and the optical axis may be regarded as equal. The luminous flux f/J passing from P to L is then

f/J = LSOJ, where Sis the area of P and OJ the solid angle at P, subtended by L. The luminous flux f/J' that would be emitted by P' to L if the luminance of P' were L' is

f/J' = L'S'OJ',

where S' is the area of P' and OJ' is the solid angle at P' subtended by L. Since f/J = f/J',

LSOJ = L'S'OJ';

hence L S'OJ' L' = SOJ. (XI-5)

Now, OJ= SLfu2 and OJ'= SLfv2, where SL denotes the area of the lens, u the object distance and v the image distance. The values of S' and S are proportional to the square of the linear magnification, i.e. to v2fu2• For (XI-5) we can therefore write:

SL L v 2 v2 L' = uz X S L = I.

uz

Hence the luminance of the image is indeed equal to that of the object. If the angles included by the image-forming rays of light are so wide

198 PROPERTIES OF OPTICAL SYSTEMS [XI

that the sine, tangent and angle between the rays and the axis cannot be regarded as equal, the formula M = vfu, which corresponds to the

I 1 I simplified lens formula - + - = -, cannot be used. It is then necessary

u v t to take into account the sine condition, which may be expressed as

sin 8 M=--

sin 8''

in which 8 and 8' are the angles included in the object and image spaces by the optical axis and the rays emerging from a point on the axis. In the case of an infinitely thin lens it is permissible with small angles 8 and 8' to say that sin 8 =tan 8 and sin 8' =tan 8'. Then

sin 8 tan 8 M = -- becomes M = --.

sin 8' tan 8'

If we denote the radius of the lens by r, tan 8 = r fu and tan 8' = r fv,

tan 8 so that M = -- = vfu.

tan8' In Fig. 101, H and H' represent the two positive principal planes of a lens system. The half-aperture angles are 8 and 8'. P, of which the luminance is L, is reproduced as an image at P' with a luminance of L'. The luminous flux emitted by P towards the lens is then

l/J = nLS sin2 8 (XI-6)

and the luminous flux l/J' which P' would emit to the lens if its luminance were L' is

l/J' = nL'S' sin2 8'.

Equating l/J with l/J' we then obtain:

nLS sin2 8 = nL'S' sin2 8',

from which it follows that

L S' sin2 8' L' =. S sin2 8 .

According to the sine condition, the linear magnification is equal to sin 8/sin 8', so that S' JS = sin28fsin2 8', and substitution in the equation

L for L' thus gives

XI-3] LUMINANCE OF IMAGES, LENSES AND MIRRORS 199

L L' =I.

That lenses have the same luminance as the iiilage of the light source can be shown in the following manner. The luminous flux emitted by the lens system in the direction of P' is equal to that radiated by P towards the system. In the case of images p:roduced by paraxial rays (see Fig.lO.Ol,it may be said that the luminous intensity of L is 5LLL, LL being the luminance of the lens. The illu-

5L mination at P' is therefore ~; thus the luminous flux incident at v2

P' is 5LLL x 5', and this must be equal to the luminous flux if> emitted v2

by P towards L, which, as we have seen above, is L. 5. w.

L5 =5LLL5' (J.) 2 ' v

Hence

so that L 5L 1 5' LL ;2 X-;;; X S'

where w · 5Lfu2 and 5'/5 = v2 ju2 •

Substitution then yields:

L 5L u2 v2 -=-x-x---1 LL v2 SL u 2 - ·

In the formation of images produced by non-paraxial rays the reproducing system consists of a .circular light source of luminance LL which illu­minates P' (see Fig .I 01). According to formula (IX-4) the illumination at P' is: nLL sin2 @', and the luminous flux emitted towards P' is:

if>' = nLL5' sin2 @'.

The luminous flux if> emitted by P to the reproducing system is cal­culated above as

if>= nL5 sin2 e. Equating if> and if>' we thus have

nL5 sin2 6J = nLL5' sin2 6J'; from which follows, once again,

L -= 1. LL

200 PROPERTIES OF OPTICAL SYSTEMS [XI

The relationship between the luminances can be similarly derived for optical systems consisting of mirrors. Let us denote the transmission factor of a lens system by -r; then

L' = -rL, and LL = -rL.

If we similarly denote the reflection factor of a reflecting system by e,

L' = eL, and LM = eL,

where LM is the luminance of the mirror.

XI-4. Optical systems as light sources; the exit pupil of optical systems

We have just seen that optical systems serve as light sources, not only for the image itself, but also for every point within the image space that receives light from the system. Apart from the consideration of the illumination in the image space (whereby we may regard the system as a point source or a non-point source), the luminous flux emitted by an optical system may also be of interest. This must also be discussed in the following sections. The considerations put forward are theoretical and refer in general to the ideal, perfectly formed, image; in practice, of course, it is often necessary to make allowances for imperfections. None the less, these theoretical considerations have their uses, since they serve for estimating the behaviour of optical systems and, conversely, for the approximate design of such systems to meet given requirements from a light-technical point of view. In this respect lenses generally have the advantage over mirrors in that they provide a much nearer approach to the ideal image than mirrors. In principle, the same considerations apply to lenses and mirrors alike. In the following we shall deal in general with lenses, taking these to be axially symmetrical, the axis of symmetry of the system coinciding with that of the beams of light. In so far as it is necessary we shall then proceed to mirrors and certain types of system which are not axially symmetrical (cylindrical parabolic mirrors), or of which the axes of symmetry of the beams and the optics do not coincide (drum lenses). Let us commence by ascertaining what we should regard as the light source with respect to points on the optical axis of an optical system

XI-4] OPTICAL SYSTEMS AS LIGHT SOURCES 201

comprising a simple lens and an object. This may be termed the equivalent light source of the system. In Fig. 102 L represents a thin lens. An image of an object Pis produced

at P', this being brought about by reason of the fact that every point on

G L emits a pencil of rays

Fig. 102. A lens L forms a real image P' of an object P. Since P' is smaller than L, P' is the exit pupil of the system for points on the axis

beyond G-

to P'. The observer's eye is sup­posed to be at A and only this point A is then seen as a bright spot. If the eye is moved along the optical axis within

the image space, i.e. to the right in the figure, it will be among beams of light ceming from points in the vicinity of A, so that flashed circles are then seen. For instance, when the eye arrives at D it will be just on the edge of beams coming from the points C (C1 and C2 and all other points on the circle of which the radius is AC). What we then see is the face of the lens, limited by the circle CcA--C2 , flashed. AtE, the eye is just at the inner edge of the beams proceeding from the periphery of the lens, and the whole lens is flashed. From that point onwards we can therefore compute the illumination E on the optical axis, by regarding the whole lens as a light source. It depends on the accuracy with we wish to evaluate E whether we employ the inverse square law or the formula E = nL sin2 e. If the eye be now moved further, we continue to see the whole lens flashed, until the point G is reached, as we are then at the outer edge of the beams emitted from the periphery of the lens. Beyond that point we see that flashed rings at the edge of the lens disappear, i.e. they become dark. When the eye is removed still further from P', the outer edges of the beams wherein the eye is still just stationed approximate to a cylinder and at infinity do form a cylinder. It will now be seen that the beam of light falling on the eye is limited beyond G by the periphery of P'; this means that the image P' may be substituted for the flashed area of the lens, or that, in other words, as far as points beyond G are concerned, P' serves as equivalent light source of the optical system; it also means that, at a sufficiently large distance, the illumination can be computed from the luminous inten~ity of the image P'.

202 PROPERTlES. OF t>PTICAL SYSTEMS [XI

Let us now turn to the optical system depicted diagrammatically in Fig. 103. Pis once more the object, an image P' of which is produced by a lens L: P' is greater than the area: of the lens. If the observer's eye be now imagined to be tnoving along the optical axis towards the right, the lens, commencing from the point A at which the lens is completely flashed, will continue to serve as light source for all points on the axis. Since the beam illuminating the eye is always sma1ler at P' in cross­section than P' itself, P' can never take over the function of light source in its entirety; from the point D for instance only the part C1C2 of P' can be seen. For all points on the axis beyond A, there­fore, the lens functions

Fig. 103. A lens L forms a real image P' of an object P. Since L is smaller than P', L is the exit pupil of the

system for points on the axis beyond A

as light source with the same luminance as that of the object. Ac­cordingly it may be said that the image P', with respect to points beyond Gin Fig. 102, and the area of the lens for points beyond A in Fig. 103, ta.ke the place of the object P and the lens L. In the first instance the image P', and in the second the lens L is known as the exit pupil of the optical system under consideration. In general, the exit pupil of an optical system may be defined as the flashed area as observed from any point in the image space of the system, and accordingly serving as equivalent light source for that point. When the exit pupil is formed by the image of a light source of which the luminous intensity in the direction of the axis is known, the luminous intensity of the image can be quite easily determined without reference to the luminance of the light source. If we denote the linear magnification by M, the area of the image will be M2 times that of the light source. Since the luminance of the image is the same as that of the source, the luminous intensity of the image will therefore be M 2 times the intensity of the source. When an image of the filament of an incandescent lamp of 500 cd luminous intensity is obtained with a linear magnification of 3 x , the luminous intensity of the image in the direction of the optical axis is 32 x 500 = 4500 cd (This value should of course be reduced by ·the extent of the losses in the optics).

XI-5] THE EXIT PUPIL OF AN OPTICAL SYSTEM 203

In closing this section we may mention the fact that the highest illu­mination on the axis occurs at point E in Fig.l02 and at point A in Fig.l03, for it is at these points that the solid angles of the illumination attain their highest values. HenGe the maximum illumination is not produced at the points where the images occur, as is sometimes thought.

XI-5. Determination ofthe exit pupil of an optical system; entrance pupil We now come to the problem of determining the exit pupil of optical systems. In this we shall in general limit ourselves to the exit pupil in respect of points on the optical axis,although the same considerations will in most cases apply to points in the near vicinity of the optical axis. To determine the exit pupil for points ftuther removed from the optical axis it is necessary to invoke geometrical-optical considerations going beyond the scope of this book. Certain qualitative considerations, nevertheless, will be gone into. According to our definition, the optical system includes not only the optics, but also the object. It is possible to determine the exit pupil of the optics by assuming that the object would be located at one of the foci of the optics. In Figs.l02 and 103 the object and lens constitute boundaries for the rays of light, these boundaries being of course physical. Apart from these the figures show certain boundaries at the images P' of the objects P, as formed by the lenses, and these boundaries may also consist of virtual images which have to be considered as lying within the image space. All the areas within these boundaries emit light of the same luminance as that of the object. From within the image space we observe only those boundaries which lie within the image space. If we direct our eyes from within this space, towards the optical system, we see only the smallest boundaries of the rays in the image space flashed. Let us illustrate this with reference to Fig.l04. In this figure an image P' of an object Pis produced by a lens L1. This image is then reproduced by a lens L2, giving an image P". In this system there are three physical boundaries for the rays, viz. the object and the two lenses. Further there are three non-physical boun­daries, these being the images P' and P" and image L1' of the lens L1

as formed by L2 . Three of these boundaries lie within the image space, viz. the lens L2 and images P" and L1'; one of these is the exit pupil. From the point A, L1' is seen to be the smallest boundary, and this is therefore the exit pupil of this system for A.

204 PROPERTIES OF OPTICAL SYSTEMS

A

Fig. 104. Determination of the exit pupil of an optical system. Of the boundaries of the rays in the image space (P•, L 1 ' and L 2) the image L 1 ' is seen from point A the smallest. Thus the image L 1' is the exit pupil of the

system, for point A

The above shows how the exit pupil can be ascertained.

[XI

The images of all the boundaries to the rays are determined within the image space; among the boundaries thus found, as well as among any physical boundaries within the space, that one which is seen to be the smallest is the exit pupil. A demonstration of this may be given in the following manner (see Fig. 105). In the image space a number of diaphragms of different diameter are depicted (i.e. boundaries for the rays). Behind these there is a large area whose luminance is that ·of the light source. Now, sup­pose that we are looking towards the diaphragms from a point in the image space; only the smal­lest of these is seen flashed and this one is the exit pupil. . If the observation is made from a point removed from the op­tical axis, the exit pupil may be bounded by portions of two diaphragms; in Fig. 105 this is illustrated at the point P, in

A p

Fig. 105. For the point P some distance from the axis of diaphragms A, B and C the exit pupil is bounded from above

by B and from below by A

respect of which the exit pupil is formed by the upper edge of diaphragm B and the lower edge of A. In the object space, the exit pupil corresponds to an object which in the words of geometrical optics is conjugate to it, and this is called the entrance pupil of an optical system. The entrance pupil is accordingly determined by the image of the exit pupil in the object space. In Fig.l02 P is the entrance pupil and in Fig. 103 it is L. The significance which can be attributed to the entrance pupil is

XI-6] OBJECT AT THE FOCUS OF A LENS 205

demonstrated in Fig. 106, in which an image P' of an object Pis produced

L, p'

~----

Fig. 106. An image of an object P is pro­duced at P' by a lens L 1 • At P' there is another lens L 2 • L 2 is then the exit pupil. The entrance pupil of this system is that part of the object P which corresponds to the size of L 2 • P can therefore be reduced to the size of this part without decreasing

the luminous intensity of L 2

by a lens L1• At P' there is a second lens 'L2 which is smaller than the image P' itself, and the exit pupil of the whole system is thus the lens L2•

The conjugate entrance pupil occurs at P, but it is smaller than P. It will be seen, then, that P can be reduced to the actual size of the entrance pupil (this being the image of L2

in the object space) without varying the exit pupil, i.e. without varying the luminous intensity of the system in the direction of the optical axis.

XI-6. Object at the focus of a lens; angle of divergence When an object is placed at the focus of a lens an image of it is formed at infinity. In such cases the exit pupil is the lens itself, and it is im­portant to ascertain the point from which the whole area of the lens actually functions as exit pupil. Let us turn to Fig. 107. An object P is situated at the focus of a lens L; we can then say that every point on L emits a beam of light. The apex angles of all these

Fig. 107. A parallel beam formed by the lens L of the object P located at the focus of the lens. It is only beyond G that the lens is wholly flashed, G is called the beam

cross-over point

beams are the same (cp). We again trace the development of the illu­mination along the optical axis. Starting frorp A, from where only the point A is seen flashed, and moving along the axis, we observe larger and larger areas of the lens flashed, until at G the whole lens becomes

206 PROPERTIES OF OPTICAL SYSTEMS [XI

flashed. Beyond G, then, we may consider L as a light source whose luminance is that of P. Lis therefore the exit pupil of the system. When the distance AG is sufficiently long compared with the size of L, so that L may be regarded as a point source, it is permissible to speak of the luminous intensity of L. Should we wish to ascertain the luminous intensity of L by measuring the illumination at the axis, we should have to be careful to locate the point of measurement beyond G. In photometry this point G is known as the beam cross-over point, which is further referred to in Part II, (section XVI-3). The distance AG can be calculated by means of the following for a circular object P. AG = R cot }rp, where R is the radius of the lens. Cot tiP= ffr, where f is the focus of the lens and r the radius of the object. We thus see that

AG = R .f. r

Let us now also see what the illumination is for points not on the axis, at a large distance from L, i.e. the luminous intensity of Lin directions other than that of the axis. In other words, let us examine the light distribution of the system. Within the cone-shaped space BGC the lens is seen completely flashed; L is therefore the exit pupil for this space. The apex angle of this cone is equal to rp. From Fig.l07 it is seen that, outside the cone BGC the lens will not be flashed when at relatively short distances from L. At an infinite distance, however, the parallel lines DB and FH, as also FC and DE, will coincide and it may be said that the light beam is bounded by a conical surface of apex angle rp, and that the lens, which is then infinitely small compared with the distance, is the apex of this cone. In practice we may regard the beam as tapering at an angle of rp even at finite distance, provided only that the distance is great compared with the size of the lens. The luminous intensity of the beam in a certain direction is then equal to 1he apparent area of the lens in that direction, multiplied by the luminance of the objec!. If the angle between the particular direction and the axis be denoted by IX, the apparent area will be S cos IX, where S is the actual area of th~ lens. The luminous intensity is thus:

I = -rLS cos IX,

XI-7] THE EXtT PUPIL OF LENS SYSTEMS 207

where L denotes the luminance of P, and l' the transmission factor of the lens. Apart from a certain amount of diffused light from the lens, no light falls beyond the lines FH and DE, and the apex angle of the beam is qJ.

The half-apex angle of a beam is known as the angle of divergence of the beam, but, as this name is sometimes given to the whole apex angle, it is advisable to indicate the angle of divergence as twice the half-apex angle, e.g. 2 X 5°, or 2 X 8.5°. In our example we have assumed that an ideal image is formed, that is, one without aberration and, in so doing, we have regarded the beams of light as sharply defined. In practice, however, there will often be errors in the formation of the image (in reflecting systems invariably), in consequence of which the beams of light are not sharply P,efined. In such systems the luminous intensity will exhibit a maximum, falling off towards the periphery. Angles of divergence are given for the beam in such cases, but these are then the angles at which the luminous intensity"(or average luminous intensity) is 50%. 25% or 10% of the maximum. Reference is often made to parallel beams and these are illustrated by drawing rays from the focus which emerge from the lens parallel to the axis; in such cases qJ = 0. The angle of divergence is determined by tan t(/J = rff. Since all light sources have finite dimensions and the value of r is accordingly always finite, tan l(/J must also always have a finite value. There is, then, really no such thing as a "parallel" beam in the literal sense of the word. A certain amount of spread must always be taken into account, however small this may be.

XI-7. The exit pupil of lens systems; vignetting

When an object is placed at the focus of a simple lens the exit pupil is the lens itself. However, if the system comprises a number of lenses and possibly one or more diaphragms, if must not at once be assumed that the area of the last lens is the exit pupil of the system; it is ne­cessary to ascertain the exit pupil in the manner outlined above; We shall now proceed to do this as applied to two widely used kinds of lens system, i.e. two objectives. These are systems of lenses used for projection purposes. Errors in the formation of the image, such as occur with single lenses, are in such cases more or less fully corrected by com­bining a number of single lenses having different characteristics.

208 PROPERTIES OF OPTICAL SYSTEMS [XI

Amongst other applications, objectives are employed in picture pro­jectors, photographic apparatus and microscopes. Objectives may consist of separate single lenses, i.e. with air spaces between them, or of lenses cemented together, or combinations of these. Diaphragms, or stops, may be interposed between the separate com­ponents. Let us now ascertain the exit pupil of: a) an objective comprising two components separated by an air space, and b) an objective having a stop between the components. The components of such objectives usually consist of one or more single lenses cemented together, but for our purpose such compound lenses can be regarded as single lenses. a) Exit pupil of objectives consisting of lenses with space between them

(see Fig. 108) H'H

I I I I

Fi' ·- -· ---,,-----1;: -------S.a

: l fxit., f I pup!

!--------''----....;

Fig. 108. Exit pupil of an objective without internal stop. L 1 and L 2 are the rear and front lenses respectively. H and H' are the positive principal planes in the object and image spaces respectively. L 1' is the image of L 1 , formed by L 2• L 2 is the

exit pupil

L1 and L2 are the component lenses of an objective the foci of which are F and F'; the foci of the components L1 and L2 are respectively F1, F1 ' and F2, F2'.

We now have to define the image L1' of Lv as formed by L2• Since L1

lies between F2 and L2, this image is virtual. If this image is greater than L2, L2 will be the exit pupil, but, if it is smaller than L2, the exit pupil will be the image L1'.

Now, in most practical objectives, L2 is always smaller than L1' so that in practice the exit pupil of an objective without built-in stop is always the front lens. When the exit pupil is the front lens, this means that the other boundaries to the rays are at least so large that they will pass the light emerging from the exit pupil in the axial direction. The only other boundary present in the system depicted in Fig.l08 is the rear lens. Let us now see how large this lens must at least be in. order to meet this condition. This can be done in two ways, viz: -

XI-7] THE EXIT PUPIL OF LENS SYSTEMS 209

I) we can construct the size of L1 such that the image L1' is just as large as L2, or

2) we can construct the paths of the rays emerging from L2 as a beam parallel to the axis.

The latter procedure has been followed in Fig.l08 in accordance with the method employed in geometrical optics; it is thus found that L1

may be appreciably smaller than L2•

In actual practice, however, L1 is usually made larger, the reason for this being that parallel rays emerging obliquely from L2 require a larger rear lens in order to permit the whole of lens L2 to function as exit pupil. This is seen from the beam S1S2 in Fig. 108, which converges at the point S in the focal plane. In order to be able to pass all oblique rays such as 511 52, the lens should have a cross-section equal to 2 X MA.

b) Exit pupil of objectives consisting of 2 lenses with a stop between them (see Fig. 109)

If a stop D be placed within the objective of Fig.l08 in the manner

Li t

shown in Fig. 109, we must, in order to determine the exit pupil, also construct the image of this stop as produced by the front lens.

·-·-·-·- In the case depicted in Fig. 109 it will be seen that this image D' is smaller than L2, so that this virtual image is the exit

Fig. 109 Exit pupil of an objective with internal stop D. The image D' of D formed by L 2 is the exit pupil of the objective. Other references as in

Fig. 100

pupil, not L2• The practical method of ascertaining the size of the image D', i.e. of the exit pupil, is discussed in Part II, Light measurements (section XIX-4).

Let us now ascertain by means of an example the exit pupil of an objective for directions other than the axial. Fig. 110 shows the same objective as that depicted in Fig. 108. According to what has been said in section XI-5 in regard to the boun­daries of the rays, the objective shown in Fig. 110 has two flashed stops, viz. the lens L2 and the image L1' of L1 as produced by L2•

If the objective be observed in the direction of the axis, L2 is the smaller diaphragm and is thus the exit pupil. However, if we look at the ob­jective from a direction deviating from the axial, L2 will be the exit

210 PROPERTIES OF OPTICAL SYSTEMS [XI'

pupil for those directions that fall within the conical surface which can be drawn through the peripheries of L1 ' and L2 •

The extreme directions are represented in the figure by the line E'C. Geometrical-optical con­struction shows that the s point on the object which r~=::-:~==__!,~~~~~~====---corresponds to the ex­treme direction in which L2 is still the exit pupil, is the point S. The emer­gent beam from this point is bounded by AS1

and CS2, where CS2 is the production of E'C.

Fig. 110. Vignetting in an objective. The rays from point T on the object at the focus do not quite fill the front lens L 2 because the image L 1 ' is not large enough to serve as background for the whole area of L 2 in the· direction of the

parallel emergent rays T (T1 and T 2)

For points outside the circle of radius FS, for example T, at which the emergent beam (ATcBT2) makes a larger angle with the axis than that from S, L1' does not offer a completely flashed background, so that L2

will not be completely flashed. If we draw a line E'B from E' in the direction of the beam from T, this line will intersect the lens L2 at B. In this manner the lower boundary of the emergent beam from S is

f Fig. 111. Front view of an objective in which vignetting occurs. The points A, B, C and D' are the same as those similarly indicated in Fig. 110

found for the plane of the drawing*). If L2 should be the exit pupil for T, the radius of L1 would have to be extended to the point G, and L1' would be cor­respondingly larger. What we actually see of the exit pupil is depicted in Fig. 111, in which both the diaphragms L1' and L2 are shown as we see them displaced with respect to each other. The points A, B, C and D' in this figure correspond to those in Fig. ll 0. The hatched area of L2 remains dark and the exit pupil is therefore smaller than L2•

L1' and L2 are not drawn as circles, but as ellipses. Due to the oblique viewing

*) The same result can be obtained with the aid of another geometrical-optical cons ruction.

XI-7] THE EXIT PUPIL OF LENS SYSTEMS 211

direction we see that the circles are deformed to ellipses, whose minor axes are equal to the diameters of the circles, multiplied by the cosine of the angle of emergence (angle T1AF1).

The above described phenomenon is known as masking or vignetting. This may be further elucidated with reference to Fig. 112a and b. On the left hand side of each of the diagrams, two stops D1 and D2 are shown in front of a surface L emitting diffuse light. The stops are of equal size, but the spacing is not so great in Fig. 112b as in 112a.

a

b

,.,--,,\ I I I I \ I ' .... __ _.,.//

p

p Q

,,..---.... ~ I ' I \

I I

\ / ',~' ' ' ' ' ' ',.., __ ... ,-'

R

.... / '

I '

\~) ~ \ I ........ __ ,,

R

Fig. , 12. Vignetting by two diaphragms D 1 and D2 filled with lighf oy a diffuse light-emitting surface L. With the longer distance between D 1 and D 2 (a), the vignetting is more pronounced than with the shorter distance (b). The figures P, Q and R shown on the right illustrate the flashed portions as seen from the directions

of P, Q and Ron the left

Now, if we look in the direction of the axis (from P) towards L, we see D2 wholly flashed with light from L, as shown by P on the right hand side of the figures a and b. When the direction of vision is removed from the axial, the stop D2

is seen to be fully flashed up to a certain direction Q. The direction in which D2 can still just be seen to be fully flashed is closer to the axis in Fig. 112a, with its greater spacing DcD2, than in Fig. 112b. At wider angles of observation with respect to the axis, vignetting occurs because a part of the background against which D2 is observed

212 PROPERTIES OF OPTICAL SYSTEMS [XI

is now formed by the side of D1 remote from L. This vignetting effect sets in more quickly with the wide separation of D1 and D 2 than with the closer spacing. In the right hand section of Fig. ll2a and b, R shows what is seen in the two cases when the observation is made in the direction of R. The direction of R in Fig. 112a is the same as in 112b. This clearly demon­strates the difference in the vignetting effect, which is more pronounced in the case of the more widely separated stops D1 and D2•

Objectives intended to cover a wide angle of the field of view, such as most photographic objectives, are therefore made as short as possible.

XI -8. The luminous flux of optical systems. Aperture of lens and mirror systems

We shall now show how the luminous flux emerging from an optical system may be ascertained. Let us commence with a single, thin, lens with an object at the focus. If we once more disregard the loss.es in the lens, the luminous flux emitted by the lens will be equal to that reaching it from the object, this being calculated in the following manner. Fig. 113 shows the object P at the focus of the lens L. A small area LJS p of P lying on the axis has a luminous intensity m the

Fig. 113 Illustrating the method of computing the luminous flux emitted

by a lens

direction of the axis of ! 0 = LJS p X L (L =luminance of P). The luminous flux ifJ LISP emitted by LJS p towards the lens is

ifJLisp = L15p X nL sin2 e.

As long as P is not too large compared with the focal length, we may assume as an approximation that each small part of P emits the same luminous flux towards L, so that the total flux reaching L is

(XI-6)

This could be, for example, the luminous flux that reaches a projection screen (ignoring losses in the lens), when an image of the light source is produced a great distance away (which may be put equal to infinity). If the light travels through the lens in the reverse direction from infinity,

XI-8] THE LUMINOUS FLUX OF OPTICAL SYSTEMS 213

an image is formed at the focus and the illumination E of the image is E = nL sin2 e, L being orice more the luminance of the object which is reproduced. This is, for instance, the illumination received at the centre of a photo­graphic film or plate with the lens at a great distance from the object to be photographed; this illumination determines the exposure time. In the first instance the luminous flux, and in the second the illumination, is determined by the angle e. Now, cote= 2/fD, where I is the focal length andD the diameter of the lens. The luminous flux and illumination thus appear to be dependent on the ratio of the focal length to the lens diameter, but independent of their absolute values. This ratio is known as the aperture, or /-number, of a lens. An /-number of n is designated as ffn, or f: n, e.g. //1.5, or f: 8. Having dealt so far with simple lenses, let us now see what takes place when the lens system comprises a number of components, as is the case with most objectives. In the case· of the objectives depicted in Figs. 108 and 109 it will be seen from the geometrical arrangement that the angle e is determined by the exit pupil EP and the focal length, viz. that cote= 2ffEP. Here, then, the /-number is the ratio of the focal length to the diameter of the exit pupil. If the directions of the rays be reversed, the exit pupil shown in the diagram functions as entrance pupil. In photographic objectives, there­fore, the /-number has to be defined as the ratio of focal length to en~ trance pupil. With a degree of accuracy that is sufficienffor photographic purposes we can say that in practice sin2 e is equal to tan2 e. Since the /-number is proportional to cote, we may also say that the illu­mination on the sensitized material, which is proportional to sin2 e, is inversely proportional to the square of the /-number. Hence, as an approximation, the exposure time is directly· proportional to the square of the /-number. In photographic objectives it is .usual to mark the different stops in such a way that each setting corresponds roughly to one half, or twice, the .exposure time for the adjacent stop. The /-numbers thus constitute geometrical series with a ratio of roughly y2 ~ 1.4 (//4.5, 6.3, 9, 12.5, 18, or // 5.6, 8, II, 16, 22). The above remarks relate to the condition whereby the object or image is at the focus of the lens or objective, i.e. the lens is "focussed to in­finity". In the case of a camera objective it will be seen that when the camera

214 PROPERTIES OF OPTICAL SYSTEMS [XI

is focused on a nearby object, the image moves back, producing a smaller .angle e than when the focussing is set at infinity and, hence, that the exposure time needs to be longer. In extreme cases this must be taken into account. Suppose that under-exposure to the extent of 25% is permissible; no correction should then be necessary for distances more than 10 times the focal length of the lens. The foregoing remarks apply to all objectives in general, but if a com­parison is to be made between one objective and another, it must not be forgotten that the illumination is proportional to the transmission factor of the objective and that this may differ considerably. The trans­mission factor of a simple lens is about 0.9; that of an objective com­prising a number of lenses might be as low as 0.5.

XI-9. Some remarks on mirror systems

The considerations outlined in the preceding sections apply in principle to all types of optical system. The examples given relate to lens systems, but mirror or reflecting systems, as well as combinations of lenses and mirrors, can be dealt with in the same way. Whilst not pursuing this further it will nevertheless be useful to note one or two points peculiar to mirror systems only. Whereas with lenses (with corrected lenses at least) one obtains exact images of objects, free from spherical aberration, mirrors give an exact image only within certain limits. \Vith concave mirrors of the three different types most in use, spherical aberration is absent only in the following cases: a) in spherical mirrors: when rays are emitted from the centre. These

are returned to the centre. b) in parabolic mirrors: when rays are emitted from the focus (or pass

through the focus; these are reflected parallel to the axis. In this case one can speak of an image at infinity.

c) in ellipsoidal mirrors: when rays are emitted from a focus (or pass through a focus): these converge at the other focus.

When the /-number is high (i.e. relatively small mirror diameter), it is in practice usually possible to anticipate a perfect image of objects within the paraxial zone.

XI-10. Drum lenses

In lighthouses use is made of cylindrical lenses of the type shown in the diagram in Fig. 114, in which b is the horizontal cross-section through

Xl-10] DRUM LENSES 215

the line AB in the vertical section a. These lenses are known as drum

c

Fig. 114. Drum lens

lenses. The vertical cross-section is the same as that of an ordinary spherical lens; this rotated about a vertical axis through the focus F gives the cylindrical form. When a light source is placed at F the rays are concentrated in the vertical direction; horizon­tally no concentration occurs and the beam emerging from the lens is accordingly fan-shaped, this being particularly suitable for lighthouses. The luminous. intensity of an optical system of this kind can be computed in the following manner. If we look at the lens from a

considerable distance in the direction of the line AB, we see a part of the lens flashed. Vertically this area extends from top to bottom, but in the horizontal direction it is no wider than the width of the light source, since there is no concentrating effect in that direction and the light source is neither magnified nor reduced. We shall denote the luminous intensity of the light source in the direction of the lens by I, the height by hand the width by b. Let H be the height of the lens. The area of the flashed part is then b X H. The luminance is that of the light source multiplied by the transmission factor -r: of

. -r:.I the lens, that 1s: --.

b.h

The luminous intensity of the lens is then

-r:.I I h = b . h X b . H = -r: . h . H.

In this expression it is seen that the width b of the light source does not occur; hence that the luminous intensity of the lens is not dependent on this width. The luminous intensity is ascertained by multiplying the height of the lens by the quotient of the luminous intensity of the

216 PROPERTIES OF OPTICAL SYSTEMS [XI

source divided by the height of the source, so the candelas per em or inch of height. It is evident that the important thing for lighthouse lamps. is not the luminance but the luminous intensity per em or inch of height.

XI -11. Cylindrical mirrors Here we shall deal only with cylindrical-parabolic mirrors. Used in conjunction with linear light sources such as the high-pressure mer­cury vapour lamp and the halogen lamp, such mirrors have assumed practical importance, in particular for flood-lighting. Fig. ll5a is a front view of a mirror of the kind used with a linear light source; Fig. 115h shows the cross-section. The form of the mirror is developed by displacing a pa­rabola parallel to itself; the focus is thus moved along a line perpendicular to the plane of the parabola. This line is known as the focal line of the mirror.

Fig. 115. Cylindrical-parabolic reflector with linear light source

Here again, concentration takes place in a vertical direction, but not horizontally. The height of the area seen to be flashed in the horizontal direction, perpendicular t'o the axis, is equal to the height of the mirror, and the width is equal to that of the light source. Directly in front of the mirror, therefore, the luminous intensity is equal to this area, multiplied by the luminance of the source (and the reflection factor of the mirror). Since the luminance of the arc in the mercury vapour lamps employed with these mirrors is not uniform over the width of the tube, the luminous intensity per c:m of height can not be used here, so that we have to ascertain what luminance should be taken for the purposes of calculation. The light that we see comes from the direction of the focal line of the mirror and, hence, from the centre of the apparent area of the arc. At this point the luminance is at its maximum, and this is therefore taken as basis for our calculation. This is valid of course only when the form of the mirror is a true parabola in cross-section, but in practice this is never actually the case. Some divergence from the true form must accordingly be taken into account~ this resulting in the fact the light reflected horizontally emanates partly

XI-12] FACETTED MIRRORS 217

from areas in the discharge which are outside the centre and conse­quently of lower luminance. Owing to such divergences, parts of the mirror may ev·en be entirely dark, since the reflected rays in these parts are emitted away from the horizontal. Even though the above may reveal a similarity between drum lenses and cylindrical-parabolic reflectors, there is also an important difference. In drum lenses the luminous intensity is the same on all sides in the horizontal direction, provided that the width of the light source is also the same all round, as is the case with cylindrical incandescent sources. With the cylindrical-parabolic mirror,' however, •the luminous intensity decreases as we move from directly in front to towards the sides; in principle, this decrease is proportional to the cosine of the angle between the direction of observation and the direction straight ahead. It con­tinues to the point at which the intensity suddenly falls off to that of the light source alone; the angle at which this takes place is governed by the lengths of the mirror and the light source.

XI-12. Facetted mirrors

A third kind of mirror will now be discussed, this being of special constr!lction, known as facetted. Such mirror-reflectors consist of a number of small contiguous mirrors; they may be plane or curved, and are usually so shaped that the basic form is that of a paraboloid of revolution, for example. The facets will then constitute tangent planes to the paraboloid. Optically, facetted mirrors differ from mirrors having the same basic shape but having a continuous curve in the greater spread which they produce; this is accompanied by a lower luminous intensity of the beam. Whereas continuously curved mirror reflectors often produce patchy or streaked beams, this is not the case with the facetted mirror, since the individual mirrors of which it consists each function as a separate source of light, the solid angle of the emitted light being bounded by the facets. Facetted mirrors are employed in place of the usually more costly, continuously curved, mirrors in all applications where their increased spread is not a drawback, or may even be desirable. The luminous intensity of facetted mirrors is equal to the sum of the luminous intensities of the facets and, if these can be calculated, the intensity of the whole reflector can of course be determined. Such a calculation will be simple and reasonably accurate only when the facets

218 PROPERTIES OF OPTICAL SYSTEMS [XI

are plane and the whole of the reflected image of the light source can be seen from the direction in which the luminous intensity is to be as­certained. Such conditions are illus­trated in Fig. 116. The three facets of the mirror F produce 3 images L1 , L2

and L3 of the light source L. In the direction of the axis these images are all fully

F

Fig. 116. Facetted mirror

visible and their luminous intensity can be computed. If we denote the reflection factor of the facets of the mirror by (}, the luminous intensity I in the direction of the axis will be

I= n X (} X I 1 + I 1,

if I 1 is the luminous intensity of L in all directions. In order to secure the effect of the greater beam spread andjor higher uniformity of the beam, reflectors are also made of which the con­tinuously curved surface is more or less uniformly indented (display reflect{)rs).

XI-13. Optical systems for the projection of images Optical systems for producing images of h; ->cts or figures are known to all; these include the old-time magic ~.tl ern, projectors for stills or moving pictures, and microscopes. The last-mentioned will not be dealt with here. Let us now consider one or two aspects of projection apparatus, mainly from the point of view of the lighting technology involved; the geome­trical optics will be discussed only in so far as is essential to the dis­cussion. Projection systems for both still and moving pictures comprise the following components: a) the reproducing element which projects the image on to a screen.

This can be a simple lens, i.e. a bi-convex or meniscus lens, but, as an image of greater sharpness is usually required than that which a single lens is capable of producing, corrected lens systems (ob­jectives), are generally employed.

b) means of illuminating the object of which an image is required. When the object is illuminated from the rear (transmitted light) we speak of diascopic projection and the apparatus may be referred to as

XI-14] DIASCOPIC PROJECTION 219

a diascope. If the object is illuminated from the front, i.e. with incident light, the projection is episcopic and the apparatus concerned is known as an episcope. Units designed for both systems of projection are called epidiascopes. In diascopic projection the objective projects a shadow-image of the object, so that it is necessary to provide a bright background for the object; in episcopic projection the light projected is reflected from the object.

XI-14. Diascopic projection Fig. 117 depicts an object P, which may be a lantern slide or film strip, of which an image is projected by a lens system; for the sake of sim­

o,

Fig. 117. In order to flash a projection lens 0 which is to project an image of an object P, a background is required that will be at least equal to C1C2 at A,

or D 1D 2 at B

plicity the lens system (the ob­jective) is represented by an infinitely thin lens 0. Although projection is effected at a finite distance, this is in general so large compared with the focal length of the lens that we may assume for our purpose that P is at the focus of the lens. Hence, of every point on P, 0 forms an image at infinity. Jhe

rays emitted from each point on P leave the lens as parallel rays, of which it may be said that they meet at a point at infinity. From each point on P a beam travels towards 0 as shown in Fig. 117 for points P1 and P 2, from which lines are drawn to Q1 and Q2 on the periphery of 0. P1Q1 and P1Q2 are then the peripheral rays of the beam from P1• If we now project these lines in the reverse direction we have the peripheral rays of the pencil of light that has to travel to Pv and, if the same be done for P2, we have the boundary of the whole beam that has to reach the object in order that every point on P will fill the whole lens with light. The figure thus shows what size the background of P has to be in re­lation to the point at which this background is placed. If this be at A, the diameter must be at least equal to C1C2; if at B it must be at least DtD2. It will be seen at once from the figure that this background is much too large for it to consist of the filament of an incandescent lamp or the crater of a carbon arc. Such would be uneconomical, moreover, as the

220 PROPERTIES OF OPTICAL SYSTEMS [XI

light from the background is required only within a limited solid angle. Now, lenses and mirrors provide the means of concentrating the light from a source within a wide solid angle into a smaller angle, whereby the luminance, apart from losses due to reflection and absorption, remains the same. Concentrating systems of this kind are called condensers, and they may consist of lenses, mirrors, or a combination of both. Fig. 118a, b, c illustrates the principles on L

which projection systems are --tP-1<t=::::C ,...----..:::,:r..--

based in conjunction with a condenser, a mirror and a com-bination of lens and mirror, c respectively. The luminous intensity of such systems can be computed for the direction of the optical axis, and, for this purpose, according to section XI-4, it is necessary to know the size of the exit pupil, the luminance of the light source and the losses due to reflection and absorption in the lenses and mirrors.

Fig. 118. The three main systems of projection

In most cases it is difficult to calculate the luminous intensity

a) system for projecting lantern slides.

in any other direction than that of the axis, since, as we have already seen in section XI-7 b) on the subject of objectives, the dependence of the intensity on the geometrical-optical charac­teristics of the optical system is much more complex.

The diapositive P is large compared with the objective 0. The condenser C is placed just behind P. The nar­rowest point in the path of the rays is the objective

and c). Systems for projecting standard film. The film gate Pis small compared with the objective 0. The narrowest point in the path of the rays is the film gate. In b) a mirror condenser is shown; in c) a condenser com­prising a parabolic mirror and a lens

Calculations can also be made to determine the luminous flux emitted by the objective. Such calculations of both the luminous intensity and the luminous flux of projection systems enables us to determine to a fairly high degree of accuracy whether a system as designed will meet the requirements imposed on it. The size and position of the exit pupil are ascertained from the geo-

XI-15] SLIDE PROJECTION 221

metrical-optical characteristics of the system. These depend on the ratio of the size of the object to that of the objective, whilst the dimen­sions of the objective, condenser and light source are in turn governed by limits set by the technique, or by economic factors, in other words by the cost of the equipment. In the following sections we shall consider two examples, viz. 1) object large compared with diameter of the objective; 2) object small compared with diameter of the objective. The first of these conditions is found in the projection of lantern slides (e.g. 3tx3t" and 24x36 mm), and we shall accordingly classify this kind of projection as slide projection. The second category includes the projection of standard 35 mm film, and this may be referred to as standard-film projection. The projection of sub-standard film (such as 8 and 16 mm) will be discussed thereafter.

XI-15. Slide projection We shall be dealing with this using an example with given dimensions. Fig. 119 shows a slide projection system. The pictures to be projected measure 23 x 35 mm2, their diagonal being 42 mm.

~-&-~~-·-·-·-·-·~·-Fig. 119. Illustrating the method of calculating data for the optical elements of a system for lantern slide projection.

The condenser lens, located immediately behind the slide, has a free aperture 50 mm in diameter. The condenser system consists of two lenses, an aspherical one close to the lamp and a spherical one close to the slide. This condenser system forms an image of the filament of the lamp

222 PROPERTIES OF OPTICAL SYSTEMS [XI

in the exit pupil of the objective. The focal length of the objective is madej3.5 so that exit pupil diameter is 100 : 3.5 ~ 29 mm. The linear magnification of the image of the filament in the object lens is 3.6. With these data we now want to calculate the luminous intensity at the axis of the beam emerging from the objective using two different projection lamps, assuming that the total transmittance r of the system is 0.5. The lamps we use are two Philips projection lamps, the data on them that are important to us being listed in the table below.

filament lum. flux horiz. lum. mean luminance bxh intensity mm lm cd bs

llOV-lOOW 6x5 1730 182 600 220V-500W lOX 9.5 11400 1650 1740

It is noticeable in a comparison of the luminance that that of the 500 W lamp is much greater than that of the I 00 W lamp. The main reason for this is the arrangement of the coils in the I 00 W lamp in one plane

~rD r-0 r;:

~ v-a b

Fig. 120. In slide projection, the objective can be filled by the image of the filament of the projection lamp in two ways, viz: a) the image falls entirely within the exit pupil of the objective; b) the image pr~jects beyond the exit pupil.

XI-15] SLIDE PROJECTION 223

next to each other, involving, for reasons of lamp manufacture, rather large gaps between them. (These can partially be compensated by the use of a hemispherical auxiliary reflector by images of the filaments.) In the 500 W lamp the coils are arranged one close behind the other in two planes. The gaps between the parts of the filaments in the foremost plane are filled by the filaments in the rearmost plane. This means that the "concentration" of the filaments in the 500 W lamp is much greater than in the 100 W type, resulting in a much greater average luminance. We shall now be calculating the luminous intensity of the objective with the two lamps in succession.

1. 110 V-100 W lamp The image of the filament in the exit pupil of the objective measures

3.6 X 6 = 21.6 mm (breadth) and 3.6 X 5 = 18 mm (height).

The diameter of the exit pupil of the objective is 29 mm, so that the image of the filament falls wholly within it (Fig. 120a). In this case the image of the filament is the exit pupil of the entire system and the luminous intensity at the centre of the beam is the luminous intensity of this image.

We calculate this in the following way: The linear magnification is 3.6, so that the luminous intensity of the image is 3.62 x the luminous intensity of the lamp. We still have to take the transmission factor of the optic (0.5) into consideration, and the luminous intensity in the light beam is

l = 3.6 2 X 182 X 0.5 cd R:i 1180 cd.

2. 220 V-500 W lamp The image of the filament in the exit pupil here measures 3.6 X 10 = 36 mm (breadth) and 3.6 X 9.5 = 34.2 mm (height).

Because the dtameter of the exit pupil of the object lens is 29 mm, the image of the filament projects beyond the exit pupil of the object lens on all sides. The lens is thus completely flashed by the luminance from the lamp, and the luminous intensity of the beam can be calculated by multiplying the luminance of the lamp by the exit pupil (bearing in mind the transmission factor of the optic, of course).

224 PROPERTIES OF OPTICAL SYSTEMS [XI

The size of the exit pupil is t n x 2.9 cm2 = 6.6 cm2. The luminous intensity of the beam is then

1 = 6.6 X 1740 X 0.5 cd ~ 5750 cd.

If we now compare the luminous intensities of the beams with the hori­zontal luminous intensities of the lamps, we find that, while the luminous intensity ratio of the lamps is 1 : 9, the luminous intensities of the beams are in the ratio 1 : 5. If we take an objective with an aperture off: 2.5, so that the exit pupil diameter is 100 : 2.5 = 40 mm, the image of the filament of the 500 W lamp also falls entirely within the exit pupil of the objective, and we can calculate the luminous intensity of the beam in the same way as with the 100 W lamp:

I= 3.62 X 1650 X 0.5 cd ~ 10600 cd.

Of course, the ratio between tile luminous intensities of the beams is then exactly the same as that between the horizontal luminous intensities of the lamps.

From this table it will be seen how important it is to have the image of the light source wholly or almost wholly within the objective, in order to make the most effective use of the luminous flux from the lamp (and also of the wattage consumed by the lamp). We can now calculate from the luminous intensity along the axis of the system the illumination at the centre of the projection screen. If a certain illumination is required at a given distance from the projector, the aperture of the objective and the light source itself provide the means of approximating closely the required value. There is, however, a limit to which such requirements can be carried, as imposed by tech­nical and economic considerations. As regards the screen illumination elsewhere than on the axis of the system, the following may be noted. In principle, the screen illumination is directly proportional to the 4th power of the cosine of the angle between the axis and the direction in which the illumination is to be evaluated, but in many cases vignetting will occur at oblique directions (see section XI-7). This effect will be all the more pronounced according as the image of the light source fills the objective to a greater extent. In practice, however, such vignetting rarely needs to be taken into account in lantern slide projection, since the rays do not travel through the objective very obliquely. If the image of the light source is not symmetrically disposed in the objective, i.e.

Xl-16] STANDARD-FILM PROJECTION 225

if the source is not properly centred, different degrees of vignetting will occur at the sides, or top and bottom, resulting in irregular illumi­nation on the screen (e.g. higher at the top than at the bottom).

XI- I 6. Standard-film projection

In standard-film projec­tion the projected object, bounded-by the film gate, is usually smaller than

~::f::::~1~1-l"""""==::::=:=:=~ the exit pupil of the ---r objective; an example of this arrangement is de­picted in Fig. 121. Fig. 121. In order to fill the exit pupil of the

objective 0 with light from every point in the film gate AB in standard film projection, the solid angles w 1 , w 2, w3 must be filled with light. The rearward projections of the boundary lines of the solid angles determine the size of the con­denser necessary to ensure the optimum flashing

of the objective (e.g. C)

In this diagram AB is the diagonal of the film gate (24.8 mm for the standard sound-film size of 15.2 X 20.9 mm2 , with

rounded corners). The objective 0 has a focal length of 12.5 em and an angular aperture of f/2, so that the diameter of the exit pupil is 62.5 mm. Proceeding from the exit pupil of the objective, the figure shows the path of the rays from a few points in the film gate. If the exit pupil is to be fully flashed, the solid angles w1 , w 2 , w 3 etc must be filled with light. Now, if we try to achieve this in the same way as for the slide projector l;>y placing a condenser just behind the film gate, say at C, to produce an image of the light source in the exit pupil of the objective, it is found when a standard type of projector lamp or carbon arc is used, that the source has to be magnified to such an extent - i.e. that the focal length must be so short - that it is not practicable to make a condenser of the required diameter. On the other hand, for a condenser having the smallest /-number that is technically possible, the light source would have to be so large that the whole unit would become uneconomical and therefore unacceptable. There is a solution, however, that entails neither of these objections, and this consists in producing an image of the light source at the point where the cross-section of the beam passing through the optical system is smailest. According to Fig. 121 this is the film gate. If, however,

226 PROPERTIES OF OPTICAL SYSTEMS [XI

an image of the filament of an incandescent cinema lamp is formed in the film gate, this image is projected on the screen by the objective, with consequent unpleasant irregularity in the screen illumination. But, if the image of the spiral be made to lie between the film gate and the objective, the image of the spiral in the film gate itself is so blurred as to ensure reasonable uniformity in the luminance. The spherical aberration present in most condenser systems also has a blurring effect on the image. In any case the luminance of the light source must be as uniform as possible. The magnification required to fill the film gate with the (unsharp) image of the light source presents no difficulty as far as the dim~nsions of the condenser are concerned. The condenser usually takes the form of a spherical mirror even though, owing to spherical aberration, this is not the ideal form for a mirror condenser*). The arrangement is now as shown in Fig. 122, in which M is the mirror.

M

Fig. 122. To determine the exit pupil of a pro­jector for standard film with mirror condenser M. In the image space there are three boundaries to the rays, viz. M', that is the image of the mirror M formed by the objective 0, L 2 , the front lens of the objective, and L 1', the image of L 1 as formed by L 2 • Under the conditions as shown, L 2 is the smallest and is therefore the exit pupil

of the system for points on the axis

either L2 or M' will be the exit pupil. Two possibilities then arise, viz.:

AB the film gate, and 0 the objective. Let us determine the exit pupil of this system. In the image space there are three boundaries of the rays, L2, L1 ' and the image M' of the mirror M. The last mentioned lies in front of the ob­jective. L1 ' is always greater than L2, as al-ready mentioned, so that

1. M' smaller than L2• In this case M' is the exit pupil. 2. M' larger than or equal to L2• L2 is then the exit pupil. To calculate the luminous intensity of the system, the luminance of the light source must be known in either case. If a 15 V, 50 A Philips cinema lamp is used, the luminance is about 3000 sb. For the luminance of the exit pupil we take about half this value, viz.

*) V.ie shall not go into the geometrical optics of projection. For this the reader is referred to a series of articles by Naumann published in "Die Kino­technik" 2).

XI-17] SUB-STANDARD-FILM PROJECTION 227

1500 sb; the loss is due in the first place to reflection and absorption in the system, and in the second to the fact that, owing to the spherical aberration of the mirror, the exit pupil is not entirely·flashed with the luminance of the light source. If we now suppose that (Fig. 122) the exit pupil of the system is that of the objective (diam. 6.25 em), the luminous intensity of the system in the direction of the axis will be

I= in X 6.252 X 1500 cd ~ 46,000 cd.

The illumination of a screen 20m from the objective is then 46,000/202 = 115 lux. At points not lying on the axis of the system the illumination is lower, and the amount of the reduction as against the axial direction depends on the degree of vignetting by the lens mountings, as well as on the relative position and size of M'. If the exit pupil ~f the system is smaller than that of the objective, the axial illumination on the screen is correspondingly lower.

XI -17. Sub-standard film projection The leading sizes of sub-standard film are the 16 mm (gate size 7.16 X

9.6 mmz; diagonal 11.6 mm) and the 8 mm (3.3 x 4.4 mm2 ; diagonal 5.3 mm). In general the ratios of film gate dimensions to objective diameter follow the same principle as for standard film, and the smallest cross­section of the beam therefore mostly occurs in the film gate. At first sight, then, the obvious thing to do would be to adopt the same system of illumination as that used for standard film, but there is an important difference. As pointed out in the previous section, the use of a condenser just behind the film gate would necessitate prohibitively large lamp filaments in order to fill the exit pupil of the objective with light, but in sub-standard film projection this method can be employed; the filaments of sub­standard-film projector lamps usually require a linear magnification of 3 X in order to fill the objective, and this can be obtained with condensers of standard !-numbers. In sub-standard film projectors the condenser is therefore usually placed just behind the film gate. In principle, this is the same arrangement as employed for the projection of slides, and the calculations are accordingly wholly analogous to those applicable to slide projectors, for which reason we shall not give any further examples. Sometimes the condenser is placed slightly further away from the film

228 PROPERTIES OF OPTICAL SYSTEMS [XI

gate, so that the image of the filament lies somewhere between the gate and the objective; in certain cases this gives a higher luminous flux from the objective, but, if the image is brought too close to the film gate the gain in luminous flux is obtained at the expense of the uniformity of the screen illumination. When this system is employed, another factor, the non-uniform distri­bution of luminance of the lamp filament, is involved. If the image of the filament occurs in or near the film gate, an image of the parallel spirals of which the filament usually consists is projected more or less clearly on to the screen, as we have already seen in the preceding section; the illumination of the screen is then noticeably irregular. Owing to the fact that the geometrical-optical conditions differ somewhat from those relating to standard-film projection (this difference will not be enlarged upon here), the effect referred to is more pronounced than in 35 mm film projection. All this does not imply, however, that when the condenser is placed just behind the film gate the non-uniformity of the luminance of the filament is unimportant. On the contrary, if the filament is too irregular the screen illumination will also be irregular, since the vignetting effect reduces the luminous intensity of the exit pupil in oblique directions more or less in stages, the more so according as the image of the filament more completely fills the objective. In the projection of slides, too, uniformity of filament luminance has its importance.

XI-18. The luminous ftux emitted by projection systems Equation (XI-6) enables us to calculate the maximum luminous flux that will reach a projection screen from a projection system with a given objective and lamp. In this formula we have the means of verifying the quality of the illuminating system. Formula (XI-6) states that:

(/) = SnL sin2 8; in this case S is the area of the gate, L the luminance with which the objective illuminates, i.e. the luminance of the light source multiplied by the transmission factor of the whole optical system, and 8 the semi­aperture angle of the objective. The latter is computed from the /-number of the objective. When making use of equation (XI-6) we must ensure that the whole solid angle, of which the semi-apex angle is 8, is filled with luminous

XI-18] THE LUMINOUS FLUX EMITTED BY PROJECTION SYSTEMS 229

flux with respect to every point in the gate (we thus assume that, for points beyond the axis, the solid angles are equal to those of the point on the axis, which is approximately true). Let us now compute the maximum obtainable luminous flux in a given case and compare this with that of the light source. In practice, in order to check the quality of the projection system, this is compared with the actual luminous flux from the objective. Take the case of a standard-film projector having as light source a 15 V, 50 A cinema projector lamp with spherical mirror, and an objective with an /-number of f : 2. (It will be seen from formula (XI-6) that the focal length of the objective is immaterial). The transmission factor of the system may be 0.6, a value which is frequently met with in practice. The luminance of the projector lamp is 3000 sb; therefore the luminance of the objective is 1800 sb. f: 2 corresponds to an angle e = 14°2', so that sin2 e = 0.059. The area of the gate in the case of standard film is 3.18 cm2. These values in equation (XI-6) give a luminous flux of

f/> = 3.18 X n X 1800 X 0.059 ~ 1050 lumens.

Of the 20,000 lrn (approx.) of the luminous flux of the lamp, the objective under consideration thus passes at most 1050 lm, i.e. about 5% reaches the projection screen. In practice, the actual luminous flux reaching the screen is somewhat less than that. In sub-standard film projection the efficiency is generally even lower, and a luminous flux of 2% from the objective is by no means a rarity. When the objective is completely filled with light, the luminous flux from the lens is proportional to the luminance of the light source. An increase in the size of the source does not affect the luminous flux emitted by the objective; only an increase in the luminance will do this. For this reason the lamps used in sub-standard film projectors have often much larger filaments than are required according to the geometrical­optical characteristics, the luminance being considerably higher, however, than that of lamps with smaller filaments, consuming less power. To increase the luminance, spherical mirrors are employed, the filament being placed at the centre of curvature so that the image of it will coincide with the filament itself. The mirror should preferably be so adjusted that the images of the separate sections of the filament fall between the sections themselves, thus ensuring greater uniformity of the luminance of the luminous area. Another means of increasing

230 PROPERTIES OF OPTICAL SYSTEMS

and equalising the luminance con­sists in making the filament in two sections, each comprising a row of vertical spirals, these being so assembled that the spirals in the one lie behind the gaps in the other (Bi-plane lamps).

XI -19. Episcopic projection

M

--------L--------V

[XI

The principle of the episcope is illustrated in Fig. 123. The subject to be projected, e.g. a book or drawing, is illuminated by a num­ber of lamps L, with or without the use of reflectors, and projection is effected by means of an objective 0. An image of the horizontal subject is projected on to a vertical screen with the aid of a mirror M

Fig. 123. Diagram illustrating the principle of the episcope. The object V is illuminated by lamps L and an image is formed by the objective 0. A mirror M throws the vertical rays

on to a projection screen

placed at an angle of 45° from the vertical. If a glass mirror is used, the specular layer should be applied to the front surface of the glass in order to avoid double contours in the image. To calculate the illumination of the screen it is necessary for us to know the luminance of the subject. If we assume for the sake of simplicity that the reflection is uniformly diffuse, we must know the illumination of the subject, but it is not usually possible to calculate this from the illumination system; we shall therefore assume that the illumination is known and denote this by E v· The luminance of the subject is then

L = e X Ev sb v :rt X 1()4

(e =reflection factor of the subject, Ev expressed in lux). If we know the area (S cm2) of the exit pupil of the objective, we can compute the luminous intensity from the following equation:

e. Ev I = S X -- X T {Xl-7) :7l • 1 ()4

(-r =transmission factor of the objective). Let us suppose that we wish to project a drawing on paper of which

XI-19] EPISCOPIC PROJECTION 231

{! = 0.7, using an episcope, the exit pupil of the objective being 14 em diameter and the transmission factor -r = 0.8. The reflection of the plane mirror might be em= 0.9. Suppose that a screen illumination of 25 lux is required at a range of 4 metres. From these particulars we can calculate the required illumination of the drawing. The range, the screen illumination and the reflection factor of the mirror enable us to find the required luminous intensity of the objective:

Ed2 25 X 42

I = - = 0.9 ~ 440 cd. f!m

Ev is evaluated from equation (XI-7), viz:

here

E __ I X n X 1()4 v-

Se-r:

n I= 440cd, S = 4 x 142 = 154 cm2, e = 0.7 and -r = 0.8,

hence . 440xnx 104

Ev = 54 0 0 8 1ux = 160,000 lux. I X .7 X .

This example will serve to demonstrate the order of the illumination required to ensure reasonable illumination on the screen. It is obvious that the objectives used for episcopes must have a small /-number, i.e. large diameter. A practical objective size, on which the above cal­culation was based, has a focal length f of 50 em and an aperture of f/3.5, which means a diameter of 14 em. Apart from a concentrating- type of optics, light sources of very high luminous flux must be used, i.e. taking a great deal of power. This constitutes a problem in view of the heat generated in the episcope, but with some forms of light source the excess energy can be dispersed by means of water-cooling. A light source having outstanding qualities from the point of view of its use in episcopes is the water-cooled super­high-pressure mercury vapour lamp (such as the Philips types.SP), which combines a high luminous flux with small physical dimensions, the water-cooling for the dispersal of the surplus energy being an integral part of the design.

232 PROPERTIES OF OPTICAL SYSTEMS [XI

REFERENCES 1) Some books on geometrical ootics:

B. K. Johnson: "Optics and optical instruments", 3rd edition, London 1960 B. K. Johnson: "Optical Design and Lens Computation". London 1948 D. H. Jacobs: "Fundamentals of Optical Engineering". New York 1943 R. W. Ditch burn:· "Light", 2nd edition, 1963 M ii 11 e r-P o u i 11 e t s: "Lehrbuch der Physik", 2er Band, 1e Halfte, Bruns­wick 1926 M. Be reck: "Grundlagen der praktischen Optik". Berlin 1930

2) H. N au m.a n n: die Kinotechnik, 11, 1929, 311-315. "Zusammenhange zwischen Spiegel, Bildfensterbeleuchtung und Lichtleistung bei der Kino­projektion" Idem. 11, 1929, 651-657. "Ober den Durchmesser von Projektionsobjektiven" Idem. 12, 1930, 10-13. "Zur Kinoprojektion mit kurzbrennweitigen Ob­jektiven"

CHAPTER XII

THE PHOTOMETRIC MEASURING-UNITS SYSTEM

XII-I. Introduction In chapter I we said that when we had dealt with the photometric system and all that this embodies, we should see in how far the defi­nitions of the conceptions and units employed in illuminating engineering would have to be extended in scope to include coloured light, and that, until then, the reader should regard the considerations dealt with as referring to white light only. Before going into this problem we will review.. what we have so far considered in the field of photometric units, and see how far this fulfils our purpose, viz. the measurement of radiation evaluated according to the visual sensation produced. In so doing we may confine ourselves to the basic unit, the candela, from which the other photometric units are derived. The candela has been defined as I /60th of the luminous intensity of_ I sq. em of the black body radiating at the temperature of solidifying platinum, meas­ured perpendicularly to the surface. This means that a certain luminous intensity is attributed to a given power (W) per steradian of a radiation having a given spectral distri­bution. Neither here, nor in the definitions of the other photometric quantities and units, however, has the visual sensation produced by the radiation been introduced. In fact, we have gone no further than to arrive at a physical conception in which the eye plays no part. In addition to defining the candela, it was tacitly assumed that luminous intensity and the other photometric quantities are proportional to the power of the radiation (e.g. we have defined luminous efficiency as the luminous flux in lumens divided by the power in watts and it was thus assumed that each watt yields an equal number of lumens). If there were only one kind of light, with a standard spectral distribution, it would be possible to express luminous intensity equally well in W Jsterad instead of in candelas ( = lmfsterad) and luminous flux in watts instead of in lumens, and the measurement could be carried out in a purely physical manner. There are, however, different kinds of light and here enters the problem

234 THE PHOTOMTERIC MEASURING-UNITS SYSTEM [XII

of evaluating all these kinds "according to the visual sensation they produce". We must, therefore, relate the physical phenomenon "radi­ation" to the physiological effect "light". The expression "visual sensation" is vague and indeterminate, and requires to be further· defined; hence we have to ascertain what character­istic or criterion is the deciding factor in our visual evaluation of radiation. As will be shown in the next section, the sensation of light is a sensation produced by the luminance of the objects observed, not by the luminous intensity or the luminous flux. This physiological sensation must not be identified with the physical quantity which produces the sensation. It is, therefore, necessary to distinguish the two conceptions by different terms. As we have seen, the physical quantity is called luminance. The visual effect is denoted by the terms subfective brightness or lum,·nosity. Thus we may say that luminance produces luminosity. We will call the criterion which enables us to evaluate radiation accord­ing to its visual effect the luminance criterion.

XII-2. The luminance criterion Illuminated objects, or light sources, are perceived by reason of their form, colour and luminance. The fact that we perceive a surface by its luminance and not by its luminous intensity or luminous flux will be understood if we know something of the principle on which vision is based (see Fig. 124). As far as the optical con­struction is concerned, the eye may be compared to a c<.mera; the lens L forms an image of objects within the field of vision (e.g. an image P'Q' of an object PQ) on a "screen" which corresponds to the frosted glass or film in the camera. This "screen" lies against the rear wall of the eye and is called the

PL----t a 84219

Fig. 124. Cross-section of the human eye in diagram. R retina; L lens; I pupil; C cornea; A anterior chamber; V vitreous humour; N optic nerve. An object PQ directly in front of the eye produces an image P'Q' on the fovea

centralis F

retina (R, Fig. 124). As in the camera, too, the eye has a variable stop, the iris or pupil (1). The retina is made up of light-sensitive elements which transmit stimuli

XII-2] THE LUMINANCE CRITERION 235

when light falls on them and these stimuli are· passed through the optic nerve to the brain where we are made aware of a sensation of light. An image is formed on the retina, then, of a certain size and illumination; the size depends on the dimensions and distance of the object observed, and the illumination, according to section XI-4, is proportional to the luminance of the object as well as to the entrance pupil of the lens, i.e. the iris. Now, the pupil of the eye is not constant in size; it varies between 2 and roughly 8 mm in diameter, and it adjusts itself automa­tically to the luminance and distribution of luminance in the field of yision. When the luminance of the field is low the pupil is fully dilated, and it closes gradually as the luminance is increased.

Investigations by S t i 1 e s and C r a yv ford 1) have shown that the more the light rays entering the eye are directed towards the periphery of the pupil, the less they contribute to the subjective brightness (Stiles­Crawford effect). No explanation of this effect has yet been found. Before the effect was discovered it was held that the luminosity was goverrted entirely by the illumination on the retina, and retinal illu­mination units based on this premise are mentioned in the literature. One such unit is the luxon, defined as the retinal illumination produced by a surface having a luminance of I cdfm2 when the pupil area is lsq.mm. This unit as such should not be used.

The illumination of the retina sets up a stimulus which by way of the optic nerve produce& a mental sensation of light. This effect, related only to the luminance and not the luminous flux or intensity, is named the subjective brightness or luminosity. In short, we may say that apart from form and colour, we perceive luminance. Now, the luminosity depends not only on the retinal illumination, but also on the sensitivity conditiot'\ of the retina. When we undergo a change from a visual field of high luminance to one of low luminance, the luminosity is at first not constant; this is partly due to the slow dilation of the pupil, as a result of which the retinal illumination is increased, and partly to a variation in the sensitivity of the retina. In other words the eye adjusts itself to the luminance of the new field, and this adjust­ment in the sensitivity is known as the adaptation of the eye. Once the eye has completed the process of adaptation we say that it is fully adapted. Use is made of the terms light and dark adaptation when the eye has adapted itself to visual fields of high and low luminance respectively. Light adaptation is accomplished quickly, but dark adapta­tion is a much slower process, as everyone will have noticed at one time or another when passing from light to dark places, and vice versa. In photometry adaptation is an important phenomenon, since reliable visual measurements are obtained only when the eyes are fully adapted.

236 THE PHOTOMETRIC MEASUKING-UNITS SYSTEM LXII

The eye is unable to tell us anything about the intensity of the luminosity in terms of dimensions or quantities and no means are at our disposal for effecting such measurement. It is also not capable of giving any information regarding degrees of luminance, other than that luminance is either higli or low; if two fields of different luminance are observed simultaneously we can only say that one is brighter than the other. In any appraisal of quantitative differences in luminance the eye cannot help us; it can do no more than inform us that two fields observed simultaneously give an equal or different luminosity. And this brings us to an important definition for our measuring system, viz. that when two fields simultaneously produce the same luminosity, the luminances of the fields are held to be equal. To this must be added that we regard the luminance as being proportional to the power; by definition, n times as much power yields n times as much luminance. Given a luminance A (proportional to power a), if this luminance be increased to a luminance B of the same kind of light (proportional to power b), the sum of these luminances A and B (proportional to a+ b) will give the same luminosity as the luminance (of the same kind of light) that is proportional to c, if a + b = c. We may therefore say that the eye adds up the luminances. As we have seen above, this does not refer to the luminosities, but to the luminances themselves, as based on the equality of the luminosities. From the fact that the eyes are capable only of a comparison of lumin­ances it follows that, if we wish to take visual measurements of other quantities used in illuminating engineering, these must be converted to luminances by one means or another.

XII-3. Luminance of coloured light In the preceding section we have said briefly that our measuring system is based on an equality of luminosity thus assuming that such equality, even among the most widely diverging colours of light, can always be perceived. If luminances. being compared differ only slightly in colour or not at all (homochromatic comparison) this is readily acceptable, but it is not so acceptable when the kinds of light differ considerably (heterochromatic comparison). As this may require some explanation let us turn to Fig. 125 which shows diagramatically a form of apparatus for the comparison of two luminances. The lines AB and BC represent two white surfaces having equal reflective properties, and both are observed by the eye 0. Let us suppose that AB receives blue light and BC red

XII-3] LUMINANCE OF COLOURED LIGHT 237

light, the lummance of AB being kept constant whereas that of BC can be varied. Comparison between the luminances of AB and BC is

Fig. 125. Arrangement for comparison of two luminances. AB and BC are diffusely reflecting surfaces, separated from each other at B. Both AB and BC are observed

from 0.

facilitated by a dividing line at B, between the two fields. The observer is asked to adjust the luminance of the red field so that the luminosity pro­duced equals that of the blue; according to the definition given in the previous section, the luminances of the red surface will then be the same as that of the blue surface. So much red light can now be allowed by the observer to

fall on BC that he can clearly judge the luminosity produced to be greater than that of the blue. If he then decreases the quantity of red he arrives at a point where he cannot say with any certainty whether the luminosities and thus the luminances are equal or different. With still less red he will quickly be able to see quite clearly that the luminance of the red is lower than that of the blue. Notwithstanding the great difference in colour he will thus have observed a difference in luminance, first in favour of the red, and subsequently in favour of the blue. Within the region of uncertainty, therefore, the difference in luminance must have been reversed. If the observer adjusts his apparr;~.tus a large number of times to the point where in his opinion there is no difference in luminosity and thus in luminance, a statistical analysis will show that it can be said of one certain adjustment that, according to the observations, the luminance of the red is equal to that of the blue. In comparisons of markedly heterochromatic light it is difficult and sometimes impossible for an unskilled observer to effect such an ad­justment for equal luminosity, and the adjustments whereby he balances the luminosities, and thus the luminances themselves, may vary widely. Nevertheless, it has been found that observers can become practised in "forgetting" the colours, so that they succeed fairly well in responding only to the luminance components of the fields compared, as distinct from colour components. Fortunately, however, there are other methods of heterochromatic photometry which demand less of the observer than the method of

238 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

direct comparison and which make possible much greater accuracy in measurement. It has been said above that adjustments for equal luminosity are not difficult when the differences in colour are sufficiently small, and use can be made of this fact also to compare luminances· between widely differing colours. Instead of making a direct comparison between the red and blue as in our example, we now compare the blue with another blue colour which is not so very different from the first, but contains also a little red; this colour is then compared with another containing slightly more red. The colours can be so selected that they constitute a gradual transition from blue to red, e.g. by increasing the red content each time and reducing the blue, thus giving a change over from blue to red in stages. In the last stage the red is compared with another red having a small blue component, but so little that it is not disturbing when the luminosities are being balanced.

This method of comparison is useful only if it appears that the final result is independent of the series of intermediate colours used (S c h r 6-d in g e r 2)). This has been investigated and verified by K 6 n i g and others.

By means of this step-by-step method, then, it is possible accurately to determine luminance ratios of all kinds of coloured light. If in place of the blue in our example we use light of the standard spectral distribution from which the candela is derived, we can thus in the same way compare the luminance of red light with the standard light. The luminosity produced by the red light being matched with that of the light of standard distribution, the same value (e.g. in stilbs) is attrihuted to the red as to the white luminance. Another method of heterochromatic photometry is the flickn method, whereby a direct comparison is made. Suppose once more that a red field is to be compared with a blue one; if the two fields are alternated rapidly the effect is a flickering of the field of view and, if a start is made once more at the point where the luminosity produced by the red is clearly greater than that of the blue, the flicker will be very marked. If the luminance of the red be then gradually reduced, the flicker will become less pronounced, or even cease altogether; a further reduction of the red luminance then again increases the flicker.

The flicker method is based on the property of the eye to react more slowly to colour stimuli than to luminance stimuli. It is possible so to adjust the speed of the alternations that the eye perceives changes of luminance, but not changes of colour. It may also be said that there is a colour flicker (1), and a luminance flicker (2). At a certain frequency (1) disappears and (2) remains.

XII-4] THE RELATIVE LUMINOUS EFFICIENCY OF RADIATION 239

If the luminances of the red and blue fields at the moment of minimum flicker should be measured by the above described step-by-step method, the luminances should prove to be the same. The accuracy with which the point of minimum flicker, and therefore also equality of luminance, can be adjusted, is very much higher than can be obtained by direct comparison of stationary fields. Here, then, we have a method of measuring coloured light by direct comparison, this being the one employed with the flicker photometer to be discussed in Part II (section XIV-4). ~he cumbersome step-by­step method is therefore not essential in procuring sufficiently accurate measurements of coloured light.

XII--4. The relative luminous efficiency of radiation From the preceding section it would appear that the problem of measur­ring coloured light, i.e. of expressing the various quantities as applied to coloured light in the conventional units, has been effectively solved. Using a photometer, we merely compare the luminance of the coloured light with that of light of the standard composition and, by balancing the luminosities, we should be able to attribute to the coloured light a certain value in stilbs, candelas, lumens etc. Unfortunately, however, it is not quite as easy as that, for, if a number of different observers are asked to measure in this manner the same quantity of coloured light (i.e. a certain quantity of radiant energy of a given, but arbitrarily selected, spectral distribution), the results will generally differ among the various observers. Luminosities produced by the same amount of energy at different wavelengths of the spectrum differ between one person and another, and are not even constant for any one observer in the course of time. As it is not acceptable that the results of measurement should depend on accidental peculiarities of the eyes of the observer, efforts have been made to lay down characteristics for the "average eye" with respect to the evaluation of light. The means to this end are based on a deter­mination of the sensitivity of the eye to radiant energy of different wavelengths, the relative luminous efficiency of radiation. To define this let us turn once more to Fig. 125. We will assume that the left hand ·field is illuminated with light of wavelength A0 , and that the energy *) reaching the eye and inducing a certain luminosity is E 0 • This can be expressed in watts. The right hand field may be

*) Where in this and following sections the term energy is used, it is intended to represent energy per second (power).

240 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

illuminated with light of wavelength A, this being so adiusted as to produce the same luminosity as the other. The energy of the light of that wavelength A that will then have to enter the eye may be denoted byE>... Now suppose that, in order to obtain the same luminosity, three times as much energy of wavelength A is needed as that of wavelength A0 ;

we then say that the relative luminous efficiency of radiation of the wave­length A is lf3rd of that of wavelength A0 • Hence the relative luminous efficiency of radiation is inversely proportional to the energy required to produce equal luminosities. In the experiment described, the wavelength A can be successively replaced by a number of others and, by ascertaining each time the power required to induce the same luminosity as that induced by the reference wavelength A0 of energy E0 , we can determine the ratios of the relative luminous efficiency of radiation in relation to the wave­length A0 for different wavelengths. The values thus obtained can then be plotted as a function of A. As it is here a question of ratios, it is customary to multiply these ratios by a factor such that the maximum point in the curve represents unity. They are denoted· by the symbol V>... The relative luminous efficiency of radiation of a large number of persons has been measured by various workers and, in 1924, the C.I.E. standard­ized the average of these VA measurements. Formerly these standardized values of VA were called the International Luminosity Factors. Later, others have carried out further measurements of VA but, although many of these were found to diverge from the internationally 1.0

accepted values, the C.I.E. have o,a up to now not considered it necessary to modify the curve 0.6

originally adopted. ot4 The values of the relative lu­minous efficiency of radiation °·2

have been plotted in Fig. 126, o and the respective values of the

lfl ( \ :I I \ - 1-· I '\

J \ v '\

~ - I-). 400 500 600 m~ ?00

wavelengths, in stages of 5 mt-t Fig .. 126. The. internation;~J !elative lu-. . T bl VI ( ) mmous effic1ency of rad1atlon ( V>..). are g1ven m a e p. 414 .

It will be seen from the curve that V.\ reaches a maximum at 555 mt-t (or more precisely at 554 mt-t) and falls off towards each end of the spectrum. For red or blue light, therefore, much more power is required to induce a given luminosity than for green or yellow light.

XII-4] THE RELATIVE LUMINOUS EFFICIENCY OF RADIATION 241

It should be pointed out here that the curve in Fig. 126 is valid only for luminances above 3 cdfm2 (approx.); what happens to the relative luminous efficiency of radiation at lower luminance levels will be seen in section XII -7. Using VA we can now derive the conditions to be met by the radiant energy in order to induce balanced luminosities. Let us denote the relative luminous efficiency of radiation at wave­lengths A0 and A1 by V0 and V1; then

(XII-I)

So far we have considered only those kinds of light which consist of a single wavelength, but in practice we are nearly always concerned with 1ight ·that comprises numerous components, of different wave­lengths. In such cases it would of course be possible to proceed along the lines given in section XII-3, but this would not be practical in view of the very large number of possible combinations. There is another course open to us, however. It is found that it is per­missible to assume that the eye adds up the luminosities of the different components (summation law*), and we can accordingly lay down the conditions for balanced luminosities for heterochromatic kinds of 1ight in this way. We substitute for the light of wavelength ..t1 another consisting of a combination of wavelengths A2 and A3 whose energies are respectively E 2 and E3; these energies are so adjusted that the resultant luminosity is equal to that of the light of wavelength A0 •

Then, assuming from the summation law that the luminances due to the components ..t2 and ..t3 are added up by the eye, the expression V 0 E0 = V1E1 is now replaced by

(XII-2)

If the light has numerous components ..t2, ..t3, A4, Ali .... whose energies are E 2, E3 , E 4 , E 5 •••• , the conditions of balanced luminosities will be:

V0 E0 = V2E 2 + V3E3 + V4E4 + V5E5 •••.

or

If individual differences in relative luminous efficiency of radiation are

*) referred to again in section XII-5.

242 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

to be eliminated, the internationally adopted values of V>. must be used in place of V2, V3 etc., so we usually write:

(XII-3)

Where the spectrum of the light is continuous it may, without in­curring any appreciable error, be regarded as being composed of small bands of wavelengths e.g. 10 mJ-t

£'~~) in width, and the radiant power r aA.

at each wavelength within these t bands can be taken as being that of the mean wavelength of each band. Fig. 127 depicts such a continuous spectrum with radiant energy plot­ted against wavelength. In this figure the total energy is repre­sented by the area enclosed within the curve and the abscissa; this area can be divided into strips LIA. in width (e.g. 10 mJ-t) and the area of each strip will then re­present the energy radiated between the wavelengths A.1 and A.l + LIA..

0.5 0,6

Fig. 127. Distribution of spectral energy of a light source. The area between the curve, the abscissa and the ordinates of 0.4 andO. 7 ~represents the energy radiated in the visib1e part

of the spectrum.

In applying equation (XII-3) we assume this energy to be radiated at a wavelength of A(= A.1 + !LIA.).

This can be stated with more mathematical precision by taking in place of strips of finite width .H, strips of infinitely small width d>.. The ex­pression .EV;..E>. is then replaced by J V;..E';..d~, in which E'>. is the energy per wavelength unit (~~). In practical calculations, however, the integral is replaced by the summation as above.

XII-5. The summation law The validity of the summation law in the measurement and evaluation of luminance is of great importance for photometry and illuminating en­gineering, since it enables us to add up the values of luminance, lu­minous flux etc. of kinds of light which differ from one another. For example, given an incandescent lamp of 1000 lm and a mercury vapour lamp of 500 lm, we may accordingly assume a total luminous flux of 1500 lm for the blended light.

XII-6] PHOTOMETRIC QUANTITIES AND UNITS BASED ON VA. 243

This law may also be formulated in another manner; when a luminous flux t;/J1 (spectral distribution 1) is equivalent to a luminous flux t/J2

(spectral distribution 2), a blended luminous flux comprising

is also equivalent to t/J1 and t;/J2 {otis an arbitrary value between 0 and 1). The same formulation will also apply to luminous intensity, luminance etc. It is not usually realised when the quantities employed in photometry are added up that use is being made of an important law, the validity of which was established only after considerable investigation (in certain circumstances it is not actually valid at all). This becomes all the more apparent if we look for a moment at the consequences of employing another criterion than the luminance cri­terion, e.g. the amount of light that might be required for the execution of a certain 'visual task, e.g. the visual acuity.

By visual acuity is meant, expressed in simple terms, the capacity of the eye to distinguish fine details in the field of view.

The summation law does not hold good when based on such criteria, however, and a system of measurement based on them would lead to great practical difficulties. This means, then, that quantities of light which are equivalent according to the system of measurement founded on the luminance criterion may in other respects not be equivalent. A striking example of this is to be found in the far higher visual acuity experienced with sodium light than with the same quantity of light from tungsten lamps.

For further information regarding the requirements to be met by measuring systems, and the extent to which our photometric measuring­units system satisfies these requirements, the reader is referred to a very interesting work by H. Konig: "Der Begriff der Helligkeit", p. 40 et seq. a)

XII-6. Definitions of photometric quantities and units based on V,\

In section XII-4 we have expressed equality of luminosity and, by definition, also equality of luminance as equation (XII-3):

VOEO = 1: V,\E,\.

By analogy we can express equality of luminous flux as: V0G0 = 1: V,\G.\, G being the power of the luminous flux. In this the terms V 0G0 and 1: V ,\G,\ represent the luminous flux, but in a different unit from the

244 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

lumen, which is the unit so far employed for luminous flux. If G be expressed in watts, the unit in which E V,xG,x is expressed is called the light-watt; we therefore write:

l/J = E V,xG,x (light-watts). (XII-4) To convert the value of the luminous flux in light-watts to the usual unit employed in photometry, viz. the lumen, it must be multiplied by a- for the moment unknown- factor, C. Equation (XII-4) then becomes

l/J = C E V,xG,x (lumens). (XII-5)

Analogous formulae can be constructed for the other photometric quantities in which the constant C will appear as the ratio of the candela (lmfsterad) to light-watt/sterad, the lux (lmfm2) to light-wattfm2 and

( lm ) light-watt the stilb --fcm2 to fcm 2• ,stj:!rad sterad

watts For example, if E in equation (XII-3) is in--dfcm2, the terms V 0 E 0 stera

light-watts and E V,xE,x will represent the luminance in fcm2• The lu-

sterad minance L in stilbs can then be written:

L = C E V"E". (XII-6)

In this way the photometric quantities and units are based on the international relative luminous efficiency of radiation V ..\· It now remains to evaluate C, and this can be done experimentally or by calculation. Experimentally, C may be determined by measuring on the one hand the luminous intensity of a light source as compared with a source calibrated in candelas and, by ascertaining on the other hand the absolute values of the spectral distribution of energy. By calculation, C may be evaluated with the aid of a formula introduced by P 1 a n c k, by computing the spectral distribution of the black body at the melting point of platinum, that is, the number of W fcm2

radiated at each wavelength. This, divided by 7£, gives values in Wfcm2

per steradian perpendicular to the surface of the black body. The values thus obtained are inserted in equation (XII-6), in which L = 60 sb (cdfcm2); in this equation V,x is known and the value of C can easily be computed. The two methods of evaluating C should of course yield the same results, but so far the extent to which the empirical result agrees with the "theoretical" value of C has not proved very satisfactory.

XII-6] PHOTOMETRIC QUANTITIES AND UNITS BASED ON VA. 245

The discrepancy may be put down firstly to errors inherent in the experiment, and secondly to uncertainty regarding the values of the constants occurring in P I an c k's formula, in consequence of which the calculation tends to yield unreliable results. So far, all the experiments carried out in order to determine the value of C have given a value in the region of 630. Calculations based on P 1 an c k's formula, applying therein the values "of the constants which are commonly considered to be the most accurate, yield a value of 680. In a paper read before the CIE meeting in Vienna in 1963 4), Preston put forward a new proposal for the basic unit of light. He suggested that the original standard of the candela and its definition be replaced by a definition of C. With the known absolute spectral energy distribution of a light source, the luminous flux can then be calculated by means of this agreed value. This method presupposes accurate measurement of the spectral energy distribution, which, however, is very difficult to do. The difficulties with the accurate spectral energy distribution can be circumvented by measuring a light source with a calibrated radiation meter (a radiometer) with a filter interposed, the spectral transmission of which is similar to VA,. The paper describes the method and a few tests made with their results. Preston found the mean value of C to be 680 lmfW using standard lamps from the National Physiral Laboratory at Teddington, Britain, the photometric data on which were, of course, known. Further tests and measurements in other laboratories will be necessary to find any differences in the measurement of energy and to eliminate them. If this were to be successful, the original standard of the candela as the basic photometric unit could be abandoned.

,\-5 In P 1 an c k's formula, E (,\, T) = c1 ,\T , th~re are two constants,

ec,; -1 c1 and c2, which must be known. At the same time, the exact temperature for which the spectral distribution of the black body is to be computed must also be known. In our case this is the melting point of platinum (T, 1 = 2043 °K). Now, there is some uncertainty in regard to the value of c1, as well as to the melting point of platinum; T , 1 is calculated on the basis of the melting point of gold (Tau). Until recently it was thought that the accepted value of Tau (1336 °K) was quite accurate, but latterly some doubt has been cast upon its accuracy and some investigators are inclined to the view that, of the three constants c1, c2 and T au• the last mentioned is the least reliable 5). Further experiments will have to show whether or not Tau (and there­fore also T 211) is some degrees higher than the generally accepted value, thus eliminating the discrepancy between the experimental and calculated values. The argument may also be reversed, and from the discrepancy between Ctheor- = 680 lmjW and C•xv· = 630 lmjW the conclusion might be drawn that Tau is equal to 1342° K (this being perhaps the most reliable

246 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

determmation of Tau)· The adoption of th1s value for Tau would at the same time eliminate another discrepancy, viz. that between the "theoretical" value of the Stephan-Boltzmann constant u (5.67 X IQ-18 Wfm2 °K 4) and the "experimental" value of the same constant (5. 74- 5. 79 X lQ-1&), the experimental value then being 5.65- 5. 70 X IQ-18Wfm2 oK4. New measurements in Germany, whose results however are not published, seem to show that the melting point of gold lies indeed some degrees higher than it was accepted up to now. The temperature found, however, is not so much higher that the above-mentioned discrepancy can be completely explained.

Experimental photometry is effected by comparison with the candela (and units derived from it). Let the luminous intensity of a light source be n candelas; we can then write:

(cd) (XII-7)

where .E V~E~ and .EV~E~,ca are the luminous intensities in light­watts/sterad of the light source, and 1 cm2 of the black body at the freezing point of platinum respectively. C can be eliminated from both sides of the equation (XII-7) and need not therefore be known; this is not necessary, since we have created a light source for which the value of 1 (cd) has been attributed to the expression C .E V~E~,ca· Reverting to formula (XII-5):

cJ> = C .E V~G~

this may also be put in the form:

cJ> = .E CV~G~.

We will now consider one of the terms of the summation, e.g. for a wavelength which we shall denote by 1, then

cJ>1 = CV1G1.

We can now calculate the value of CV1 for this wavelength, for example, and thus obtain a constant for the latter (usually denoted by K~), in­dicating the ratio of the luminous flux (lm) to the power (W) radiated as monochromatic light at that wavelength. K>.. is thus given in lmfW, and is known as the luminous efficiency of radiation at wavelength A.. For A.= 555 mfl (V~ = 1) K~ reaches its maximum (Km). This is then numerically equal to the constant C in equation (XII-5), and, in this formula, Km (or· K) is usually employed instead of C. We shall accordingly employ K in place of C in the following. K or Km (680 lmfW) is thus theoretically the highest possible value of the luminous efficiency of a light source. If we regard K as the ratio

XII-6] PHOTOMETRIC QUANTITIES AND UNITS BASED OV ·VA. 221-7

of lumens to light-watts (both units of luminous flux), K will be an abstract value. But, if K ( = Km) be regarded as one of the values of KJ..., so that K becomes the quotient of the luminous flux (lm) by the energy (W) emitted at a wavelength of A.= 555 mp, where KJ... = Km and V" = I, then K must be expressed in lmjW. The reciprocal of K or Km is known as the mechanical equivalent of light and is denoted by M. This is expressed in Wjlm. Assuming that K = Km = 680 lmjW, M = 0.00147 Wjlm.

The expression "mechanical equivalent of light" is more or less obso­lete and its use is not to be recommended.

The definitions of the photometric units, and with them the quantities used in photometry, as based on the international values of VJ..., have completely ohanged the face of photometry. For many decades those interested in photometry were concerned with the question which method of photometry was the best, that is, the best suited to the characteristics of the average eye. The answer to this is now: "a method of photometry is correct ff it yields results which are compatible with formula (XII-6) L = K E EJ... VJ...". Since there are few observers whose spectral sensitivity correspond exactly to the international values of V J..., it is a fortunate circumstance that in the last decades we have found in the photo-electric cell a means of determining the value of E E" VJ... along purely physical lines, thus eliminating the human eye as an element in photometry. This is possible because the spectral sensitivity of these cells can be matched with the international relative luminous efficiency of radiation; further reference is made to this point in Part II. In section XII-3 we have already mentioned the term employed for photometry by comparison of light sources of different colours, viz. heterochromatic photometry. The spectral distributions of the kinds of light to be compared are thus necessarily different. When the kinds of light to be compared have the same spectral com­P9Sition and are of the same colour we speak of homochromatic photo­metry; in this case the results of measurement are not dependent on the spectral sensitivity of the observer's eye or of the photo-electric cell. Where the colours of the two kinds of light are similar but the spectral composition is different (and this is quite possible), the results of measure­ment are then indeed dependent on the spectral sensitivity of the observer or photo-electric cell. This is known as subjective homochromatic photometry.

248 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

The term pseudo- homochromatic photometry is sometimes employed when the difference in colour is so slight that it does not interfere with the accuracy of the balance and when the difference in spectral com­position is so small that the measurement is practically independent of the spectral sensitivity of the observer or photo-electric cell. Every photometric value expressed in units oflight corresponds to a radia­tion oxpressed in units of energy, and, indeed, every photometric value is the corresponding radiation value assessed in relation to VA.. The radiation quantities have also been given appropriate terms. The values that correspond to one another are given in table VIII (page 416) for the purposes of comparison.

XII-7. Vision at high and low luminance levels; Purkinje effect We have mentioned in section XII-3 that the internationally adopted values of v,\ are valid only for luminance levels above 3 cdfm2 (ap­prox.). We shall now see how spectral sensitivity is affected at lower luminance levels. Let us first consider the construction of the eye and in particular that of the retina. The latter consists of light-sensitive elements of two kinds, between which a clear distinction in form can be made, viz. rods and cones; in addition to this difference in form there is a very much greater difference, viz. in the manner in which they function. Practically speaking, at very low luminance levels only the rods operate, whereas at luminance levels above about 3 cdjm2 only the cones are called into play. Between these two regions there is a zone in which the luminosity is due to both rods and cones. A second difference is that the cones enable us to perceive colours, whereas the rods are not capable of distinguishing differences in colour. With the rods alone, everything would appear to us as in a black and white photograph, that is, we should perceive only differences in lu­minance. The transition from cone-vision to rod-vision when the lu­minance is reduced is the reason for the fact that the perception of colours of everything around us fades away when darkness is falling.

The cause of the difference in colour sensitivity between the rods and cones· must be sought in the light-sensitive (photo-chemical) substances present in the rods and cones. In the rods this is the visual purple, of which the photo-chemic.al properties are f~.i~ly accurately_ k!lown. The substances responsible for the photo-sens1t1ve charactenstlcs of the cones have so far not been identified with any certainty.

A third important difference between the rods and cones is to be found in the local distribution of the two different kinds of photo-sensitive elements in the retina. Around that point in the retina where the image F (Fig. 124) is formed of an object viewed directly by the eye i.e. which we see by looking straight ahead, there is a small practically circular

XII-7] VISION AT HIGH AND LOW LUMINANCE LEVELS 249

area about 0.25 mm in diameter, which contains only cones. This is the fovea centralis and the point where it occurs reveals a slight de­pression in the retina. The fovea centralis lies in the centre of an area of the retina about 2 mm in diameter known as the macula lutea, or yellow spot, so called because it is of a slightly yellow tint. The number of cones per unit area gradually decreases from the fovea towards the edge of the yellow spot and, at the same time, there is an increase in the number of rods. This gradual transition from cones to rods continues beyond the yellow spot so that, away from the centre (at the periphery), the concentration of cones is only low and that of the rods relatively high.

One of the many peculiarities of vision that can be explained by this distribution of the rods and cones is the following: if an effort is made to fix the gaze on a small object which, owing to its low luminance, is barely visible (e.g. a very faint star), it seems to disappear. The image is then formed on the fovea, which contains no rods, but, if the direction of vision is shifted through a small angle, say 10°, the object again becomes visible because the image is then formed at a point on the retina where rods do occur.

The fourth difference between rods and cones lies in their respective spectral sensitivities. In the same way that the relative luminous efficiency of radiation above 3 cdfm2 has been determined (region of photopic vision), the spectral sensitivity has also been measured in that range of luminances where only the rods function (region of scotopic vision). The curve thus obtained is reproduced in Fig. 128 (curve V,x'), in which

1,0 v

0,8 l 1 0,5 I

I .

0,4

I v ./_

0 400 1---'

J v 'l '~

I I \

1/ I \

I

I L I

I

500

' v,: \

\

\ \ % \

\. ...... ....... 500 mp 700

-).

Fig. 128. V,x'. The international relative luminous efficiency of radiation of the

dark-adapted eye (Scotopic vision). V,x. The international relative luminous efficiency of radiation of the light-adapted

eye (Photopic vision).

the international curve for pho­topic vision has been included for comparison (V,x). The V,x' curve has been adopted by the C.I.K in 1951 (Stockholm). The values of V ,x' for values of wavelengths in stages of 5 mp. are given in Table VII (p. 415). A noticeable difference is to be seen between the two curves. In cone vision the maximum occurs at 555 mp. (yellow-green), whereas for rods the maximum is at 507 mit (blue green); the curve for scotopic vision is

displaced with respect to that for photopic vision in the direction of the shorter wavelengths. There is a pronounced difference particularly

250 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

at the red end of the spectrum; the photopic vision curve (VA) extends much further into the red than the scotopic vision curve (VA'). In the region between pure cone vision and pure rod vision the relative luminous efficiency of radiation lies between V ;~ and VA' dependent on the luminance level. In this intermediate region the curve shifts from V;~ to VA'· In consequence of this difference in spectral sensitivity the ratio of the luminosity of one coloured object to that of another varies when the radiant power is considerably reduced, but is maintained at the same relative levels for the different colours. Let us illustr.ate this by means of an example. Suppose that one half of the field of view is a surface radiating yellow light of 581 mp, with high luminance, and that the other half radiates green light of 530 mp,. From Fig. 128 it will be seen that VA for both wavelengths is the same. If we now make the luminosities equal we shall also have balanced the respective energies. If we then reduce the luminance levels by reducing both of the energy values by ·a factor of, say, 1000, the energy at each side will still be the same, but the relative luminous efficiencies which we can read from curve V >.' in Fig. 128 will now be different, viz. 0.114 and 0.81. This means that, after reducing the luminance level we should have to make the energy of the yellow light seven times as much as that of the green to achieve similar lumi­nosities. Hence, for the same amount of energy, the green light gives a much greater luminosity than the yellow. The effect when the experiment is carried out with red and blue spectral colours is even more marked. Something very similar is experienced in nature. When we look at red flowers among green foliage by twilight, the flowers appear almost black, because the rods in the retina are so insensitive to red light (curve VA', Fig. 128). This phenomenon is clearly perceptible, although not quite so prominent as in monochromatic light, seeing that the objects also reflect other rays which usually tend to lessen the effect. In general, this peculiarity of the eye is known as the Pur kin j e effect, after the Czech physiologist P u r k i n j e who was the first to describe it in the literature. The cause, that is, the displacement in the VA curve, whictl in Pur kin j e's time was not yet known, is often alluded to by the same name. The occurrence of the Purkinje effect is the reason why in visual photometry on the basis of the international VA curve we have to ensure that the luminance of the surface perceived by the eye is in the zone of cone-vision, i.e. above 3 cdfm2 (approx.).

XII-8] EQ.UIVALENT LUMINANCE 251

After extensive investigations B o u m a') has ascertained that the summation law applies with all accuracy in the region of pure rod vision.

XII-8. Equivalent luminance

Equality of luminosity has been formulated in (XII-3) as

VOEO = E V>.E>..

By definition, it is held that wherr the luminosities are balanced, the luminances are also equal. V0 E0 and E V>.E>. are expressions for the luminance, the unit being light-watt ---d~fcm2; the luminance can be expressed in stilbs (XII-6) as stera

L = K EV>.E>.. This definition is based on the international V>. curve; it applies also to the luminance region (below about 3 nt) where the rods function either partially or exclusively and regarding which we have seen in the previous section that the luminous efficiency of radiation is not the same as it. is above 3 nt. This means that, in that particular region, equal luminances in accordance with expression (XII-6) are no longer accompanied by equal luminosities, as already shown in the previous section, in which a comparison was made between yellow light of 581 m.u and green light of 530 m,u. At the higher luminance level the luminosities were balanced, whereas at the lower level the balance no longer exists. Although it is not possible, outside the photopic region, to relate sub­jective brightness to luminance in a unique manner, it is of interest to have some scale in order to be able to evaluate subjective brightness. To this end the conception of equivalent luminance has been introduced. In 1891 K 6 n i g formulated the definition of equivalent luminance (named by him Helligkeit) as follows: Put for a light of a certain spectral distribution the equivalent luminance proportional to the energy (and thus to the luminance) for all values of luminance; in order to ascertain the equivalent luminance of another kind of light this is compared with that of the reference spectral distri­bution and the equivalent luminances are said to be equal when the two luminosities are balanced. The choice of the reference spectral distribution i!> arbitrary. K 6 n i g chose, as reference, light of 535 m,u and determined the equivalent luminance for the other wavelengths over a large range of luminances.

252 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

After K 6 n i g several other kinds of reference sources have been proposed. In 1951 the C.I.E. agreed upon standardizing as the reference spectral distribution that of the light emitted by a full radiator at the freezing point of platinum (2042° K). The results of measurements of equivalent luminance with light of the C.I.E. spectral distribution as reference light are shown in fig. 129.') Here the luminance (L) and the wavelengths have been plotted on the ordinates. Each point of the area of the diagram thus represents a physically defined starting point (defined by L and A.), and at each of these points the relevant result of the measurement of the equivalent luminance can be indicated. If, thereafter, the points with equal equi­valent luminance are connected by curves, then fig. 129 is obtained. The dotted line connects the wavelengths for which the luminance is equal to that of an apparently equally bright field with light with the spectral distribution of the black body at 2042° K. The Purkinje effect can clearly be edt,.

IO•r---t:--,...---.,..----r----!-.,0 seen in the figure. If, for instance, Q

blue light of 475 mp, is com-pared with red light of 625 mp,, then it is to be seen that in order to obtain an equivalent

1---+---+--.;-+--..... "1.~,.

luminance of 10, the same 10"'1------+----b_,."""""+---::::l

luminances are required (points P and Q). If, however, an equi­valent luminance of 10-3 must be produced, the luminance required is for the red light (Q') about 100 times as great as for the blue light (P'). Points P and Q lie in the region of pure cone vision whereas at P' and Q' the luminosity is produced· by nearly pure rod vision. If we denote the relative luminous efficiencies of ra­diation at a certain (low) lu­minance level by Vo' and V1'

for wavelengths A0 and A.1

650mp

Fig. 129. The equivalent luminance L' as a function of wavelength ,\ and luminance L. Points P, Q and P', Q' demonstrate the

Purkinje effect.

XII-8] EQUIVALENT LUMINANCE 253

respectively, then balanced luminosities can again be expressed by:

Here V 0 ' E 0 and V1' E 1 are expressions for equivalent luminance, ana­logous to the expressions for luminance. To bring these equivalent luminances on the above adopted scale and to express them in e.g. cdfm2, we have to match them with the reference light, for which the equivalent luminance is equal to the luminance or K' .EV >.'E.\ = K .EV >.E >.· For the equivalent luminance of the light A.0 we have the equation

K'V0 'Eo = K' .EV>.'E>. = K.EV>.E>..

Here K' is again a luminous efficiency of radiation. Its value changes with decreasing brightness levels till in the region of pure rod vision, where V>.' become~ constant, it also takes a constant value. It is only in this region that the equivalent luminance is related by a constant factor to the luminance which factor, however, depends on the spectral distribution of the light considered. The value of K' depends on the choice of the reference spectral distri­bution. This can be seen from the following argument: If V 0 ' E 0 = V1' E 1 and A.1 is the wavelength of the reference light, then the equivalent luminance of the light with wavelength A0 can be ex­pressed by

K'Vo'Eo = K'V1'E1 = KV1E1

in which V 1 is the relative luminous efficiency of radiation of A.1 in the photopic region. If the equivalent luminance is defined as above by

K' must be given another value since V1'E1 and .EV/E>. are equal but, consequently, V1E 1 and .EV>.E>. are different. In the region intermediate between the pure cone vision and pure rod vision, doubling K' V 0 ' E0 means by definition doubling the equivalent luminance but not of the energy since the energy governs the luminance level and thus also V >.'. From the point of view of the lighting engineer this means that at low luminance levels twice as many lamps will give to an object twice the luminance, but not twice the equivalent luminance.

254 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

XII-9. Minimum perceptible luminance difference and sensitivity to luminance difference

We have shown in sections XII-2 and 3 that the only quantitative assessment of luminosity that the eye is capable of making is one of equality in respect to two fields. If the experiment described in section XII-3 be carried out with the wedge depicted in Fig. 125 with the same kind of light on each side, and we then balance the luminosities, the luminance on one side can be varied slightly without producing a visual sensation that the balance has been disturbed. The threshold value of the differences in luminance, i.e. the lowest value of the difference in luminance, that is only just perceptible, is called the minimum per­ceptible luminance difference. If we denote this difference with respect to a luminance L by LJL, the variation which is only just perceptible will be LJLfL. Hence high sensitivity to luminance differences is accompanied by a low value of LJLfL. In order to be able to indicate greater sensitivity by means of a higher numerical

LfAL value, it is more convenient, J

however, to employ the re­ciprocal LfLJL, and this is known as the sensitivity to luminance difference. The con­ception is often called contrast sensitivity but since different kinds of contrast occur, it is preferable to use a more ex­plicit term. Values obtained for sensitivity to luminance difference depend very largely on the conditions under which measurement is carried out, and in particular on the luminance distribution in the field of view. It is therefore not surprising that the values

1 I fib

120

I 100

~ 0 v :7 B

I v 60 /

1/ I/ ~ 1'\ j /

20 I l ...... ,....v I\ , 0 Q01 0,1 1 10 1001 fOJ 104 J -:a -1 0 a 3 4

Fig. 130. Sensitivity to luminance difference Lf.dL.

a) dark background, b) and c) bright background.

published by different workers vary considerably among themselves. Fig. 130 illustrates the results of three measurements; as the range of luminances over which the measurements extend is so very great, the luminance L has been plotted on a logarithmic scale. Curve a in this figure represents the average results of measurements

XII-9] SENSITIVITY TO LUMINANCE DIFFERENCE 255

by Konig, B Ian chard and Bouma, and all of them are in fairly close agreement; curve b shows measurements by S t i I e s B), and curve c was obtained from a large number of tests carried out by Schuhmacher 9). Curve a is based on comparison fields covering only a small part of the field of view, the rest of the field being dark. The conditions under which curves b and c were produced were roughly the same; the field of view, filled with light of uniform luminance L subtended an angle of about 40°, and the luminance of a small part of the field was varied to an extent LfL. The difference between curves band c, which were nevertheless obtained under very similar conditions, may probably be put down to slight differences in the measuring conditions on the one hand and individual differences among the observers on the other. The curves should not be looked upon as average for a large number of observers, seeing that S t i I e s employed two observers and S c h u h­m ache r only one *). In view of the position of the results of S t i 1 e s' last two measurements the form of the end of curve b is somewhat uncertain. The last point of all indicates a bend in the curve, but the measurements do not reveal exactly where· the bend commences. Curve a is particulary important in photometry, since the conditions under which it was obtained are entirely similar to those under which most visual photometers are operated. From curve a we see that with luminance values between 10 and 5000 cdfm2 the sensitivity to luminance difference is between 50 and 60, which means that in vimal photometry, in the range of luminance concerned, the difference .between any two levels may be approximately 2% without implying that one of . the adjustments is incorrect. If a precise result is required, this can be obtained by readjusting the luminance a number of times in succession and taking the average of the results. The curve refers to homochromatic comparison. With heterochromatic comparison the sensitivity is lower and the accuracy of the measurement becomes lower according as the luminous fields in the photometer differ more widely in colour. Curves b and c are also of interest in connection with photometry, not that it is likely that measurements would ever be carried out under

*) The results obtained by both investigators are the outcome of a series of comparative tests in which the absolute values of the measured sensitivities were of minor importance.

256 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

precisely the same conditions as those under which these curves were taken and thus yield the degree of accuracy inherent in these curves. The curves do teach us, however, how the accuracy of the balance can be increased by the conditions under which measurement is carried out. It may be deduced from the diagram that the sensitivity to luminance difference, and therefore also the accuracy of the matching, may be increased by giving the surroundings of the photometer fields a certain luminance, instead of leaving them dark. The curves shown in Fig. 130 are also of interest to the lighting engineer, since conditions will arise in his field of activity which are the same as, or in any case very similar to those under which the curves were obtained. For example, the conditions met with in street lighting are quite com­parable with those under which curve a was determined; conditions in rooms with light-coloured walls will often be similar to those relating to curves b and c, or somewhere between curve a and curves b and c. Apart from the effect of other physiological and psychological elements in vision, it is the influence of the luminance distribution in the field of vision and the sensitivity to luminance difference which have been largely responsible for prompting the lighting engineer to concentrate more on the distribution of luminance in his projects. In this connection the term brightness engineering, or better luminance engineering has become current. When the luminance of the surroundings, partly or wholly, is higher than that of a field in which small differences in luminance are to be observed, the sensitivity to luminance difference is greatly reduced; such circumstances give rise to what is known as glare. The effect of a light of great luminance upon our powers of observation in general and not only of differences in luminance, is only too well known. Measurements of sensitivity to luminance difference were carried out even in the earliest days of lighting technology, for in 1729 B o u g u e r stated that it was possible to distinguish one field of luminance from another only when the difference in luminance is as I : 64. Others including Weber (1834) came to the conclusion that sen­sitivity to luminance difference in the (relatively small) luminance range within which they made their observations, was roughly constant, and from this arose Weber's law, which states that Ljt1L is constant for all values of L. Fig. 130 shows how much truth there is in this. It is only under the conditions relating to curve a (dark surroundings) that the law holds, for a certain range of luminance and then only as a rough approximation.

XII-9] SENSITIVITY TO LUMINANCE DIFFERENCE 257

Curve b shows that under other conditions there is no question of any constancy in Lf AL.

Weber's law is often referred to as the law of Fe c h n e r-W e be r but this is not accurate, for the law was formulated by W e b e r in 1834. Fechner (1851) drew from Weber's law the conclusion that the luminosity is proportional to the logarithm of the luminance L. but the validity of this has been the subject of much controversy. The proof is as follows. The minimum perceptible contrast Cmtn is expressed as Cmtn = ALJL (The fraction iJLfL (the minimum perceptible luminance difference) is also known as the Fechner fraction). If AL/L is constant a number of luminance values, which form a geo­metrical progression, correspond to a number of just distinguishable levels of luminance. This suggests that the luminosity (h) is proportional to the number of steps between the level considered and the lowest perceptible level •(or another standard level) giving

h = C1 log L + C2

which is the mathematical equivalent of Fe c h n e r's law. There are objections to this equation, however, particularly from the point of view of practical illuminating engineering. For example, the mathematical interpretation and treatment of the expression ALJL is inaccurate as it implies that AL would have this meaning that, if the luminance of a given part of the field of view be increased from L to (L + AL) the luminosity with respect to that part would be just per­ceptibly higher. This would in turn imply that the variation in the luminosity refers to a certain part of the retina, and this is not the meaning to be attached to AL in Weber's law. Under this law AL is the difference in luminance between two adjacent parts of the field of view that will produce a perceptible variation in the luminosity. AL thus relates to the sensation arising from two adjacent parts of the retina, which is very different from the assumption in Fechner's law. For, if the luminance of a part of the field of view is varied, the adaptation of the eye varies with it and so also does the luminosity of the whole field, including that part of which the luminance has not actually been altered. Wlien the luminance of part of the field of view is increased, the lu­minosity resulting from the rest of the field is reduced accordingly. Cmtn should "therefore properly be expressed as the difference between luminance levels, e.g. L 1 - L 2• Even though Fe c h n e r's law must be discarded in face of indisputable proof, we must nevertheless concJude from the results of a test as des­cribed below that, for one and the same adaptation of the eye, the luminosity is approximately proportional to the logarithm of the lumi­nance. An observer is given three contiguous fields, the outer two of which are of constant but different luminance (L 1 and La). The luminance of the central field (L2) is variable. He is then asked to vary L 2 so that the difference in the luminosity between L 1 and L 2 will be the same as that between L 2 and La. The adjustment is not made so that L 2 = ! (L 1 +La), but that, roughly, L 2 = V L 1 X La; in other words, whereas the lumi­nosity induced by L 2 is the arithmetical mean of the sensations in respect to L 1 and La. luminance L 2 appears to be the geometrical Mean of L 1

and La. From L 2 = V L 1 x-La it follows that log L 2 = l (log L 1 + log La).

258 THE PHOTOMETRIC MEASURING-UNITS SYSTEM [XII

Whereas we measure the luminance on a linear scale, the luminosity follows a more or less logarithmic scale, and this can be verified roughly by comparing two rows of grey fields arranged in order of increasing luminance. The luminance values of the successive gradations in the one row differ by a certain factor, and those of the other row differ each time by a certain value of the luminance, so that for the one we write:

LI L2 La r = r = r = .... = constant 2 3 4

and for the other: L 2 - L 1 = L3 - L 2 = L, - L3 = . . . . = constant.

The variations in luminance in the first row are therefore logarithmic, and those in the second arithmetic. The logarithmic row will give an impression of a uniform variation in luminance; the other one does not, the sensation being that the steps are too great at the darker end of the row and too small at the lighter end to suggest uniform variation. A more precise idea of visual evaluation of luminance or, more properly of luminance ratios, is found in the work of M u n s e II in his system of colour ranging, and also in N e w h ill's investigations. M u n s e II painted a scale in 10 steps of luminance, inducing equal hlminosity gradation, from the "blackest black" to the "whitest white" that he could produce. Photometry has shown that the relative values of the luminances of these steps V are as follows: v = 1 2 3 4 5 6 7 8 9 10

1.18 3.05 6.4 11.7 19.3 29.3 42.0 57.6 76.7 100 2.6 2.1 1.83 1.65 1.52 1.43 1.37 1.33 1.30

Below the relative luminance values are shown the ratios of each step to the previous one , and from this it will be seen that the ratios become smaller as the luminance values increase. If the variations were logarithmic the terms would be:

1.18 1.94 3.2 5.25 8.45 13.85 22.7 37.2 61.0 100 with a constant ratio of 1.64. Although there is an obvious depatture from a logarithmic evaluation of the luminance ratios, the gradual decrease in the ratios among the steps of the M u n s e II chart seems to explain the fact that when a variable luminance located between two constant luminances is adjusted for equal luminance differences, the ratios are found to be roughly logarithmic. The divergence from the logarithmic variation will be greater according as the relative difference between the two constant luminances is increased. The results of M u n s ell's investijations are confirmed by investiga­tions carried out by New hi IJlu),who submitted to a number of observers a series of surfaces of which the reflection factors were tak;m at random. The observers were then asked to give the luminance of each surface a relative value between 0 and 10 in tenths of unity. By plotting the results in a graph, the reflection factors corresponding to the steps 0 to 10 could then be determined. The scale thus obtained was found to agree very closely with the M u n­s e II scale.

XII-9J SENSITIVITY TO LUMINANCE DIFFERENCE 259

REFERENCES 1) W. S. S t i I e s and B. W. Crawford, Proc. Roy. Soc. B 112, 1933, 428.

"The l~minous Efficiency of Rays entering the Eye Pupil at different Points" See also: W. D z i o be k, Das Licht 4, 1934, 150-153. "Der Stiles-Crawford­Effekt und seine Bedeutung fiir die Photometrie"

2) E. S c h rod in g e r, Ann. Phys. 63, 1920, 481-520. "Grundlinien einer Theorie der Farbenmetrik im Tagessehen III"

3) H. Konig: "Der Begriff der Helligkeit". Neuchatel 1947 4) J. S. Preston, "A radiometric basis for the unit of light", Paper P.63.13 for the

C.I.E. meeting in Vienna (1963) S) H. Moser, l.J. Stille and<..:. Tingwaldt, Optik4, 1948/49,463-471.

"Zweite Strahlungskonstante und Goldpunkt" W. de Groot, Physica 16, 1950, 419-420. "The Radiation Constants and the Light Equivalent of Energy" W. de Groot, Ph. Res. Rep. 8, 1953, 401-410. "Some Remarks about the so-called Crova Wavelength"

6) P. J. Bouma, Proc. Kon. Akad. Wet. Amsterdam 38, 1935, 35-45, 148-161. "Grundlinien einer allgemeinen Theorie der Farbenmetrik", I und II

') W. de Groot, Ph. T. R. 15, 1953, 182-187. "Photometry at Low Luminance Levels"

8) W. S. S t i I e s, Proc. Roy. Soc. London, B 104, 1929, 322-351. "The Effect of Glare on the Brightness Difference Threshold"

9) R. 0. Schumacher: "Die Unterschiedsempfi.ndlichkeit des helladap­tierten menschlichen Auges". Thesis Berlin 1940. See also: Das Licht 11, 1941, 134-135

IO) S.M. New hi 11, Amer. Journ. of Psych. Austin 63, 1950, 221-228. A Method of Evaluating the Spacing of Visual Scales

PART II

MEASUREMENT OF LIGHT

CHAPTER XIII

GENERAL CONSIDERATIONS

XIII-I. Introduction. Principles of visual and physical photometry In Part I (Ch. XII) we have already defined photometlry and the prin­ciples upon which it is founded. Let us briefly review these once again. Photometry is the measurement of radiation on the basis of the visztal sensation induced by it. This visual sensation is the luminosity or subjective brightness; the eye receives visual sensations of colours foo, but the evaluation of colours lies beyond the scope of photometry; it is dealt with in colorimetry. Qualitatively, we can only judge one luminosity to be equal to another when the two are observed at the same time. Of the actual intensity of a luminosity or of the luminance of the object that produces the sensation, the eye is unable to tell us anything. By definition, the luminance is proportional to the energy. Comparisons of different kinds of light, i.e. light of which the spectral compositions differ, must be carried out with the aid of the international relative luminous efficiency of radiation V .\· For measurements of luminous intensity, for example, this is expressed by equation (XII-7):

K E V.\E.\ = nK E V.\EA,ca cd.

As the value of 60 candelas has been attributed to a certain light source (1 sq. em of the black body at the freezing point of platinum), it is not necessary in photometry to know the value of K in the expression K E V.\E.\,ca· Photometers must be so designed as to be capable of yielding results in conformity with equation (XII-7) and with the corresponding equations for the other quantities used in photometry. There are two kinds of photometer, based on different principles, viz. those in which the eye as a light-sensitive element forms an integral part of the measuring system (visual photometers), and those which make use of the physical or chemical action of the light, whereby the eye has no other task than the reading of electrical instruments (physical and photographic photometers).

264 GENERAL CONSIDERATIONS [XIII

When dealing with the two main types of photometer, viz. visual and physical, the question immediately arises as to which of the two is to be preferred? In general it is the physical photometer, the advantages of this system as compared with the visual being as follows. I. The elimination of the visual element and differences among the

spectral sensitivities of individuals, this being particularly important in heterochromatic photometry.

2. The possibility of obtaining greater accuracy. 3. The possibility of effecting technical measurements (i.e. those which

need not conform to such lligh standards of accuracy), using only simple instruments requiring no great skill in operation and main­tenance.

For many types of measurement, therefore, visual methods of photo­metry are no longer employed; apart from in special instances, photo­metry has now reached the stage where all homochromatic and technical heterochromatic measurements may be effected with physical photo­meters. Visual measurement has long been supreme in the field of hetero­chromatic precision photometry, but it has nonetheless had to give way here, too, largely to the objective methods. The instruments required for this are rather expensive, which has delayed their general intro­duction into photometric laboratories. In the meantime, since visual photometers are still used, even in homo­chromatic photometry, we must devote some considerable space to this subject. In the meantime, since visual photometers are still so widely used, even in homochromatic photometry, we must devote some considerable space to this subject. We shall not deal with the special branch of photometry known as spectro-photometry, by which we understand the step-wise comparison of the spectra of two light sources, over a range of wavelengths, and the plotting of energy/wavelength curves. This includes, for example, measurements of the spectral transmission of coloured filters 1).

XIII-2. Standard light sources By "measurement" is meant comparison with a standard to which a certain value, expressed in a particular unit, has been attributed. Just as length is measured against a standard such as a rule, standard lamps of which the value is known, and expressed in one of the units employed in photometry, are used in photometry. Such standards are all derived from a primary standard.

XIII-2] STANDARD LIGHT SOURCES 265

As standard of measurement of length we have the international metre, whic~ is maintained in Paris in the form of a platinum rod; for the standard light source we use the black body at the freezing point of platinum, as described in section 111-2. This last mentioned standard can be reproduced at any given time or place and satisfies the requirement that it is always constant, both in luminous intensity and spectral composition. At the same time, it is not quite such a simple matter to reproduce the actual apparatus required; owing to its form and size, and the limited solidifying time of platinum, this standard is not suitable for use in routine photometry. Moreover, it is only a luminous intensity standard (also used as a luminance standard), whereas standards of luminous flux are also required. Only a limited number of laboratories possess the primary standard of luminous intensity. They include: the National Bureau of Standards (N.B.S.), Washington, U.S.A.; the National Physical Laboratory (N.P.L.), Teddington, England; Laboratoire Central d'Electricite, Paris; Physikalisch-TechnischeBundesanstalt(P.T.B.),Brunswick,Germany,and the National Research Council (N.R.C.), Ottawa, Canada. For daily use, technical light sources are employed as standards, although these do not satisfy the requirement that they shall be constant for an indefinite period. The luminous intensity and flux of all such standards depreciate during their life, and also the spectral composition of the light may change in consequence. For this reason the following system is applied: some of the lamps are calibrated by comparison with the primary standard. This is generally done in one of the Standard La­boratories. These . sub-standards (also called secondary standards) are in tum used for the calibration of working standards; since little use need be made of the sub-standards, which can therefore be regarded as constant for a considerable time, they can be kept in service longer and it is not necessary to invoke the somewhat costly services of the standard laboratory so often for re-calibration purposes. The working standards are intended for every-day use, and are regularly re-calibrated against the sub-standards. Incandescent and gas-discharge lamps both display a fairly rapid change in luminous flux during. the initial part of their life (in incandescent lamps an increase, in discharge lamps a decrease). After this a very gradual deterioration in the luminous flux sets in, and it is only then that we can speak of a "constant" luminous flux (and constant electrical characteristics).

266 GENERAL CONSIDERATIONS [XIII

All incandescent and gas discharge lamps on which light measurements must be taken must therefore be allowed to bum until they became reasonably constant, this being known as ageing the lamps; filament lamps with a normal life of 1000 hours are thus aged for about 24 hours at rated voltage, and gas discharge lamps for about 100 hours. The construction of standard lamps must be such that during the time that they are used no variation shall occur in their characteristic photo­metric values as a result of changes in the geometrical conditions, in connection with which the following may be noted.

a) Incandescent lamps For standards of luminous flux, vacuum lamps with straight or coiled filament may be used, or, in the higher ratings, commercial gas-filled, single coil lamps. The use of coiled coil lamps as standards is not to be recommended since the risk that at different times varying numbers of turns of the coil may be short-circuited, and the luminous flux thus modified, is greater than with single coils. Gas-filled lamps must always be burned in the same position. Changes in the burning position alter the flow of the gas in the lamp and hence also the cooling conditions of the filament, and this necessarily affects the luminous flux of the lamp. The construction of standard lamps for the measurement' of luminous intensity must conform to higher requirements than that of those in­tended for luminous flux measurement. In the latter only a constant total luminous flux is required; the distribution of the flux in space is less critical. In the case of luminous intensity standards, however, not only must the total luminous flux be constant, but so also must be the luminous intensity within a certain solid angle. This is necessary firstly because the size of the acceptance surface of the photometer is finite and therefore subtends a finite solid angle with the lamp, and secondly because - of even greater importance - any slight deviation from the correct direction of measurement in the set-up must otherwise be taken into account. For these reasons the following conditions must be avoided, or steps taken to eliminate the undesirable effects which they produce: a. cords or blisters in clear bulbs; these function as small lenses and,

especially when the filament is small, tend to produce irregularities in the light distribution;

b. images of the filament formed by the rear wall of a (clear) bulb and coinciding partially or wholly with the filament itself; if small differ-

XIII-2] STANDARD LIGHT SOURCES 267

ences in the direction of measurement occur, the masking effect of the filament on the image may vary considerably and, with it, the luminous intensity;

c. straight coils; the projection of the luminous area of a spiral increases when the line of observation is moved from the direction perpendicular to the axis of the spiral in slightly divergent directions in a plane through the axis;

d. ring-shaped filaments; owing to gradual sagging of the spiral sections between the "pigtails", the form of the luminous area, and hence also the light distribution, varies.

It is not always possible to avoid such contingencies in the design of standard lamps for luminous intensity. For example, in the higher ratings the use of a coiled filament is unavoidable; the adverse con­sequences must then be avoided by taking suitable precautionary measures. As examples of standard lamps for which this has been done we may mention the following, which have proved reliable in practice.

Fig. 131. Straight filament vacuum lamp. Can be used as luminous inten­sity standard when

rotated.

1. Straight filament vacuum lamps (for low ratings up to about 50 cd) a. With the filament

suspended in the form of a cylinder (Fig. 131) ;such lamps must be rotated when in use to eliminate incon­stancy due to differ­ences in the mutual masking of the various sections of the fila­ment and their ima­ges. Since there is no gas in the lamp there

06467

Fig. 132. Straight fil­ament vacuum lamp with filament sections mounted in a single plane. Can be used as luminous intensity standard with-

out rotation.

can be no eddies to disturb the constancy of the luminous intensity at different speeds of rotation.

b. With filament suspended in a single plane (Fig. 132).

2. Gas-filled lamps with single-coil filament and opal glass bulb. Such lamps are operated horizontally; the luminous intensity is measured in the axial direction of the lamp. The opal glass bulb ensures that the

268 GENERAL CONSIDERATIONS [XIII

luminous intensity is practically independent of possible minor changes. in the geometrical conditions of the filament. In many cases standard lamps for luminance measurements take the form of tungsten ribbon lamps 9) (see Fig. 133), the light emitting part consisting of tungsten rolled into a thin strip. This results in a relatively large area of almost completely uniform luminance.

66461

b) Gas discharge lamps Although the performance of gas discharge lamps in general is not as constant as that of tungsten filament lamps, their use as luminous flux standards some­times has so many advantages compared with in­candescent lamps that the inconstancy is accepted. It is essential always to use these lamps in the same burning position and, in the case of sodium lamps, it is important to leave the lamp in the photometer until it is quite cold, since displacement of the metal on the wall of the bulb, which may occur if the lamp is moved whilst the sodium is still in the molten Fig. 138. Tungsten

ribbon lamp state, results in appreciable differences in the luminous flux. Even when these precautions are taken it is

advisable to re-calibrate the standards much more frequently than is done with tungsten lamp standards. Since the wattage consumed by gas discharge lamps burning on a certain of the ballast, there is no point in quoting the luminous flux of a gas discharge lamp for a given supply voltage. Another reason why it is useless to state the luminous flux for a certain lamp voltage or current is that the wattage .consumed at such values depends on the form of the current, which determines the apparent lamp power factor. The luminous flux of gas discharge lamps can therefore only be given in respect of the wattage consumed by the lamp; for a given wattage the luminous flux is independent of the voltage, current and power factor that determine the actual wattage consumed, within fairly wide limits.

XIII-3. Simultaneous and substitution methods

Comparisons of quantities for measurement purposes can be effected by the following methods.

a. by direct, i.e. simultaneous comparison with a standard in the photo­meter (simultaneous method),

XIII-4] SOME PRACTICAL HINTS 269

b. by comparison in the photometer of the value to be measured with that of any constant light source (comparison lamp), after which the light to be measured is replaced by a standard which is in turn compared with the comparison lamp (substitution method).

The substitution method is always employed when the photometer used is calibrated. In photometry preference is usually given to this method, since those parts of the photometer which correspond to the two comparative fields need not be identical. With the simultaneous method a second measurement with the photometer fields reversed is usually necessary in order to render the final result independent of the optical asymmetry of the photometer.

XIII-4. Some practical hints For the purposes of measurements in general - and this of course includes photometry- it is essential that we do in fact measure only the quantity that we are interested in, neither more, nor less. This appears to be stating the obvious, but it is not so obvious that this always does in fact take place. Unless care is taken, it many be found that inac­curacies result due to more light entering the photometer than should be there, owing to the presence of stray light, i.e. D:s D2 0, light reaching the photo- P ---+---- ·-1---- t

---- --- L meter other than directly ., t-i:::-::. =-:-.=-.=:--. -:--. =.=-. .::::::~:.... ~.~;:;;;_;~,..,~~ from the sourc.e being meas,. ~ 1----------r--=.---

~ --- -ured. ~-

Such stray light may be Fig. 134. Arrangement for screening-off stray radiated from objects, walls light or the ceiling in the la-boratory, or direct from other light sources, but may also be caused by scattering in compound lens systems (objectives). More often than not such light can be excluded in the manner shown in Fig. 134. A number of screens with diaphragms (stops) is placed between the light source and the photometer in such a way that the acceptance surface of the photometer is completely filled by the light of the source to be measured and the sides of the stops facing the photometer. In a later section we shall see how stray light may be avoided in certain special instances. Stray light is quite the most prolific source of error in photometry and

270 GENERAL CONSIDERATIONS [XIII

it would be useful if all photometric laboratories were to display notices reading:

"LOOK OUT FOR STRAY LfGHT''.

In most cases the presence or absence of stray light can be detected by screening the light to be.measured at one or two suitable points (usually close to the light source); any light then measured in the photo­meter will be stray light. It is even possible in this way to ascertain roughly the degree of error entailed when stray light is admitted and, from this, to decide whether further screening against stray light is necessary, or whether conditions are compatible with the required degree of accuracy. It is also possible to obtain less light in the photometer than that cor­responding to the quantity to be measured. This may be due to accidental screening of the light, or it may be thought that an exit pupil is com­pletely flashed with light whereas in fact it is not. This should be checked. Mention has already been made of the necessity of ageing standard light sources, and this applies equally to other light sources which have to be measured, unless a value relating to the source prior to ageing is ex-pressly required. It is generally advisable to effect calibration with at least two standards; should one of them have undergone a change for some reason, this will become apparent if a second standard is used. Should the differences revealed by two standards exceed the anticipated degree of inaccuracy, the calibration can be checked against a third standard to show which of the other two is faulty. In order to ensure the highest attainable accuracy, the value of the standard should correspond as closely as possible to the value to be measured. If the values differ by too mnch for comparison on one and the same scale on the photometer, one of them must be reduced, and this inevitably introduces an element of inaccuracy in the measurement. In view of the fact that the luminous flux of almost all electric light sources is largely dependent on the supply voltage and current, it is important to make sure when measuring lamps that this is carried out at the correct values of current and/or voltage. In the first place this necessitates sufficiently accurate instruments, and in the second place a mains supply that is sufficiently constant, so that the lamp can be operated long enough at the specified voltage or current to permit the necessary readings to be taken. In the third place, it is

XUI-4] SOME PRACTICAL HINTS 271

essential that the voltage or current as read from the instruments is actually that at which the lamp is operating, and that no error is in­troduced by contact resistance. To ensure a correct measurement of the voltage at the lamp terminals, a special lampholder having four con­tacts is recommended. Such a lamp­holder is shown diagrammatically in Fig. 135. Current is supplied by means of contacts I and 3, this being measured by an ammeter A; contacts 2 and 4 connect the lamp to a volt-meter V. If contacts I and 3, which 6fJ47o

carry the current, were also connected to the voltmeter, any contact resistance between the larrip cap and the contacts would result in a reading higher than the actual voltage; the difference

Fig. 135. Diagram of lampholder for use in photometry. Two special contacts (2 and 4) are provided for the voltmeter leads; contacts 1 and 3 are for the current supplyc

consists of the voltage drop across the contact resistance, which is proportional to the current flowing through this resistance. If the voltage is measured between contacts I and 3, the voltage drop is proportional to the lamp current, but if we measure the voltage across contacts 2 and 4, it is proportional to the voltmeter current and is consequently negligibly small, provided that the contacts have been

L well designed . ..---(::::--..;;..--- The use of separate voltmeter con-

Fig. 136. Circuit for measurement of the power consumed by gas discharge

lamps.

tacts is particularly necessary when the lamps to be measured carry a heavy current or are for use on low voltage. It is advisable, when using standard lamps, to measure the current after adjustment of the vol­tage and to compare this with the calibration current of the lamp, in order to have a check on the effect of any possible contact resistance in the circuit. It should be remembered that when

the current is being measured, the voltmeter current is also flowing through the ammeter; to obtain the exact value of the lamp current,

272 GENERAL CONSIDERATIONS [XIII

therefore, we deduct the voltmeter current from the reading of the lamp current. To eliminate errors due to contact resistance, incandescent lamp stand­ards taking large currents andjor for use on low voltage (e.g. tungsten ribbon lamps) should preferably be calibrated and employed at a certain current. The best circuit that can be employed for adjusting a gas discharge lamp to a required power consumption is shown in Fig. 136. Here, the voltage coil of a wattmeter W is connected in parallel with the lamp L and curren.t coil A. The wattmeter thus indicates the wattage taken by the lamp, plus that absorbed by the current coil of the wattmeter. If the latter amount of power cannot be disregarded it must be deducted from the meter reading. In order to avoid too much distortion of the alternating lamp voltage, the resistance of the voltage coil should be such that the current flowing in the coil is not more than 5% of the lamp current. Details of suitable electrical measuring instruments for use in the photo­metry of gas discharge lamps may be found in a publication by Mar­t erst o c k s). Further details on the measurement of the electrical data on fluorescent lamps will be found in I.E.C. Publication No 81 4). The recommendations in this Publication should also be observed in the photometry of these lamps. See also XVII-5.

REFERENCES 1 ) See for instance:

P. J. Bouma 1) Ch. I, p. 144 et seq. J. W. T. W a 1 s h: "Photometry", 2nd edition, Chapter XI, London 1952

~) Ph. T. R. 5, 1940, 82-87. "Tungsten Ribbon Lamps for Optical Measurements". Compiled by J. V o o g d ..

3) J. Marterstock, Lichttechnik 2, 1950, 177-181. "Uber die Eignung elektrischer Messinstrumente fiir die Photometrie von Gasentladungslampen"

4) "International specification for tubular fluorescent lamps for general lighting service", I.E.C. Publication 81

CHAPTER XIV

VISUAL PHOTOMETRY AND PHOTOMETERS

XIV- I . Principle of the visual photometer As we have seen in section XII-2, for the visual measurement of other quantities than luminance, these must be converted to luminance. The manner in which this is done, however, will be only an incidental point in our discussion. In this section we will deal with photometers in the limited sense of the word, viz. those parts of the system by means of which the comparison and adjustment of the luminances are effected. Seeing that in visual photometry two luminances have to be compared in every case, the visual photometer must in the first place provide two photometer fields (usually referred to as the photometer field), one part of which derives its luminance from the quantity to be measured, the other from a standard or comparison source. An example of this has already been mentioned in section XII-3, viz. the wedge depicted in Fig. 125. Once the photometer has been adjusted for equal luminosities, that is, in the photopic region, also for equal luminances, we say that photo­metric balance has been obtained. In order to ensure optimum accuracy of adjustment the two parts of the photometric field must be contiguous. It is generally accepted that the dividing line between the two parts of the field should be as sharply defined as possible for maximum accuracy of adjustment, although investigations by K r u it h of 1), together with those of Middleton have shown that, under certain circumstances, some lack of distinctness of the dividing line does not affect sensitivity to luminance differences unfavourably. In the design of photcmeter fields the makers usually endeavour to make the dividing line as fine as possible. A small space between the two sections will easily reduce the sensitivity to luminance difference by a factor of 5. In the second place, the photometer must permit of adjustment for equal luminosity. To this end, at least one of the parts of the photo­meter field should be variable in luminance to a measurable degree. In order to give a more general picture of the process and to assist

274 VISUAL PHOTOMETRY AND PHOTOMETERS [XIV

in the further description of photometers let us first consider a particular photometric system. This is depicted in Fig. 137, and is of a type that

_[ __ ~ __ ;lf _ _:_ ____ l __ a JJ7 X

Fig. 137. Equipment for the measurement of luminous intensity. PF Photometer field; X light source to be measured; CL comparison lamp;

EP eyepiece; T tube; 0 observer's eye.

is suitable for the meas-uring of luminous in­tensity. For measurement of the luminous intensity I., of a light source X the source is placed at a distance of a metres from the photometer field PF, which in this case is the wedge shown in Fig. 125.

The illumination produced on the right hand side of PF results in a luminance that is proportional to the illumination. The wedge reflects diffusely and the luminance is accordingly observed by the eye 0. The left hand side of PF is illuminated by a comparison light source CL (luminous intensity ICL); the resultant luminance of this side of PF is also observed by the eye. The distance between CL and PF is variable and is now adjusted so that the two halves of the field induce the same luminosity. Let the distance of CL as read from a scale be d., metres. The light source being measured is now replaced by a standard lamp of known luminous intensity I, and the distance between CL and PF is again adjusted to obtain photometric balance. Let this new distance be d metres. From these values we can now compute the luminous intensity of X in the following manner. With the initial adjustment the luminances and also the illumination values of the two halves of the field were equal, so that

For the second adjustment we have:

I IcL a2 = -;p:·

Dividing the first term of the first equation by that of the second, and proc ~eding similarly for the second terms we then have:

XIV-1] PRINCIPLE OF THE VISUAL PHOTOMETER

from which it follows that

I., I

d2 I =-X I.

"' d.,2

275

Hence the luminous intensity I CL of the comparison lamp and the distance a are immaterial provided that the luminance of the photo­metric field remains above the Purkinje region, i.e. above 3 cdfm2. In the type of photometer just described the method used for obtaining the variation of luminance is based on the inverse square law; this is the method generally employed in photometry, but there are others which will be dealt with later. · Between the photometer field and the eye of the observer a tube is usually provided (Tin Fig. 137) for the purpose of excluding extraneous light; the end of the tube is covered with a disc (eye-piece, EP) provided with an 8 mm aperture, the size of the pupil of the dark-adapted eye. That part of the photometer which contains the photometer field and the eye-piece is known as the photometer head. In the arrangement shown in Fig. 137 the light to be measured is collected by a photometer field which in this case is simultaneously the comparison surface; in most photometers the photometer field is distinct from the comparison surface and consists of diaphragms mounted in front of the comparison surface. In such cases the tube usually includes a lens which is adjustable with the tube itself so that each and every observer can adjust the field to the distance at which his eye is best accommodated, thus avoiding fatigue. In this way, too, the field of view is limited, the eye is not distracted and measurement is unaffected by the sur­roundings of the photometer field. The adjustable lens, moreover, provides a means of varying the angle under which the photometer field is observed. In the photometer field in Fig. 137 the distances from the light source to the various points on the field are not equal; hence, especially at small distances, the illumination and luminance of the two halves will not be perfectly uniform. For a precision photometer, however, this must be so, and the light is then accordingly made to fall perpendicularly on to the comparison surface. If the comparison surfaces are not sufficiently diffuse, so that the luminance factor varies with the direction of observation, this direction must be fixed.

276 VISUAL PHOTOMETRY AND PHOTOMETERS [XIV

For accuracy in a photometer we can therefore lay down three con­ditions, viz. I. The photometer fields must be as close together as possible; the

dividing line must not display more then a limited lack of sharpness. 2. The light to be measured must fall on the comparison surface at

right angles. 3. The comparison surface must be sufficiently diffuse, or the direction

of observation must be fixed. In the following sections we shall deal with the most common forms of photometer fields and the most common methods used for obtaining the variation of luminance, thereby restricting our review to a few of the many photometer designs that have appeared in the course of time. Once the form and functions of the more important components and of a few different kinds of photometer are known, the method of functioning of other types will present no difficulty. Moreover, many publications on the subject are available. 2)

XIV -2. Forms of photometer field Although the oldest forms of photometer field went out of use long ago, it may nevertheless be instructive to say something about them,

5

L, ~

I I I I I I

A

I I

I

i f

f

f

L2 X I

f

Fig. 138. Bouguer photometer

if only to demonstrate the wrong way of doing things. The oldest form of photometer field is depicted in Fig. 138. The apertures in the screen S are covered with translucent paper and are illuminated by the lamp to be measured and by the standard light source (L1 ~nd L2 respectively). Screen A serves to ensure that each

aperture receives light only from its own respective source. This ar­rangement was designed by B o u g u e r (published in 1760). It does not in any way meet the requirement that the fields should be contiguous, and the attainable accuracy was accordingly very low. In this respect the design of Lambert (1760) and Rumford (1792) as illustrated in Fig. 139 was an improvement. Here the two light sources L1 and L2 illuminate a plane F. To prevent the light from one source overlapping with that from the other, two screens sl and s2 are arranged so as to be capable of rotation about a vertical axis; they

XIV-2] FORMS OF PHOTOMETER FIELD 277

are so adjusted that the edges of the shadows which they throw on the surface F are contiguous. The eye of the observer is at W and ac­cordingly receives reflected light; (B o u g u e r employed transmitted light). An objection to this photometer is that the light to be measured is not incident at right angles and so does not uniformly illuminate the comparison surface (in this case simultane­ously the photometer field). Furthermore, the demarcation of the shadows is not sharply defined, a fact that has been regarded

--....,.....,..........---F I /I\ \

/~//~, ~/1 I "'\ /1 i \ \

/ :klw \\ x \\ L, I \~

I ~ I \

i 66474 \ L2

as a serious defect in the design. This sup- Fig. 139. The Lambert Posed shortcoming has led to various and R u m f 0 r d photo­

meter other designs which went a long. way towards meeting the objections. All these were based on the principle of Ritchie's wedge (1826) (Fig. 140). The sides of the wedge consist of mirrors M1 and M2 which receive light from the two sources under comparison respectively, and reflect it on to transparent paper P. At a later stage the mirrors were replaced by matt white paper, the luminance of which was observed directly. This gave the photometer fields depicted in Figs. 125 and 137.

p --.1.---...J--

Fig. 140. Ritchie wedge

A subsequent im­provement was in­troduced by Bun­s en (1843) in the form of his grease­spot photometer; the photometer field was of opaque white paper, a small spot on which was

rendered trans­lucent by impregnation with oil or wax (Fig. 141). The Bunsen photometer head can be employed in ways, viz.

Fig. 14l. Photo­meter field in the B u n s e n grease-

spot photometer

several different

1. The illumination on one side of the field can be kept constant by a comparison lamp, whilst the other side is illuminated successively by the lamp on test and the standard source (i.e. by the substitution method). In both cases the lamp distances are so adjusted that the luminance of the translucent spot is seen to be the same as that of

278 VISUAL PHOTOMETRY AND PHOTOMETERS [XIV

its surroundings. The simple proof that the luminous intensities are then in inverse proportion to the square of the distance to the translucent spot need not be given here.

2. The light sources to be compared can be placed on both sides of the screen in tum and the distances determined at which the spot disappears in both instances. This gives four positions (two for each source), from which the required luminous intensity can be computed.

3. The lamps can be placed as before on both sides of the screen and so positioned that the contrast between the grease-spot and its sur­roundings is the same as seen from both sides, the arrangement then being as shown in Fig. 142. Mirrors M1 and M2 enable the screen to be observed from both sides simultaneously. Here again the luminous intensities are inversely proportional to the square of the distances.

It should be noted that it is not correct to effect the adjustment for

Fig. 142. Photometer head in which the translucent spot can be observed from both sides simultaneously.

disappearance of contrast when only one side of the field is observed. In the case of the third method it must be noted that the adjustment is not carried out for equal luminance, but for equal contrast. It has been found that under favourable con­ditions the eye is capable of registering equality of contrast more accurately than equality of luminance. This principle is encountered again in one of the most accurate types of modem photometer head.

A disadvantag.e of the grease-spot photometer is that, since the spot receives light from both sides, sensitivity of adjustment is lower than with photometer h~ads in which each half of the field receives light from only one of the sources. This is to say that a displacement of, for example, 1 em of one of the light sources involves a smaller variation in the ratios of the luminances of the comparison surfaces with the Bunsen photometer than with the photometer heads where each half of the field is illuminated by one of the light sources only. We come now to the types of photometer field which go the farthest towards satisfying the requirements formulated above, viz. the prisms of Lummer and Brodhun (1889).

XlV-2] FORMS OF PHOTOMETER FIELD 279

One construction is depicted in Fig. 143; this comprises two 90-degree glass prisms. The plane hypotenuse AC of one of them (ABC) makes angles of 45° with the sides of the right angle, and light perpendicular to one of the sides of the right angle is totally reflected by the PS2

hypotenuse and emerges at right angles from the other side. The hypotenuse of the other prism DEF is convex, with a part of the centre ground flat; this part in­cludes angles of 45° with the sides of the right angle. The flat part lies against the hypotenuse of prism ABC in such a way that it makes optical contact with it, i.e. there is no intervening air space, and

I I I I I --r--- -IH

p

f 6647(}

light passes through the pair as though they were a single piece of glass. Hence light falling on GH is not reflected by AC, but passes straight through it.

Fig. 143. L u m m e r B r o d h u n equality-of-luminance prism

If a comparison surface PSI be placed behind DE, light passing from it to G H goes straight through. Light from a comparison surface PS2

66479

Fig. 144. Image of the photometer field in a Lummer Brodhun

prism

placed behind AB, falling on parts AG and HC of the hypotenuse AC is totally reflected and emerges from BC; light falling on GH. passes through towards the right. When the prism is viewed from the direction P a figure is seen as shown in Fig. 144. The luminance of the figure in the centre is derived from PSI and that of the outer ring from PS2• If the prism is carefully prepared, the line of division between the two figures is so fine and sharp that it can be made to disappear entirely when the luminances are balanced. An example of a complete photometer head with Lum­mer-Brodhun equality-of-luminance prism is depicted in Fig. 145. In this, PS is a matt white screen (consisting of,. say, a Plaster of Paris plate a few millimetres in

thickness), of which the two sides serve as comparison surfaces. MI and M2 are mirrors to reflect the light from both sides of PS into the

280 VISUAL PHOTOMETRY AND PHOTOMETERS [XIV

Lummer-Brodhun prism P. The tube T contains a lens as described in the previous section. EP is the eye-piece.

I I

--------~~+~ ----------p~~,7~

'" "' T

''" ''" :-..' ,~,

EP' Fig. 145. Photometer head with L u m mer­

B r o d h u n prism

equal contrast.

The photometer head is usu­ally capable of rotation through 180° about an axis XY, so that the comparison surfaces can be reversed with respect to the light sources under comparison. In this way any differences in the reflecting properties of the two sides of the Plaster of Paris plate can be checked and corrected. The other kind of Lum~

mer-Brodhun prism is il­lustrated in Fig. 146; this admits of adjustment for

In this case the hypotenuses of both of the prisms ABC and BCD are plane, these being placed together along BC. In the hypotenuse of BCD a figure is etched or ground so that its face is not in optical contact with the hypotenuse of ABC. The form of this figure is shown by the hatched portion of Fig. 147. When the line of vision is m the

PS2 direction of P, light from the com-parison surface PS1 (Fig. 146) is only transmitted by those parts of BC which are in optical contact, whilst only that light from PS2 is seen which is reflected in front of the recessed figure. The figure as observed is depicted in Fig. 14 7; the luminance of the hatched portion is derived from PS2 and that of the non-hatched part from PS1•

As so far described, the prism is still an equality-of-luminance prism, but it may easily be converted to a contrast

c I I I I I I

!PI I I

t 66481

Fig. 146. L u m me r-B rod hun prism for measurement by equality

of luminance or contrast

XIV-2] FORMS OF PHOTOMETER FIELD 281

prism by placing two plane pieces of glass at E and F. These lie in the paths of the rays transmitted and reflected, respectively, by the two trapezoids in Fig. 147, and the luminance of these trapezoids is then 8% lower than that of the surrounding area. Photometric balance is established when the contrast of the left-hand trapezoid with respect to the left-hand semi-circle is equal to that of the right-hand trapezoid and corresponding semi-circle.

66482 Fig. 147. Image of the photometer field in a L u ro­me r-B rod hun contrast In some photometer heads the glass plates

are hinged so that the prism can be em­prism

ployed as an equality-of-luminance or an equality-of-contrast prism. It has already been mentioned in our description of the Bunsen grease­spot photometer that balancing of contrast has the advantage over balancing of luminance in that the eye can better appraise the former PSt than the latter.

With contrast photometers the adjustment for balanced con­trast sometimes presents difficulties, but these can often ~e avoided if the photometrist is made aware of the fact that 1t is easy to induce the impression that the trapezoids lie above the semi-circles when they are darker, and below them when brighter than the semi-circles themselves. The contrast on the two sides is balanced when the two trapezoids give the impression that they are at equal distance above the semi-circles.

Today, Lummer-Brodhun prisms have superseded the Bunsen grease-spot photometer for laboratory work.

A~+-+---+-+-'78 Of the remaining types of photometer prism we shall

I

1\ I \

I \ I I \

I + \ \

66483

Fig. 148. F res n e 1 bi-

prism

mention one other, as this will be encountered in one of the types of photometer to be discussed, viz. the Fresnel bi-prism, depicted in Fig. 14.8. This is an isosceles prism ABC. Two comparison surfaces PS1

and PS2 are placed in front of it and, of the light from PS1 only that part is refracted in the direction of the observer which passes through face BC; of the light from PS2 only that part is seen which passes through AC. The two fields AC and BC are divided by a sharp line of demarcation at C.

282 VISUAL PHOTOMETRY AND PHOTOMETERS [XIV

XIV -3. Methods used in photometers for obtaining the variation of luminance

In order to enable the luminance of photometer fielgs to be balanced, the luminance of one of them is usually made variable. This generally applies to the light from the comparison lamp. A distinction is made between step-wise variation and continuous systems. T~e first kind is used for the approximate levelling out of pronounced differences in the luminance of the fields, the second serving for the more accurate adjustment. It is almost always essential that the. method of variation, in its operation, shall not modify the spectral composition of the light. Let us now discuss some of the methods commonly employed.

1. Step-wise variation of luminance Rotating shutter. This is a rotating disc from which a sector has been

Fig. 149. Rotating sector disc

ex is thus 360.

cut (Fig. 149). It is placed in the path of the rays illuminating the comparison sur­face or between the comparison surface and the photometer field. If the disc is rotated slowly, a flickering of the field is observed, but if the speed is sufficiently high the flicker disappears and the luminance appears constant. The central angle of the sector being denoted by ex, the observed luminance according to

ex T a 1 bot's law (1834) is 360 times that

without the shutter. The attenuation factor

The shutter may have either a fixed or a variable attenuation factor, and some shutters are even continuously variable during rotation. It is better, especially for low attenuation factors (small central angle) to determine the actual factor by calibration and not by computing it from the dimension of the angle. Rotating shutters naturally satisfy the condition that the spectral com­position of the light must not be modified.

XIV -3] METHODS FOR OBTAINING THE VARIATION OF LUMINANCE 283

Absorption filters. It is very difficult to produce a perfectly neutral filter, for which reason it is not advisable to use neutral filters for at­tenuation factors of less than 0.1. Filters of greater density usually cause an alteration in the colour which is in most cases not acceptable. Good neutral filters are supplied by various manufacturers. An exposed photographic plate will in some cases prove useful, but care must be taken that it is uniformly exposed, and allowance should be made for the Callier effect (See Section X-12) that such plates exhibit. A variable attenuation factor can be ob-tained by means of a neutral wedge, which is a filter of graded density, usually obtained by tapering the material (Fig. 150). In order to compensate the refraction, this wedge is combined with a clear wedge having as nearly as possible the same refractive index, or with a second neutral wedge. Neutral filters of which the density varies in stages are also on sale.

l ! ~I

66485

Fig. 150. Neutral wedge with transparent wedge to compensate the refraction

of the neutral one

Diffusing glasses. Sometimes an opal glass is inserted in the path of the rays to serve as secondary light source for illuminating the com­parison surface of the photometer. In such cases care should be taken that the glass is sufficiently non-selective, since many kinds of opal glass transmit appreciably more in the green than in other parts of the spectrum; others, again, transmit more red. Reflecting glasses. Attenuation can also be obtained through reflection from an optically fiat glass plate. With perpendicular incidence and e = 8% the attenuation factor is then 0.125. High requirements are put on the flatness of the glass surface. When polarised light is attenuated, and when variable adjusting systems employing polarised light are adopted (see 2 below) it should be remem­bered that light falling on the glass obliquely will be polarised. Diaphragms. When lenses are included in the path of the rays from the light source to be measured, diaphragms can also be incorporated. They can also be employed in conjunction with diffuse glass for con­trolling the luminous intensity of the glass as secondary light source.

2. Continuously variable systems for luminance variation Based on the inverse square law. An example of this kind of system has already been given in section XIV-I (Fig. 137). The principle of this and the actual design are so simple that little need be said about them;

284 VISUAL PHOTOMETRY AND PHOTOMETERS [XIV

it is only necessary to note that in some photometers the comparison lamp is not movable as in Fig. 137, but :fixed, the comparison

surface instead being adjusted during measurement. Based on the use o.f polarised light. When a ray of natural light (I in Fig. 151) falls on a prism cut in a certain manner from calcite or quartz it is split by the prism into two rays 2 and 3, the po­larisation planes of which (see Section I -3) are perpendicular to each other; in other words the two rays of light are polarised at right angles to each other. As the polarised rays follow different directions in the prism, such prisms are

Fig. 151. Double-refracting known as double-refracting prisms. In .Prism these, then, we have the means of split-

ting a beam of natural light into two rays which are polarised in planes perpendicular to each other. Prisms of the kind shown in Fig. 151 are not so suitable for this purpose, in that the emergent rays are parallel. A prism that is quite suitable in this respect is the Wollaston prism (Fig. 152), which consists of a double-refracting or a glass prism I and a .double-refracting prism II, so proportioned and combined that the emergent rays, which are polarised perpendicular to each other, are symmetrical. For photometric purposes the most widely used prism is the Nicol, as depicted in Fig. 153. It is so constructed that one of the polarised beams is transmitted whereas the other is disposed of laterally, to be absorbed by the black housing of the

I

prism. '2

The Nicol comprises two half-prisms se-parated by a layer of cement. An incident

][

ray I is split into two rays 2 and 3 which Fig. 152. W o 11 aston prism

are polarised at right angles to each other. At the interface between the two half-prisms ray 3 is totally reflected, passes laterally through the prism and is absorbed in the manner

XIV -3 J METHODS FOR OBTAINING THE VARIATION OF LUMINANCE 285

described. Ray 2 passes unchanged through the interface and emerges from the underside, parallel to the direction of incidence. To serve as a means for luminance variation in a photometer two identical Nicols can be placed one behind the other (Fig. 154). Nicol I alters the ray of natural light A into polarised light. Now, if Nicol II is positioned exactly as Nicol I, the ray B will pass through it (parallel Nicols); in Nicol II there is no longer a ray which, as ray 3 in Fig. 153, is absorbed. But, if the second Nicol is rotated goo on its axis, no light at all passes it (crossed Nicols); all the light in ray B is reflected and absorbed in the manner of 3 in Fig. 153. If the second Nicol be rotated to an intermediate position, the rays are split as in the first Nicol, so that a part of the light emerges from II, the rest being suppressed. When the second Nicol is slowly rotated on its axis from the parallel to the crossed position, the amount of transmitted light is gradually Fig. 153. Nicol

prism reduced from maximum to zero. Between the lumin-ance of the incident light (L;) and that of the transmitted light (L,.) the following relationship exists:

(XIV-I)

where oc. is the angle between the polarisation planes of the two Nicols. When high attenuation is employed (oc. ~goo) great accuracy is needed in the calibration and reading of the angle oc., for which reason it is in such cases often preferable to employ three Nicols, one behind the other. Initial attenuation is obtained by rotating the first of these to a giYen position, for which the attenuation factor can be determined.

Fig. 154. Two Nicols in combination

The exact adjustment is then effected by turning the third continuously rotatable Nicol as required. In some photometers use is made of polarised light with Nicols in the

286 VISUAL PHOTOMETRY AND PHOTOMETERS [XIV

following manner. The kinds of light to be compared are split in a double­refracting prism, such as a Wollaston prism, into beams polarised at right angles to each other. Only one beam of each kind of light is utilised for the measurement, these being beams which are polarised at right angles to each other. Each fills one half of the photometer field, which is observed through a Nicol. The zero position of the Nicol is taken to be that in which one of the fields (e.g. field I) is completely dark, the luminance of the other being then at a maximum. When the Nicol is rotated through 90° the situation is reversed. In practice the Nicol is actually adjusted for equal luminance of the two sections of the field. If the luminance of fields I and II be denoted by L1 and L 2 ,

and if with balanced luminance the Nicol has been rotated through an angle oc from the zero position, the following relation ship applies:

(XIV-2)

Variable diaphragms. Continuously variable diaphragms can also be employed for variation of luminance, these being usually operated by means of a screw movement. An example is given in Fig. 155.

664{)()

Fig. 155. Diaphragm with screw adjustment

Other systems for luminance variation than those described above are also in use, some of which will be mentioned in our discussion of photometers.

XIV -4. Photometer bench. Some examples of visual photometers

Photometer bench One of the most useful and important pieces of equipment in a photo­metric laboratory is the photometer bench. It is particularly useful because

XIV-4] PHOTO)'dETER BENCH 287

it may be used for so many different kinds of measurement and is thus a universal instrument. At the same time photometer benches are restricted in their use in that they are not transportable-. A photometer bench (Fig. 156) consists of supports with rails A (usually 12 to 20 feet in length) on which the various accessories required for the measurement, such as the photometer head B, lamps C, and screens D, can be moved backward and forward. Often these are mounted on carriages, if necessary with pointers, and graduations are provided along the length of the bench so that the relative positions of the lamps

Fig. 156. Photometer bench

and photometer head can be noted. Different photometer heads can be used on the bench. Together with the head and the screens, the bench then comprises a complete photometer. To overcome the drawback of non-transportability of the photometer bench various transportable photometers have been designed, the system for luminance variation and the comparison lamp being contained within the unit.

The Weber photometer Among transportable photometers the W e b e r photometer is one of the most convenient and widely used. It consists of two tubes (A and B, Fig. 157) mounted at right angles to each other. P is a Lummer­Brodhun (equality or contrast) prism whose fields are observed through a lens L and eye-piece EP. The end of the tube B carries a detachable lamphouse with incandescent comparison lamp CL. The comparison lamp illuminates a movable opal glass PS1 inside the tube B,

288 VISUAL PHOTOMETRY AND PHOTOJ',fETERS [XIV

this being the comparison surface for the light from the comparison lamp. The luminance of the opal glass is inversely proportional to the distance d from the lamp, this distance being measured off on a scale. The scale is frequently graduated with the inverse squares of the distances, thus facilitating the computation involved in the measurement. The comparison surface for the light to be measured may be an opal glass PS2 placed in the square holder C, or a Plaster of Paris plate mounted at an angle of 45° in front of the tube A. Filters can also be included in the holder C.

c

A

1.:.

Fig. 157. Weber photometer

The Weber photometer is used on a stand or, if mounted on a carriage, can be included on a photometer bench.

Martens Polarisation photometer In this photometer the system for luminance variation makes use of polarised light. The principle is illustrated in Fig. i58. S is the com­parison surface (e.g.a Plaster of Paris plate) for the light to be measured. By way of the totally reflecting prisms P1 and P2 an image of this sur­face is produced at A by lens L, and the luminance at A is thus pro­portional to that of the surface S. The comparison lamp CL is mounted in a fixed position and illuminates an opal glass M (luminance LB). Behind M there is a diaphragm B. The comparison is made between A and B, the light from which passes. through a lens with a Wollaston prism W which splits each beam into

XIV-4] PHOTOMETER BENCH 289

two, polarised at right angles to each other. A Fresnel hi-prism is mounted against the Wollaston prism, and this again divides each of the four beams into two others. In the plane of the eye-piece EP, therefore, 4 images each of A and B occur. The optical system is so designed that of the eight images 6 are screened off by the eye-piece and only two reach the eye, one each from A and B, their polarisation planes being at right angles to each other. Between the hi-prism and the eye-piece there is a rotatable Nicol N; if that position of the Nicol be taken as zero where­by the luminance of B is at a maxi­mum whilst A is totally dark, the luminance LA of A when the Nicol is rotated through an angle oc to establish photometric balance, in accordance with (XIV - 2), is

i I I I I t t I . ,; 1/ ,f

EP

Fig. 158.. Martens polarisation photometer

The Be c h stein Flicker photometer In Chapter XII (section XII-3) we have already seen that direct comparison of kinds of light of widely differing colours cannot be carried out with any great degree of accuracy, but that accurate results can be obtained by illuminating the fields in rapid alternation with the two kinds of light. This is the principle on which the various flicker photometers are based. Of the many types available we shall deal only with the instrument devised by Be c h stein, details of which are depicted in Fig. 159. A and B are the comparison surfaces of the photometer. The light passes through prisms P 1 and P 2 and lenses L1 and L2, and falls on prism P. This prism, which . forms the photometer field, rotates on its axis XX'. It is in two parts, the outer, annular section C (see Fig. 160) and central section D being tapered in the manner shown. The photo­meter is also equipped with the usual tube fitted with lens and eye-piece.

290 VISUAL PHOTOMETRY J\ND PHOTOMETERS [XIV

When prism P assumes the position shown in Fig. 160a, the light from the comparison surface A, shown here as a Ritchie wedge, is refracted by the annular section of the prism C so that it travels to the eye parallel to the axis. The light from B reaches the eye after refraction in the central part of the prism D. The field of view may be represented by the figure E in Fig. 160a, in which the horizontally hatched part denotes the light from A and the part with the vertical hatching the light from B. When the prism is rotated through 180° it will be in the position shown ih Fig. 160b; refraction by the central part D now directs the light from A towards the eye, and the ring C the light from B; the field of view then assumes the form shown by

6M04 diagram F. A comparison of the diagrams E and F Fig. 159. Be c h- shows that the position of the two light sorts have

s t e i n flicker now been reversed. photometer Rapid rotation of prism P produces flickering, which

is at a minimum when the luminances of the two kinds of light are equal, hence the instrument is adjusted for m .nimum flicker. At high speeds of the prism the flicker disappears entirely and the photometer field exhibits a mixed colour; if the speed be increased still further the flicker disappears even when the luminances are not the same. To ensure the highest possible accuracy, therefore, it is neces­sary to adjust the flicker frequency so that a certain amount of flicker remains, or so that it just disappears, but reappears with the slightest alteration in either of the luminances to be. compared. The optimum flicker frequency is dependent on the luminance of the photometer field, of the colours of the two kinds of light being compared, and on the observer himself. It has to be found each time afresh, for which reason flicker photo­meters are equipped with means of adjusting the frequency. If a D.C. mptor is used for the drive, the speed can be controlled with a rheostat.

M~ I I I I I I I If If I I I

\ ~ ), I I I I I I I

c~·c 101 I IDI I I I 1 I I I I I I I I I

E •• F

b 66495

Fig. 160. The prism in the Bechstein

flicker photometer

XJV-5] HOMOCHROMATIC AND HETEROCHROMATIC PHOTOMETRY 291

XIV -5. Homochromatic and heterochromatic photometry All the photometers described in the foregoing can of course be em­ployed for homochromatic and quasi-homochromatic photometry. In homochromatic photometry a photometer field of 6° to 7° is recommended for the most accurate results. Flicker photometers were designed especially for heterochromatic photometry; other systems for the latter have already been mentioned in Chapter XII (section XII-3), viz. the direct comparison and step­by-step methods. The direct comparison method is not employed because of its inaccuracy, except possibly for fundamental research into the peculiarities inherent in the method of comparison itself. The step-by-step method is reasonably accurate, but, as it is time­consuming, it is not used for routine photometry. As fourth method we must now mention the filter method, which is indeed used for routine work. By this method the colour difference between the two kinds of light is reduced by means of filters to such an extent that the measurement becomes homochromatic, or at least quasi-homochromatic, and can accordingly be effected by the use of normal photometers as for homochromatic comparison. If the light from a coloured light source such as a sodium or mercury vapour lamp is to be measured, the comparison lamp used is an in­candescent lamp burning at a certain temperature. The spectral distri­bution of the light of the latter can be measured or computed; the photometer is calibrated with a standard incandescent lamp. To measure the coloured light a filter is placed in the path of the rays from the comparison lamp so as to match the light with that of the lamp to be measured. The transmission factor can be determined for each wavelength by means of a spectrophotometer, and these data, together with the spectral distribution of the light from the comparison lamp, are used to compute the total transmission factor of the filter for the light from the comparison lamp. This transmission factor can also be ascertained by measurement in a flicker photometer. The filter method can also be employed in flicker photometry; the matching of the colours of the light increases the accuracy of the photo­metric balance obtainable. To ensure a high degree of accuracy it is essential always to us.! the filter at the same temperature, since the spectral transmission of the filter varies with the temperature. In heterochromatic photometry, selectivity of the comparison surfaces

292 VISUAL PHOTOMETRY AND PHOTOMETERS [XIV

and photometer fields may prove a source of error; this should be borne in mind when making a choice of these components. If all the literature ever written on the question as to which of the above mentioned four systems of photometry should be employed to obtain the best results were collected together, it would fill volumes, and we should find ourselves with a collection of exchanges of opinion and friendly controversy extending over many decades. Now that the relative luminous efficiency of radiation has been fixed in the form of an objective standard, this, in conjunction with the definitions of photometric units, has made the above question super­fluous and we can say that any method is correct which gives results in conformity with the definitions mentioned, i.e. in accordance with the V A-curve. Two conditions may be laid down that must be satisfied by any visual system of heterochromatic photometry likely to give satisfactory results, viz. 1. The luminance of the photometer field must be above the zone in

which the Purkinje effect occurs (> 3 cdjm2);

2. The photometer field should be of such a size that only the area of cone vision in the retina is involved in the observations, i.e. that the field of vision subtends an angle of not more than 2°. Some investigators prefer not to exceed 1.5°, but an objection to this is that the balancing, if made at smaller angles than 2°, is very fatiguing.

For the rest, the luminous efficiency of radiation of the eye of the observer and consequently the choice of observer is very important. More will be said about this in the next section.

XIV -6. Choice of observer If we apply the name standard observer to one whose spectral sensitivity curve agrees exactly with the International V A-curve, it can be said at once that no such person exists. This fact constitutes a serious handicap in photometry, but fortunately we have the means of limiting the drawbacks of having to work with non-standard observers, and in many cases to overcome them altogether. Only perpons with normal trichromatic vision are suitable as observer, i.e. only those who have a normal colour sense, without any form of colour-blindness. Whether or not a person is a normal trichromat can be determined by means of colour tables, for example that of S t i 11 in g. This consists of a number of rectangular panels containing rows of coloured spots

XIV-6] CHOICE OF OBSERVER 293

or circles, some of which together form a letter or figure. This latter and the surrounding spots are of different colours, a;nd the differences between the colours of the rectangles and figures ar~ sufficiently pronounced for persons with normal colour vision to distinguish all the latter without difficulty. To persons showing some form of colour­blindness, some of the cyphers appear to be of the same colour as that of the surroundings, so that they are unable to read them. Even though an observer may thus prove to have normal colour vision, his spectral sensitivity will differ from the internationally accepted values, in some instances considerably, but to some extent invariably. In order to be able to ascertain the correct values from results obtained by non-standard observers I v e s and K i n g s b u r y 3) have devised a simple system, although this is limited to the photometry of light sources having a continuous spectrum such as incandescent lamps combined with filters. Buck 1 e y 4) has shown that this system is not suitable for use with light sources having a line spectrum, such as gas discharge lamps. I v e s and K i n g s b u r y employ two liquid filters consisting of a yellow and a blue solution in plane-parallel-sided glass envelopes giving a liquid thickness of exactly 10 mm. The yellow liquid .is a solution of 72 g potassium bichromate in 1 litre of distilled water, and the blue liquid 57 g copper sulphate in 1 litre of water. These solutions have been selected such that for a standard observer their transmission factors at a temperature of 20° C are exactly the same for light from a vacuum incandescent lamp burning at a colour temperature of 2077°· K. We say then that, to the standard observer, the yellow-to-blue ratio (Y I B) is unity. An observer whose Y I B ratio is > 1 has higher luminous efficiency of radiation at the long-wave end of the spectrum, and is less sensitive at the short-wave end than the standard observer. The reverse holds for an observer when his YIB ratio is < 1. It has been found empirically with heterochromatic measurements on light sources, the spectrum of which is continuous, that there is a linear relationship between the results obtained by different observers, and their Y j B ratios. It is therefore possible to ascertain the correct values from only a small number of observers whose YjB ratios are known. The spectral sensitivity of the individual is not constant; it changes slowly in the course of time and it is also subject to periodic variation. Dr e s 1 e r 5) has ascertained that for some observers the spectral

294 VISUAL PHOTOMETRY AND PHOTOMETERS [XIV

sensitivity varies according to the season, and he suggests as a possible explanation that this is related to seasonal dietary conditions. For this reason it is necessary to check the Y I B ratio from time to time. The selection of observers for measuring light with line spectrum is rather more of a problem; in this case the Y I B ratio of the observer is of little assistance. This may be explained by the fact that the YIB ratio is a measure of the spectral sensitivity over two wide spectral zones as a whole, whereas sensitivity within smaller zones in the spectrum tends to vary considerably among persons having the same YIB ratio. For example, the results obtained by two such observers in measuring sodium lamps, of which the light consists almost exclusively of radiation of 589 mp, may be found to differ by an appreciable percentage. The following system gives satisfactory results in the measurement of light with a line spectrum. Observers are selected whose YIB ratio is in the region of 1 (say between 0.98 and 1.02). There will then be every chance that these observers will correctly evaluate certain kinds of monochromatic light. Whether this is so, and which of the observers is the most suitable, is ascertained experimentally by having them take measurements from standard lamps with line spectra (sodium and mercury vapour lamps) previously calibrated at one of the standard laboratories. It is quite likely that one or more of the observers will give a value that agrees with the calibration. The results obtained by the others will usually exhibit a definite deviation from the standard and, once this has been ascertained for light from different kinds of lamp, their measurements can be corrected accordingly.

REFERENCES 1) A. M. K r u it h of, Ph. T. R. 11, 1950, 333-339. "Perception of Contrasts

when the Contours of Details are blurred" 2) For the study of the different types of photometer the reader is referred to

the various handbooks in existence, of which we will mention here the books of W a I s h and S e w i g listed on page 388

3 ) H. E. I v e s and E. F. Kingsbury, Trans. I.E.S. New York 10, 1915, 203-208. "On the Choice of a Group of Observers for Heterochromatic Measu­rements" The same authors, Trans. I.E.S. New York 10, 1915, 259-270. "A method of Correcting Abnormal Color Vision and its Application to the Flicker Photo­meter" H. E. I v e s, J. Frankl. Inst. 188, 1915, 217-235. "The Photometric Scale" See also: K. S. Gibson, J.O.S.A. 9, 1924, 113-121. "Spectal characteristics of Test Solutions used in Heterochromatic Photometry"

4) H. Buck 1 e y, Ill. Eng. London 27, 1934, 118-122 and 148-157. "Hetero­chromatic Photometry with Particular Reference to the Photometry of Lum-inous Discharge Tubes" ..

5) A. DressIer, Das Licht 10, 1940, 79-82. "Uber eine jahreszeitliche Schwan­kung der spektralen Hellempfindlichkeit"

CHAPTER XV

PHYSICAL PHOTOMETERS AND PHOTOMETRY

XV--1. Principles of physical photometers and photometry In physical photometers the energy of the light rays falling upon the surface of a light-receptor is transformed into another form of energy. Since "energy", or better "energy per second" i.e. "power", in terms of lighting technology refers to "luminous flux", the flux falling on the light-receptor, i.e. the illumination is measured. Whereas in visual photometry all the photometric quantities are con­verted to luminance, in physical photometry they are converted to illumination. The principal types of physical photometer are the following: 1. Photo-emissive cells 2. Photo-voltaic cells (or barrier-layer cells)

Both of these convert the radiant energy to electrical energy, which can be measured with a voltmPter or ammeter.

3. Bolometers Here a wire is heated by the radiant energy; the variation in the resistance of the wire is a measure of its temperature and therefore also of the incident radiation.

4. Thermo-couples In these the temperature of the junctions of a number of thermo­couples is raised by the radiant energy, and the e.m.f. thus generated is a measure of the incident radiation. To the above types of photometer, must be added a type of photo­meter in which use is made of the chemical action of light:

5. Photographic photometers Here the radiant energy is utilised to expose a photographic plate. The density of the negative is a criterion of the exposure (illumination X exposure time).

Of these five types, photo-emissive and photo-voltaic cells are by far the most important, and the only ones employed in routine photometry. The other types arc used for special purposes only. In general, physical photometers are calibrated with a standard light

*) Besides the light and radiation sensitive types of receiver 1 to 5, there are also nowadays "photoresistors", generally made of cadmium sulphide, the electrical resis­tance of which drops as the illumination increases. We should also mention photodiodes and phototransistors, made of semiconductive silicon or germanium. The feature of these is that, on being. illuminated, they pass, amplify or supply an electric current. These photoelectric components are at present mainly used in electronic switching and control, since their stability as receivers in photometry is usually inadequate or can be achieved only under special conditions.

296 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

source in order to provide the relationship between the reading of an electrical measuring instrument and the illumination. This calibration then serves in tum to determine the illumination produced by other light sources. In some cases a balance method is used, the photometer then being illuminated alternately by the light source under examination and a standard or comparison light source. An electrical measuring in­strument then indicates when the illumination produced by the two sources is equal.

XV- 2. Photo-emissive cells When exposed to light or other electromagnetic radiation, many sub­stances have the property that they then emit electrons. Under certain circumstances these electrons can be measured as electric current. Such effects are grouped under the heading of photo-electric phenomena.

When the electrons are emitted from the ir-radiated surface we speak of external photo-electric effect; this takes place with many metals. Two other kinds of photo-electric effect will be in­troduced in the next section, which deals with

K photo-voltaic cells.

66531

Fig. 161. Photo-emis­sive cell. A anode,

K cathode

Use is made of the external photo-electric effect in photo-emissive cells 1) to measure light. The cell consists of a bulb with a thin metallic layer on the inside (the photo-cathode K, Fig. 161). The bulb may be either exhausted (vacuum cells), or filled with a rare gas (gas-filled cells). A part of the bulb is left uncoated so that light can enter, and when the light falls on the photo­cathode electrons are emitted in all d~rections

within the bulb. To cause an electric current to flow it is n~cessary to include a second electrode (the anode, A, in Fig. 161); the electric circuit outside the cell must be closed with a resistance, for example that of a milliammeter. The current

then flowing is very small indeed, however, since only a small part of the electrons liberated reach the anode, so that, in order to "capture" more or, if possible, all the free electrons, the anode must be given a positive potential with respect to the photo-cathode. A serviceable photo-current is thus produced only when the cell is incorporated in a circuit such as that shown in Fig. 162. The battery B

XV-2] PHOTO-EMISSIVE CELLS 297

supplies the positive voltage for the anode; a meter (M) is included for measuring the photo-current. If at a given illumination level the anode voltage be raised slowly from zero, the current is seen to increase. The higher the voltage the greater the number of emitted electrons drawn towards the anode; vacuum and gas-filled cells behave quite differently, however, with increasing anode voltage.

K

A

l-

In the case of vacuum cells a condition is reached at a certain value of the voltage whereby all the electrons liberated from the cathode are caught by the anode. A further increase in voltage beyond that point is not accompanied by a consequent rise in the photo-current, the current having reached a saturation point (see curve V

Fig. 162. Simple circuit for the measurement of photo­current. A and K are res­pect vely the anode and cathode of the photo-emis­sive cell. B battery for anode supply, M meter

in Fig. 163). When the anode voltage is maintained above the saturation level the photo-current, for a given illumination, is constant. In gas-filled cells ionisation occurs when a certain voltage (the ionisation voltage of the gas) is reached, which means that electrons leaving the cathode and colliding with the gas molecules, liberate electrons from the latter in far greater numbers than those of the colliding electrons themselves. In this way the photo-current is amplified (see curve G, Fig. 163). When the voltage on a gas-filled cell is raised still further, increased

l

tl---t----------f------1

v

XJO 150V i6531

Fig. 163. Photo-current as a function of the anode voltage: V in vacuum cell, G in gas-filled cell. In contrast with the gas-filled cell no further rise in photo-current takes place in the vacuum

cell beyond a certain voltage

ionisation finally leads to break­down, that is to say a luminous gas discharge occurs, the current flowing through the gas being then independent of the illu­mination. There are also other differences between vacuum and gas-filled cells. The photo-current in a vacuum cell is strictly propor­tional to the luminous flux en­tering the cell, whereas with gas­filled cells the relationship between the two quantities is not linear;

298 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

the photo-current rises more steeply than does the luminous flux. Again, in vacuum cells the photo-current follows all :fluctuations in the luminous :flux without any -measurable delay, whereas the current :flowing in gas-filled cells shows a certain inertia with respect to fluctu­ations due to the time taken by the ionisation process. Another important point of difference between the two types is the stability; the photo-current of vacuum cells is very constant, not only during the short time in which measurements are carried out, but also over longer periods. Gas-filled cells are not so stable. Comparing the peculiarities of the two kinds of photo-emissive cell as outlined above we see that the vacuum cell is obviously the more suitable for photometric purposes, and in the following, therefore, we shall consider this kind of cell exclusively. The composition of the photo-cathode is important from the point of view not only of the manufacturer, but also of the user of the cell in photometers, seeing that upon this factor depends the magnitude of the photo-current and also the spectral sensitivity. It is possible by means of quantum theory to compute for photo-cathodes of pure metals the extreme wavelength at which the cell will be capable of delivering current. For some metals the upper limit lies in the ultra­violet region, with others, namely the alkali metals (lithium, sodium, potassium, rubidium and caesium), ~his limit (known as the red limit) occurs in the visible spectrum, or :tnay even be in the infra-red. It has been found possible to displace the red limit towards higher wavelengths by making use of special cathode compositions. Here, the most important elements are antimony and caesium. Photo-cathodes with different proportions of these materials give a large number of different spectral sensitivity curves. The spectral sensitivity curves of a few photocells are given in Fig. 164. Besides those for a caesium and a caesium-antimony cell is that for a potassium type developed as the first photocell thirty years ago and long used primarily for photometric purposes. For purposes of comparison, the international spectral sensitivity (curve 4) is also shown in the figure. The scale has been so chosen that the curves pass through the same point at 555 nm. Further reference is made to the spectral sensitivity of photo­emissive cells in section XV -6. The construction of photocells has changed in the course of time. The first potassium and caesium cells looked roughly like the one shown in Fig. 161. The whole of the inside of the glass bulbs was coated with a thin layer of metal except for the part through which the light was to be allowed to enter the cell.

XV-'2] PHOTO-EMISSIVE CELLS 299

Modern photocells consist of a cylindrical glass bulb in which the photo­cathode is arranged in one of a variety of ways. It may be located directly on the cylindrical wall, on a bent metal plate or on the flat top of the bulb (frontal cathodes). In the latter case, the photocathode must be semi-transparent. There are photocells with quartz tops, so as to permit the measurement of ultra­violet radiation. Before proceeding to a discussion of the methods employed in measuring the photo-current it is necessary to consider one or two other properties of photo-emissive cells, which are of interest in photometry.

~~--~--~----~--~--~ i

300 1500

Fig. 164. Spectral sensitivity. 1 potassium vacuum cell, 2 caesium vacuum cell. 3 ceasium-antimony cell. Curve 4 is the V"A curve. The scales are such that the curves pass through the same point at the wavelength at which visual sensitivity is highest, (555 mJ.t)

First of all comes a warning. Photo-emissive cells must never be operated at temperatures exceeding 50° C. The sensitivity varies a little from one point to another in a photo­cathode and it is accordingly advisable to carry out measurements in such a way that the same area of the cathode is utilised each time, e.g. by placing a diffusing glass in front of the window. In frontal cathode photocells, of course, a shutter can be placed directly in front of the cathode to provide constant limitation. Photo-emissive cells may exhibit a dark current, that is, a current which can flow even in the absence of light in the cell. This may be due to el~ctrons leaving the cathode under the influence of temperature, or to leakage as a result of inadequate insulation between anode and cathode. Such defects in the insulation may occur either on the inside or the

300 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

outside of the bulb and, in order to avoid them, the space between the anode and the cathode is usually made as large as possible. For example, the cathode is taken upwards out of the bulb .and the anode downwards. External leakage current is prevented by applying a conductive band or ring round the bulb and connecting this to earth. If the cathode is also earthed the leakage will then be to earth and thus does not give rise to errors. At constant temperatures dark current of thermal origin does not usually give rise to any difficulties, especially when the current is small com­pared with the photo-current. If it is of the same order as the photo­current itself it may present some difficulty. This may be avoided by reducing the temperature of the cell. Alternatively, with the cell at a constant temperature, compensation may be provided for the dark current. Measurement of the photocurrent Formerly the photo-voltaic cells were not as stable as they are to-day. This is why photo-emissive cells were almost exclusively used for all really precision light measurements, i.e. for measuring both stationary and rapidly changing light phenomena. Nowadays, the photo-voltaic cell has largely replaced the photo­emissive cell i~ the field of the measurement of stationary light phenome­na, that is to say,. those especially that relate to lamps that burn steadily. Furthermore, photo-voltaic cells have the advantage over the photo­emissive cells that it is easier to adapt' them to V.~. with the aid of filters. The current generated by the photo-voltaic cell cannot, because of its slow response speed, follow fast changes in illumination, like the lighting up and extinguishing of ·a "Photoflux" lamp. The field of the photo­emissive cell is therefore almost totally restricted to the measurement of such phenomena. We thus propose here to ignore photo-emissive cell circuits for the measurement of constant luminous flux. A few circuits for the measurement of very rapidly developing light phenomena are dis­cussed in connection with the measurement of quantityoflight (XVII-7). Photocells with secondary electron amplification Finally, we must deal with another kind of photocell developed over the last twenty years which, because of its special properties, has its own place among the above mentioned types. We are referring here to photo­cells with secondary emission, also known as secondary electron multipliers or SEM's2). Here, the electrons emitted from the photocathode are attracted by an auxiliary anode capable of emitting electrons itself under the effect of the electron bombardment. The auxiliary anode emits five to ten

XV-2] PHOTO-EMISSIVE CELLS 301

electrons for every incident one, thus amplifying the original photocurrent five to ten times. If this stream of electrons is allowed to fall on another auxiliary anode, once again at a higher positive potential, the same cycle is repeated. If, therefore, several auxiliary anodes are connected or arranged one behind the other, the photocurrent will be considerably amplified. These cells, then, permit the measurement directly with a pointer instrument of photocurrents that would otherwise require an amplifier However, the greatest illumination that can be measured with SEM's is still relatively small. The limit is set by the heaviest current permissible between the final auxiliary anode and the final stage anode. In its turn, this current is determined by the greatest loadability permissible in view of the heat developed between these two anodes. Furthermore, it must be borne in mind that any emission of heat (dark current) from the photocathode is also amplified, and finally the sensitivity of this cathode changes, making the accuracy of measurement no greater than with a vacuum cell. A better range should therefore be used in measurement. Because the degree of amplification rises steeply with the voltage, the latter must be maintained constant within at least 1% and, for greater precision, to within 1 Ofoo. Initially, the stability of SEM cdls was not very high either during a series of measurements or over a longer term. It was also governed by periods of load and rest, and thus where rather scrupulous measure­ment was required, repeated calibration was necessary. These cells have, however, been improved to such a great degree now that they are quite usable for higher-precision measurements, provided that the calibration of the instruments is regularly checked. SEM cells are used where ordinary photocells or photo-voltaic cells supply photocurrents that are too weak, e.g. in spectral photometry, in astronomy to measure the luminous intensity of stars and, lately, for measurements dealing with street lighting. There are no serious problems involved in the measurement of the photocurrents of SEM cells. If the photocurrent is passed through a large resistor, the voltage generated across the resistor can be measured with an amplifier, a sensitive diode voltmeter or an oscillograph. A sensitive current measuring instrument can also be inserted in the circuit between the anode and the final· auxiliary anode, where, in this case, there should also be a protective resistor. The greatest difficulty is the current supply because, as already stated, high requirements are made of the stabilisation of the stage voltages.

302 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

XV-3. Photo-voltaic cells This type of cell is capable of converting incident light into electrical energy without the aid of the auxiliary voltage required by the photo­emissive cell. The property responsible for this is the barrier-layer photo­effect, by which light causes electrons liberated from a semi-conductor to migrate via a banier layer towards an adjacent metallic layer where they can return to the semi-conductor by way of an external resistance e.g. a milliameter *). Without any other apparatus, then, a photo­current of measurable magnitude can be obtained, a fact that has gone far towards promoting the use of photo-voltaic cells in photometry. On account of the presence of the barrier layer, the photo-voltaic cell's are often called barrier-layer cells.

~"'"""'""""",.,.,.Jl"".,..,l'!l"" ., 1

Fig. 165. Diagram showing construction of a) front-wall cell, b) rear-wall cell. 1 semi-conductor, 2 barrier layer, 3 metal layer, 4 metal layer in contact with

semi-conductor

Photo-voltaic cells 3) thus consist in effect of a metallic layer and a semi-conductor separated from each other by a thin insulating layer, the barrier layer. The layer of metal or counter-electrode must be so thin that it will transmit light which can then penetrate into the semi­conductor and there liberate electrons. A support (base) of metal, in contact with the semi-conductor must also be provided. There are two kinds of barrier-layer cell, viz. those in which the barrier layer is either in front of, or behind, the semi-conductive layer, the electrons leaving the latter from that side on which the light enters (front-wall cells), or from the other side (rear-wall cells). The two systems are illustrated diagrammatically in Fig. 165. In these diagrams 1 is

*) In these cells another phenomenon occurs besides the barrier layer effect, i.e. the internal photo-effect, which consists in an increase in the conductivity of certain semi-conductors (including selenium) due to the liberation of electrons by the incident light. Use is made of this fact for technical purposes. A selenium cell consisting in principle of a piece of selenium with two electrodes is placed in a closed circuit under tension. When light is allowed to fall on the cell the resistance drops and the current can be used to operate a relay. Photo-electric cells of this kind are sometimes referred to as photo-conductive cells; due to their inaccuracy however, they are not suitable for the measurement of light.

XV-3] PHOTO-VOLTAIC CELLS 303

the semi-conductor and 2 the barrier layer; 3 is the metal electrode to which the photo-electrons migrate from the semi-conductor, through the barrier layer. The metallic layer, in contact with the semi-conductor to provide a return path to the latter for the electrons, is denoted by 4. Light enters in the direction of the arrows, so that in front-wall cells the metal counter-electrode 3 and the barrier layer 2 have to be trans­parent, or in rear-wall cells the metal base 4. As will be seen from Fig. 165 the :flow of electrons in front-wall cells is against the direction of the incident light, and in rear-wall cells in the same direction as that of the light. Two different kinds of barrier-layer cell are made, namely cuprous oxide, and selenium cells, of which the former are produced as either front or rear-wall cells, whereas the latter are front-wall cells only. For photometric purposes the selenium cell is employed almost ex­clusively, and we shall accordingly concern ourselves with this type only. In any case, the characteristics of the two kinds of cell are roughly the same. As mentioned above, the selenium cell is a front-wall cell, and the order of the layers is as shown in Fig. 165a. From the point of view of the current and voltage delivered, photo­voltaic cells behave very differently from photo-emissive cells. In the latter, at any rate in vacuum cells, the photo-current is strictly propor­tional to the incident luminous :flux. In photo-voltaic cells the current delivered is only in one special case proportional to the incident lumi­nous flux. Apart from its functioJ? as photo-electric cell the system also works as a rectifier, which is a good conductor when the metallic layer 3 (Fig. 165)

Fig. 166. Equivalent circuit of a barrier-layer cell. The cell can be regarded as consisting of a barrier layer S, a capacitance C and a

resistance R

is negative, and a poor conductor when the semi-conductor 1 is nega­tive. Illumination renders the me­tallic layer 3 negative; hence an electric current will :flow from 3 to 1, i.e. in the opposite direction to that of the photo-current. The greater the external resistance the higher the voltage across the barrier layer (between 1 and 3), and the stronger the current opposed to the photo-current. The photo-voltaic cell may be re­presented by the equivalent circuit

304 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

depicted in Fig. 166. S is the barrier layer, functioning as recti­fier, R is the resistance of the semi-conductor in which the cur­rent flows in the opposite di­rection to that in the barrier layer; R is dependent on the incident light. Re is the external resistance, e.g. a milliammeter. The transparent metal film and the selenium with metal baseplate, separated by the thin barrier layer (th·e resistance of which is high), together constitute a capacitor as denoted by C in Fig. 16B. Whereas the current generated by the barrier layer is propor­tional to the luminous flux (curve a in Fig. 167), the curve of the inverse current arising from the voltage produced by the photo­current across the external re-

700

600

500

300

66539

Fig. f67. Photo-current i as a function of the illumination E on a photo-voltaic cell. a) photo-current flowing in the absence of external resistance; b) counter current produced by the voltage across the barrier layer. c is the total current

to.n.

100.11.

soo.n.

resulting from a and b

sistance is more or less as shown by curve bin Fig. 167. The total current flowing in the external circuit is as shown by curve c.

2500Lux 665(0

The photo-current character­istic depends upon the value of the resistance in the ex­ternal current circuit in the manner illustrated in Fig. 168. With a small resistance the current is very nearly pro­portional to the illumination on the cell; with shorted resistance it is perfectly linear. With increasing resistance, the characteristics become more

Fig. 168. Cell-current i as function of the illumination E for different values of the

e:x;ternal resistance

XV-3] PHOTO-VOLTAIC CELLS 305

and more curved at high illumination values. If the ext.ernal resistance be made sufficiently high the current will show no further rise at a

c c

R

!2 Fig. 169. The measuring range of a photo­voltaic cell with meter should not be increased with resistance in series (a), but with resistance

in parallel (b)

certain illumination owing to the opposing current. In practice this means that the mea­suring range of selenium­cell photometers should not be increased by con­necting a resistor in series with the meter, but by shunting the meter with a resistor (Fig. 169). Fig. 170 shows the volt­age developed by a given

cell for different values of the external resistance. The current characteristic is accordingly only linear with low external resistance and low illumination. As a general rule it may be taken that there will be sufficient linearity as long as the cell voltage remains below 15 mV. For accurate measurements, however, calibration is essential in all measuring ranges. Photo-voltaic cells are not quite cells. They need rather frequent

so stable as vacuum photo-emissive re-calibration, the frequency of this

operation depending on the desired degree of accuracy. These cells, moreover, are not so constant over short pe­riods as photo-emissive

V (volt)

cells and they are there­fore less suitable for pre­cision measurements. At the same time, selenium cells are now being mar­keted, the properties of which are so good that these cells can be used for all routine photo­metry.

., I 0,1

500 1000 1500

ff000.11. 2000.11. 1000.11. 500.11.

I ~E, 2000 2500 3000 Lux

o&S42

Fig. 170. The voltage V developed across different resistances as a function of the illumination E

In a humid atmosphere barrier-layer cells tend to absorb the moisture,

306 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

to the possible destruction of the cell, as manifested by a considerable deterioration, or even a complete absence of photo;-sensitivity; these cells should accordingly never be used or stored in too damp an at­mosphere. Barrier-layer cells are now available which are completely sealed in an enclosure consisting partly of glass and partly of metal. This enclosure is filled with an inert gas. In this manner a perfect protect­ion from moisture is afforded which considerably extends the life of the cells. Another peculiarity of photo-voltaic cells is that they may be subject to fatigue, that is, the current drops when once the illumination is applied, and takes some minutes to become constant; the drop in value amounts to some few per cent. A typical spectral sensitivity curve for a selenium cell is reproduced in Fig. 171 together with the VA curve. The va­rious makes of cell differ in their sensitivities, and the cells also tend to

75 vary in this respect in-dividually. From Fig. 171 it is seen that the spec­tral sensitivity of the selenium cell extends beyond the visible region of the spectrum, so that it is possible by means of coloured filters to match the sensitivity more or less perfectly with the VA curve. We

Fig. 171. Spectral sensitivity of a selenium cell (full line) compared with the VA curve (dotted

line).

shall have more to say about this in Section XV-6. The spectral sensitivity of selenium cells also varies somewhat with the illumination, that is to say the current versus illumination curves differ slightly according to the wavelength of the light. For technical measure­ments this is of no great importance, but for precision work allowance should be made. The current generated by a selenium cell is dependent on the temperature. In cases where measurement and calibration are effected at a constant temperature this has no significance, but, if after prior calibration a sele• tium cell photometer is to be used at different temperatures, this

XV-3] PHOTO-VOLTAIC CELLS 307

fact must be taken into account if the degree of accuracy required justifies it. This variation in the current may amount to 0.5% per degree centigrade. It is dependent on the particular kind of cell and the external resistance. In measuring the extent to which the cell is dependent on temperature it should not be forgotten that the resistance of the meter has also a temperature coefficient, which is, however, much lower than that of the cell itself. For precision measurements the temperature coefficient of the cell and meter should be determined together. Very high illumination levels (> 10,000 lux) should be avoided, as they 1, 0

0, 9

,8 0

0 ,7

0, 6

o. 5

0 ,4

0, ..J

0, 2~

0, 1

0 d'

Fig.

......_ ~

"~ \

-·

!\. ,~

\ \ \ ~1

\ ,,. K2J 1\' 1'\

' 5o"

~ \ 900 66544

172. Angular response of barrier-layer cells

10

8

6

5

4

2

1,5

1,25 1,1

I v

I I ~3

v / 2

.,//

0 0 20° 40° 60° 80° 66545

Fig. 173. Curves for correction of the reading error due to dependence on angle of incidence of the cell

responses shown in Fig. I 71

tend to raise the temperature of the cell too much and thus reduce its life. It is better not to exceed 1000 to 2000 lux (100-200 fc). The sensitivity of the working area of the cell is not usually the same at all points, and the illumination should therefore be distributed as evenly as possible. If illun1ination of only a part of the area is unavoidable, the masked cell should be calibrated for accurate results. An important feature of barrier-layer cells is the dependence of the photo-current on the incident angle of the light, in other words: their angular response. When a cell is illuminated by a light source of constant luminous intensity and at a constant distance from it, the actual illu­mination varies with rotation of the cell in accordance with the cosine of the angle turned through (curve I in Fig. 172). The variation in the meter reading follows a rather different curve, i.e. curve 2 or 3, Fig. 172.

308 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

The construction of the cell has a great bearing on the shape of the curve; a cell with a flat rim will give a curve similar to curve 2, but one with a rather high rim which begins to introduce a masking effect at relatively small angles of incidence will produce a characteristic like curve 3. If a protective glass window or coloured glass filter is fitted in front of the cell the effect of variations in the reflection factor of the glass will also have to be taken into account. Readings in respect of directions of incidence other than normal must thus be multiplied by a factor which can be derived from Fig. 172, from which charts of the kind depicted in Fig. I 73 can be plotted. These charts show more clearly than Fig. 172 the amount of error in the uncorrected reading. Without a correction, therefore, the cell is not suitable for measuring light falling at larger angles of incidence than about 45°. The necessary correction can be computed from curves such as those shown in Fig.l72, but this is usually not very convenient, seeing that the angle of incidence of the light to be measured must then be known. When light enters from numerous directions at the same time it is quite impossible to calculate the appropriate correction.

2

I

Fig. 174. 1 cell, 2 internally blackened tube, 3 spherical opal glass

:::~ 66547

Fig. 175. Two 'cells placed one above the other. With perpendicular in­cidence of the light the upper cell completely masks the lower one. With oblique incidence the lower cell is also partly or wholly illuminated and thus makes good the deficiency in the current supplied by the upper

cell

Needless to say many investigators have sought means of counteracting this dependence of barrier-layer cells on the direction of incidence; of the rather numerous suggestions put forward we may mention the following, which most nearly answer the purpose. I. The arrangement shown in Fig. 174, comprising a tube 2 mounted on

the cell I and supporting a spherical opal glass 3. The height of the

XV-3] PHOTO-VOLTAIC CELLS 309

tube and the radius of curvature of the glass depend on the trans­mission factor and absorption of the opal glass and must be deter­mined empirically*).

2. Two cells, one above the other (Fig. 175). With perpendicularly incident light the upper cell only is illuminated; the wider the angle of incidence the more the lower cell takes part in the measure­ment. Hence the lower cell supplies that part of the current which the upper cell fails to deliver. The distance between the two cells can be calculated approximately from the angular response of the two cells, but should, however, be determined empirically.

3. The device shown in Fig. 176 which is a cover with a central hole, fitted over the cell. This cover is painted white on the inside. With oblique angles of incidence, light reflected by the cell is reflected back, in part, from the inside of the cover. The correction thus

66548

Fig. 176. Cover with central hole placed over cell

6&549

Fig. 177. Cell with lens

obtained is incomplete and, moreover, results in an appreciable reduction in the measuring sensitivity (roughly proportional to the ratio of cell area to size of hole). For results which need not be too accurate, however, it -is useful and fairly simple to construct.

4. A flat opal glass plate, frosted on the upper face, placed over the cell. The correction obtained is scarcely complete, but the method is simple and serviceable where only a rough correction need be made.

5. A thin lens cemented to the cell (Fig. 177), permitting almost com­plete correction, with negligible reduction in sensitivity. 4 )

6. Another suggestion is to use a sphere with white interior as shown in Fig. 178a. Since the wall of the sphere will never reflect uniformly diffusely, the cell receives relatively more light with the direction of incidence shown in the figure than with other directions, and over-compensation is thus obtained in this and adjoining directions.

This drawback can be overcome by using the arrangement of Larche

*) This description is taken from an unpublished report of the Illuminating Engineering Laboratory of the N.V. Kema, Arnhem.

310 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

and S c h u 1 z e 5) designed for the measurement of ultra-violet radiation, but which can be used also for measuring visible radiation. L a r c h e and S c h u 1 z e provided the sphere of Fig. 178a with a cone-shaped screen (Fig. 178b). The diameter of the screen must be somewhat greater than that of the photo-cell, whereas the entrance opening must be much greater than the photo-cell. With this arrangement a practically perfect correction for the effect of the angle of incidence can be obtained.

Measurement of the photo-current The current from a photo-voltaic cell can usually be measured with a moving-coil in­strument. If the current is too small, however, a mirror galvanometer may be used. With very small currents ( < 1 o-s A) a sort of creeping effect occurs, that is, the current takes some time to assume a constant value, owing to the fact that the internal resistance of the cell is then very high compared with the resistance of the galvanometer. Since it is not possible with photo-voltaic cells to increase, as with photo-emissive cells, the voltage obtained across a high resistance, there is nothing to be gained by using an amplifier with the former type of cell. Since the short-circuit current is proportional to the illumination, a circuit for measuring this current is important. A suitable circuit, suggested by S e 1 en y i, is shown in Fig. 179. The cell current I. is taken through resistors R1 and R2• Current (I) from the battery B flows through R2 in the opposite direction to that of the cell current I.. By means of

Fig.· 178a. Sphere pain­ted matt white, as sug­gested for counteracting the dependence on angle of incidence of photo­electric cells. Owing to inadequate diffuse reflec­tion from the white paint the counteracting effect is not likely to be suffi-

cient

b. Sphere of fig. 178a, corrected according to Larche and Schulze.

S conical screen .

a rheostat R3 current I is so adjusted that there is no potential difference between the points P and Q, and no current then flows through the galvanometer (centre zero) G. This instrument is connected across the terminals of the cell and: when no current is passing, the cell voltage is zero and I • is the short-circuit current of the cell.

XV-3] PHOTO-VOLTAIC CELLS

According to Kirchhoff's law:

from which: Ic.R1 = (I-Ic)R 2

R2 I=I---

c R1 + R2

311

Hence, if R 1 be made many times larger than R 2, I will be many times higher than I c and the current I can thus be made so large that it can be measured with a simple meter, e.g. a milliammeter.

p -

Q

Fig. 179. Circuit for measuring the short­circuit current of photo-voltaic cells (ac­

cording to S e l e n y i)

Ftg. 180 is a diagram of a perfectly practicable circuit for measuring the short-circuit current of a photo-voltaic cell. The short-circuit current flows through resistors R7, R 8 and R 10, R 7 + R 10 corresponding to R 1 in Fig. 179 and Rs to Rz. R 10 is a decade resistor chain with which the factor by which the short­circuit current of the cell is amplified can be altered. That is to say that R 10 can be used to regulate the range of measurement of the arrangement. Points P and Q correspond to points P and Q in Fig. 179. The current compensating the short-circuit current in R 8 is faken from the battery B via the potentiometer Rt and Rz. Rt is for coarse and R2 for fine adjustment. A is the ammeter for measuring the battery current. The potentiometer R 4 + R 5 can be used to obtain a whole-number ratio between the indication of A and the value of the calibration lamp to be measured during calibration. R 6 serves to compensate the temperature effect on the internal resistance of ammeter A. R3 is a limiting resistor to prevent undesirably heavy currents from the battery.

312 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

+

Fig. 180. Circuit diagram of an amplifier for the measurement of the short-circuit current of photo-voltaic cells A = milliammeter, class 0.5, 0-l rnA B = 4.5 V flashlamp battery G = 0 - l pA light-beam galvanometer, R = 1000 n R 1 = 5 k!l, 3 W wire-wound potentiometer R 2 = 500 n, 3 W wire-wound potentiometer R 3 3.7 kn, l W carbon resistor R 4 l kn, l W wire-wound resistor R 5 = 1 k!l, 3 W wire-wound potentiometer R 6 = l 00!1, l W wire-wound resistor R 7_ 8 = 1000!1 wire-wound resistors, R1 = 900 nand Rs = 100 n R 9 = 25 kn, 3 W wire-wound potentiometer R 10 = Decade resistor, divided into thousands and tens of thousands and possibly

also hundreds of ohms Sk1 = 6 A switch Sk2 = telephone switch

The sensitivity of the galvanometer G can be controlled with the poten­tiometer R9. Sk2 is a "telephone key" with which the photo-voltaic cell and the gal­vanometer G can be short-circuited in the inoperative state. The compensation of the short-circuit current of the cell can be auto­mized and the current proportional to the current of the cell can be read-off directly. This automation can be improved so that the measuring results can be read-off with an digital device, or be recorded on paper with an electric type-writer, directly in numbers. An example, an auto­matic photometer for measuring the luminous flux of incandescent lamps, is mentioned in XVII-5.

XV-4] BOLOMETERS AND THERMO-COUPLES 313

XV-4. Bolometers and thermo-couples In bolometers and thermo-couples the surfaces rece1vmg the radiation are blackened to render them non-selective. For the comparison of light of different spectral compositions it is accordingly necessary to match the spectral sensitivity more or less with VA (see Section XV-6). The essential part of a bolometer is the receptor, which consists of a metal wire or strip, this being very thin in order to limit its thermal capacity. This element is heated by absorption of the radiation falling on it, ana the electrical resistance thereby increases. This variation in resistance can be measured, and is a direct indication of the amount of the radiation absorbed. In order to avoid difficulties due to irregular cooling of the element by air currents, the element is mounted in a bulb which is preferably exhausted. To render the measurement independent of variations in the ambient temperature, bolometers are made which have two identical receptors, one of which is screened from the incident radiation. Each of the receptors is connected to one arm of a Wheatstone bridge, and the variation in the resistance of the irradiated receptor is thus quite simply measured. With thermo-couples use is made of the property that two different metals placed in electrical contact with each other develop an e.m.f. when the temperature at the junction is higher than that at the extremities. The junction is blackened and thereby absorbs the incident radiation, so raising the temperature. In photometry, and measurement of radiation in general, theM o ll thermo-pile 6) (see Fig. 181.) is the most widely used. This consists of a number of thermo-couples connected in series. The thermo­couples themselves are blackened strips (300 x 6,u) of the alloys constantan and manganin, joined with silver solder; these strips are made by soldering together

66552

Fig. HH. The M o 11 thermo-pile

fairly thick plates of the two metals and then rolling them out to a thickness of 6 microns. Here, again, the strips are made as thin as pos­sible in order to keep the thermal capacity low. The ends of the elements

314 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

are attached to a copper plate, electrically insulated but making effective thermal contact; these ends thus have a high thermal capacity and remain cool during the measurement, whereas the soldered ends which remain free from the copper plate are heated. The working area is limited, by means of a plate with a hole in it, to the extent of a circle 6 mm in diameter. The voltage from the thermo-pile is measured directly by means of a galvanometer or millivoltmeter. At low illumination levels the e.m.f. generated is proportional to the illumination, but at higher levels this no longer holds, because the blackened strips lose a portion of their absorbed energy by radiation; this loss is relativelygreaterat higher temperatures, and the rise in tempera­ture is then no longer directly proportional to the energy absorbed. Sensitive instruments (galvanometers) must be employed with bolo­meters and thermo-couples, which are accordingly suitable for laboratory use only. Because of their fairly high sensitivity, bolometers and thermo­couples are particularly useful where the radiation intensities are only low, e.g. in the spectral analysis of light (spectrophotometry).

XV-5. Photographic photometry The density of an exposed photographic plate or film is a function of a great variety of factors, such as the illumination, the e:)\posure time, the method of development, the spectral composition of the light and, of course, the characteristics of the plate or film; the latter differ widely between one kind and another, and may even vary- albeit not to a considerable extent - from batch to batch of a given kind. In first approximation, the density of most films and plates is a function of the product of the illumination and the exposure time, that is, the exposure. If the material is exposed in a camera with objective, the illumination, as we have seen in Section XI-8, is proportional to the luminance of the object photographed. It can accordingly be said that the plate or film measures luminance. If it can be made possible during measurement for all the variables apart from the luminance to be held constant, the density will be a function only of the luminance, and we thus have a means of measuring luminance photographically. This can actually be done by making an exposure, not only of the luminance to be measured, but also of a series of known luminances of the same spectral composition as that of the

XV-6) PHOTOMETERS FOR HETEROCHROMATIC PHOTOMETRY 315

first-mentioned luminance, preferably using the same exposure time and lens stop. A comparison can then be made between the densities in respect of the light under test and the reference densities, by means of a densitometer. In so doing, allowance must be made for the effects of possible vig­netting of the objective, of the decrease in illumination towards the edges of the plate due to reduction in the apparent area of the exit pupil of the objective, and to the fact that the beams do not strike the plate at right angles. These effects can be determined for every aperture of the particular objective by using a photo-emissive cell or photo-voltaic cell to measure the relative illumination values at various points on the plate or film with constant luminance of the object. Allowance must also be made for the characteristics peculiar to photo­graphic material in general, viz. the different "effects" known in photo­graphy (e.g. Callier effect (section X-12), Eberhard effect). Since the density increases with the exposure time (the effect on plate or film is cumulative), it is possible photographically to measure much smaller luminances than by physical methods. The photographic method of photometry is employed for the measure­ment of the intensity of spectral lines, i.e. in spectrophotometry, and also of the distribution of luminance on artificially illuminated roads (see Section XIX-3).

XV-6. Physical photometers for heterochromatic photometry We have seen from the previous section that none of the physical photo­meters described possesses an inherent spectral sensitivity corresponding to VA· For the comparison of kinds of light of different spectral com­position the following procedures can be followed: a) In front of the light receptor one or more filters can be placed whose

spectral transmission is such that the spectral sensitivity of the combination more or less equals VA (filter method).

b) Correction factors can be ascertained for certain kinds of light. c) The light under test can be broken down into its spectrum by means

of a monochromator; in the plane of the spectrum a template is placed which has the form of the V .\ curve (geometrical method).

d) The energy v. wavelength curve (spectral energy distribution) of the light can be plotted with the aid of a spectrophotometer, the energy value being then multiplied by the appropriate value of V .\ for each wavelength. In effect, the value of the expression .EV.\E~

is computed (algebraic method).

316 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

This last method is difficult and cumbersome, and is adopted only in fundamental research; in photometry it is of little importance. For routine measurements the methods based on the use of filters and correction factors are both used; for precise results the filter and geo­metrical methods are mainly employed. The more the two kinds of light differ in composition, and the higher the required accuracy of the results, the better must be the agreement with V "; this determines whether and how many filters should be used. It should be borne in mind, however, that the correction for measure­ment of light the spectrum of which is continuous need not be so precise as for light with line spectrum. With light of a single wavelength, any difference with respect to V" at that wavelength is carried through in its entirety into the final result. Should the divergence, say, at 590 mtl (roughly the wavelength of the light from a sodium lamp) be for example 10%, the error in the measurement of sodium lamps will be 10%. If in a certain region of the spectrum the spectral sensitivity of the cell diverges from V -'• then in measuring light having a continuous spectrum, only that part of the light comprising wavelengths occuring in that region will be wrongly measured. Since the remainder is measured correctly, the divergence from V" is only partly reflected in the overall measurement. Since each and every filter used represents a source of absorption of light and hence a depreciation in the sensitivity, as few filters must be used as possible, commensurate with the purpose in hand. Filters are made of glass or gelatine, or they may be of liquid. As the spectral transmission of such filters is to some extent dependent on temperature, it is necessary, for precision measurements, to maintain a temperature that is constant within narrow limits. Amongst gelatine filters there are some that gradually decolorize when exposed to light, a fact that should not be overlooked. Compared with the other kinds, liquid filters have the advantage that the spectral transmission can be modified by varying the concentration of the solution. These filters of course lend themselves only to laboratory work, and not to transportable photometers. In using the filter method it should be remembered that the spectral sensitivities of individual cells vary; when precise results are required, therefore, the appropriate filter should be found for each cell and, as the characteristics of both cells and filters are liable to change in time, the combination should be subjected to a regular check.

X. V-6] PHOTOMETERS FOR HETEROCHROMATIC PHC'T'OMETRY 317

It is also to be borne in mind when using filters that light p<'.ssing obliquely through a filter has a longer path to travel than perpendicularly incident rays and that the absorption thus varies with the direction of incidence. It is necessary, therefore, in precision measurement, to carry out the calibration and the measurements with the same direction of incidence. Let us now consider the adaptation by filters of the different receptors discussed in the foregoing.

1. Photo-emissive cells There are special filters on the market for caesium-antimony cells pro­viding good adaptation to VA,. For exact matching to VA, the correct filter, or rather the correct combination of filters, must be found for each individual cell. Caesium cells can be corrected for use with the usual temperature radiators by means of a blue filter; for more effective correction K 6 n i g7)

has prepared a combined filter consisting of eight glasses.

2. Photo-voltaic cells Cell-filter combinations, the sp~ctral sensitivities of which approximate to V" are marketed by a number of firms. For technical purposes these can usually be employed without a correction factor, but for precision work the necessary factor should be determined for each cell and each kind of light. When this is done it should be remembered, however, that the spectral sensitivity of photo-voltaic cells tends to vary slightly with the illu­mination. When filter cells are used, the variation due to the effect of the direction of incidence on the filter has also to be taken into account.

3. Bolometers and thermo-couples For matching non-selective thermal receptors with V" Gibson, Tee 1 e and Keegan s) have developed a filter composed of one glass and one liquid filter, the spectral transmissivity of which is for all practical purposes equal to V". However, as this combination trans­mits some infra-red radiation, T e e 1 e 9) used it in conjunction with a red filter that transmits no visible radiation but which has a high transmission in the infra-red. T e e 1 e takes two measurements, one with, and one without this extra filter; the difference between the two results is then the required value.

318 PHYSICAL PHOTOMrTERS AND PHOTOMETRY [XV

It is possible to determine and employ correction factors for kinds of light of known spectral distribution e.g. from sodium lamps, mercury vapour lamps working at different vapour pressures, neon lamps, fluorescent lamps etc. If the composition of the light to be measured is not known, however, use must be made of receptors whose spectral sensitivity is matched with V ,~.. Wide use is made of correction factors for simple illumination photometers with barrier-layer cells. The makers of such photometers usually supply correction factors for the more generally used technical light sources. Whereas in principle the filter method is an approximation method for matching receptors with V ,~.,the geometrical method is, in principle, exact. Since we are dependent on the physical constants (spectral transmission) of materials, in the manufacture of filters, it is not possible to produce a filter the transmissivity of which, wavelength for wavelength, will guarantee a perfect match with V ,~.. At the same time this is not such an obstacle as it would seem, for it has been found possible by means of combinations of filters to secure a match with V >.that is sufficiently accurate for precision measurements. In the geometrical method, agreement with V >. is not d~pendent on the physical constants of the material employed, but on the degree of accuracy with which we are able to make a template of a given form, and on the accuracy with which the spectral sensitivity of the receptor can be measured.

Fig. 182. Schematic diagram of photometer based on the geometric principle

We shall now describe the geometriCal method in reference to Fig. 182 which is taken from an article by V o o g d 10), in which be describes a physical photometer working on the geometrical principle. The light to be measured, passing through a slit S, is concentrated as a ,parallel" beam by a lens L1 and is then analysed into its various

XV-6] PHOTOMETERS FOR HETEROCHROMATIC PHOTOMETRY 319

wavelengths by a prism Pr. The rays of different wavelength, which are parallel also after refraction by the prism, are brought together by another lens L2 in its focal plane, to form a spectrum. The height of the spectrum is the same at all points and corresponds to the height of the slit S. The light from the spectrum is passed through a lens L3

to ·the receptor R. · Whereas in the filter method each wavelength is so absorbed by the filter that the spectral sensitivity of the filter-receptor combination as nearly as possible equals V .\. this is achieved in the geometrical method by screening off a part of the radiation of each of the projected adjacent wavelengths, a template D being used for this purpose, in the plane of the spectrum. The shape of the template can be computed from the spectral sensitivity of the receptor (G.\) and V .\· At every point on the template the height must be proportional to V .\/G.\. Nevertheless, the method is not so simple in realisation as would appear at first sight. In the first place it is not easy to make a template exactly of the required form. Further, owing to the finite width of the slit, the wavelength is not reproduced in the spectrum as a line, but as a band, the width of which equals that of the image of the slit. These bands therefore overlap, in consequence of which the transmission of a wave­length A. is not determined by the height of the template at the po­sition D, but by the average height of the template corresponding to the image of the slit. The smaller the slit the greater the accuracy, but the lower the sen­sitivity of the photometer. The average transmission for a given wave­length will the more precisely agree with the computed transmission according as the form of the template more nearly approaches the linear. Again, the more the spectral sensitivity of the receptor approximates to V .\. the more linear the template will be; for this reason K 6 n i g 11)

has suggested that the sensitivity of the cell be roughly matched with V.\ by using one or more filters (F, Fig. 182). The first photometer based on the geometrical principle was constructed in 1915 by I v e s, the receptor being in this case a thermo-couple (photo-electric cells were then unknown).

320 PHYSICAL PHOTOMETERS AND PHOTOMETRY [XV

REFERENCES 1} M. C. Teves, Ph. T. R. 2, 1937, 13-17. "The Photo-electric Effect and

its Application in Photo-electric Cells" A. S o m m e r: "Photoelectric Cells". London 1946 R. Sew i g: "Objektive Photometrie". Berlin 1935 V.·K. Zworikin and E. G. Ramberg "Photoelectricity and its application," l~ew York 1949 H. Carter A. M. I.E.E. and M. Donker, "Photo-electric devices in theory and practice", Philips Technical Library, 1963.

2) M. C. Teves, Ph. T. R. 5, 1940, 253-257. "A Photocell with Amplification by Means of Secondary Emission"

3) W. C h. van Gee I, Ph. T. R. 8,. 1946, 65-71. "Blocking-Layer Photocells" 4~ G. B. Buck, Ill. Eng. 44, 1949, 293-302. "The Correction of Light-sensitive

Cells for Angle of Incidence and Spectral Quality of Light" 5) K. Larche and R. S c h u I z e, Zts. f. Techn. Phys. 23, 1942, 114-117.

"Uber ein Ultraviolett Messgerat mit Vorsatzkugel fiir Strahlungseinfall unter grossem Winkel"

6) L. S. 0 r n s t e i n, W. ] . H. M o 11 and H. C. B u r g e r: "Objektive Spektralphotometrie". P. 4 et seq. Brunswick 1932

7) H. K 6 n i g, Helv. Phys. Acta 16, 1943, 421-422. '"Praz1sions-Photometrie mit Caesiumzelle und Kombinationsfilter"

S) K. S. Gibson, R. P. Tee 1 e and H. ]. Keegan, J.O.S.A. 29, 1939, 144. "An Improved Luminosity Filter"

9) R. P. Tee 1 e, J.O.S.A. 31, 1941, 696-704. "A Physical Photometer" 10) ]. V o o g d, Ph. T. R. 4, 1939, 260-266. "Physical Photometry" 11) H. K 6 n i g, Helv. Phys. Acta 7, 1934, 433-453. "Beitrage zum Problem

des Vergleiches verschiedenfarbiger Lichtquellen"

CHAPTER XVI

MEASUREMENT OF LUMINOUS INTENSITY

XVI-I. Principle of luminous intensity measurement In Part I two definitions of luminons intensity were introduced. Ac­cording to the first of these, luminous intensity is the quotient of lu­minous flux divided by the solid angle, with infinitely small solid angle, that is, the luminous flux per steradian (see Section III-1), and ac­cording to the second it is the limit of the product Ed2 when d --+ oo (Section IX -1). The second definition can be employed for all practical forms of light source, i.e. of which the dimensions are finite. In the case of the ideal point source the second definition gives way to the first. It has already been stated in Section IX-1 that luminous intensity is always measured as a quantity answering to the second definition, i.e., the product of the illumination and the square of the distance from the light source to the photometer*). Sufficient has already been said about the measuring distance for diffuse light sources in Section IX-12, and the reader is therefore referred to that section for details. In effect, luminous intensity measurements amount to measurement of illumination. As mentioned in Section XII-2 this is converted to luminance in visual photometry very simply by using a diffusely transmitting or reflecting screen as comparison surface in the photo­meter. In our description of a luminous intensity measurement in Section XIV-1 we have already indicated the general method of carrying out such measurements. Instead of the primitive photometer, however, a well-constructed instrument will usually be employed, of the kind described in Section XIV -4. With physical photometers for the measurement of illumination, lu­minous intensities are measured in the same way, a light-receptor being used instead of the comparison surface of the visual photometer.

*) The method of measuring luminous intensity discussed in Section X\'I-4 is an exception.

322 MEASUREMENT OF LUMINOUS INTENSITY [XVI

XVI-2. The photometer bench A description of a photometer bench has already been given in Section XIV-4, so that here we need only deal with its use for the measurement of luminous intensities. Figures 183a, b and c show the different ways in whifh the bench can be used with visual photo­meters. In Fig. 183a the positions of the lamp on test X and comparison lamp CL are fixed, the photo­meter P being adjusted on the bench to establish photometric balance. Fig. 183b shows the lamp on test X as fixed; the photometer P and comparison lamp CL are at a fixed distance from each other and can be moved together on the bench for photqmetric balance. The arrangement shown in Fig. 183c differs from b in so far that P and CL-are fixed and X is adjustable.

-OL

1 ~p X

1 g

-CL - X

1 ~p 1 !2

CL -~p

-x

1 l Fig .. 183. Possibilities of the photometer bench for the measurement of luminous

intensity CL = comparison lamp. P = photometer head. X = light source to be measured a) CL and X fixed, P adjustable b) X fixed, CL and P adjustable c) CL and P fixed, X adjustable

If a physical receptor is to be employed, the substitution method is usually adopted. It is then possible to dispense with calibration of the cell and hence to disregard its characteristics (important in the case of photo-voltaic cells) by so adjusting the distance of the standard lamp, after measurement of the lamp on test, that the cell delivers the same current for the same external resistance, i.e. that it receives the same illumination. The luminous intensities will then be proportional to the square of the distance from lamps to cell. In this case the cell functions, as it were, as a null instrument. Other null methods can also be employed, for example a photo-emissive cell may be illuminated rapidly in turn by the lamp on test and standard lamp. The distance from the cell to one of the lamps is adjustable. If the illumination on the cell differs between one lamp and the other the cell current varies, and this results in a pulsating current. For equal

XVI-2] THE PHOTOMETER BENCH 323

illumination values and cell currents the total cell current is constant. If the cell current be measured with the aid of an A.C. amplifier the

-=

66555

Fig. 184. Null method of luminous intensity mea­surement with two photo­voltaic cells. When the illumination is the same on each cell the meter

does not deflect

-Fig. 185. Luminous intensity mea­surement with the aid of two lenses. Lens L1 produces an image of the light source in L 2• The adjustable diaphragm D enables the image to be isolated almost completely from its background, thus effectively screening

off any stray light

output meter will indicate no current when photometric balance is established. In another null method 2 barrier-layer cells are used, these being placed back to back on the photometer bench with a meter in series with them in the manner shown in Fig. 184. If the sensitivities of the cells are equal (or matched by placing a filter or stop in front of one of them) the meter will not deflect when photometric balance is obtained. Fig. 185 shows an arrangement which offers two advantages, viz. an increase in sensitivity of the cell employed, and the fact that it is practically independent of extraneous light. The light to be measured falls on a lens L1 which forms an image of the source in lens L2• The focal iengths of L1 and L2 are such that L2

produces an image of L1 in the celi C. A variable stop placed in front of L2 makes it possible to isolate the image of the light source from the image of its background, so that only light from the source enters the cell. The light collected by L1 is distributed evenly over the image of L1

in the cell, and the same part of the cell accordingly always receives uniform illumination. If the diameter of L1 is made considerably larger

:: } D D D D 66557

Fig. 186. Tube K with screens D for ex­cluding stray light

than that of the cell, the photo-meter is rendered more sensi­tive than when measurements are carried out with the naked cell. Another device for excluding stray light, which is, however,

324 MEASUREMENT OF LUMINOUS INTENSITY [XVI

a little less effective than the system of lenses, consists of a long tube placed before the cell (Fig. 186). This tube is painted matt black on the inside and is provided with diaphragms as photometer screens (D).

XVI-3. Apparatus for measurement of light distribution The preceding seGtion deals in particular with the measurement of the luminous intensity of a light source in only one direction. To measure the light distribution of a source it must be possible to ascertain the intensity in various directions, and some form of equipment is required by means of which this can be done quickly and easily. The simplest method, at any rate one that can be carried out using the simplest apparatus, consists in measuring the illumination at a number of points on a plane surface upon which the light from the source is allowed to fall. Formula (VI-5) (E =I cos3 r~../h2) can be used to determine the luminous intensity, in the different direction5, from the illumination. Needless to say this method is suitable only where the measuring surface need not be too large, i.e., when the beam is quite narrow or the measuring distance not too great (e.g. car and cycle lamps). In all other instances specially designed apparatus is generally used. The construction of apparatus for measuring light distribution is governed by the following factors: a) the minimum distance between photometer and light source, com-

patible with the desired accuracy; b) the dimensions of the light source; c) whether or not the burning positions of the light sources may be varied. a) In Chapter IX we have seen that the minimum measuring distance, in order to obtain a reasonable accuracy, for circular diffuse light sources is five times the diameter of the light source, and that this distance for linear sources is twice the length of the source. This means that with conventional fittings the measuring distance will be 4 to 6 metres (15-20 ft). As applied to mirrors and lenses it is shown in Section XI-6 that the point of measurement of the illumination must lie beyond the beam cross-over point. The distance from the beam cross-over point to the projector varies.according to the type of optical system and light source used in it *) and, for large searchlights, may be several miles. The distance for a motor-car headlamp will be as much as 7 to 10 metres (25-35 ft),

*) Further details in Section XVI-4.

XVI-3] MEASUREMENT OF LIGHT DISTRIBUTION 325

but two or three metres (7-10 ft) is sufficient for cycle lamps owing to their small dimensions. In any determination of the minimum measuring distance for a beam type source the beam cross-over point is not the only factor to be taken into account, however; it should be remembered that at points beyond the beam cross-over point a projector functions as a luminous disc (the limiting area of a projector is the exit pupil). Hence the same con­siderations apply to the minimum measuring distance required to yield a certain degree of accuracy of Ed2 as for diffusely radiating sources. In practice the distance from projector to beam cross-over point is generally much greater than the minimum distance necessary for a sufficiently close approximation to the limit value of Ed2•

R e e b 1) makes a distinction between the optical limit distance and the photometric limit distance. By the former is understood the minimum measuring distance as determined by the geometrical characteristics of the projector, that is, the distance from projector to beam cross-over point. Reeb's definition of photometric limit distance is the minimum measuring distance necessary to ensure a sufficiently close approximation to the limiting value of Ed2•

b) This calls for little comment, since it is obvious that large, heavy fittings, searchlights etc. will necessitate more robust constructions than small, light, sources. c) With many light sources, e.g. gas-filled incandescent lamps and sodium lamps, the luminous flux is to some extent dependent on the burning position.

The above considerations have led to the design of two types of ap­paratus for the measurement of light distribution, based on different principles. Both types are made in a large number of variations. Stated briefly, the difference in principle amounts to this, that in the one type the light source is stationary and the photometer is moved, whereas in the other the reverse is the case. The first kind is employed for light sources which can be measured at short range, and in cases where the burning position of the source may not be changed during measurement. The second is more suitable for the measurement of sources requiring such a long measuring range that it is not possible to move the photometer around the source. In practical forms of the first mentioned type of apparatus the photo­meter is, nevertheless, usually stationary; a system of mirrors arranged to rotate in a meridian plane around the source so as to r.eflect the

326 MEASUREMENT OF LUMINOUS INTENSITY [XVI

light into the photometer produces the same effect as a photometer moving round the source. For measurements in other meridian planes the source is rotated on its axis, leaving the burning position as before. Fig. 187 depicts an arrangement of this kind. 6)

The mirrors M1 and M2, which are mounted on an arm, are rotated together about the axis AB. Light from the source L travels via M1

and M2 to the photometer P. For measurements in other meridian planes the source is rotated about axis CD. Facilities for rotation are an advantage in measuring the average distribution (Section IV -1). The speed of rotation, de­pendent on the asymmetry of the light distri­bution, is from 100 to 200 r.p.m. and, where large lighting fittings are concerned, this im­poses high requirements on the robustness of the apparatus. The mirrors "fold up" the measuring distance, and the supporting arm can thus be very much shorter than if the photometer itself were moved. Visual photometers must, of course, be kept stationary, but with cell-photometers the cell mounted on a bracket, can be moved round the light source, in which case the mirrors are unnecessary. This works quite well with photo­voltaic cells, but not so well with photo­emissive cells, since movement of the leads may cause interference with the measurements. Types of equipment used for the measurement of projectors are depicted in Figs. 188 and 189 2).

p

A

c

·-'-· L . B

\ I? ' . . I \ .

\ I \ I . I \ . \ I I \·

Fig. 187. Apparatus for measurement of light distribution. The light from a source L is re, fleeted towards the photometer by mirrors M1 and M2 mounted on a bracket which pivots

about AB.

Fig. 188 shows a frame capable of rotation about the fixed horizontal axis AB. The projector is mounted in the frame in such a way as to rotate about axis YY, which follows the movement about AB. In the system shown in Fig. 189 the fixed axis CD is vertical and the axis of rotation XX is in the initial position perpendicular to the direction of measurement. The motion is applied through reduction gearing, the position of the projector being read from a graduated arc fitted, if desired, with a vernier. A different kind of instrument is depicted in Fig. 1903). This consists of two gimballed rings A and B, in the inner one of which the projector

XVI-3] MEASUREMENT OF LIGHT DISTRIBUTION 327

is mounted. The inner ring also carries an arm C with slide D which can be moved along an arc E. This arc can be rotated about the horizon­tal axis ZZ (through worm gearing), and the inner ring with the projector is able to follow this motion by reason of the gimbals. In the three types of apparatus described the measuring direction is in each case adjusted in a different manner, that is, in each the direction is determined by different systems of co-ordinates. If the results of measurement are to be plotted as an isocandela diagram it is necessary to ascertain what spherical co-ordinates are to be included in the chart (see Ch. V). In the latter we plot the angular displacement from the axis of the projector in respect of which the measurement is taken. For the measurement, however, the photometer is stationary and the

y

'R

A B

66559

Fig. 188. The frame R rotates on the fixed horizontal axis AB; the projector P rotates about axis YY which follows

the motion of R about AB

c

R

Fig. 189. The frame R rotates on the fixed vertical axis CD; the projector is pivoted on axis XX which follows the

movement of R about CD

projector is rotated, so that when it is turned to the right and upwards the measuring direction with respect to the axis of the beam is displaced to the left and downwards. If in Fig.l88 the projector is rotated through a certain angle IX about the fixed axis AB, and is then rotated about YY so the the measuring direction maintains an a,ngle IX with the normal to the axis YY, the measuring direction will describe a parallel (line of latitude). For the direction of measurement, the rotation about YY is thus rotation in longitude, and that about AB one in latitude. The system of co-

328 MEASUREMENT OF LUMINOUS INTENSITY [XVI

ordinates for this apparatus accordingly has a vertical pole axis, and is therefore the Y-system, whilst that for the apparatus depicted in Fig. 189 is the X-system. With measuring directions which do not diverge more than 10 to 15° from the axis there is no need to make any distinction between the one system of co-ordinates or apparatus, and the other, seeing that up to such angles the me­ridians of the one run very close to the paral­lels of the other. y.; e now come to the apparatus depicted in Fig. 190. Here, on rotation about axis ZZ the directions of measurement des­cribe conical surfaces about the axis, ZZ being then the pole axis. Dis­placement along the circle E is displacement in latitude and rotation about ZZ is the motion in longitude, so that

A

Fig. 190. The projector is mounted in the inner measurements are ef- gimballed ring of A and B. The axis of the projector fected with co-ordinates is deflected from the initial position a long a graduated

arc E. Rotation of the arc E enables the luminous of the Z-system. intensity of the projector to be measured in directions This instrument is used lying on the surface of a cone.

to advantage for the azimuthal spherical projection, since the parallels of latitude described when the instrument is rotated about ZZ are reproduced as circles. Furthermore, this apparatus facilitates the deter­mination of the average luminous intensity of a source at similar angles from the axis in cases where measurement in four directions is not enough to ensure an accurate average. The four directions can be easily reproduced with the systems shown in Figs. 188 and 189 by rotating upwards, downwards, to left and right.

XVI-4] MEASUREMENTS ON PROJECTORS 329

XVI-4. Measurements on projectors A few supplementary remarks must be added to what has been said in the previous section about measurements on projectors 4).

In section Xl-6 the beam cross-over point is defined as the point at which the exit pupil of an optical system is seen to be completely flashed. Only as from this point may the inverse square law be applied. If we are to determine the luminous intensity of projectors as being the product Ed2 , E must be measured for a value of d that at least equals the distance g from the beam cross-over point to the projector, in addition to which this distance must be great enough to permit of a sufficiently close approximation to the limiting value of Ed2• In

&&562

Fig. 191. Figure illustrating the calculation of the location of the beam cross-over point (distance g) of a parabolic mirror with spherical light source

at the focus

general, this condi­tion will be met at the beam cross-over point. In section XI-6 the distance g is com­puted for circular diffuse light sources placed at the focus of a lens. Let us now calculate this for a parabolic mir­ror with spherical dif­fuse light source at

the focus, with reference to Fig. 191. If the beams reflected from in­dividual points on the mirror be drawn, it is found that g is determined by the inner boundary rays of the beam as reflected at the edges of the paraboloid. From Fig. 191 it is found that:

g = yjtan <5.

If r ~ y, we may write for tan <5: tan <5 = rjv, where v is the vector radius of the edge of the parabola. Now, v =I+ x, where I is the focal length and x the depth of the paraboloid; hence:

y (f + x) g=--­

r (XVI-I)

Similar calculations can be made for other .forms of light source, but there will not usually be much object in so doing, seeing that in practice

330 MEASUREMENT OF LUMINOUS INTENSITY [XVI

numerous divergencies from the theoretical conditions occur. The significance of the derived formula for g lies thus mainly in the fact that it tells us upon what elements the position of the beam cross-over point depends. It may be said that g is greater according as the light source is made smaller and the mirror larger. In practice the beam cross-over point is found to be nearer to the projector itself than is indicated by theory, and to an increasing extent as the mirror departs more from the theoretical form. If there is any uncertainty whether the measuring distance is sufficiently large, or if it has to be determined in advance, the best method is to measure the luminous intensity at different distances to find the distance as from which Ed2 retains the same value. Some typical measuring distances occurring in practice have already been given in Section XVI-3. In the case of search lights the distance from projector to beam cross­over point is often so large (sometimes several miles) that absorption and scattering of the light by the particles of solid matter suspended in the atmosphere between the projector and photometer must be taken into account. This is particularly important where measurements are taken in the open. Such atmospheric effects can at all times be ascertained by setting up next to the projector a diffuse light source whose luminous intensity can be measured at such close range that the effects of the atmosphere on the result can be disregarded e.g. in the photometric laboratory. Each time that measurements are taken from the projector - or less frequently if stability of the atmospheric conditions permits - the in­tensity of the diffuse source is then also measured and the results ob­tained from the projector are corrected with the aid of those taken from the diffuse source. n necessary the latter source can be placed closer to the photometer. In this case the corrections of the projector measurements by means of the measurements from the diffuse source must be carried out with the aid of the absorption formula (X-12). The light spot produced by a beam on a screen can be projected at a short distance by placing a positive lens in front of the projector (Fig. 192) 5).

If we imagine a screen at an infinite distance from the projector B (in practice this would simply mean a long distance), the image on the screen would be reproduced (obviously reduced in size) in the focal plane of the lens. If the screen be placed in the focal plane F of L, this will show on a reduced scale what the beam looks like at infinity. To project

XVI-4] MEASUREMENTS ON PROJECTORS 331

the cross-section of the beam at other distances by means of the lens, the lens formula enables us to compute the distance from the lens at which the screen should be mounted. This method can be employed for measuring purposes as well, but two factors should then be taken into consideration. In the first place the lens reflects light back to the projector and this is once more thrown back by the latter. This light is superimposed on the actual beam, but

F

L

Fig. 192 .. , Measurement of the light distribution of a projector at short range. The lens L produces at the focal plane F an image of the beam projected to infinity by the projector. The effect of light reflected by L can be almost completely eliminated

by a filter Fi

its light distribution is entirely different, so that the distribution of the projection of the beam differs from that of the beam it-self. Carried out in this way, measurements would yield faulty results, and a means of correcting this consists in introducing a filter (Fi in Fig. 192) at an angle of 45°

with the optical axis, between the projector and the lens. The light to be measured than passes once through this filter and is attenuated by a factor r. The light reflected by the lens passes through the filter three times and is accordingly attenuated to the extent of r. The filter is mounted at an angle of 45° so that light reflected from it is defleeted from the direction of the axis and does not enter the projector. The second point concerned is the deficiencies inherent in single lenses (mainly spherical aberration). For the degree of accuracy usually demanded in such measurements the aperture of the lens should not be greater than f : 8. This system can be successfully employed for motor-car headlamps for which a lens 25 em (10") in diameter, with a focal length of 2 metres (7') is suitable. There is another method that can be used in certain cases to attain a measuring distance that is less than the distance between projector and beam cross-over point. If it is possible to split the light-emitting surface of the projector into precisely defined elements (e.g. by means of screens), the distribution of the light-elements can be measured separately. The luminous intensity of the whole projector in a given

332 MEASUREMENT OF LUMINOUS INTENSITY [XVI

direction is then obtained by adding up the intensities of the elements in that direction. From the formula (XVI-I) for the distance from projector to beam cross-over point it is seen that g is proportional toy, that is, the distance from the edge of the projector to the axis. When elements of the projector are measured, y is the distance from the edge of the element to the axis of the element and is accordingly much smaller than the y that relates to the whole projector. Therefore, g can also be very much smaller. This method is quite suitable for the measurement of cylindrical-pa­rabolic mirrors and drum lenses.

XVI-5. Photometer with Maxwellian view In the visual photometers described in Section XIV-4 the pupil of the observer's eye is not reduced by an artificial pupil and is thus wholly flashed with the luminance of the comparison surface. Sometimes, however, measurements of luminous intensity can be effected by means of a photo-meter in which the

pupil of the eye is only ~Lar~::::::::::.:::==::==:=~(~ partially flashed with · -·-·--·-·-· the light by producing in it a reduced image of the light source to

Lz R

v

be measured (MalCwel- Fig. 193. 'Diagrammatic representation of a photo-lian view). meter with Maxwellian view

The principle of such photometers will now be explained with reference to Fig. 193, which shows the paths of the rays of light measured. For the luminance of the photometer field of the comparison source the usual system with completely flashed pupil can be used. This section is not included in the figure. La is the light source the intensity of which is to be measured. A reduced image of La is formed by the lens L1 in the lens of the eye L2• PF is the photometer field, e.g. a Lummer-Brodhun prism. The eye is accommodated to PF, hence an image of PF is produced on the retina R. Let us now compute the illumination E R on the retina as produced by La, disregarding the losses in the lens L1 and in the eye. The following notation will be used:

XVI-5] PHOTOMETER WITH MAXWELLIAN VIEW

I = luminous intensity of La in the direction of the photometer, u = distance from La to Lv

333

v = distance from L1 to L2 (PF is considered as coincident with L1),

f = focal length of Lv v0 = distance from L2 to retina R, 5 = area of L1 effective in forming the image of La in L2 ( ~-= area

of PF), <PR = luminous flux falling on the retina, 5' = area of retinal image of S.

<J>R Then ER = 51.

The luminous flux <PR falling on the retina is equal to that which 1s received from La by area 5 of L1; hence

I <J>R =- 5. uz

The area 5' of the retinal image of 5 is reduced by a factor equal to the square of the quotient of image and object distances of 5 and 5' with respect to the lens L2, so that

We can therefore write:

v 2 5' =-i-S. v

I -S

<P u2 I v2 ER=~=--=- X-.

5' v 2 v 2 u 2 ....!!_5 0

v2

1 1 1 Introducing the lens formula -:- =- +-

1 u v uz f2

we find that v2 = --· (u- /)2'

I f2 ER =-X .

vo2 (u- /)2 hence

The retinal illumination is thus inversely proportional to (u- /) 2 and we therefore again have a square law, in which, however, the distances are measured up to the front focus of L1• Usually the distance u is so great compared with the focal length that it can be measured right up to L1•

With such photometers only the substitution method is employed; for the comparison luminance, as already mentioned, the normal system with wholly flashed pupil is used.

334 MEASUREMENT OF LUMINOUS INTENSITY [XVI

If the retinal illumination be computed for a particular light source and distance as found by the two methods, it is found that with the Maxwellian view E R is generally thousands of times greater than that with completely flashed pupil. The lower measuring limit by the first­mentioned is therefore very much lower than by the second. Nevertheless the method whereby the pupil is completely flashed is to be rreferred wherever possible, and the other method is only used when the first is impracticable owing to the luminance of the photometer field being too low. The method with Maxwellian view has several disadvantages. Since the dividing line between the photometer fields never disappears entirely, but remains visible as a light or dark strip, the accuracy of the balance is lower than when the pupil is wholly filled. Again, the image of the light source must be exactly adjusted in the centre of the eye-piece. Owing to the Stiles-Crawford effect (see Section XII-2) any displacement of the eye during measurement results in a variation in the retinal illumination and therefore also of the luminosity. The use of a chin-rest is therefore recommended. In order to minimize the effects of inevitable small displacements, care should be taken that the image of the lamp on the pupil of the eye is small, preferably not larger than about 1 mm in diameter. Maxwellian view is employed in photometers when light sources of relatively low luminous intensity have to be measured at large distances, and also for the measurement of very low intensity sources at normal distances, e.g. cycle rear lamps and reflectors.

REFERENCES I) 0. R e e b, Optik 9, 1952, 254-273. "Zur Frage der photometrischen Grenzent­

fernung" 2) T h. H. Pro j ector, Ill.,Eng. 48, 1953, 189-191. "The Use of Zonal Con­

stants in the Calculation of Beam Flux" The same author, Ill. Eng. 48, 1953, 192-196. "Versatile Goniometer for Pro­jection Photometry"

a) J. Bergman s and H. A. =· Keitz, Ph. T. R. 9, 1947, 114-122. "Deter­mining the Light Distribution and Luminous Flux of Projectors"

') J. M. W a 1 dram, Trans. Ill. Eng. London 16, 1951, 187-207. !'The Photo­metry of Projected Light"

") A. B 1 on de 1, Comptt . .:- Rendus 188, 1929, 1464-1467. "Sur une methode nouvelle pour !'etude en laboratoire des faisceaux des appareils optiques" P. C i b i e, Comptes Rendus 200, 1936, 2136-2138. "Methode de contr6le en laboratoire des projecteurs de lumiere des automobiles"

CHAPTER XVII

MEASUREMENT OF LUMINOUS FLUX, QUANTITY OF LIGHT, AND LUMINOUS EMITTANCE

XVII -I. The Ulbricht sphere photometer Towards the end of the last century it was gradually realised that the correct quantity characterising a light source is not the luminous in­tensity, but the total luminous flux. Until then no other method of determining luminous flux was known than that of computing it from the light distribution. This method is not straight forward, however, and efforts have quite naturally been made to devise an instrument by means of which luminous flux could be measured in a simple manner. Various suggestions were put forward, but none of these gave satis­factory results until in 1900 U 1 b r i c h t 1) solved the problem by introducing his sphe1'e photometer, or integrating photometer.

According to (III-4) q, = Jldw, and it can be said that this photometer determines the integral Jldw.

The photometer consists of a hollow sphere, the interior of which is painted matt white. The light source is suspended inside the sphere. Now, if a part of the interior be screened from the direct rays from the source, the illumination of that part will be proportional to the luminous flux of the source, provided certain conditions are fulfilled. These con­ditions will be discussed later. In this way it is possible to measure an unknown luminous flux, once the photometer has been calibrated with a standard source. The sub­stitution method is almost invariably employed, the source under examination being suspended at the same point as the standard source. It is only necessary to measure the illumination at the measuring point on a relative scale and, as a rule, we do not measure the illumination as such, but another quantity that is proportional to the illumination. This can be done either visually or physically. Figs. 194a and b depict two methods of effecting the measurement. In both fig11res K is the sphere, L the light source, M the measuring point, and S the screen that masks the direct rays from L in the direction of M. The difference between the two diagrams lies in the manner in

336 MEASUREMENT OF LUMINOUS FLUX [XVII

which the illumination of M, or more properly the quantity proportional thereto, is measured. In effect, in the first case (Fig. 194a) M is measured from the rear, and in the second (Fig. 194b) from the front. With the arrangement shown in Fig. 194a a hole is provided in the wall of the sphere at M, into which a light-receptor (photo-electric cell or opal glass) is inserted. The luminance or luminous intensity of the opal glass is measured, this being proportional to the illumination on the inside face of the glass. Alternately a photo-electric cell is placed im­mediately behind the opal glass.

11

---o ---------- ----- ,--- ------- ,- ....... -------

Fig. 194. Principle of the U 1 b rich t sphere. K sphere painted matt white; L light source to be measured; S screen; M measuring position. With the ar­rangement shown in a) a quantity that is proportional to the illumination at M is measured behind M (the sphere window); in b) a similar quantity (the luminance)

is measured from the front through aperture 0 in the sphere

Since an opal glass is often fitted at the measuring point (invariably, during the period when no other form of photometry than the visual was known), the measuring point is referred to as the sphere window, or, for short, the window. In Fig. 194b the luminance of M - which is proportional to the il­lumination - is measured through an aperture 0 in the wall of the sphere. This measurement is generally effected by the visual method. It is only rarely that this arrangement of the photometer is employed.

XVII-2. Illumination of the sphere window in the "ideal" integrating photometer

Let us now compute the illumination E of the sphere window *) as a *) We shall also refer to the sphere window as applied to Fig. 194b, as the following considerations apply to both arrangements.

XVII-2] IDEAL INTEGRATING PHOTOMETER 337

function of the luminous flux in the "ideal" sphere photometer. By this is meant a sphere photometer the interior of which is uniformly diffuse, it being assumed, moreover, that the screen, light source and reflective properties of the window do not disturb the distribution paths of the rays in the sphere, and that the reflection factor is in­dependent of the wavelength of the light, i.e. that reflection in the sphere is not selective. Let us denote the reflection factor of the sphere wall by (!, the luminous flux by (/> and the radius of the sphere by R *). A small area L15 of the sphere wall receives a luminous flux equal to L1$. Hence L1S reflects a luminous flux of eL1$. This area is a uniformly diffuse light source which illuminates the whole interior of the sphere. According to section IX-4 the sphere is a McAllister equilux sphere, that is to say the illumination of the sphere as produced by L1S is the same at all points. The luminous flux incident on the sphere wall being eL1$ and the area of the sphere 4nR2, the illumination L1E 1 of the sphere wall, and hence also that of the sphere window, produced after reflection by L1S, will be:

Every part L1$ of the luminous flux falling on the sphere wall con­tributes an element L1E1 towards the total illumination E 1 of the sphere as a result of the initial reflection so that

The total reflected luminous flux e$ is again reflected by the sphere wall, and the illumination E 2 of the window arising from this second reflection is found in the same way as Ev viz:

(!2(/J

E2 = 4nR2.

Proceeding thus for the 3rd, 4th, 5th ... nth reflection, we find for the total illumination E of the window, resulting from the 1st ... nth reflections together:

*) The following is taken largely from an unpublished article by the Lighting Engineering Laboratory of N.V. Kema, Arnhem (1944), to whom the author's acknowledgements are due.

338 MEASUREMENT OF LUMINOUS FLUX

E = Et + £2 + Ea + ... +En= er/J e2r/J earp enrp

= 4nR2 + 4nR2 + 4nR2 + · · · + 4nR2 =

(/)( ) (/) e = 4nR2 e + e2 + ea + ... + en = 4nR2 X t - e.

Let k be the quotient of E divided by r/J; then:

E = kr/J.

[XVII

(XVII-I)

(XVII-I a)

If there were no screen in the sphere, light direct from the source would fall on the window; the illumination on the window would then depend on the luminous intensity of the light source in the direction of the window and thus on the light distribution of the source and the position of the source with respect to the window. It is therefore dependent on variable and casual factors, and is not proportional to the total luminous flux, as is the case with the illumination produced by the reflected light. The screen is therefore provided betw~en the source and the sphere window in order to eliminate the effect of this illumination which is not proportional to the luminous flux. From the above derivation of equation (XVII-I) it appears that in the case of the ideal, uniformly diffuse reflecting sphere the illumination of the window, produced after the luminous flux has been reflected at least once (the indirect illumination), is independent of the distribution of the flux over the sphere wall, i.e. is independent of the light distribution and position of the light source.

XVII-3. Illumination of the window of a non-ideal integrating photo-meter

For a hollow body of arbitrary form, of which the reflection factor is the same at all points, we have already derived a formula that will give the mean illumination at the wall (section X-14, equation X-9). This is:

(/) 1 Em= S. 1- e'

where S is the area of the hollow body. In deriving this equation we proceeded from the assumption that the light from the source reaching the wall is not screened. In integrating photometers, however, the direct rays in the direction of the window are screened off, so that we have now to consider the illumination produced by light that is reflected

XVII-3] NON-IDEAL INTEGRATING PHOTOMETER 339

at least once. For the mean indirect illumination Ei , then, equation m (X-9) becomes

E. =~.-e-. 'm S 1- (} (XVII-2)

If the illumination of the window be related to Ei by a factor p, we m

may write:

Now, if we put: p_ _e_ =k s ·1-e '

we find - as for the ideal sphere - that

E = kf/>.

(XVII-3)

In the derivation of equation (XVII-3) we have made no assumptions regarding the shape of the hollow body. What we have assumed is that e is independent of the place at the sphere wall, and is the same for all reflections, i.e. that e is independent of the angle of incidence, the factor p being independent of the place on the sphere wall and of the light distribution of the light source~ In practice, however, there are certain departures from these ideal conditions, viz.: 1. The reflection from the sphere wall is never uniformly diffuse. 2. The reflection factor of the sphere wall is not exactly the same at

all points. 3. The screen masks not only the light source, but also a part of the

sphere wall. 4. A part of the reflected light falls on the light source, the suspension

device and the screen, and is partly absorbed by these. 5. The window, or the receptor, is not subject to the cosine law in

transmitting or absorbing light. Glancing rays are not evaluated by the photometer to the same extent as rays in directions more closely approaching the normal.

6. The reflection factor may differ slightly at different wavelenghts, i.e. the sphere wall may reflect selectively.

In consequence of these deviations from the ideal conditions k = E (f/> is in practice dependent on the position, the light distribution, size

340 MEASUREMENT OF LUMINOUS FLUX [XVII

and absorption factor of the light source, as well as on the spectral distribution of the light.

XVII-4. Measures to be taken with non-ideal integrating photometers to approximate to the ideal sphere

Let us now see what limitations are imposed by the above divergences in the use of the integrating photometer, and what steps can be taken - or in some cases must be taken - to correct the effects of these de­viations. The following cases may occur in practice: 1 (a) equal relative light distribution of the light sources to be compared,

(b) light distributions not equal, 2 (a) absorption of the light travelling to and fro within the sphere

the same for both light sources, (b) absorption not the same,

3 (a) spectral distributions of the light sources identical, (b) spectral distributions not the same.

Combinations of Ia orb, 2a orb and 3a orb are, of course, always met with in practice.

Ia and b. Light distribution of sources to be compared If a narrow beam of light were to be thrown successively towards different points in the ideal integrating photometer with receptor answering the cosine law, exactly the same reading would be obtained for each direction of the beam. The photometer reading would thus be inde­pendent of the distribution of the luminous flux in space (the light distribution) and it would be possible to compare light sources of widely differing distribution patterns without the slightest error. If this test be carried out in a practical form of integrating photometer, different readings are obtained for different directions of the beam. Fig. 195a shows the result of such a test in an Ulbricht sphere; the divergence from the average value of k as a percentage is here plotted as a function of the point where the beam, coming from the centre of the sphere, strikes the wall. The average value of k is the average value as computed over all the solid angles with apex at the centre of the sphere. The curve exhibits two dips, one at A as produced by the presence of the screen, the light falling on the window being thus reflected at least twice (screen error), and another at B, this being due to the fact that the luminous flux reflected by that part of the wall which lies

XVII-4] APPROXIMATION TO THE IDEAL SPHERE 341

opposite the window is partly masked by the screen (screen shadow error). The presence of the screen is not the only source of divergence from the ideal conditions, however; departures from uniformly diffuse reflection also play a part. These differences between the ideal and practical forms of sphere result in an error in the comparison of light sources of which the light distri­bution differs.

s p

® --

Fig. 195. Evaluation by the photometer P of the luminous flux as a function of the part of the sphere on which it falls. The curves show as a per­centage the divergence from the average in integrators a) of pure spherical form, b) semi­regular 14-sided body, c) of the form shown. S indicates the 'location of the screen

342 MEASUREMENT OF LUMINOUS FLUX [XVII

The measures that can be adopted to minimize this error as much as possible will be discussed later. Seeing that a true sphere, with its doubly curved surface, is more difficult and more expensive to make than a body with plane boundary faces, many integrating photometers are made in the form of more or less regular polyhedrons; a much favoured pattern is a semi-regular tetradecahedron or fourteen-sided body which is, in effect, a cube with flattened corners. When curves are plotted for an integrating photometer of this type along the same lines as those in Fig. 195a they will appear as shown in Fig. 195b (two curves are given, because measurements in two planes of symmetry have been taken. Comparison of Fig. 195awith 195b shows that, in the sphere, k is practi­cally constant over a large range, whereas in the polyhedron k is not constant in any ranges; the error involved when the light distributions are not the same is thus greater than in the case of the sphere. It is possible, however, to construct a polyhedron that is very much more satisfactory in this respect than the conventional patterns, and the results obtained from a model are illustrated in Fig. 195c, together with the form of the polyhedron itself. Compared with figures 195a and b, the curve in Fig. 195c is seen to correspond much more closely to those of the sphere. From the above it will be clear that the true ~;phere is preferable to polyhedrons when the light distribution of the sources shows a marked difference. It will be seen from Figs. 195a, b and c that the evaluation of the lu­minous flux radiated in various directions by a source will vary from one type of photometer to the other. The illumination on the window, or in the case of the non-ideal receptor the reading obtained from the photometer, is a function of the location on the wall of the sphere of the luminous flux to be measured. If we write:

E = k<P,

it follows that k = f (oc, {3), where oc and {3 indicate the position on the wall of the sphere in angular co-ordinates. If the light is distributed over a number of points:

E = L:k. LI<P (XVII-4)

if LI<P denotes the luminous fluxes incident at the various points of the sphere wall.

XVII-4] APPROXIMATION TO THE IDEAL SPHERE 343

In case Ia (light distributions equal) in every term k . Lf«P the value of Lf«P varies in the same proportion (say: a) when the standard light source is replaced by the light source to be measured, so that we obtain for the illumination or the reading of the photometer for a light source X:

E.,= 1: k. a. Lf«P =a. E.

In this case the dependence of k on the position in the sphere is imma­terial which means that, with equal relative distribution of the sources to be compared, the geometry of the photometer and uniformity of the reflection factor are no longer sources of error. From (XVII-4) it follows, moreover, as already discussed above, that if the relative light distribution of the sources is not the same, the variation in k may have a pronounced effect on the results obtained. In order to reduce this effect as much as possible the following measures can be taken: a .. The reflection factor of the paint applied to the sphere must be made

as high and as uniform as possible. This implies that the inevitable soiling of the bottom of the sphere will necessitate frequent re­painting.

b. The sphere should be large and the screen as small as possible. c. The light source must be so arranged in the sphere that the greatest

Fig. 196. The light sources placed in the integrator should be so arranged that the greatest possible part of the luminous flux falls in the areas of constant k, which im­plies that linear sour­ces should lie per­pendicular to both

screen and window

possible part of the luminous flux falls on the areas of the sphere for which the value of k varies the least from the average value in equation (XVII-4), i.e. on those parts which lie outside the zones of screen and screen shadow error. Linear light sources such as tubular fluorescent lamps and fittings should accordingly be sus­pended in the sphere with the axis at right angles to the screen (see Fig. 196). Beams of light should always be made to fall on areas outside the zones of screen and screen shadow error. d. The screen should be diffusely translucent,

with a transmission factor of about 3%. e. A receptor, or window, should be used which,

as far as is possible, also follows the cosine law at glancing incidence.

To the above may be added the following ex­planatory remarks.

a. The fact that e should be as high as possible

344 MEASUREMENT OF LUMINOUS FLUX [XVII

may be explained as follows. The value of k as a function of the position on the sphere wall depends in the first place on the contribution, after initial reflection, as made by each point towards the illumination of the window. As photometer paints are reasonably matt, the luminous flux is nearly enough uniformly distributed over the sphere wall after the initial reflection, and the contributions of the 2nd and 3rd, and subsequent reflections, towards E are thus dependent almost exclusively on the geometry of the integrator. The contribution to E arising from the first reflection from a certain point in the sphere may be said to be proportional to k1f/J. The sum of the contributions due to subsequent reflection is propor­tional to (e2 + e3 + e4 + ... e") = e2/(I -e). When e is increased the term e2 I ( 1 - e) increases in value more steeply than the term k1e, so that, when e is high, the effect of k1 and its variations is reduced. In practice, a lack of uniformity in the reflection factor will be mainly the result of the heavier soiling at the bottom than at other parts of the integrator, in consequence of which k is lower at the bottom than elsewhere. The effect of this on the measurement is the more marked according as a larger part of the flux is allowed to fall on the bottom part. Particularly in the measurement of light beams radiating vertically downwards should care be taken to maintain a sufficiently high degree of uniformity; this can be done by accurately ascertaining (e.g. from the light distribution) the luminous flux of a concentrated type of source with constant luminous flux (e.g. a lamp with silvered bowl), and sub­sequently taking measurements from this source in the integrator by comparing it with a standard lamp, which radiates in all directions. If the value of the luminous flux thus obtained is found to be below a certain limit (say 97% of the original value), it will be time to repaint the integrator. In order, in the case of narrow beams, to be less dependent on variations in e, the directly illuminated part of the integrator can be enlarged by mounting the projector in the wall of the sphere instead of at the centre, the position and size of the screen being adapted to suit. b. Zones in which the screen and screen-shadow errors occur should be made as small as possible. To this end the screen itself should be made as small as possible compared with the sphere. At the same time, this is determined mainly by the size of the light source, which means that the sphere should be as large, and the screen as small, as the di­mensions of the light source will permit.

XVII-4] APPROXIMATION TO THE IDEAL SPHERE 345

c. Requires no further explanation. d. The screen error can be almost fully corrected by employing a diffuse translucent screen. The necessary transmission factor can be calculated; it is found that for all practical purposes -c = 3% is satisfactory. It is not possible by such means to eliminate error due to the screen shadow. e. Light entering the receptor or sphere window at glancing incidence comes from the zone of the screen error, so that, if the cosine law is not sufficiently satisfied by the receptor, this results in an increase in the screen error.

2a and b. Absorption by the light sources under comparison If the light travelling to and fro in the sphere is absorbed to the same degree by each of the sources under comparison, the illuminations of the window and thus the reading taken from the photometer are affected to the same extent, hence no correction is necessary. However, if the light-absorbing properties are dissimilar, an endeavour must be made to ascertain the effect on the final result. The absorption which affects the accuracy of the result is that of the light which is reflected at least once. According to H e 1 w i g 2) the influence of absorption of the indirect light by the light source under examination can be measured by placing

Fig. 197. Integrator with auxiliary lamp H pro­posed by H e 1 w i g to eliminate error arising in the comparison of light sources whose absorption

differs

in the sphere a constant indirect light source, which emits a constant indirect luminous flux into the sphere. If the il­lumination on the window produced by this indirect flux be measured successively with the two source to be compared extinguished in turn, and the difference is found to be say, no/o, this means that the one source absorbs n% more of the indirect light than the other. This correction of n% is applied to the readings obtained from the sources compared. A constant indirect luminous flux can be obtained by placing in the sphere an auxi­liary lamp screened in such a way that only indirect light, i.e. reflected at least once,

falls on the sources under comparison (see Fig. 197). The whole procedure of the luminous flux measurement is then carried out as follows:

346 MEASUREMENT OF LUMINOUS FLUX [XVII

Standard source on, auxiliary lamp out: reading a5

out, Source to be measured on,

" out,

on: h5

out: ax

on: hx

Disregarding the indirect light absorption we find that the unknown luminous flux is

ax cpx =-X cps•

as where cps is the luminous flux of the standard. This is corrected for absorption of the indirect light by multiplying it by the quotient of the readings taken from the auxiliary lamp (hsfhx)· Thus the required luminous flux is

ax hs <l>x = - X - X cps·

as hx

Particularly for the measurement of lighting fittings, the absorption of which can be anything up to 40% more than that of the naked lamp, He 1 wig's method employing an auxiliary lamp is a very useful means of ensuring accurate results. An objection to the method, which does not, however, weigh heavily against its advantages, is that the result is obtained from four readings instead of oniy two, so that the probable error is v2 times greater.

3a and b. Spectral distribution of the light sottrces under comparison When the spectral distributions of the light sources to be compared are the same the spectral reflection from the sphere wall has no effect on the results of the measurement. If the composition is not the same, however, even relatively slightly selective reflection may have a pronounced effect on the results. The illumination of the window is proportional to e/(1 -e) and, if the values of (! for two wavelengths are equal to 80 and 81%, respectively, this difference in (! will give a relative difference of about 6% in the illumination of the window. In consequence, thus, the photometer paint should be as little selective as possible and, of equal importance, should remain so in use. Paints which tend to turn yellow are therefore to be avoided at all costs, for which reason oil-bound paints are never used; size-bound paints are employed instead, these being applied as aqueous suspensions. Various recipes for such paints are to be found in the literature, but we

XVII-4] APPROXIMATION TO THE IDEAL SPHERE 347

append two, one of which has been adopted by the National Physical Laboratory, corresponding to the specification in British Standard 354: 1961 3), and another, taken from the German D .. I.N. 5032 4).

~pecification to B.S. 354: 1961 1. Pre-treatment and primer on the wall will not generally affect the

optical properties of the whole of the coating system, but should ensure proper keying to the basic material and provide protection against cor~osion and peeling.

2. The recommended top-coating consists of two layers: 2.1 a white primer and 2.2 a water-soluble, matt layer that can be washed off when soiled and

replaced by a fresh coating.

2.1. It is desirable for the white primer to have a matt surface and to be proof against yellowing in time. The colouring material should therefore contain .a sufficient proportion of a strong covering pigment, e.g. titanium oxide, dispersed in a non-yellowing medium. Two-component-coatings of the polyvinyl acetate emulsion type are to be recommended.

2.2. The washable, matt white topcoat can be made from fine.precipita­ted barium sulphate ("blanc-fixe") dispersed in a water-soluble, yellow­ing-resistant medium, e.g. sodium carboxymethyl cellulose that is as free from fibres as possible. There are other colloids with similar properties. A recommended mixture consists of 1000 parts by weight of "blanc-fixe" (barium sulphate); 25 parts by weight of sodium carboxymethyl cellulose (low viscosity); 1000 parts by weight ofwater. The mixture can be further improved to give the most non-selective reflection possible. The slight decrease in the spectral range of the white pigment (barium sulphate) in the blue region can be compensated by the addition to the mixture of an aqueous paste of a carbon pigment that emphasises the blue, e.g. ivory black. For further details on the preparation of the mixture, the British Standard should be consulted.

2. Recipe to DIN 5032, July 1957 For the undercoating of the integrator it is recommended: On the primer of Nitro-binder and zinc-oxyde a thick magnesium-

348 MEASUREMENT OF LUMINOUS FLUX [XVII

oxyde solution with 0.5 w% gelatine (for 200 g MgO 1 g gelatine) diluted with water to a sprayable, or paintable thickness must be applied at least two times. Attention should be paid that no drip occurs. Such coating is almost aselective and its reflection factor is independent of the temperature of the ball. The renewal of the integrator coating should take place regularly, at least once a year.

The results of measurement depend not only on the spectral reflectivity of the sphere wall, but also on the spectral sensitivity of the operator in visual photometry, or of the photo-electric cell in physical photo­metry. In the latter instance matching of the spectral sensitivity of the cell with V.\ gives us the means of correcting any selective reflection in the integrator. The influence of the spectral reflectivity of the sphere wall and the spectral transmission of the window can be determined spectrometrically by suspending a lamp in the integrator, then ascertaining the spectral composition of the light transmitted by the window, and comparing this with the spectral composition of the source itself 5).

An incandescent lamp can be employed for the long-wave region of the spectrum. In order to ensure sufficient light in the blue, it is useful to carry out the test also with an incandescent lamp with blue bulb, or with a "daylight" (U.S.A.) or "colour matching" (G.B.) fluorescent lamp. A curve is then drawn giving the calibration of the integrator as a function of the wavelength. This curve can be combined with the spectral sensitivity curve of the photo-electric cell (by multiplying the values of the two curves at each wavelength) to produce a curve representing the sensitivity of the integrator and cell at every wavelength. From this it is possible to derive the spectral transmission curve to which a filter placed before the cell would have to conform in order to ensure readings from the cell that would be wholly in accordance with V.\. Such an ideal filter can be approximated as closely as possible by a filter or combination of filters as obtainable on the market. The system of using correction factors can be employed for integrators, just as in the case of photo-electric cells; as applied to integrators the correction factors are determined for the combined integrator and photo­electric cell and, even when filters are used in the manner described above to approximate to the ideal measurement (to match V.\), it may be necessary to determine the correction factor. The difficulties arising from selective reflection from the wall of the integrator when luminous flux measurements of coloured sources are

XVII-4] APPROXIMATION TO THE IDEAL SPHERE 349

made by comparison with standard incande~cent lamps can be sur­mounted by making a homochromatic instead of a heterochromatic comparison; to this end the photometer is calibrated with standard light sources of the same kind as the source under test. This restricts the difficulty to one of measuring the standards. In the routine photometry of, say, gas discharge lamps, the difficulty can be entrusted to a standard laboratory. A method of routine photometry for gas discharge lamps by which calibration is effected with lamps of the same kind as that under test has been given by 0 ran j e 6), who describes integrators of cubic form with barrier-layer cells roughly matched with VA as light receptors. For a rapid check on the integrator, and also in order to save the standard gas discharge lamps, the standard lamp in the integrator is replaced by an incandescent lamp suspended at a fixed point. The reading thus obtained ftom a meter indicates a certain number of lumens, which represent an "apparent" lumen value of the filament lamp, since the light distribution of the latter may differ considerably from that of the discharge lamp, and also because the whole integrator does not give a measurement that is strictly in accordance with the V ,~ curve. The incandescent lamp is then used as sui--standard for the measurement of other gas discharge lamps of the same type and need only be com­pared with the standard lamp periodically. Although the luminous flux of gas discharge lamps is not as constant as that of standard incandescent lamps, this method nevertheless gives more reliable results than direct comparison with standard incandescent lamps, a fact that is of considerable importance in routine photometry. We have now dealt with the influence of the light distribution, absorption by the light source, and the spectral distribution. It remains for us to consider the effect of the location of the source in the integrating photo­meter. The curve in Fig. 195a (k as a function of the point where the light strikes the sphere wall) was plotted for perpendicular, or almost per­pendicular, incidence, that is, with the source at the centre of the sphere. The more obliquely the light strikes, the more pronounced the change in the form of the curve, seeing that the sphere is not a uniformly diffuse reflector. When the light distribution is the same for each light source, the result of the measurement is independent of the variation in k, and it makes no difference at what point the sources are suspended, provided that this is the same point in each case. When the light distribution of the sources differs, the two sources are

350 MEASUREMENT OF LUMINOUS FLUX [XVII

suspended in such a way that the curve k (ex) shows the largest possible area in which k is constant, or nearly so; that is to say in the centre of the sphere (projectors as we have already seen, can be mounted outside the sphere so that the beam falls on the opposite wall). All this holds good as long as the sources to be compared are the same size or can be regarded as such. But, if a comparison is to be made between a large and a small source and it is not possible to regard the large one as small compared with the dimensions of the integrator, it must be remembered that the various parts of the larger source occupy different parts of the sphere. Such is the case with tubular fluorescent lamps which are made in lengths of several feet. In order to avoid having to make integrators which would be very large in all directions, th"s kind of lamp can be measured in cylindrical integrators. A number of photo-electric cells are mounted in the wall of the cylinder, these being connected in parallel so that the readings will be independent of the part of the lamp whence the luminous flux is received. The number and positions of the cells are determined experimentally by moving a small tubular lamp to and fro within the integrator. Calibration can be effected with a standard tubular lamp having the same light distri­bution as an element of the fluorescent lamp. If it is possible to calibrate with a lamp of the same type as that under test, any integrator can of course be used. The luminous flux of an image-projection objective can be measured in an integrator in the same way as that of a beam projector, through an opening in the integrator. In the case of such narrow beams, however, it is advisable to calibrate the integrator with a similar beam of known luminous flux. The luminous flux from objectives can also be determined from the screen illumination measured at uniformly distributed points in the image. This will give the average screen illumination, and the total luminous flux is found by multiplying this average by the area of the image. The luminous flux of the beam used for calibrating the integrator can be ascertained in the same way. To conclude this section, we should also like to mention an automatic photometer for measuring the luminous flux of incandescent lamps, as described by Van Gorcum and van der WaaP). With this photometer, all that need be done once the lamp is inserted is the pressing of a button. The photometric and electrical data on the lamp are then automatically recorded.

XVII-5] LUMINOUS FLUX OF FLtrORESCENT LAMPS 351

It has been considered useful, in view of the special properties of fluores­cent lamps, to add some remarks on their measurement and that of fittings using them. These considerations apply not only to measurements of the luminous flux but to all photometric measurements made on gas discharge lamps. First of all, a few remarks pertaining to all gas discharge lamps. The power consumed by the gas discharge lamp is determined by the voltage applied and the impedance connected to the lamp in series (the ballast). A choke or a choke in series with a capacitor is generally used as a ballast. In order to obtain comparable and reproducible values for the luminous flux of the lamp, we can use two methods: 1. We can supply the lamp with its rated power with the aid of some choke by adjusting the voltage across the lamp and the choke. 2. We can connect the lamp in series with a choke with standardised electrical characteristics The lamp and choke must then be connected to a fixed voltage. The International Electrotechnical Commission (I.E.C.) has chosen the second method and has standardised, in its Publication 81, chokes for a series of fluorescent lamps, these chokes being referred to as reference ballasts. This series is being supplemented by specifications for reference ballasts for other gas discharge lamps. The actual luminous flux of the lamp(s) does not affect the measurement of the efficiency of a fitting. Nevertheless, the lamp must provide the same luminous flux when both the naked lamp and the fitting are measured. This condition can be met by the use of the same, arbitrary choke in both cases. The most important remarks on the measurement of mercury vapour and sodium vapour lamps have now been made. We have not yet finished with tubular fluorescent lamps, however. In contradistinction to mercury and sodium vapour lamps, the luminous flux and the electrical properties of tubular fluorescent lamps depend greatly on the temperature of the surroundings in which the lamp is burning. To put it more precisely, the luminous flux is determined by the temperature of the coldest point on the lamp (which is generally the centre of the underside of the tube). This means that not only the ambient temperature, but also the movement of the air in the measuring room determine the coldest point. An addi-

352 MEASUREMENT OF LUMINOUS FLUX [XVII

100 ;t%) t 90

80

70

60

50

40 ~~~~~~~~~~~~~~ 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 ---rube wall temperature(•c)

-8 -¥ 3,5 ~ 14; 2p ~ 3,2 38 4fo Sf' ~ 62 6( 7,3 -25-Ambient temperature ("C)

a

100 ; (o/o) t 90

80

70

60

so

40 +---;!n;';::;;;~-=:n:;t-;c:;-+;:~;-;::\;~";;!::-;;t:-:"1=-:;f=-10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 ---Tube wall temperature(°C)

-13-'7,5 -i 4 10 1,6 21 I~ 34J!l,S~55}.557,563,5fi9.5 -25-.Ambient temperature(oc) b

Fig. 198 Dependence of the luminous flux of Philips fluorescent lamps on the ambient temperature. a) TL40W, b) TL65W.

tional condition for measurement, therefore, is a draught-proof measuring chamber. The I.E.C. has specified an ambient temperature of 25 °C. The conditions for measurement in the case of fluorescent lamps are there­fore clearly specified: reference ballast, voltage, ambient temperature (25 aq and a draught-proof measuring chamber. Figs. 198a and b are the typical tube wall temperature/luminous flux curves for Philips TL fluorescent lamps of 40 and 65 Win colour 33 (white). In the case of the 40-watt lamp (Fig. 198a) the optimum for the luminous flux lies at 39 ac; this is the temperature attained by the lamp in a draught-proof room at an ambient temperature of 25 °C. The luminous flux decreases at both higher and lower temperatures. With the 65-watt lamp (Fig. 198b) the optimum is also at 39 °C. This temperature is already attained with a room temperature of some 20 ac because, compared to the forty-watt lamp, the output rises more rapidly than the temperature of the surface of the fluorescent lamp (for the same tube diameter, the output is about 60% greater, while the tube is only some 25% longer). At the prescribed ambient temperature of 25 ac, the luminous flux of the 65-watt lamp is about 1% lower than at the optimum temperature. The measurement of high-power fluorescent lamps, e.g. the Philips TL120W, is particularly fraught with difficulties. Because the power in this case is almost double that of a 65-watt lamp, while the diameter is slightly smaller and the length is the same, the ambient temperature for

XVII-5] LUMINOUS FLUX OF FI,.UORESCENT LAMPS 353

the optimum luminous flux is about -5 oc unless special measures are tp.ken. However, artificial aids can raise the ambient temperature at which the best possible luminous flux is obtained. In the case of the Philips TL120W lamp, a bulge is fashioned at the centre of the underside of the tube. This locally increases the cooling surface and lowers the temperature at the coldest point on the tube. The difficulties in measure­ment are caused by the fact that these lamps require a very long stabilisa­tion time. If such a lamp is measured after having been moved, it can take up to eight to twenty-four hours before the luminous flux and elec­trical data are constant. If, however, the lamp is then left in position, and this is usual in a lighting installation, although hardly possible for the purposes of photometry, two to three hours will generally be sufficient for re-stabilisation. *) In the measurement of such a lamp in a fitting where adjustment has to be made to an entirely different point on the characteristic, stabilisation takes a great deal of time. If measurements are to be made in the morning, the lamp should be switched on the previous evening. The curve of the luminous flux as a function of temperature means that, in fittings and particularly in closed fittings in which the temperature is higher than room temperature, the luminous flux of fluorescent lamps is reduced. The efficiency of such a fitting is determined not only by the absorption of the reflective and transmissive parts of the fitting, but also by the temperature increase brought about by the enclosing of the lamp(s). We now have to find out how to define the efficiency of a fluorescent lamp fitting, bearing the above remarks in mind. Because a lighting installation must be calculated from the efficiency and the luminous flux data provided by the manufacturer for the lamps, the obvious thing to do is to take for the luminous flux of the naked lamps that found from a measurement of the luminous flux made in the draught-proof room at 25 °C. If the efficiency is to be properly deter­mined, the naked lamp must be operated with the same ballast as that in the fitting. *) The difference in the stabilisation time between, for instance, the 40 and 65-watt lamps and the high-power lamps may be explained in the following way: While the lamp is not burning, there are droplets of mercury in the tube, which can move about in the tube during transport. When the lamp is ignited, the mercury droplets begin to evapo­rate. The vapour must diffuse inside the tube. As soon as it has reached the coldest part of the tube and the vapour pressure at this point is established, the luminous flux i_s stabilised. We have spoken of the coldest point in 40 and 65-watt lamps, but, in fact, the temperature along the length of the tube is almost the same and the diffusing mercury vapour reaches a point oflowest temperature almost immediately. In the 120W lamp the lowest temperature prevails only in the bulge in the tube. It can therefore take a very long time before the diffusing mercury vapour reaches this point, and the stabilisation time can be very long.

354 MEASUREMENT OF LUMINOUS FLUX [XVII

The essential condiuon here is that the power provided by the ballast for the lamp shall not be too far from the rated value. If, for instance, the ballast supplies a 40 W lamp with only 30 W, calculation of a lighting installation on the measured efficiency with the rated luminous flux of the lamp as 100% would provide only 75% of the calculated illumina­tions in the installation. The condition that the power of the lamp be not too far from the rated value may be considered satisfied if the ballast meets the specifications of I.E.C. Publication 82 9).

Of course, 25 °C should be taken as the ambient temperature during measurement. The definition of the efficiency of a fluorescent lamp, which we refer to as the operating efficiency, can, then, be given in the following form: the quotient of the luminous flux of the fitting and that of the lamp mea­sured in a draught-proof room at 25 oc, with each lamp operated naked and with its appropriate ballast. When using the operating efficiency thus defined, the lighting engineer must remember that the efficiency will change if the temperature is very different from 25 °C. By and large, the efficiency initially increases as the room temperature drops, but subsequently decreases. This will be of no great significance for indoor lighting, but very important to lignting installations in the open. In some cases the effect of low temperatures must be included in calculations: The cooling effect of wind must also be taken into account. There are two ways of determining the losses occurring in the fitting for purely optical reasons (by absorption) when stating the efficiency, viz: 1. A thermoelement can be attached to the centre of the underside of the lamp(s) before the luminous flux of the fitting is measured, and, after the final temperature lias been attained, the temperature of the lamps can be measured. The percentage decrease in the luminous flux of the lamp due to the temperature increase in the fitting can then be read off from a graph like the one in Fig. 198. The "optical" efficiency can easily be calculated from this value and the operating efficiency. 2. If the lamp in the fitting is switched on after attaining the ambient temperature of 25 oc, the temperature of the tube wall will attain the value that it assumes on burning naked at 25 °C within two to three minutes. The enclosure by the fitting then has no effect. If the l11minous flux is now measured, the value with the lamps at 25 oc ambient tempera­ture, and thus the "optical" efficiency is measured. The moment at which the value must be read off is fairly sharply

XVII-6] LUMINOUS FLUX FROM LIGHT DISTRIBUTION 355

defined. After being switched on, the lamp heats up rapidly to the temperature that it attains at an ambient temperature of 25 °C. This is accompanied by an increase in the luminous .flux. This process finishes after two to three minutes. Because the rise in temperature due to the heating of the fitting is a much slower process, the luminous flux of the lamp after stabilisation at 25 oc remains constant for sufficiently long for measurement. This method cannot be used for lamps with a long stabilisation time, like the overloaded "TL" 120 W.

XVII-6. Determination of luminous flux from the light distribution or from an isolux diagram

The luminous flux of axially symmetrical light sources can be determined by means of the Rousseau diagram in the manner described in Section IV-1. When the light source is asymmetric the luminous flux can be ascertained from the average distribution with the aid of a Rousseau diagram or from an isocandela diagram, provided this is drawn on a spherical projection proportional to the area (Section V-6). The method of deriving the luminous flux from the isolux diagram has already been given in Section VII-9. Determination of the luminous flux from the light distribution has its uses in some cases, because, owing to selective reflection from the interior of the integrator, direct measurement of the luminous flux of coloured light is more difficult than a luminous intensity measurement. To determine the luminous flux of standard coloured light sources, therefore, the light distribution is sometimes measured. If the luminous flux only is required, irrespective of the light aistribution, it is not necessary to take into account the rules for the measuring distance. The illumination E on a spherical surface surrounding the light source can be measured at any desired or suitable distance, and multiplication of E by the area of the sphere then gives the luminous flux. For this purpose, use may be made of a Rousseau or an isolux diagram plotted on a spherical projection proportional to the area. This amounts in fact to what is done when the luminous flux is ascertained from the "actual" light distribution, since the luminous intensity is always computed from the illumination.

XVII-7. Measurement of quantity of light The quantity of light emitted by long-life light sources is determined graphically from the kind of chart shown in Fig. 12, the points as plotted

356 MEASUREMENT OF LUMINOUS FLUX [XVII

being obtained by measuring the light source at different times during the burning period. With short-lived sources such as flash bulbs the quantity of light produced

c

can be measured in the manner described by v a n L i e m p t and d e V r i en d 10) (see Fig. 199). The flash bulb is suspended in the integrator K with matt

66572 glass window M in front of which

Fig. 199. Arrangement used by van L i e m p t and d e V r i e n d for the measurement of the quantity of light produced by flash bulbs. The capacitor C, charged from an external source, is partially discharged by the photo­current from a cell P, induced by the flash bulb in the sphere. The voltage drop across the capacitor is proportional to the quantity of light from the flash

bulb

a vacuum photo-emissive cell P is placed. Included in the photo­current circuit is a capacitor C which is charged up -- with cell in darkness -- to some 300 V D.C. as read from an electrostatic voltmeter V. When the flash bulb is fired a photo­current flows in the circuit and the capacitor voltage functions

as anode voltage for the cell. The photo-current discharges the capacitor. Provided that, when the flash has taken place the voltage on the capacitor is still higher than the saturation voltage of the cell, the voltage drop across the capacitor will be proportional to the quantity of light delivered by the flash bulb. A final voltage in excess of the saturation voltage of the photo-electric cell can be ensured by a suitable choice of capacity.

p

l ~--------------------------~+

Fig. 200. Circuit for measuring the quantity oflight in lighting phenomena that change abruptly.

XVII-8] MEASUREMENT OF LUMINOUS EMITTANCE 357

Another circuit permitting the determination of the luminous flux cycle of the flash bulb while the content is burning is shown in Fig. 200. While there is no light falling on photocell P, capacitor C is charged to a certain voltage, e.g. 200 V. When the light from the ignited flash-bulb falls on the photocell, the photocurrent partially discharges the capacitor. Once burning is over, the residual voltage can be read off. The voltage drop across the capacitor is proportional to the quantity oflight that has fallen on the photocell. If a resistor R with an oscillograph in parallel with it is inserted in the circuit, the luminous flux cycle can be displayed on th.e oscillograph and recorded by photog•aphy.

XVII-8. Measurement of luminous emittance As we have seen in Section VIII-5, luminous emittance, in the case of a surface that does not radiate light of itself, is identical with illu­mination, and its measurement is accordingly a measurement of illu­mination. Methods of carrying out the measurement are described in the following chapter. In principle it should be possible to measure the luminous emittance of radiating surfaces by placing an illumination photometer on the surface to be measured, if it were not for the fact that reflection from the comparison surface of the photometer increases the emittance (repeated reflection). As a rule the luminous flux must be determined by means of an integrator or from the light distribution, the emittance being then arrived at by dividing the value of the luminous flux thus obtained by the area of the radiating surface. With near-uniformly diffuse radiating surfaces it may be sufficient to measure the luminous intensity perpendicular to the surface, and to calculate from this the luminance and multiply the result by n.

REFERENCES

1) R U 1 b rich t, E.T.Z. 21, 1900, 595-597. "Die Bestimmung der mittleren raumlichen Lichtintensitat durch nur eine Messung" R. U 1 b r i c h t : "D~s Kugelphotometer". Munich and Berlin 1920

2) H.-J. H e 1 wig: "Uber lichttechnische Messungen mit Hilfe von Hohl­raumen". Thesis Berlin 1933. See also: das Licht 4, 1934, 115-118, 135-140 and 156-159. "Uber li<;:.httechnische Messungen mit Hilfe von Hohlraumen" and 5, 1935, 33-34. "Uber praktische Erfahrungen mit der neuen Messmethode fur die Ulbrichtsche Kugel"

3) British Standard 354: 1961. Recommendations for photometric integrators

358 MEASUREMENT OF LUMINOUS FLUX [XVII

4) DIN 5032. "Photometrische Bewertung und Messung von Lampen und Be­leuchtung". Juli 1957 See also: W. E. K. Mid dIet on and C. L. Sanders, Ill. Eng. 48, 1953, 254-256. "An Improved Sphere Paint"

5) M. H. Gab r i e 1, C. F. Koenig and E. S. Steeb, Gen. El. Rev. 54, 1951, no. 9, 30-37 and no. 10, 23-29. "Photometry". This article gives a survey of the use of the integrator for measuring luminous flux of lamps.

6 ) P. J. 0 ran j e, Ph. T. R. 5, 1940, 166-170. "Technical Photometry of Gas­Discharge Lamps"

7) A. H. van Gorcum and B. v.d. Waal, Electrotechniek, 41, 1963, (335-342), "Automatische gloeilampenfotometer met digitale registratie'' (in Dutch) (Eng.: Automatic incandescent lamps photometer with digital recording) The same authors: "Automatic photometer with digital recording for checking quality of incandescent lamps", Control, 11, 1967, 158-162 and 236-240.

8) H. J. J. van Boort and D. Kolkman, Philips Techn. Review, 19, (1957 /58) 333-337, "The double-flux "TL"-lamp, a fluoresent lamp of high output per unit

length". 9) I.E. C. Publication 82. "Recommendations for ballasts for fluorescent lamps".

10) J. A . .M. van Lie m p t and J. A.~ de V r i end, Rec. Trav. Chim. Pays Bas 52, 1933, 163. "Studien tiber die Explosion von Schwefelkohlenstoff-Stick­stoffmonoxyd -Gemischen''

CHAPTER XVIII

MEASUREMENT OF ILLUMINATION

XVIII -1. Introduction Illumination is measured either in order to know the illumination itself, i.e. to obtain a value of this quantity expressed in lux or footcandles, or as an intermediate stage in ascertaining values of other photometric quantities, in which case it is usually sufficient to know the illumination as a relative value only. In the foregoing sections on the measurement of luminous flux and luminous intensity, examples of this procedure have already been given. In principle there is no difference between the one kind of measurement of illumination and the other, but the distinction has to be made seeing that special instruments have been designed for use where the objective is the measurement of the illumination, these being adapted to the necessity of measuring this quantity in lighting installations. Portable illumination photometers, from which a direct reading of the illumination in lux or footcandles can be taken, need not, in general, be highly accurate, but reliable and compact in size; the cost should also be reasonable, so that this need not form an obstacle to the wide use of these very handy meters among lighting contractors, architects and so on. Apart from such "pocket" illumination photometers, more accurate meters are also obtainable which are larger in size and can be used down to lower levels of illumination. Illumination photometers can of course be employed for other measurements as well, but the question then arises as to whether the instrument is accurate enough for the purpose envisaged. Naturally, illumination can be measured either visually or physically. In visual meters the illumination to be measured must once again be converted to luminance, but this can be done quite simply by introducing a diffuse reflecting or translucent screen in the plane in which the measurement is to be taken. This screen serves as comparison surface for the photometer. With physical illumination photometers the field of the photometer

360 MEASUREMENT OF ILLUMINATION [XVIII

(photo-electric cell) is in itself light-sensitive and it is only necessary to place the cell at the point where the illumination is to be measured.

XVIII-2. Measurement by means of laboratory photometers In our example of a luminous intensity measurement (Fig. 137) it has already been seen that the measurement is in fact one of illumination. In that case the measurement gives the luminous intensity as the product Ed2 , although the numerical value of E does not appear in the ultimate calculation. All conventional visual photometers which work with a comparison surface are suitable for the measurement of illumination; if an absolute value is required the photometer must of course be calibrated, this being done the most simply on the photometer bench with the aid of standard lamps of known luminous inten.sity. The value of the illumination is obtained from the inverse square law E = I jd2 when a value in lux or footcandles is desired. After what has been said in the previous section and in Chapter XV regarding photo-electric cells there is no need here to enlarge on the use of physical photometers. Calibration is effected in the same way as for visual photometers.

XVIII-3. Visual illumination photometers When illuminating engineering ultimately arrived at the stage in its growth where it was evident that illumination levels and uniformity of illumination were important factors for effective seeing conditions, the need arose for simple, portable and not too costly instruments capable of measuring the illumination produced by installations reason­ably accurately. At that time photo-electric cells were not known, and visual illumination photometers were accordingly designed and marketed in a variety of types. Because of the lower requirements to be met by such instruments compared with those imposed on laboratory equipment it was not necessary to employ costly optical devices such as the Lummer-Brodhun prism, nor would it have been possible to reconcile such designs with the need for cheapness. Again, from the point of view of size and weight and also the price, large voltmeters or ammeters could not be employed in the instruments, and accuracy was accordingly not so high. Since the importance of such illumination photometers has declined

XVIII-3] VISUAL ILLUMINATION PHOTOMETERS 361

owing to the appearance of physical meters, only one specimen of the former will be described; this is the Osram luxmeter 1), which was one of the last designs to be produced. A sketch of this instrument is shown in Fig. 201, partly cut away to reveal the interior.

14 12

2-f¥WWi~~~~ 9--ff--------"' ...... """1

7 5 3 4 If

Fig. 201. A visual illumination photometer (Osram luxmeter)

The receptor for the light on test is section l of an opal glass sphere 2 which projects slightly above the top cover of the instrument. This arrangement ensures that only a small error is involved when the light enters at glancing incidence (this being more or less the same device that is used for barrier-layer cells (Fig. 174). The sphere works on the same lines as the Ulbricht sphere and functions as comparison surface for the photometer. A second opal glass sphere 3, which is illuminated by the comparison lamp 4, serves as comparison field. The photometer field comprises a glass plate 5 mounted at an angle of 45° between the spheres; this plate is divided into narrow bands which are alternately silvered and clear. To balance the luminance the observer looks through a window 6 in the cover plate, towards this glass plate. He then sees the non-silvered bands with sphere 3 as background, and, reflected in the silvered bands the luminance of sphere 2. Seen thus close together, the alternate bands of luminance from the two sources greatly facilitate observation and ensure accuracy of balance. The variation of luminance is obtained by rotating the cylinder 7, the wall of which is cut away obliquely. On rotation, the wall of the

362 MEASUREMENT OF ILLUMINATION [XVIII

cylinder thus cuts off a larger or smaller portion of the aperture 8, through which the light from the comparison lamp passes. The cylinder is ad­justed by means of a knurled flange 9 attached to it and projecting through the case of the instrument. The scale, calibrated in lux, is read through a window in the cover (10). A pocket lamp battery II supplies the lamp, and the current is adjusted to the correct value by means of a rheostat (operated by knob 12), as read from an ammeter 13. This illumination photometer has two measuring ranges, one of which runs from 0.4 to 400 lux. The other is obtained by sliding a filter having an attenuation factor of I 0 in front of the comparison lamp; the maximum illumination that can then be measured is thus increased to 4000 lux.

XVIII-4. Physical illumination photometers Visual illumination photometers have now been rendered almost com­pletely obsolete by the introduction of the photo-voltaic cell which permits of designs that do not need any external sources of voltage and which can be used by any layman in photometry. The photo­emissive cells manufactured before the appearance of barrier-layer cells did not lend themselves well to portable illumination photometers, seeing that they require an auxiliary voltage; only one model appears to have been marketed, and this has been superseded by the barrier­layer cell type. The introduction of physical illumination photometers brought with it not only the advantage of simplicity, but also a further advantage over the visual type. The latter were of course designed only for direct comparison, but when gas discharge lamps with their coloured light made their appearance measurements by direct comparison became difficult and unreliable. By employing correction factors or by .placing correcting filters in front of the <>ells even the layman will have no difficulty in taking reliable measurements of coloured light. For technical measurements in the illumination range from 50 to 1000 lux (5-100 fc) com­mercial illumination photometers are available in a size of about 3" x 2" x If' complete with cell and meter, these being typical pocket in­struments (see Fig. 202). In this type of instrument the cell is usually recess-mounted and is often · protected from

Fig. 202. Pocket size physical illumination

photometer

XVIII-4] PHYSICAL ILLUMINATION PHOTOMETERS 363

damage by a glass window. This system involves considerable risk of the instrument being wrongly used, since even at fairly steep directions of incidence the meter begins to depart from the cosine law, this be­coming the more marked as the direction is made more oblique. An example of this has already been given in Section XV-3 (Figs. 172 and 173). 'The use of these illumination photometers for modern lighting installations with their numerous large size light sources (particularly fluorescent lamps) may well result in errors. To illustrate this point we have calculated the error in the case of the two cells whose angular response is shown in Fig. 172, in respect of measurement under two different conditions of illumination. For the one example we have assumed an infinitely large, uniformly-diffusely radiating surface, and for the other an infinitely long linear source, also giving uniformly diffuse radiation. It is further assumed that the luminance of each source is uniform at all points. We assume also that the cells (Fig. 172) are calibrated with light en­tering with perpendicular incidence; with the aid of the curves shown in the figure it is then possible to compute the error made when measure­ments are taken of illumination produced by the sources on a surface parallel to them. The illumination produced by a part of the source radiating towards the cell at an angle of, say, 40° appears to be evaluated 4% too low by cell 1 and 36% too low by cell 2. To find the total error in the reading from the cell in respect of the total illumination we must first calculate the relative illumination values Ea. produced in the cell for every value of the angle ex; added together, these elements will give the total illumination. The illumination elements for all the angles of incidence must be multi­plied by the evaluation factor to be derived from Fig. 172 to give the cell-reading for each such element; these values added together then give the cell-reading for the total illumination. It can be computed that for the diffuse surface, Ea. is proportional to sin ex cos ex, and, with a linear light source, to cos2 ex.

The proof of this is to be found in the derivation of equations (IX-4) and (IX-17), respectively, by means of the infinitesimal calculus. The next stage is to effect a graphical integration of the expressions E = J sin a cos rxda and E = J cos2 ada. If we denote the evaluation factor of the cell by Pa. and the cell-reading by A, the second summation must be the graphical integration of the equation A = f Pry, sin a cos rxda and A = J Pa. cos2 a.da. respectively. Division of A by E then gives the error in the cell-reading.

364 MEASUREMENT OF ILLUMINATION [XVIII

The elements of the illu- E« mination and cell-readings can be added by plotting Ea. and A a. as a function of ex

fo.SJ--t--+-~~t--+---+---1

(Figs. 203a and b); in these o,3l----t---,~-+-+--+~&-....lftl.:--~-l figures the curves 1 relate to Ea. and curves 2 and 3 to 0•

cells 1 and 2, respectively. The sum of the values of

!l

Ea. and Aa. is obtained from the areas enclosed between the curves and the abscissae. The total areas indicate the fcc values of the infinitely large I 1

I

-~ or long light source, i.e. for o. a= goo.

8 '-.. ~ ~ 5

_.-I

" ~ ft ' ~ 1((3

"' ~

In the case of light sources q of finite dimensions, if the 0

point.of measurement lies on the central perpendicular to q the source, the error in the reading can be ascertained

0 ~

by measuring the areas in Figs. 203a and b lying be­tween the ordinates of 0° and ex, where ex is the angle tween the rays emitted from the boundaries of the light sources and the normal at the measuring point.

0° 10° 20° 30° 40° 50° 50'" 70° 80° 90° -oc

66966

Fig. 203. Illumination elements Ea and their evaluation by the barrier-layer cell shown in Fig. 172 as a function of the angle of incidence a. a) with uniformly diffuse radiating surface, b) with uniformly diffu~e linear so~rce. In a) E~ is proportional to sm a. cos a.; m b) to cos a.. The values of sin a. cos a. and cos1 0c are plotted a!'

ordinates

The errors for ex= 45° and ex= goo have been computed along these lines from Figs. 203a and b, and the results are given in the table on page 365. Although under practical conditions of illumination the situation will not usually be such as to entail quite so much error as is shown in our examples, these will nevertheless serve as a warning against the use under certain conditions of barrier-layer cells having an angular response which diverges too much from the cosine law. If the cell is fitted with a flat correcting filter, oblique incidence of the light introduces another source of error in that the path travelled by

XVIII-4] PHYSICAL ILLUMINATION PHOTOMETERS 365

I I Circular

\

Linear light source light source

0°-45° I% <I% Cell I

0°-90° 4.5% 2%

0°-45° 5.5% 3% Cell 2

0°-90° 24% 9%

the light is longer than with perpendicular incidence, with the result that the absorption is increased. Larger portable illumination photometers are also made, usually with a separate cell connected to the meter by a flexible lead; these provide for a lower measuring limit and greater accuracy. The better instruments of this type can also be used for laboratory work; they can be obtained with multi-range provision, achieved by connecting shunts across the meter; some models contain two or even four cells in parallel in order to secure a lower measuring threshold. The lower measuring ranges are required not only for laboratory pur­poses, but also for measurements on street lighting. Even though the cells in this type of photometer are not usually covered with a glass, and have only a low rim which does not produce a screening effect, care should be taken when they are used for lighting installations comprising large size sources, or in street lighting. For measurements on roadways at points some distance from the actual light source, i.e. where the light enters at wide angles to the normal, the difficulty of the angular response can be overcome by measuring the vertical instead of the horizontal illumination, in planes perpendicular to l"!l.e road axis. As mentioned in Section VI-4 the horizontal illumination can be computed from this quite easily. Owing to the impression of greater accuracy produced by the design of these instruments (which is certainly justified when they are used correctly) there is nevertheless a tendency to attribute rather too much accuracy to the results obtained under all conditions. The problem of correction of the cells for use in illumination photometers, both with respect to the angular response and to the absorption by filters without too much reduction in sensitivity has been dealt with in detail by B u c k 2), who suggests the use of filters which are thicker at the edges than in the centre. In conclusion we repeat here the precautions mentioned in Section

366 MEASUREMENT OF ILLUMINATION [XVIII

XV-3 to be observed in the use of barrier-layer cells with a view to preserving the cells and avoiding faulty measurement: a) Have the photometers re-calibrated at regular intervals; b) Keep the cells away from damp as much as possible.

XVIII-5. Calibration of illumination photometers Illumination photometers can be calibrated on the photometer bench by means of luminous intensity standards. If the calibration is to be precise it should be effected for various distances. With the physical illumination photometers this is essential in order to lay down the devia­tion from linearity which the cells ex­hibit. In pocket photometers it is usually sufficient to check the scale at a number of points. If many such photometers are to be calibrated it is useful to employ a specially constructed apparatus of the kind shown in the sketch in Fig. 204. Here L is a lamp below which a frosted glass disc G is mounted. Diaphragms (B) of different aperture can be inserted under the glass G in order to vary the illumination on the photometer LM, which is placed on a table with vertical adjustment so that the distance d from the diaphragm to the cell will always be the same. Calibration is done with the aid of an accurate illu­

(l)L G

I tk I I I I

d

I I j 1LH , ..

'1!11' LJ

l '/

66577

Fig. 204. Sketch of appa­ratus used for calibrating

illumination photometers

mination photometer calibrated with standard lamp on the photometer bench.

REFERENCES 1) Li.u.La. 1932, 183-184. "Der neue Osram-Beleuchtungsmesser" 2) G. B. Buck, see 5) Chapter XV

CHAPTER XIX

MEASUREMENT OF LUMINANCE

XIX-L Direct visual measurement of luminance Since luminance, among the photometric quantities, is the only one that can be perceived as such by the eye, only luminance can be measured directly by comparison with another luminance. We shall term this method the direct visual measurement of luminance, in contrast with the indirect methods in which luminance is measured with the aid of another quantity. Since photometric units are all based on the unit of luminous intensity, i.e. the candela, the value of the standard luminance with which the photometer has to be calibrated must be determined by measuring the luminous intensity of a certain area and dividing this by the area itself, unless the comparison be effected with the luminance of the black body at the temperature on which 'the candela is based. By means of such a standard luminance a photometer can be calibrated by flashing one of the photometer fields with this luminance, the other field obtaining its luminance from the comparison lamp. When the measurement has been made the standard luminance is replaced by the unknown luminance, photometric balance being then obtained between this and the comparison lamp. The photometer field can be flashed with the luminance to be measured in various ways, viz. 1. The photometer can be so positioned in front of the surface X of which the luminance is to be measured that

X PF

the photometer field PF is wholly flashed (Fig. 205). The surface

under examination then functions as com

~~;:::=~=~~~~~~~~~~5P parison surface for the f photometer. As will

Fig. 205. Direct visual luminance measurement by means of a photometer. X = surface the luminance of which is to be measured. PF = photometer field. L = photometer lens. P = pupil of photometer

be seen from the fi­gure, the surface under examination must be of a certain size if this method is to be used.

368 MEASUREMENT OF LUMINANCE [XIX

The size of the surface involved in the measurement can either be computed from the geometrical-optical details of the photometer, or deter­mined experimentally by placing a diffuse light source before the pupil P of the eye-piece and measuring the diameter of the projected light spot on the surface under examination. In many cases, however, this area will not be sufficiently large, in which case it can be enlarged artificially, by: 2. reproducing the surface under examination in the photometer with a lens. The exit pupil of this lens, flashed with the light of which the luminance is to be measured, then functions as comparison surface; it must be large enough to ensure that the photometer field and the pupil of the eye-piece are wholly flashed. According to Abbe's law (Section Xl-2) the observed luminance is then equal to the luminance of the surface under examination, apart from losses in the optical system *). The point at which the image of the surface under examination occurs is immaterial and may therefore be such as will suit the circumstances, provided that the photometer field and the pupil of the eye-piece are completely flashed. This has. to be verified, in every case, by noting the path of the rays in the photometer and lens. This can be done experimentally by placing a diffuse light source in front of the pupil, measuring the diameter of the projected light spot at the exit pupil of the lens and then measuring the diameter of the light spot projected in the surface under examination, with the lens interposed. Two examples of this method of measuring luminance are illustrated in Fig. 206a and b. In Fig. 206a a part of the surface under examination X is reproduced by the lens L1 and the magnifier L2 in the photo­meter, the image being formed in the pupil P of the eye-piece. P is in the focal plane of L2, so that X is at the focus of L1• From the path of the rays coming from the edge of P as bounded by the photometer field PF the minimum size required for the exit pupil of L1 will be seen. This also shows what part of X actually takes part in the measurement. The linear magnification is equal to the quotient of the focal length of L2 divided by that of L1, and from this, together with the diameter of the pupil, the size of the measured portion of X can be computed. Fig. 206b depicts the same photometric arrangement, but in this case a part of X is projected by L1 onto PF. (Since X and L1 are shown in

*) The measurement of the transmission factor of lenses is discussed in Section XX-2.

XIX-1] DIRECT VISUAL MEASUREMENT OF LUMINANCE 369

the same position in relation to PF as in Fig. 206a, the focal length of L1 is different from that in Fig. 206a(shorter)). The diameter of the measured part of X is now determined by the focal length of L1 and the distance LcPF. Which of the three methods described (or another in which an image of X is produced at a different point) should be used, depends on various factors. In the first place this choice depends on the size of the part of the surface

PF

PF

Fig. 206. Visual luminance measurement based on an image of the surface for measurement produced by a lens L 1: a) in the pupil of the photometer. b) in the

photometer field PF. L 2 =photometer lens

under examination that is available for the measurement. In this respect the method shown in Fig. 205 is the least favorable. The possibility of employing one of the other methods depends on the available measuring distance, the focal length and aperture of the lenses available, (for greater magnifications the method of Fig. 206b requires a larger aperture of L1).

Another factor affecting the choice of method is the degree to which small irregularities in the luminance of the surface under examination. are observed. In the method not using projection (Fig. 205) each point of the photometer field receives its light from a large part of X and, in Fig. 206a, even from every point of the measured part of X. Small irregularities in the luminance of X are thus not observed. With the method shown in Fig. 206b each irregularity is reproduced at PF and is therefore observed.

370 MEASUREMENT OF LUMINANCE [XIX

Since the luminance is independent of the path of the rays it is not generally necessary to employ corrected lenses; the use of uncorrected lenses may even be recommended for smoothing out irregularities. It may be added here that in the measurement of relatively large sur­faces care should be taken that the lens L1 is not placed too close to the surface under examination; otherwise some error may be involved in the measurement owing to the fact that light reflected by the lens L1

increases the luminance of the surface. The error can be reduced by screening off that part of the surface which is not actually included in the measurement.

1 he optical pyrometer as luminance meter Optical pyrometers are sometimes used for the measurement of luminance. These instruments also come within the category of direct x 8

visual luminance meters and r· Lt ~L La should therefore be men- (:::.....::==:;~._~&l--"""""'=::·---=-t=:: A ______ !P tioned in this section. ~ \[ ! Primarily, optical pyro­meters are intended for measuring the temperature of temperature radiators, but we shall now discuss the principle and use of this instrument as a luminance

Fig. 207. Optical pyrometer used as luminance meter. An image of the surface of which the luminance is to be measured is produced in the plane of the filament of an incandescent lampCL by means of a lens L 1• The luminances of the image and filament are compared through a

lens L 2 and pupil P

meter in reference to Fig. 207. The optical pyrometer as a means of measuring temperatures does not fall within the scope of this book. An image of the surface under examination X, possibly enlarged, is produced by a lens L1 in a plane B in the optical pyrometer. In this plane there is the filament of a comparison lamp CL, the temperature and hence the luminance of which can be adjusted by varying the current. Via another lens L2, the filament and the image of X are observed simultaneously by the eye, which is located behind the pupil P of the eye-piece. The current flowing in CL is so adjusted that the filament and the image of X appear to be equally bright, the filament thus "disappearing". The instrument is calibrated with standard luminances, from which a lamp-current versus luminance curve is plotted. At high luminance levels, which have to be reduced in order to avoid difficulty in measurement due to glare, a filter can be placed in the eye-piece. If 2 selective filter is used, the instrument must be calibrated with light

XIX-1] DIRECT VISUAL MEASUREMENT OF LUMINANCE 371

of the same spectral composition as that of the light to be measured. In some cases the red filter incorporated in the pyrometer for temperature measurements can be employed. Since the luminances are balanced at the boundaries of the filament, only a very small part of the area to be measured is actually involved in the measur·:ment.

Portable luminance meters for field-work Just as for the measurement of illumination, instruments which are easily portable are made for the measurement of luminance and are therefore suitable for use other than in laboratories. Whereas the visual type of illumination photometer has been almost completely superseded by physical instruments, this is not the case with luminance meters since the lowest values that generally have to be measured produce an illumination on the cell that is too small to be measured with simple, robust portable microammeters. In Section XIX-2 a portable luminance meter will be mentioned in which a photo-emissive cell is used.

L

R 664J/9

Fig. 208. Portable visual luminance meter designed by L u c k i e s h and T a y l o r

In the course of time numerous visual lu­minance meters have been devdoped. These can usually also be used as illumination photometers by plac­ing a matt white screen at the point where the illumination is to be measured and measuring the lumin­ance of this screen

with the meter. Calibration can be effected by means of a known illu­mination. Of the various portable luminance meters we shall mention only one viz. that designed by L u c k i e s h and T a y 1 or 1) (Fig. 208). An image of the surface of which the luminance is to be measured is produced by a lens L. The photometer field comprises this image and two small silvered rectangles (depicted in Fig. 208b). These rectangles reflect the light from a diffusing glass in front of the comparison lamp. The photometer field is observed through the pupil P 1 and the current of the comparison lamp is adjusted by means of a rheostat R, as read

372 MEASUREMENT OF LUMINANCE [XIX

from an ammeter AM. Photometric balance is obtained by rotating a disc A, and the luminance is read from a scale inside the instrument, through a pupil P 2• As reducing filters can be introduced in the path of the rays of the light to be measured, or from the comparison lamp, and, as the lamp can be adjusted for two different current values, the measuring range of the instrument is quite wide, viz. from about 0.1 cdfm2 (0.03 mL) to about 180,000 cdfm2 (50 L). The size of the field of view of the photometer is not more than 0.1 °.

XIX-2. Visual and physical luminance measurements obtained from measurement of illumination

Since in this method the luminance measurement is converted to a measurement of illumination, and as it is immaterial whether this is measured visually or physically, it will not be necessary for us to deal with the visual and physical methods separately. One such method has already been mentioned in the preceding section; the luminous intensity is measured (through the x illumination on the com- I- II j j parison surface) of a part [====A~======~~-=====~==~c of the surface under exa- r- II J L mination the size of which can be exactly determined, the lumin­ance value being ob­tained by dividing the luminous intensity by the area. In so doing it is

Fig. 209. Physical luminance measurement. The aperture A in the end cover of the tube is filled with the luminance to be measured from X and functions as light source for the photo-electric cell C. The screens S check the passage of stray

light in the tube

obviously necessary to take into account the rules relating to the ratios of measuring distance and the diameter of the light source (the area to be measured), and the size of the acceptance surface of the photometer. This method is suitable for use with a simple luminance meter with barrier-layer cell of the kind depicted in Fig. 209. The meter consists of a tube with matt black interior. The front cover has in it an aperture A which is presented to the surface under examination; the barrier-layer cell C is mounted at the other end. Screens S can be inserted in the tube to prevent stray light from falling on the cell. For the measurement of primary sources, the tube can be placed against the surface to be measured; in the case of surfaces which derive their luminance from incident light the tube is placed at a distance from

XIX-2] VISUAL AND PHYSICAL LUMINANCE MEASUREMENTS 373

the surface that will ensure that the incident light is not screened by the tube to any appreciable extent. Needless to say, when this method is employed, the area available for measurement must be fairly large.

X

L

The method which we shall now PF describe is particularly suitable

when the surface available for the _ _ _ _ _ ~ measurement is only small.

An image of the surface X to be measured (Fig. 210) is projected

Fig. 210. Luminance measurement derived from measurement of illu­mination. An image of a part of the surface X whose luminance is to be measured is produced on the comparison surface PF or photo-electric cell C. L functions as light source for the

by a lens L onto the comparison surface PF of a visual photometer or onto a photo-electric cell C. The luminance L of X is then computed from the resultant illumination in the following manner. photometer The luminance of the lens is TL

(T =transmission factor of the lens). LetS be the size of the exit pupil of L; then L is a light source of which the luminous intensity is T LS cd. The illumination E produced by L on the comparison surface for a distance L-PF equal to d is then

TLS E=-. d2

E can be measured in accordance with one of the methods outlined in Chapter XVIII, and L can then be computed from

Ed2 L=-.

TS Example:

Let the measured illumination E be 8.5 lux; d is found to be 2 m, ,. = 0.75 and S = 30 cm2; then:

Ed2 8.5 X 22 L = -:;s = 0.75 X 30 = 1.51 sb.

The size of the measured portion of the surface can be obtained from the focal length of the lens, the distance d and the diameter of the light receptor.

Suppose that the focal length of the lens in the above example is 125 mm and the diameter of the photo-voltaic cell used, 40 mm; the diameter of the measured area is then

125 . 2000 x 40 = 2.5 mm.

374 MEASUREMENT OF LUMINANCE [XIX

Conversely, the appropriate focal length and measuring distance can be computed from the size of the surface available for measurement and that of the light-receiving area. Here, a further determining factor is the threshold of measurement of the photometer. With visual photo­meters, therefore, it must be borne in mind that the luminance of the photometer field should not be less than the minimum luminance of 3 cdfm2. With this method of measurement, too, precautions should be taken that the luminance to be measuretl is not appreciably increased by light reflected from the lens. A physical luminance meter suitable for field-work and having attractive features is described by Freund 2). It contains a lens (objective f: 1.9) in focussing mount which forms an image of the region to be measured on a reticule in the optical housing. A circle in the center of this plate indicates the exact extent of the area being measured. This image is viewed magnified through a telescopic eye-piece. A partial reflecting mirror just ahead of the reticule plate reflects a portion of the light downward to the photo tube. An opaque aperture plate is located at the focus of this light beam directly above the photo tube. This plate excludes all light except the portion of the image that corresponds to that within the circle of the reticule. The light emerging from this aper­ture passes through a filter wheel and then onto the cathode of a photo­emissive cell. The photo-current is amplified, and the output current can be read from a logarithmic-response microammeter built into the luminance meter. The microammeter is calibrated to read directly the footlambert luminance of the surface measured. The scale is calibrated from 1 to 100 footlamberts. Range multipliers of 0.1, 1, 10, 100, 1000 and 10 000 are available on the range switch. The measuring range runs from 0.1 ftla (about 0.3 cdfm2) up to 106 ftla (about 350 sb). The ac­ceptance angle of the meter is 1.5°. The batteries needed for the anode voltage of the cell and for the am­plifier are contained in the meter. For continuous use over long periods the meter can be connected to an external power supply to replace the batteries.

XIX-3. Measurement of luminance distribution It may be necessary to determine distributions of luminance in cases where the luminance over a given area is not constant, but differs between one point and another. Measurements are then taken at a number of points over the surface and curves are plotted to show the

XIX-3] MEASUREMENT OF LUMINANCE DISTRIBUTION 375

distribution along a line, or in the form of iso-luminance curves. It is important that in the measurements at the individual points the area included shall be so small that the variation in luminance over that area does not make the final result more inaccurate than is compatible with the requirements. The measuring method should be adapted accordingly. In its execution the method will depend on the nature of the surface concerned, i.e. its size, and its possibility of movement. The distribution of luminance of the bulb of an electric lamp or lighting fitting, for example, can be measured with the aid of a lens and an illumination

·3-C

Fig. 211. Diagram illustrating the measurement of the distribution of luminance of a lamp bulbK, (or lighting fitting). An image of a part of the bulb is projected by lens L onto the photo-electric cell C. The bulb is moved in the direction of the axis and perpendicular thereto, a sharp image of every measuring point being

obtained

photometer by moving the bulb in a direction perpendicular to the axis of the lens (see Fig. 211). After each such movement the apparatus is carefully re-focused, for which purpose lines drawn just above or below the measuring point will be found useful. In the case of small light sources (e.g. the arc of a mercury vapour lamp) an image of the whole discharge can be projected onto a screen, and the cell of the photometer, provided if necessary with a slit-shaped diaphragm, is then moved across the image. Obviously, all such measure­ments are of the kind usually taken in the laboratory. The measurement of the luminance of street lighting systems presents quite a different problem. The aim of luminance measurements in street lighting is mainly twofold: the determination of the average road-surface luminance (as a measure of the lighting level) and of the local variations in that luminance (as a measure of the patchiness). The average value, as well as the local varia­tions in luminance are only of interest as seen by the motorist and on that part of the road which is of importance to him for his observations (about 150 to 600 ft in front of the driver). Local variations in the lumi­nance can only be measured if the field of vision of the luminance meter is small enough; the solid angle it subtends should certainly not be

376 MEASUREMENT OF LUMINANCE [XIX

greater than that corresponding to the smallest object of importance to the driver (about 3 X 3 minutes of arc). It is an obvious idea to carry out the measurements of the local luminance from one point, which is representative of the positions normally occupied by the eyes of the driver. Experience gained with a luminance meter3) designed for making local measurements of the luminance from a single point has shown, however, that very high demands must be made not only on the accuracy of the means of adjusting the instrument but also on the skill of the observer if sufficiently reproducible results are to be obtained. One important aim ofthe local measurements is the determination of the average luminance of the road surface by suitable averaging of the local values. This should be done either so that the points under measurement are so distributed over the road surface that each one takes up an equal area in the perspective picture, or, if another distribution is chosen (normally a simpler one and thus easier to plot on the plan of the road) each local value should be weighted in proportion to the corresponding area taken up in the perspective picture. An instrument for the measurement of road surface luminances which avoids the difficulties involved in measurement of local values from one point, i.e. from long distances and which, moreover, enables the average luminance to be measured directly is described by Asmussen and de Boer4). The instrument is provided with two identical objectives (01 and 02 in Fig. 212). Both produce an image of the part of the road under measurement in the plane of an exchangeable slide S. The lowest piut of the slide contains a diaphragm D. The light falling on the photo­multiplier M then comes exclusively from a part of the road of shape determined by the contours of the diaphragm D. These contours are reproduced in the form of thin black lines on a transparent window A in the top half of the slide S (taking the relative positions of the lenses 01 and 02 with respect to the part of the road in question into considera­tion). By aiming the luminance meter so that the black lines coincide with the edges of the image of the part of the road on which it is desired to carry out the measurements, one ensures that only light from this parit of the road falls on the photomultiplier M. · If a local luminance has to be measured, a slide with a rectangular aper­ture D and a corresponding rectangle on the window A is placed in the instrument, which is then placed flat on the road about 10 ft from the spot under measurement (see Fig. 213). An image of the rectangle on the window A is then projected via objective 0 2 (Fig. 212) onto a small screen

XIX-3] MEASUREMENT OF LUMINANCE DISTRIBUTION 377

placed in the middle of the area under measurement. The light for this purpose is provided by the lamp L, which is turned into the optical axis of the objective 0 2.The image of the rectangle is adjusted with the aid of two setting screws so that half of it is. visible of the screen. In this position the photomultiplier only receives light coming from the rectangle shown in Fig. 213 about 4 inches broad and 31 inches long (0.1 X 0.8 m). When the distance between the screen and the meter is 10 ft, the optical axis of the objective makes an angle of nearly 1 o with the surface of the road .

. Fig. :ll4: Luminance meter as described by Asmussen and de Boer. 01 and 02 = ob­jectives; S = slide with diaphragm D and window A; E = eyepiece; L = lamp for setting the meter for the measurement of the luminance of points on the road surface; B = push-button switch for the lamp; M = photomultiplier; C = calibration de.vice; K = hood for screening against light-scatter.

378 MEASUREMENT OF LUMINANCE [XIX

It is permissible to carry out measurements of the local luminance ex­clusively at this angle, even though the angle at which the motorist observes the part of the road of interest to him varies from about 0.5 to about 1.5°, since the influence of a variation in the angle of observation in this range on the observed luminance is negligible. For measurements of the average luminance, the instrument is mounted vertically on a stand (Fig. 214) or held vertically in the hand (Fig. 215). A slide is then introduced with a trapeziumshaped aperture in the diaphragm D and a corresponding trapezium on the window A. The lamp L, which is used to determine the area under measurement for local luminance measurements, is now turned out of the optical axis of the objective 0 2.The observer can then see through the eyepiece E the outline of the trapezium on the window projected on to the surface of the road, and can adjust the luminance meter until this outline coincides with the part of the road he is interested in. A large number ofslides with trapeziums of different shapes are provided with the instrument. The two horizontal lines always correspond to distances of 200 and 530 ft in front of the instrument, when it is held at a height of 5 ft (60 and 160m, when at a height of 1.5 m). The slanting side lines correspond to various widths of road, increasing in steps of about 1,5 ft (0.5 m) for narrow roads and about 3 ft (1 m) for wide roads if the road is observed from a height of 5 ft ( 1.5 m) in the middle of the right-hand lane (or in the middle of the left-hand lane if the slide is reversed). If it is desired to make measure­ments on a strip of road of another length, the meter should be held at a different height. The length of the strip under measurement and its distance from the meter are proportional to the height of the meter. Since it is impossible simultaneously to point the meter accurately in the right direction and to read the meter deflection, the meter is provided with a cable release, which is pressed at the moment when the meter is properly adjusted; this fixes the needle on the scale, so that the observer can take the reading at his leisure and accurately. The use of slides with diaphragms D of different areas means that the instrument has a different sensitivity with each different slide, and must thus be recalibrated each time the slide is changed. This is done with the aid of a built-in calibration unit C (see fig. 212), which contains a number of seasoned calibration lamps behind an opal-glass window, which ensures that the luminance is distributed evenly over the whole area of the window. This calibrating unit can be pushed up with the aid of a button so that the opal-glass window comes in front of the objective

XIX-3] MEASUREMENT OF LUMINANCE DISTRIBUTION 379

0 1• The voltage of the lamps in the calibration unit is stabilized by the voltage-stabilizing unit. The calibration proper consists in changing the amplification factor of the amplifier so that with the given slide the instrument shows the known luminance of the calibration unit, multiplied by a calibration factor dependent on the sort of light under investigation.

The photographic method

By the photographic method, luminance measurements are not taken from the road surface directly, but from a photographic reproduction; the measuring points can therefore be precisely laid down in the photo­graph. The latter is of course in perspective and if necessary a plan view can be derived from it by transposition. An example of the photographic luminance measurement of a road surface may be found in an article by Bouma 5). In this case a photograph of the road was taken from eye-level on a rapid panchromatic plate, a horizontal strip of which was shielded. Later, in the laboratt>ry, an exposure was made on this strip of a light box having at the front a glass plate divided into a number of bands blackened to different densities, whose luminance values were known. This exposure was made with the same kind of light as that of the street lighting system and with the same exposure time. Comparison of the density of the different parts of the road in the photograph, with the bands of which the lu­minance is known, enables the luminance of every point on the road to be ascertained. The actual method of measuring the densities is explained in Section XX-2. With this method it must be remembered that for a given luminance of the object the illumination and therefore also the density of the photograph decreases towards the edges of the plate. As shown in Section XI-14 the illumination varies as the fourth power of the cosine of the angle of divergence from the axis, provided no vignetting occurs. If this does occur the decrease in density in the direction of the edges is even greater. In methods of measurement such as the one described, this decrease in illumination has to be determined experimentally by placing in front of the camera a surface of uniform luminance and measuring the relative illumination by means of a photo-electric cell having a very small acceptance surface.

380 MEASUREMENT OF LUMINANCE [XIX

XIX-4. Determination of the size of the exit pupil of lenses For the measurement of luminance as derived from measurement of the illumination produced by lenses it is necessary to know the size of the exit pupil of the lenses. Let us now see how this can be done. The exit pupil of single lenses and of objectives without internal stops (Section XI-7; example a) are defined by the mount of the lens (or of the first lens respectively), and can be determined by measurement. It is more difficult to ascertain the exit pupil of objectivfS having a stop between the component lenses (Section XI-7; examP.le b). In most cases the exit pupil is then represented by the image of the stop as produced by the first lens. The size (and if necessary also the relative position) can be found experimentally by projecting an enlarged image of it through an objective of which the exact focal length is known. The magnification is computed from the object and image distances and the focal length; measurement of the image and the magnification then enables us to determine the size of the exit pupil.

REFERENCES 1) M. Luckie s h and A. H. T a y 1 or, J.O.S.A. 27, 1937, 132. "A Brightness

Meter" 2) K. Freund, Ill. Eng. 48, 1953, 524-526. "Design Characteristics of a Photo­

Electric Brightness Meter" 3) J. B. de Boer, Rev. Gen. des Routes et des Aerodromes 280, 1955, 83-S4 "Un

luminancemetre pour l'eclairage public". 4) E. Asmussen andJ. B. de Boer, Public Lighting 27, 1962, 136-140, "A luminance

meter for street lighting". S) P. J. Bouma, Ph. T. R. 4, 1939, 292-301. "Measurements carried out on

Road Lighting Systems already installed"

CHAPTER XX

MEASUREMENT OF REFLECTION, TRANSMISSION AND ABSORPTION

XX-1. Measurement of reflection factor The reflection factor of a surface depends on the angle of incidence and it is therefore not strictly correct to speak of the reflection factor, as such, of a material or surface. Nevertheless this is usually done, the reflection factor being taken to refer to those angles of incidence at which it is constant or nearly so. In Chapter X we have seen that the reflection factor of glass of refractive index 1.5 is practically constant up to angles of incidence from 45° to 50°, and in the case of metals, to almost glancing incidence. Since it is only angles such as these that are of interest in illuminating engineering, in general, the unqualified reflection factor is accordingly sufficient for most purposes. In the design of instruments for the measurement of reflection factors we have usually a choice, therefore, of angles of incidence between oo and ca. 45°. Besides the reflection factor at given angles of incidence there is of course also the reflection factor in respect of light incident from all sides, i.e. the diffuse reflection factor. The measurement of this neces­sitates special apparatus and will accordingly have to be included in our discussion.

Measurement of the reflection factor of light incident from one direction only

Let us first consider the measurement of reflection factors with light incident at angles between oo and 45°. Since the reflection factor e is the ratio of two luminous fluxes, we can obviously employ the two methods of luminous flux measurement, i.e. by using an integrator or by determining the light distribution. In the case of purely regular reflection by a perfectly smooth (polished) surface (metal or transparent glass), e can also be evaluated by measuring firstly the luminous intensity of a light source and secondly that of the image of the light source. A measurement obtained from the light distribution with the aid of a

382 REFLECTION, TRANSMISSION, ABSORPTION [XX

Rousseau diagram is of course only appropriate for diffuse reflecting surfaces, and then only when the distribution (i.e. the luminous intensity or luminance indicatrix) is also to be ascertained. Otherwise this method of measurement is far too cumbersome and it is much quicker to use an integrator. Measurement of the reflection indicatrix will be discussed in section XX-4. For the measurement of reflection factors using integrators many methods are given in the literature. Most of them employ a standard reflecting surface, with which the surface to be tested is compared. These methods can be described on general lines in reference to Fig. 216. KisanUlbricht sphere. A beam of light enters through an opening 0 1 and falls on the plate to be tested which is mounted at 0 2

in the sphere. To ensure a sharply defined beam of light the following arrangement can

Fig. 216. Principle of apparatus for measuring reflection factors with the aid of a standard reflecting surface

be used. A lens L1 (condenser) produces an image of the light source in another lens L2 which throws an image of L1 on the test plate and ac­cordingly produces a clear-cut light spot. A diaphragm D of any desired shape can be placed in front of L1 • If the light does not fall perpendicularly onto the plate to be measured a circular light spot can be ensured by using an elliptical diaphragm. In order to exclude stray light, screens can be interposed between L1 and L21 and it is important that L2 be as clean as possible. ScreenS ensures that no light reaches the sphere window P directly from the plate to be measured. When the photometer reading has been taken, the test plate is replaced by a standard plate and the unknown re­flection factor is then easily computed from the readings, which are directly proportional to the respective reflection factors. Either a visual or a physical photometer may be used. The precautions to be taken in connection with the spectral sensitivity when measuring luminous flux (section XVII-4) must of course also be observed here. The standard surface usually takes the form of a plate coated with magnesium oxide by deposition, i.e. the oxide produced by the com-

Fig. 213. Setting of the luminance meter for the measurement of the local luminance for a part of the road surface of I 0 X 80 mm2 (the rectangle indicated by the black lines in the centre of the adjusting slide).

Fig. 214. Measurement of the average luminance of a road surface by means of the lumi­nance meter mounted on a stand.

XX-1] MEASUREMENT OF REFLECTION FACTOR 383

bustion of a piece of magnesium wire or tape is. made to form a deposit on the plate. The reflection factor of magnesium oxide is usually given in the literature as 0.95, but if any given plate is coated with MgO without any particular precautions the ,..esultant layer of MgO will never yield a reflection factor of 0.95. "Letter circular" LC-547 1) of the Bureau of Standards contains a specification for the preparation of a standard MgO-coated plate. In this the most important feature is the method to be employed to deposit the magnesium oxide in a very finely divided condition on the plate. This results in a thick layer (about 0.5 mm) which is not liable to flake off and has a high reflection factor; for a plate prepared in accordance

X

with the specification a reflection factor of 0.97 should be obtainable. Magnesium carbonate blocks, supplied with data

8 on their reflection factor are available on the mar­ket for use as standard plates. When the surface of such a block has be­come dirty, a fresh clean

Fig. 217. The Taylor method of measuring reflection factors without the aid of a standard

reflecting surface

surface can be produced simply by grinding off the soiled layer.

Another method of measurement, which does not necessitate the use of a standard plate and which is accordingly very suitable for general purposes outside the laboratory, has been devised by A. H. Taylor2). The test plate X is placed in an Ulbricht sphere K (Fig. 217) and is iliuminated by a beam of light. A screen S prevents direct light from X from falling on the window P. Let the illumination at the window be Ex. The beam is then rotated about the axis AA' so that it strikes a part of the sphere wall, the window not being screened in this case. If the reading be now Ek, the reflection factor is found from

The correctness of this method can be shown in the following manner. If the luminous flux of the beam is 1/J lm, the plate will reflect e.,<P lm,

384 REFLECTION, TRANSMISSION, ABSORPTION [XX

this being the luminous flux which we wish to measure. Further, denoting the area of the sphere by S and the reflection factor of the sphere wall by ek, it will be seen that, as in the derivation of formula (XVII-I), the illumination at the window is

When the beam strikes the sphere wall directly, ek<l> lm is reflected. Since in this instance the luminous flux falling on the sphere window for

the first time is not screened, E k is not proportional to ~.but to 1 - (!k

Division of the expressions for E., and E k then gives e.,. With this method the light sources, i.e. the plate to be measured and the illuminated area of the sphere wall, are not situated in the same place, for which reason it is preferable to arrange for the illuminated areas to be located symmetrically with respect to the sphere window. If the plate to be measured is quite flat and gives regular reflection the reflected light will also produce a sharply defined light spot in the sphere, and the best arrangement is one in which the position of this light spot with respect to the window is the same as for the other light spots. This can be done along the lines shown in Fig. 217, in which the specularly reflected beam from X strikes the sphere wall at B. Since the distribution of the luminous flux from X to be measured may occur in many different forms (owing to the reflection being diffused to a lesser or greater extent) it is recommended that the shape of the integrator be such that the factor k mentioned in section XVII-4 varies as little as possible with ek, i.e. it should be a true sphere. Seeing that the measurement is dependent on the value of ek at a particular point on the sphere wall and, in the case of flat specular plates on two such small areas, care should be taken that ek is uniform, in other words that the sphere is repainted at the first sign of soiling. Although this method does not necessitate a standard plate it is advisable to check the photometer from time to time with a standard plate. If the plates to be tested are specular it is also good practice to take a check

XX-1] MEASUREMENT OF REFLECTION FACTOR 385

occasionally with a specular standard plate (e.g. a plate of well-silvered plate glass) which can be calibrated from time to time by means of luminous intensity measurements.

Measurement of diffuse reflection factor For the measurement of diffuse reflection sphere is again the appropriate apparatus. The sphere can then at the same time supply the diffuse illumination for the plate to be measured. The arrangement may be as shown in Fig. 218a, b or c. In Fig. 218a the sphere is illuminated by an incandescent lamp, so screened that no light from it falls directly on the plate to be measured X. In Fig. 218b a beam of light is projected into the sphere.

p

.£ Fig. 218. Three methods of illuminating surfaces for the measurement of the diffuse reflection factor

factors the Ulbricht

In both systems the illumination on X is from all sides, but not exactly with the same luminance in all directions. The third method (Fig. 218c) is better in this respect; here the light from a sphere containing three or four screened lamps passes into a second sphere in which the test plate is mounted. Many methods of measuring diffuse reflection factors are to be found in the literature, but they differ from one another only in detail and all have this much in common that, although the measurement is straight­forward, calculation of the required value of e is rather complicated. We shall now take one of these methods as an example, and any one of the systems of illumination shown in Figs. 218a, b and c will suit our purpose.

386 REFLECTION, TRANSMISSION, ABSORPTION [XX

An aperture QR is made in the wall of the sphere K (Fig. 219) and three measurements are taken at the window P in order to determine the reflection factor, viz: 1) with QR uncovered, or covered with a box lined with black velvet.

Let the illumination of the window be E0 •

2) with QR covered with a plate having the same reflection factor as the sphere wall (ek),. Illumination of window Ek.

3) with QR covered with the plate to be measured (reflection factor e.,). Illumination of window E.,.

Now if the areas of the circle QR (A) and of the remainder of the sphere (Ak) are known, the required reflection factor e., can be computed from the three measurements by making use of the fact that the combined luminous flux absorbed by the circle QR and by the rest of the sphere is equal to the luminous flux entering the sphere (4>). Let us denote the absorption factor of the sphere by ock ( = 1 - ek) and that of the plate to be measured by oc., ( = 1 -e.,). With QR uncovered, the absorption factor of QR (ot0 )

is obviously unity. Using the above data we can compute the luminous fluxes incident on the sphere wall

Q 66.507 R

Fig. 219. Principle of apparatus for measur­ing diffuse reflection

factor

and the open circle in each of the three examples. Multiplication by the respective absorption factors then gives the amount of the absorbed luminous flux and, by equating the sum in each case with tP we obtain three equations containing the unknown e.,, ek and 4>, from which ex can be evaluated. The luminous flux falling on the sphere wall and in the opening is equal to the product of the illumination and the area. Because, according to IX-4, the luminous flux of a point on a sphere that is completely diffusely reflective is distributed evenly over the surface of the sphere, i.e. the illumination is the same everywhere, the illumination at the sphere window is equal to that of the aperture, no matter what the reflectivity of the surface arranged in the aperture. This also applies when the surface in the aperture is not spherical but plane. In the first case, with QR uncovered, the illumination of the wall of the sphere (in which the sphere window is located) is E 0, as is that of the circle. There is a luminous flux (1 - p) S · E 0 lm incident on the wall of

XX-1] MEASUREMENT OF REFLECTION FACTOR 387

thesphere,partofwhich,rx.~c(l-p) S ·Eo= (1-e~c) (1-p) S ·Eo lm, is absorbed. That incident on the circle is p · S · E 0 lm, which is com­pletely absorbed. The sum of these absorbed luminous fluxes is equal to the luminous flux rt> falling into the sphere. Thus the first equation is

In the second case, where QR is covered with a plate of the same reflection factor as that of the wall of the sphere., we find the luminous flux absorbed by the wall of the sphere in a similar way to the first case, with (1- l?k) (1- p) S · E1c lm. The circle absorbs (1- ek) p · S · E1c 1m. In total, therefore, the flux absorbed is ( 1 - ek) S · E 1c 1m. This luminous flux is once again equal tort>, and therefore the second equation reads:

rt> = E~c (1 - e~c)S.

in the third case (QR covered by the plate to be measured) the luminous flux absorbed by the wall of the sphere is Ex (1- e~c) (1- p) S lm and that absorbed by the circle Ex ( 1 - e a:) p · S lm. For the third equation we then have

rt> = Ex (1 - ex) p · S + Ex (1 - e~c) (1-p) S = Ex · S [(1- ex)P + (1 - e~c) (1 - p)].

rt> and S are eliminated by equating the right-hand sides of the three equations, and Sneed not therefore be known, but only p, i.e. the fraction of the total surface of the surface of the sphere corresponding to the circle QR. Solving the three equations to Px gives

Ex- Eo E~c ex= . --. e~c E~c - Eo Ea:

In the latter expression, p .need not be known if the reflection factor of the wall of the sphere is known. It is possible to circumvent the determination of p (with ek unknown) by making a fourth measurement in which the circle is covered with a plate of known reflection factor. This then gives a fourth equation, and thus p can also be eliminated.

388 REFLECTION, TRANSMISSION, ABSORPTION [XX

XX-2. Measurement of transmission factor Transmitted light is that part of the incident flux that is not reflected or absorbed. Since the reflection and absorption factors depend on the angle at which the light strikes, the transmission factor is also dependent on incidence and must be measured at certain angles. When we speak of a transmission factor, without further qualification, this generally refers to perpendicular incidence. Whereas the reflection factor is nearly always found to be constant for angles of incidence of from oo tb about 45° and we accordingly have a certain amount of freedom in regard to the angle at which measurements are taken, no such latitude exists in the case of transmission, because at oblique angles the absorption factor tends to increase steeply owing to the longer path travelled by the light through the medium. The higher the transmissive exponent and consequently the absorption factor, the more sharply the latter increases with oblique incidence and the more the transmission factor varies with the angle. The Ulbricht sphere can be used for the measurement of transmission factors, the arrangement then being as shown in Fig. 220. The sphere has an opening in it at A through which a beam of light is projected. A screen S is placed in the sphere to prevent light from A from reaching the window P. Let us assume that the photometer reading in respect of the light entering at A is a0 •

The plate to be measured X is placed before the opening A and this produces a reading a.,. In the same way as for the measure­ment of reflection by Taylor's method we find that the transmission factor is given by the quotient a.,ja0 •

The same arrangement will serve for the measurement of transmission fac­tors for diffuse incident light, but of

==(-s 66508

Fig. 220. Principle of apparatus for measuring the transmission

factor

course with a different form of illumination; the arrangement shown in Fig. 218c may be used instead. The transmission factor of transparent materials can be measured by placing them in front of a surface of uniform luminance, the luminance of this area being measured first without and then with the plate in front of it. Care must be taken, however, that the plate to be measured

XX-2] MEASUREMENT OF TRANSMISSION FACTOR 389

is placed sufficiently far from the luminous surface to ensure that the luminance is not appreciably increased by light reflected from the plate. This method is suitable for either plane or curved objects and is there­fore suitable for determining the transmission factor of lenses and lens systems, e.g. objectives. In such cases the area of uniform luminance should not be any larger than is strictly necessary for the purposes of the measurement, seeing that due to reflection from the faces of the lenses light emanating from beyond the essential area produces a certain luminanc~ in the direction of the optical axis; if included in the measurement, this light will falsify the result obtained for the transmission factor. The transmission factor of lenses and objectives can also be measured by producing an image of a surface of uniform luminance· and measuring the illumination of this image. The luminance of the objective can then be measured in the manner described in section XIX-2; this luminance is of course less than that of the projected area, and the transmission factor is the quotient of the luminance of the objective divided by the known luminance of the projected surface.

Measurement of density Density, as measured with a densitometer, may be defined as in section X-12 as the logarithm of the attenuation in the luminous flux from a given illumination system falling on a light receptor after passing through a photographic film or plate. In view of what has been said in Chapter XI we may now say that in many cases the word luminance can be used instead of luminous flux. Densitometers are employed in sensitometry, that is to say for determining the photo-sensitivity of photographic materials. The apparatus consists of a source of illumination and a receptor, i.e. a photometer, and may be of the visual or physical type. The most obvious method is of course to take measurements with and without the plate, from which the density is then found by simple calculation, but the great objection to this is the wide difference that may occur between the luminous flux values with and without the plate (for a density of 3 the ratio of the fluxes or luminances is 1 : 1000). For this reason many densitometers compare the density of the test plate with that of a neutral wedge (see section XIV-3). Other types of densitometer work on the principle of the polarisation photometer.

390 REFLECTION, TRANSMISSION, ABSORPTION [XX

As a general rule, the area of the plate or film available for sensitometric measurements is fairly large (at least I sq. em) and neither the con­struction of the instruments nor the measurement present any problem. The illuminating and acceptance solid angles should correspond to the <;onditions under which the plate or film on test is normally used, but it should be noted here that many of the meters available on the market are somewhat lacking in this respect. In publications on the subject, moreover, this point is generally given far too little prominence. Data regarding densities as measured with apparatus of differing optical design cannot be compared one with the other. For the measurement of the intensity of spectral lines (spectrophoto­metry), as represented by lines on a photographic plate or film, micro­photometers are used. Since the spectral lines as produced on the photo­graphic plate are often very narrow, only very small areas are available for the measurement, and the precision of the instrument has to be very high indeed. These instruments are usually self-recording. The same remarks apply to photographic photometry, for which purpose microphotometers can also be used. The areas available for measurement are in this case generally so large, however, that it is not really necessary to use a microphotometer. For the measurement of densities in connection with the making of contact prints, where it is a question of measuring the total incident and transmitted fluxes, the method described at the commencement of this section is of course quite suitable, viz. measurement of the trans­mission factor by means of an integrator. Details of the many types of densitometers and microphotometers would be beyond the scope of this book; the reader is referred to the many publications already available on these subjects 4).

XX-3. Measurement of absorption In view of the fact that a. = I - (e + T), the absorption factor can be at once obtained if p and -r have already been measured in accordance with one of the methods indicated in the previous sections. Ifthe material to be tested transmits no light ( -r = 0), measurement of the reflection factor is sufficient; with materials that do transmit light the absorption factor can also be found by effecting one measurement only. The plate under examination X is suspended in the centre of an Ulbricht sphere (Fig. 221) fitted with a screen S to prevent light from X from reaching the sphere window P directly.

XX-4] LUMINANCE FACTORS AND GLOSS

Fig. 221. Principle of apparatus for measuring the absorption

factor

L A

391

66509

A beam of light L is projected through the opening A onto the plate inside the sphere, and a reading is taken. The plate is then removed from the beam (but is left inside the sphere), so that the beam strikes the wall of the sphere, and another reading is taken. In the same way as was shown in section XX-1, on the basis of Tay­lor's method of determining the reflection factor, we now find that the quotient of the readings is the absorption factor. If the reflection from the test plate is wholly or partially specular, care should be taken that the plate is not set exactly perpendicular to the beam of light, as an appreciable portion of the reflected light would then be lost through the opening A. The transmissive exponent of transparent or translucent plates (section X-15) can be computed from the absorption and reflection factors together with the thickness of the plate. In formula (X-14) Wu = W;e-ad, W; denotes the luminous flux entering the plate, i.e. the incident light less the reflected flux. W; is thus pro­portional to ( 1 - e). (/Ju represents the emergent flux and is proportional to 1 - (e + ex). Formula (X-14) can thus be written:

1- (e +ex) = (1- e) e-a4.

One of the methods outlined in the preceding sections can be employed to determine ex and e, and d can be measured with a micrometer, thus enabling a to be computed. Similarly, in formula (X-l5) Wu = W; e-ad the value of a in respect to diffuse media can be ascertained.

XX-4. Measurement of luminance factors and gloss The apparatus required for measuring luminance factors must be capable of illuminating the surface under examination from different directions

392 REFLECTION, TRANSMISSION, ABSORPTION [XX

of incidence, and of providing means of measurement of the luminance in different directions. Apparatus of this kind are known as gonia­photometers. Fig. 222 shows the principle of this apparatus. X is the plate under

' ' ' I

examination, L a beam of light capable of being moved in an arc about an axis at X, and P is a photometer, similarly mounted on a pivoted arm. If luminance factors are to be measured beyond the plane in which L can be revolved, the axis on which the arm of P rotates must be capable of rotation on an axis at right angles to the first axis. By means of such apparatus luminance Fig. 222. Principle of the gonia-

photometer factors can be measured with either reflected or transmitted light.

As we have already seen in section X-7 the luminance factor is the quotient of luminance and illumination. The latter can be measured in accordance with one of the methods described in Chapter XVIII. For the measurement of the luminance we have the choice between direct measurement on the one hand and, on the other, determination from the luminous intensity, this being divided by the (apparent) area. Either a light source without optical system, or a "parallel" beam of light may be used for the illumination, but in any case care must be taken that the part of the plate involved in the measurement is suf­ficiently uniformly illuminated. With direct illumination from the source this means that the source has to be placed sufficiently far from the plate. This also ensures that· the angles of incidence in respect of the plate do nat vary too much. When a "parallel" beam of light is employed the reduction of the il­lumination towards the edges of the beam should be borne in mind (see section XI-6). It is advisable, after first making a rough calculation of that part of the beam in which the illumination on the plate is prac­tically uniform, to check the variation in the illumination with a photo­electric cell having a small acceptance surface (using a diaphragm if necessary). If the luminanca is to be measured directly, thus not by way of a luminous intensity measurement, it is necessary to check by calculation that the area required by the photometer for the purposes of the mea-

XX-4] LUMINANCE FACTORS AND GLOSS 393

surement is fully provided by the illuminated part of the plate (section XIX-I). This is particularly important with glancing directions of incidence, since the apparent foreshortening of the plate is then con­siderable. When a luminance measurement is carried out by actually measuring the intensity of the plate in the case of transmitted light, the radiating area can be precisely limited by a diaphragm. For glancing directions of incidence, however, the diaphragm should be extremely thin, as otherwise a part of the plate will be screened (see Fig. 223). For the measurement of reflected light by way of the luminous intensity the area should be limited by confining the beam, in view of the fact that diaphragms

66511

Fig. 223. Reduction in the apparent area of a surface ac­companying oblique incidence through a diaphragm of too

great a thickness

always reflect a certain amount of light which is then included in the measurement as stray tight. In this, any lack of uniformity in the illumination is of little importance, as will be seen from the following. From section X-7 we see that the luminance factor is

L {J=­

E

(the indices are omitted for the sake of simplicity). In this, L = IJS cos IX and E = C/>jS, where I is the luminous intensity, cJ> the incident luminous flux and S the illuminated area. For (X-7) we can therefore write:

I

s cos IX I fJ = -cp- = cJ> cos IX.

s Care must be taken that the beam is properly limited so as to avoid stray light, and this can be done by using a clean lens and placing stops between the light source and the lens to prevent light reflected from the inside of the housing from leaving the lens in oblique directions. In all measurements of luminance factors it is necessary first to consider whether the result will really yield a luminance factor (as a constant of the material), whether the acceptance angle of the photometer is

394 REFLECTION, TRANSMISSION, ABSORPTION [XX

sufficiently small for the measurement of the luminance factors, and whether it is necessary to take measurements at various acceptance angles to ascertain if it is possible to measure the luminance factor (see considerations in section X-8 in which a method of varying the acceptance angle is given).

Luminance factors of road surfaces As mentioned in section X-10 the luminance factors of road surfaces play a very important part in street lighting. Measurements of luminance factors can be taken over a small portion of the road surface by means of a goniophotometer, either in situ, or from a sample of the surface in the laboratory. Alternatively the luminance distribution of the road surface can be measured as illuminated by a certain light source, in conjunction with measurement of the distribution of illumination. From the two distri­butions the luminance factor can be computed for any point on the road surface (i.e. direction of illumination). The photographic method of luminance measurement is also often employed for road surfaces, either as applied to samples 4), or with the aid of the luminance and illumination distribution in respect to the lighting system of the whole road 5).

A simple method of measurement to find the value p cos3tp has been worked out in connection with the method of calculation given in X-10 for luminances of street lighting with the aid of this value. It is possible with the luminance meter described in XIX-3 to find the value p cos3. tp both in the open, i.e. in the. street, and in the laboratory. Measurements in the open permit examinations of street and road surfaces in their natural surroundings under varying weather conditions, while laboratory measurements are suitable rather for systematic re­search, e.g. into the effect of different grain sizes or minerals. The prin­ciple of measurement is the same in both cases. A fitting with a constant luminous intensity on to the surface to be measured is arranged at a constant height. The luminances at the individual points are noted for the actual angles of incidence tp and the angle t5 between the plane of inci­dence of the light and the plane of observation.

I cos3 tp L = p = L (rn, t>). h2 .,.

Since the height and luminous intensity of the light source used are

XX-4] LUMINANCE FACTORS AND GLOSS 395

kept constant, the scale of the luminance ,meter can be linearly trans­formed into a p cos3 rp scale with the factor h2fJ.

h2 p cos3 rp = I . L = f (rp, J).

Since the p cos3 rp values are to be used for luminance measurements as in X-10, the measurement conditions must correspond to the observa­tion conditions. Objects on the road viewed at an angle of some 3 minutes of arc are just clearly distinguishable. For the most important observation distances of the vehicle driver, this corresponds to an average road surface area of0.2 X 2.0 m2. Because of the astigmatic character and discrete light distribution of a fitting, different luminances will be pro­duced at various points on this area, and they will be reflected to the observer with widely varying luminance coefficients within this area. Since differences in luminance within this area can no longer be perceived separately, it is logical to speak of this area as a "point" and of the ayerage reflective properties of the area as "point" reflection. The measuring arrangement in all measurements should be restricted to the smallest possible space, and, in laboratory measurements, too, the field of measurement should be made as small as possible by the size of the sample, which is always limited. In order to fully embrace the surface structure properties of road surfacing materials, a measuring field of 0.05 X 0.5 m2 is quite adequate. If this measuring field is to be adapted to natural observation conditions, the true scale should be converted to this model scale. In practical lighting installations, where the size of the measuring field may be 0.2 X 2.0 m2, the suspension heights are some 10 m and the area of the fittings about 0.2 m2. In the case of the model scale, therefore, of 0.05 X 0.5 m2, the height of the fittings is about 2.5 m with an illuminating area of 0.013 m2. The p cos3 rp values can be measured in this arrangement with a luminance meter. Here, however, the measuring aperture on the sample side of the meter may not. be greater than three minutes of arc. W1th these methods, the reflective properties of road surfaces are determined with the same integration limits as with visual observation. Fig. 224a is a diagram of the measuring arrangement for finding the p cos3 rp values in the open. The P cos3 rp values of the test surface F for different positions X of the fitting S are measured at an angle of 1° to the plane of the road and a measuring aperture of 3' with an objective luminance meter M (e.g. the design described in XIX-3). The values measured are shown graphically in the form of "EP diagrams" (see

396 REFLECTION, TRANSMISSION, ABSORPTION

F -

..,..---\ ,. .--· _..-t

Fig. 224: a. Finding p cos3 q~ values in the open. b. The measurement in the laboratory.

G

[XX

a

b

X-10). In laboratorymeasurement (Fig. 224b), the principle ofmeasure­ment is unaltered, but here the p cos3 <p values can be continuously recorded by a writer, dependently on the angle of rotation c5 with the aid of the turntable T or dependently on the angle of incidence of the light rp with the aid of the movable fitting S on the rail G. Fig. 225 shows the practical design of such a measuring arran{{ement.

XX-4] LUMINANCE FACTORS AND GLOSS 397

Measurement of gloss The design of glossmeters has been adapted as far as possible to the definition of gloss as formulated by the designers. In m<J,ny cases, however, they do not fully meet the requirements, i.e. the acceptance angle of the photometer may not be suitable for measuring correctly the quantities on which the definition is based. It will be clear from what has been said in section X-8 that where very glossy surfaces are encountered the acceptance angle has a very pronounced effect on the results of measu­rement, and that different instruments will yield different values for the gloss of one and the same surface even using the same definition of gloss. Descriptions of all the existing types of glossmeters would take us too far afield and it will be sufficient to refer the reader to the work by Harrison, of which mention is made in section X-9' 6). Here we shall describe briefly only a few types, which are made in various executions.

l) Meters based on the 0 s t w a l d K l u g h a r t method The sample under examination AA' is illuminated from an angle of 45° to the normal and the luminance is measured in the direction of the normal (Fig. 226a). The plate is then rotated through an angle of 22.SO (Fig. 226b), the directions of observation and illumination remaining as before, so that the luminance is measured in the specular direction. In variations of this method the plate may be turned so that the maximum luminance is reached.

~ I

"" t; ;;; A ~'?m=m'?m=:m-,A'

Q

Fig. 226. Principle of gloss measurement by the 0 s t w a 1 d K 1 u g h a r t method

From the luminance values (measured in different ways according to different methods) figures are computed which are taken to be a measure of the gloss.

398 REFLECTION, TRANSMISSION, ABSORPTION [XX

2) Polarisation glossmeters In glossmeters of this type (e.g. the Ingersoll Glarimeter) the percentage of polarised light is determined that occurs in the reflected light. Such meters are used chiefly for the measurement of the gloss of paper. In section X-3 it has already been pointed out that the light reflected specularly by electrically non-conductive media (water, glass) is po­larised. At a certain angle practically all the reflected light is polarised in one plane. The light reflected from uniformly diffusely reflecting surfaces is not polarised. On the assumption that the light reflected by paper is in part reflected specularly and is therefore polarised~ the rest being diffusely reflected (unpolarised), the percentage of polarised light is measured at the angle of maximum polarisation in one plane, and this is taken to be a measure of the gloss. Since metals in general do not polarise light on reflection, these pola­risation gloss meters are not suitable for the measurement of metallic surfaces.

3) Objective gloss meters With this type of glossmeter the surface to be tested is illuminated by a beam of light from a given direction, and the reflected light is measured in the specular direction, this being compared with the specularly reflected light from a standard surface, usually of polished black glass. Light reflected in other directions than the specular is thus ignored. The value obtained by measurement is sometimes referred to as the objective gloss, and the instruments as objective glossmeters. The measu­rement gives no indication, however slight, of the form of the reflection indicatrix.

4. Method ~I H a r r i s o n In section X-9 a method of gloss assessment by means of a gloss number, as introduced by Harrison, has been mentioned. For carry~ng out the photometric measurements needed to arrive at the gloss number, only a goniophotometer is required. Since the principle of this kind of apparatus has already been discussed in the beginning of this section we need not give further details as to the method of measurement and we may refer to what has been said in section X-9 on the subject.

Fig. 215. Measurement of the average luminance of a road surface by means of the lumi­nance meter held in the hand.

Fig. 225. Arrangement for measuring the reflective properties of samples of road surfaces.

XX-4] LUMINANCE FACTORS AND GLOSS 399

REFERENCES 1) National Bureau of Standards, Washington, Letter Circular LC-547, March 17,

1939. "Preparation and Colorimetric Properties of a Ma!fnesium-Oxide Reflect­ance Standard"

2) A. H. T a y I or, Sci. Pap. Bur. of Stand. 17, 1922, 1-6, no. 405. "A Simple Portable Instrument for the Absolute Measurement of Reflection and Trans­mission Factors" A. H. T a y I or, J.O.S.A. 25, 1935, 51-56. "Errors in Reflectometry"

3) See for instance: G. A. B o u t r y, 7) Chapter X and R. S e w i g: "Hand­buch der Lichttechnik", P. 359 et seq.

4) J. B e r g m a n s, see 2) Chapter X S) J. M. W a 1 dram, Ill. Eng. 27, 1934, 305-313 and 339-351. "Road Surface

Reflection Characteristics and their Influence on Street Lighting Practice" Also: G.E.C. Journal 6, 1935, 67-86

6) V. G. N. Harrison: "Definition and Measurement of Gloss" London 1945

APPENDIX

INTERNATIONAL CO-OPERATION IN ILLUMINATING ENGINEERING 1)

Anyone who is accustomed to employing the basic photometric quantities and their corresponding units in his daily activities will find it hard to realise that there was a time when all such matters were shrouded in complete darkness. It was at the international Congress of Electricians held at Geneva in 1896 that the system of photometric concepts still in use today was first laid down on the proposals of A. B 1 on de 1. Not long after­wards, in 1900, the gas experts, at an international meeting in Paris agreed upon the necessity for more accurate methods of determining the light-giving power of lighting gas. Arising from this the International Photometric Commission was in­augurated and met at Zurich in 1903, 1907 and 1911. This commission had the greatest interest in establishing an internationally recognised unit of luminous intensity. At that time there were fairly considerable differences among the various countries as to the conception "candle". In order to obtain legal backing, even though in one country only, the scientific co-operation of a state laboratory was essential; the Physikalisch­Technische Reichsanstalt at Charlottenburg, Germany, fulfilled this mission with the Hefner-unit lamp, upon which the "Hefner candle" (HK) was based. By 1907 the National Physical Laboratory at Ted­dington, England, the Bureau of Standards at Washington and the Conservatoire des Arts et Metiers, Paris, had agreed upon a unit of luminous intensity, which was fixed by means of a set of carefully calibrated carbon filament lamps, at a value corresponding to 1.11 HK. The fact that this unit was somewhat prematurely called the Inter­national Candle was quite naturally an obstacle to all endeavours over a number of years to establish a really universal unit of luminous intensity. The "New Candle", based on the radiation of the black body at the freezing point of platinum, was introduced on I st January 1948 by the Bureau International de Poids et Mesures; in the summer of 1948 the Commission lnternationale d'Eclairage (C.I.E.) suggested for this unit the international name "candela" (pronounced candela; abbrev. cd), and this has since been adopted in various countries.

404 APPENDIX

The Commission Internationale d'Eclairage (C.I.E.), (Eng.: International Commission on Illumination, Germ.: Internationale Beleuchtungs­kommission) was established in 1913 out of the old International Photo­metric Commission, as it was realised that illuminating engineering covered a very much wider scope than merely the photometric mea­surements and calculations which can be regarded as the backbone of illuminating engineering. The C.I.E. has set up committees to deal with the following subjects and to report their findings to the General Assembly of the C.I.E.:

Basic Quantities Vocabulary Photometry Colorimetry Colour Rendering Signal Colours Photopic, Mesopic and Scotopic Vision Visual Performance Sources of Visible Radiation Sources of UV and IR Radiation, and Mea­surement Operating Accessories Characteristics of Illuminating Engineering Materials Photometric Requirements for Luminaires Pre-determination of Illumination and Lumi­nance Causes of Discomfort in Lighting Agreable Luminous Environment Home and Hotel Lighting School and Office Lighting Industrial Lighting Mine Lighting Lighting of Public Buildings Hospital Lighting Lighting for Selling Lighting for Stage and Studio Lighting for Indoor Games Lighting in Hazardous and Corrosive Situa­tions

INTERNATIONAL CO-OPERATION IN ILLUMINATING ENGINEERING 405

Daylight Street Lighting Aviation Ground Lighting Railway and Docks Lighting Airborne Lighting and Signals Lighting for Outdoor Sports Automobile Headlights and Signal Lights Floodlighting and Advertising Signs Fundamentals on Traffic Signals Education in Schools, etc. Popular Education Lighting Codes, Regulations and Legislation

At present (1967) 38 countries take part in the activities of the C.I.E. Each country acts as the centre (secretariat committee) for one or more of the abovementioned subjects. This function includes the presentation of a summary report on the progress made since the last meeting at each meeting of the C.I.E. These meetings take place every three to four years. Sessions of the commissions are arranged in the intermediate periods too where necessary. The C.I.E. communicates with several international organisations, a few of which we should like to mention here. The work mentioned <;~-t the beginning of this book on the new unit of luminous intensity was urged as early as 1931, together with the Bureau International des Poids et Mesures. For several years now the C.I.E. has been working successfully on the illumination of airfields for the International Civil Aviation Organisation (I.C.A.O.). Moreover, together with, for example, the International Organisation for Standardisation (I.S.O.) it has been seeking an agreement or. the Uniformisation of Automobile Lighting.

One of the achievements of the C.I.E. has been the plotting of the curves of the international relative luminous efficiency of radiation for the light­adapted and the dark-adapted eye. The system proposed· by the C.I.E. known as the trichromatic system of colour classification and the four-language list of quantities and terms, with definitions, used in lighting engineering are also well known. The meetings of the C.I.E. (Paris 1921, Geneva 1924, Saranac (U.S.A.)

406 AP;I'ENDIX

1928, Cambridge 1931, Berlin 1935, Scheveningen 1939, Paris 1948, Stockholm 1951, Zurich 1955, Brussels 1959, Vienna 1963 and Washing­ton 1967) bring experts in lighting engineering from all over the world together. They thus promote the exchange ofviews and provide a survey of the state of science and technology which is recorded each time in a bulky volume, the "Comptes Rend us".

REFERENCES 1) N. A. Halbertsma, Trans. Ill. Eng. Soc. London 12, 1947,97-107. "International

relations in illuminating engineering". Folder C. I.E.: "What it is, what it is trying to do, how it can be helpful in many countries where it is not now represented", published in 1954 by the Central Office of the C. I.E:.

TABLE I

TABLE I Table of solid angles

407

The size of the conical solid angle cu with half-apex angle ex is expressed in stera­dians. The increment of cu for each increment of 5° in ex is also given.

(X "' Increment (X "' Increment (sterad) of cu (sterad) of cu

oo 0 0.0329 95° 6.83 0.543 50 0.0239 0.0716 100° 7,37 0.535 100 0.0955 0.1186 105° 7.91 0.523 15° 0.214 0.165 110° 8.43 0.506 20° 0.379 0.210 115° 8.94 0.486 25° 0.589 0.253 120° 9.42 (3 .. ) 0.462 30° 0.842 0.294 125° 9.89 0.435 35° 1.136 0.334 130° 10.31 0.404 40° 1.470 0.370 135° 10.72 0.370 45° 1.840 0.404 140° 11.10 0.334 50° 2.24 0.435 145° 11.43 0.294 55° 2.68 0.462 150° 11.72 0.253 60° 3.14 H 0.486 155° 11.98 0.210 65° 3.63 0.506 160° 12.19 0.165 70° 4.13 0.523 165° 12.35 0.1186 75° 4.66 0.535 170° 12.47 0.0716 80° 5.19 0.543 175° 12.54 0.0239 85° 5.74 0.548

I 180° 12.57 (4 .. )

90° 6.28 (2") 0.548

408 APPENDIX

TABLE II Values of coss ex cos3 fJ for a number of values of the angles ex

and fJ (see Fig. 227).

cos3 ex

cos3 fJI Olno II 1.001[)11 0.955 I 0.901 1 0.830 0.650 0.550 0.450 0.353

5° 11 0.989 0.944

10° 11 0.955 11 0.912

15° 11 0.910 11 0.861

20° 11 0.830 11 0.793

25° II 0.744 II 0.711 , __ _!tic___ __

30° 11 0.650 ij 0.621

~II 0.550 11 0.525

40° 11 0.450 11 0.429

45° 11 0.353 11 0.338

50° ·1/ 0.266 ~ 0.254

55° ~ 0.189 11 0.180

~II 0.125 11 0.119

65° 11 0.075511 0.0721

70° ~ 0.0400 11 0.0382

75° 11 0.017311 0.0166

1 0.891

1 0.861

1 0.812

1 0.748

1 0.671

1 0.585

1 0.495

1 0.405

1 0.319

1 0.239

1 0.170

1 0.113

1 0.0680

1 0.0361

1 0.0156

1 0.820 0.642 1 0.543 1 0.444 0.350

1 0.793 1 0.620 1 0.525 1 0.429 1 0.338

1 0.748 1 0.585 1 0.495 1 0.405 1 0.319

1 0.689 1 0.539 1 0.456 1 0.373 1 0.293

1 0.618 1 0.484 1 0.409 1 0.334 1 0.263

1 0.539 1 0.422 1 0.357 1 0.292 1 0.230

1 0.456 1 0.357 1 0.302 1 0.247 1 0.194

1 0.373 1 0.292 1 0.247 1 0.202 1 0.159

1 0.293 1 0.230 1 0.194 1 0.159 1 0.125

1 0.220 1 0.173 1 0.146 1 0.119 1 0.0940

1 0.157 1 0.123 1 0.104 1 0.0849 1 0.0667

1 0.104 1 0.0812 1 0.0687 1 0.0562 1 0.0442

1 0.0626 1 0.0490 1 0.0415 1 0.0339 1 0.0267

1 0.0332 1 0.0260 1 0.0220 1 0.0180 1 0.0141

1 0.0144 1 0.0113 1 0.009541 0.007791 0.00613

TABLE II 409

50° I

55° I

60° 65° I

70° 1 72.5° I

75° 1 76.5° 1 78.5° 80°

0.266 1 0.189 0.125 0.0755 0.0400 0.0272 0.0173 0.0128 0.00792 0.00524

0.263 1 0.187 1 0.124 1 0.0746 1 0.0396 1 0.0269 1 0.0171 0.0126 1 0.007831 0.00518

10.254 1 0.180 1 0.119 1 0.0721 1 0.0382 1 0.0260 1 0.0166 1 0.0122 1 0.007571 0.00500

!

1 0.170 1 0.113 I -~ 1 0.0245 1 0.0156 1 0.0115 1 0.007141 0.00472 0.239 I o.o680 o.o361

--0.220 I I o.157 1 0.104 1 0.0626 1 0.0332 1 0.0226 1 0.0144 1 0.0106 1 0.006581 0.00434

0.198 1 0.140 1 0.0931 1 0.0562 1 0.0298 1 0.0202 1 0.0129 1 0.009471 0.00590 1 0.00390

0.173 1 0.123 1 0.0812 1 0.0490 1 0.0260 1 0.0177 1 0.0113 1 0.008261 0.005151 0.00340

0.146 1 0.104 1 0.0687 1 0.0415 1 0.0220 1 0.0149 1 0.009531 0.006991 0.00436

0.119 1 0.0849 1 0.0562 1 0.0339 1 0.0180 1 0.0122 1 0.007791 0.005721 0.00356

0.0939 1 0.0667 1 0.0442 1 0.0267 1 0.0141 1 0.00961 1 0.006131 0.00450 L

0.0705 1 0.0501 1 0.0332 1 0.0200 1 0.0106 1 0.007221 0.00460 1 0.00338 a ..

0.0501 1 0.0356 1 0.0236 1 0.0142 1 0.007551 0.005131 0.003271 0.00240 {J Ip ,

1 0.0236 1 0.0156 1 0.009441 0.00500 1 0.00340 1 0.00217 -- - 4_ ___ --

0.0332

1 0.0142 1 0.009441 0.00570 1 0.003021 0.002051 0.00131 8

0.0200 -----------178

1 0.007551 0.00500 1 0.003021 0.00160 1 0.001091 0.000694 Fig. 227

0.0106 BP =h tan{J cos a:

p

88

0.00460 1 0.003271 0.002171 0.00131 1 0.0006941 0.0004711 0.000301 I E = .2 cos3 a: coss "' h2 f3

410 APPENDIX

TABLE III tanp

Values of -- for a number of values of the angles IX and p (see Fig. 214, page391) COS IX

IX

II oo

I 100

I 15°

I 20°

I 30°

I 35° 40°

p ·~ 1 1 0.984 0.966 0.940 0.866 0.819 0.766 p

oo 0 I

0 I

0 I 0 I

0 I

0 I

0 I 0 I 50 II 0.0875 11 0.08751 0.0888 I 0.0906 I 0.0931 I 0.101 I 0.107 I 0.114 I

10°11 0.176 11 0.176 1 0.179 I

0.183 I 0.188 I 0.204 I 0.215 I 0.230 I

150 II 0.268 11 0.268 1 0.272 I

0.277 I

0.285 I

0.309 I

0.327 I

0.350 I 20°11 0.364

11 0.364 1 0.370 I

0.377 I 0.387 I

0.420 I 0.444 I

0.475 I 25°11 0.466

11 0.466 1 0.473 I

0.483 I

0.496 I

0.538 I 0:569 I

0.609 I

300 II 0.577 11 0.577 1

0.586 I

0.598 l 0.614 I 0.667 I

0.705 I

0.754 I

35°11 0.700 11 0.700 1 I I

I

I I I 0.711 0.725 0.745

I 0.809 0.855 0.914

400 II 0.839 11 0.839 1

0.852 I

0.869 I

0.893 I

0.969 I

1.02 I

1.09 I

450 II 1.00 111.00 I 1.02

I 1.04

I 1.06 I 1.15 I 1.22 I 1.31

I 500 II 1.19

111.19 I 1.21 I 1.23 I 1.27 I

1.38 I 1.45 I

1.56 I 550 II 1.43 1\1.43 I 1.45 I 1.48 I 1.52 I 1.65

I 1.74 I 1-.86

I 60°11 1.73

111.73 I 1.76 I 1.79 I 1.84 I 2.00 I 2.11 I 2.26 I 65°11 2.14 ~ 2.14 I 2.18

I 2.22

I 2.28 I 2.48 I 2.62 I 2.80 I

70°11 2.75 112.75 1 2.79 I 2.84 I 2.92 I 3.17 I 3.35 I 3.59 I 75°11 3.73 13.73 I 3.79 I 3.86 I 3.97

I 4.31 I 4.56 I 4.87 I

45°

0.707

0

0.124

0.249

0.379

0.515

0.659

0.816

0.990

1.19

1.41

1.69

2.02

2.45

3.03

3.89

5.28

TABLE III 411

50° 55° 6QO 65° 70° 72.5° 75° 76.5° 78.5° 80°

0.643 0.574 0.500 0.422 0.342 0.301 0.259 0.233 L199 I 0.174

I I I I I I~ I

I 0 0 0 0 0 0 0 I 0 0

0.136 I 0.153 I 0.175 I 0.207 I 0.256 I 0.291 I 0.338 I 0.375 I 0.439 I 0.504

0.274 I 0.307 I 0.353 I 0.417 I 0.516 I 0.586 I 0.681 I 0.755 I 0.884 I 1.02

i 0.417 I 0.467 l 0.536 I 0.634 I 0.783 I 0.891 I 1.04 I 1.15 I 1.34 I 1.54 I

0.566 I 0.635 I 0.728 I 0.861 I 1.06 I 1.21 I 1.41 I 1.56 I 1.83 I 2.10

0.725 I 0.813 I 0.933 I 1.10 I 1.36 I 1.55 I 1.80 I 2.00 I 2.34 I 2.69

0.898 I 1.01 I 1.15 I 1.37 I 1.69 I 1.92 I 2.23 I 2.47 I 2.90 3.32

1.09 I 1.22 I 1.40 I 1.66 I 2.05 I 2.33 I 2.71 I 3.00 I 3.51

1.31 I 1.46 I 1.68 I 1.99 I 2.45 I 2.79 I 3.24 I 3.59 4.21

1.56 I 1.74 I 2.00 I 2.37 I 2.92 I 3.33 I 3.86 I 4.28

1.85 I 2.08 I 2.38 I 2.82 I 3.48 I 3.96 I 4.60 I 5.10

2.22 I 2.49 I 2.86 I 3.38 I 4.18 I 4.75 I 5.52 6.12

2.69 I 3.02 I 3.46 I 4.10 I 5.06 I 5.76 I 6.69

3.34 I 3.74 I 4.29 I 5.07 I 6.27 I 7.13 I 8.29

4.27 I 4.79 I 5.49 I 6.50 I 8.03 I 9.14 I 10.6

5.81 I 6.51 7.46 8.83 I 10.9 12.4 I 14.4

412 APPENDIX

TABLE IV Units of illumination and their mutual conversion factors

I lx I mlx I ph I fc ---·

II Lux= lx 1

I 103

I I0-4

I 9.29 X I0-2

Millilux = mlx II

I0-3 I

1 I I0-7 I

9.29 X w-s

Phot =ph II

104 I 107

I 1

I 929

F oat-candle = fc II

10.76 I 10760 11.076 X w-3 I 1

Fig. 228. The relations between the various units of luminance. The units are plotted

on a logarithmic scale

TAB

LE V

T

he u

nits

of

lum

inan

ce a

nd t

heir

mut

ual

conv

ersi

on f

acto

rs

I sb

I

nt

I cd

jft2

I

cdji

n.2

I as

b I

L I

Stil

b =

cdjc

m2

= sb

II

1 I

104

I 92

9 I

6.45

I

3140

0 I

3.14

I

Nit

= c

djm

2 =

nt

II IQ

-4

I 1

I 9.2

9 X

IQ

-21

6.45

X I

Q-4

1 3.

14

13

.14

xi0

-41

cdjf

t2

111.

076

X I

Q-3

1 10

.76

I 1

16.9

4 X

IQ

-31

33.8

I 3

.38

X I

Q-3

1

cdji

n.2

II I

0.15

5 I

1550

I

144

I 1

I 48

70

I 0.

487

I 11

3.18

x 10

-s1

I 2.9

6 X

IQ

-21 2

.05

X I

Q-4

1 I

I

Apo

stil

b =

asb

0.31

8 1

IQ-4

I L

ambe

rt =

L

or l

a II

0.31

8 I

3183

I

296

I 2.

05

1 10

4 I

1 I

I m

L o

r m

la

II 3.

18 X

10-

41

3.18

I

0.29

6 I 2

.05

X I

0-3

1 10

I

IQ-3

I lj

12.2

1 X

I0

-3

foot

lam

bert

=

equi

vale

nt

foot

cand

le =

3.

43 X

10-

4 3.

43

0.31

8 10

.76

1.07

6 X

I0

-3

appa

rent

foo

tcan

dle

= ft

L o

r ft

la

I I I

mL

3140

0.31

4

3.38

487

0.1

103 1

1.07

6

I ft

L

I 29

20

I 0.

292

I 3.

14

I 45

2

19.2

9 X

IQ

-2

I 92

9

I 0.

929

1

I

~ t:J:I t"'

t<l < ~ -V:l

414 APPENDIX

TABLE VI International Relative Luminous Efficiency of Radiation for Photopic Vision ( V~)

A VA A v~

A v~ (m,u) (m,u) (m,u)

525 0.7932 675 0.0232 380 0.0000 530 0.8620 680 0.0170 38~ 0.0001 535 0.9149 685 0.0119 390 0.0001 540 0.9540 690 0.0082 395 0.0002 545 0.9802 695 0.0057

400 0.0004 550 0.9950 700 0.0041 405 0.0006 555 1 705 0.0029 410 0.0012 560 0.9950 710 0.0021 415 0.0022 565 0.9786 715 0.0015 420 0.0040 570 0.9520 720 0.0010

425 0.0073 575 0·9154 725 0.0007 430 0.0116 580 0.8700 730 0.0005 435 0.0168 585 0.8162 735 0.0004 440 0.0230 590 0.7570 740 0.0003 445 0.0298 595 0.6949 745 0.0002

450 0.0380 600 0.6310 750 0.0001 455 0.0480 605 0.5668 755 0.0001 460 0.0600 610 0.5030 760 0.0001 465 0.0739 615 0.4412 765 0.0000 470 0.0910 620 0.3810 770 0.0000

475 0.1126 625 0.3210 775 0.0000 480 0.1390 630 0.2650 485 0.1693 635 0.2170 490 0.2080 640 0.1750 495 0.2586 645 0.1382

500 0.3230 650 0.1070 505 0.4072 655 0.0816 510 0.5030 660 0.0610 515 0.6082 665 0.0446 520 0.7100 670 0.0320

TABLE VII 415

TABLE VII International Relative Luminous Efficiency of Radiation for Scotopic Vision

(Young Eye) (VA') VA' is equal to unity at 507 ffil£

). VA'

). VA'

).

I VA' (mp) (mp) (mp,) I

525 0.880 675 0.0001 380 0.0006 530 0.8ll 680 0.0001 385 O.OOII 535 0.733 685 0.0001 390 0.0022 540 0.650 690 0.0000 395 0.0045 545 0.564 695 0.0000

400 0.0093 550 0.481 405 0.0185 555 0.402 410 0.0348 560 0.3288 415 0.0604 565 0.2639 420 0.0966 570 0.2076

425 0.1436 575 0.1602 430 0.1998 580 0.1212 435 0.2625 585 0.0899 440 0.3281 590 0.0655 445 0.3931 595 0.0469

450 0.455 600 0.0331 455 0.513 505 0.0231 460 0.567 610 0.0159 465 0.620 615 0.0109 470 0.676 620 0.0074

475 0.734 625 0.0050 480 0.793 630 0.0033 485 0.851 635 0.0022 490 0.904 640 0.0015 495 0.949 645 0.0010

500 0.982 650 0.0007 505 0.998 655 0.0005 510 0.997 660 0.0003 515 0.975 665 0.0002 520 0.935 670 0.0001

Qua

ntit

y of

radi

atio

n

Rad

iant

flu

x

Rad

iant

inte

nsity

Rad

iant

em

itta

nce

Rad

ianc

e

Irra

dian

ce

TA

BL

E V

III

The

rad

iati

on q

uant

itie

s an

d th

e co

rres

pond

ing

quan

titi

es i

n il

lum

inat

ing

engi

neer

ing

Q.

= lP

•. t

Qua

ntit

y of

ligh

t Q

=lP

·t

fPe

= ~·

= J l

PoA

d).

L

umin

ous

flux

lP =

? =

Km

f lP

eA ·

VA ·

d).

lP.

Lum

inou

s in

tens

ity

lP

I. =

-I=

-w

w

lP

. M

.=s

Lum

inou

s em

itta

nce

lP

H=

s

1.

lP.

Lum

inan

ce

I lP

L

o=

---=

L

=--=

Sc

osa

cosa

·S·w

S

cosa

co

sa·S

·w

lP.

Illu

min

atio

n lP

E

.= S

E

=-g

~ -a> ~ '1:1

t>l

z t:l ~

LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS 41 7

LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS OF A NUMBER OF LIGHTING FITTINGS

The fittings are classified in accordance with the classification of the C. I.E. (page 58).

f/J = total luminous flux of the fitting f/J 0 = luminous flux in upper hemisphere f/J 0 = luminous flux in lower hemisphere

Fittings for Incandescent lamps

'lb 100

80

60

4()

:JO

0 0 ./

0

v 1/

J 60 0 120 0

White enamelled metal fitting

(direct)

% 100

80

60

4()

20

0 ./ oo v

f/J 0 = f/J

v I

60 0 120 0

17111

Fig. A

1710

Street lighting fitting; steel plate reflector, inside white vitreous enamelled

(direct) <Po=f/J

Fig. B

418 %

TOO

80

60

40

20

0 0

/

./ /

0

APPENDIX

/

60 0 120 0

White enamelled fitting, louvred With bowl-silvered lamv

'*' 100

80

60

40

2 0 oo

/ v

(direct) (/> 0 = (/>

........ v

&7153

Fig. C

Street lighting fitting; steel plate reflector white enamelled, with opal glass ring

0,1, , 00

80

60

40

0 0~

/ j

.... v

(direct) (/> 0 = 0.91 (/>

Fig. D

-......... ~

&765& --"----= 0 180°

White enamelled metal fitting; open at the upper side for illumination of the ceiling

(semi-direct) (/> 0 = 0.85 (/>

Fig. E

LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS 419

OAJ 100

80

60

4()

:lO

0 0

v 0

v v 60 0

-!--

/ v

67&57 .. .-120 1800

Fitting with glass bowl; upper part opal glass, bottom frosted clear glass

% 100

80

60

4()

:JO

0 o• v

/

/

6(1.

(semi -direct) (/) 0 = 0.71 (/)

-1--v

Fitting with opal glass bowl

(semi -direct) (/) 0 = 0.68 (/)

% 100

y

80

60

4()

:JO

0 -· 0

./ !7

-· 60

1--v v

-· /800

Fig. F

6 67858

Fig. G

6 67660

Fitting with glass bowl; upper part opal glas with concave insert of frosted clear glass

(general diffuse) (/) 0 = 0.57 (/)

Fig. H

420 APPENDIX

'II> 100

80

60

40

:;o

v~-""

v

L--v 6 oo"

1.----I:Jo"

Fitting with opal glass bowl

(general diffuse) f/> 0 = 0.55 f/J

" 100-BO

60

40

10

0 o• so• 110" tBO•

17111

Fig. I

Fitting of translucent plastic material; open at the upper side

'Yo 100

BO

60

20

0 0

v 1--J....-

_g 60

(semi-indirect) f/>0 = 0.77 tP

v~"'

I/ l/

120 g

Fig. J

Fitting with reflector of translucent plastic material, open at the upper side, louvred at the bottom

(semi-indirect) f/>0 = 0.75 f/J

Fig. K

LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS 421

100

16)

~· /,.,

v / 17117

0 v 0 60

Metal fitting, closed at the bottom, open at the upper side

(indirect) t[>c = if>

Fig. L

F£ttings for two tubular fluorescent lamps

--- Light distribution perpendicular to the axis of the fitting

- - - - - - - Light distribution in the vertical plane through the axis

180°15fl 1'-fl

0

100

300

o/o 100

900 80

60

40

600 20

0

v I

j ....,~

00 60" 1:JO" 18

Trough fitting, reflector white enamelled, ends open

{direct) if> = tP 0

Fig. M

422 APPENDIX

OJb 100

80

60

40

20

% 100

80

60

/ ......

)

../ v

60 0 120 0

Fitting of Fig. M louvred

~ 1/ f----

I/ j

_........v

I ,....

(direct) (/J c = (/J

~-1--

e

Fig .. N

Fitting of Fig. M with apertures at the top, ends open

(direct) (/J c = 0.9 (/J

Fig. 0

cd

LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS 42 3

tao• t5IJO

% tOO ao 60

40

20

--

/ 0 a•

--

// '/

v 60° t20°

Fitting of Fig. 0 louvred

tao• (semi -direct) ([> 0 = 0.87 ([>

" 100 r-r-T""T""T"""T"-r-r-r-r...,.--r--,-,-,--,-,-,

ao·-60

40 hrrr.~++++++~~,hH

20hr~r+++++++~~~H

~~++++++~~goo QWLLL~~~~~~~~ o• 60" t2o• tao•

60"

e

Fig. P

White, diffusing metal fitting with cover of opal plastic material (direct)

q,o = q, Fig. Q

424

cd

so•

APPENDIX

% toor-T""'..-.-~~~~~,....-,..-,-,--,

40 mm~m:tttttw 20~++~~~~~1-+-~+4

so• t2o• teo•

Fitting with white ceiling plate and cover of opal plastic material (semi-direct)

(/) CJ = 0.85 (/)

% toor-T""'~~~~~~,....-,..-,-,....,

80~++++~+4+4~1-t-~H

so ttt=ttttttittm:$:1:~ 40• ~++~+-J......,.~I-t-1-+-+-++-l

~~++~~~~1-t-t--+-+-++-l

0~~~~~~-LLLLL~

o• so• t2o• teo•

Fig. R

Fitting with white ceiling plate and cover of corrugated clear plastic material

(semi -direct) (/)CJ = 0.8 (/)

Fig. S

LIGHT DISTRIBUTION AND ZONAL LUMINOUS FLUX DIAGRAMS 425

% too lUI

60

AO

20 !JO• 0 o• 60" 120" teo•

Fitting with metal side-screens, open at the top, louvred

'Yo 100

80

1200 60

40

20

tJ k".l:=-lt"+-+-t---1-+-4 900 00

_v

(general diffuse) fP 0 = 0.55 fP

-, v'"'" ~ v

/ 1.---

!..--

Fig. T

Fitting with side-screens of opalescent plastic material, open at the top, louvred

(general diffuse) (/> 0 = 0.50 (/>

Fig. U

INDEX

Where not otherwise indicated, the numbers refer to sections

Abbe's law. Absorption

Measurement of­-factor .....

. .XI-2 X-1, 15

Acceptance angle (of a photometer)

XX-3 . X-1 . X-7 XII-2 Adaption ... .

Albedo .... . Algebraic method . Amplitude .... Angle of divergence . Angstrom unit . . . Angular response of photo-voltaic

. X-5 XV-6

1-2 .XI-6

1-2

cells. . . . . . . . . . XV-3 Aperture of optical systems . XI-8 Apparent foot-candle ... XIII-7, X-6 Apparent surface (of a light source)

. . . . . . . . . . . . VIII-2 Apostilb . . . . . . . . . . VIII-7 Auxiliary lamp of Helwig . . XVII-4 Average luminance . . . . . VIII-I Average luminous intensity .. III-3, IV-7 Azimuthal projection . . V-6

Barrier-layer photo-cells XV-3 Barrier-layer photo-effect XV-3 Beam cross-over point . . . XI-6, XVI--4 Bechstein flicker photometer . . . XIV-4 Black body III-2, XII-6 Bolometers. . . . . XV-4 Bouguer photometer . XIV-2 Brightness . . . . . . VIII-I

Subjective-. . . XII-I, 2 Bunsen grease-spot photometer . XIV-2

Calibration of illumination photo-meters . . . . XVIII-5

Callier coefficient . X-12 Callier effect . . . X-12 Candela . . . . . 111-2 Candelas per em or in. of height. XI-10 Candelas per sq. in. . VIII-I Candle . . . . . . 111-1

Hefner - . . . . 111-2 International - . 111-2

Candle-power . 111-1 Ceasium cell . . . XV-2 Circular light sources IX-3, 4, 8 Classification of lighting fittings. IV-II Coefficient of utilization Vll-2 Cold light . . . . . 1-2 Comparison lamp . . XIV -1 Comparison surface. XIV-I Condenser . . . . . Xl-14

Cone-vision . . . . Continuous spectrum Contrast prism . . . Contrast sensitivity . Corpuscular theory .

XII-7 1-6

XIV-2 Xll-9

Correction factors of photo-cells Critical angle of reflection Cylindrical mirrors . . . . . .

1-1 XV-6 . X-3 Xl-11

Dark current . Dekalumen Densitometers Density ...

Measurement of­Depreciation factor . Diascope ..... Diascopic projection Diffuse cylinder

luminous flux of­Diffuse reflection . . Diffuse reflection factor Diffuse transmission . Diffuser

XV-2 . III-2 XX-2

X-12, XV-5 XX-2

. . VII-2

. . Xl-13

.XI-13, 14

VIII-4 . X-5 . X-5 .X-11

Perfect - . . . . . X-6 Uniform - . . . VIII-2, X-5 Luminous flux of uniform - . VIII-3

Diffusion factor . . . . . . X-13 Direct-comparison method . . . XII-3 Direct lighting . . . . . . . . IV-11 Direct visual measurement of lumi-

nance ...... . Double-refracting prism Drum lens ..... .

Edison's lamp . . . . . . Efficiency of lighting fittings Electromagnetic . . . . . Electromagnetic theory of light Emittance (see luminous emittance)

XIX-I XIV-3 Xl-10

1-5 . IV-4

1-2 1-1

Entrance pupil . XI-5 E.P. diagram. . X-10 Epidiascope . Xl-13 Episcope. . . . XI-13, 19 Episcopic projection. . XI-13, 19 Equality-of-luminance prism . . XIV-2 Equilux spheres of McAllister . . . IX-4 Equivalent foot-candle. VIII-7, X-6 Equivalent luminance . . Xll-8 Ether . . 1-1 Exit pupil XI--4, 5, 7 Exposure X-12, XV-5 Eye-piece ... XIV-I

INDEX

Facetted mirrors . . . . . XI-I2 Filters

absorption of coloured- . X-I6 Filter method ..... XIV-5, XV-6 Flicker photometer (Bechstein) . . XIV-4 Flicker photometry . XII-3, XIV-4 Fluorescent lamps I-6

Measurement of- . XVII-5 f-number . XI-8 Foot-candle . . VI-I Foot-lambert. VIII-7 Fovea centralis XII-7 Fresnel bi-prism XIV-2 Frequency . . I-2 Front-wall cell XV-3 Full radiator . Ill-2, XII-6

Gas-discharge lamps Gas-filled cells . . . Geometrical method Gloss . . . . . . .

Objective - . . . Measurement of­

Gloss number . . . Goniophotometer . . Grease-spot photometer

Harmonic vibration . Heat rays ..... Hefner candle . . . Helwig (auxiliary lamp of-) . Hemispherical candle-power . Heterochromatic photometry

I-6 XV-3 XV-6 . X-9 XX-4 XX-4 . X-9 XX-4

XIV-2

I-2 I-2

. III-2 .XVII-4 . . III-3

. . . . . . . . . . . XII-3, XIV-5 Homochromatic photometry

. . . . . . . . . . . XII-3, XIV-5 Horizontal illumination . . . . Vl-4 Horizontal luminous intensity . III-4

Illuminating engineering Illumination . . . .

Horizontal -Measurement of­Solid of-. Vertical-. Illumination with oblique in-

I-5 Ch. VI

. VI-4 Ch. XVIII

VII-6 . VI-4

cidence . . . . . . . . . . . VI-3 Illumination diagrams . . . . . VII-4 Illumination diagram (rectangular) VII-5 Illumination photometers XVIII-3, 4

Calibration of- - . XVIII-5 Physical - - . . XVIII-4 Visual-- . . . XVIII-3

Incandescent gas light I-5 Incandescent lamps . 1-5 lndicatrix of diffusion X-5 Indirect lighting IV-11 Infra-red radiation . I-2 Integrating photometer (see integra-

tor)

427

Integrator . . . . . . . XVII-I, 2, 3, 4 Approximation to ideal - . XVII-4 Ideal - . . . . . . XVII-2 Non-ideal . . . . . XVII-3

Internal photo-effect XV-3 International candle . . III-2 International luminosity factors. XII-4 International relative luminous effi-

ciency of radiation Photopic vision . Scotopic vision .

Inverse square Jaw Iris . . . . . . . Iso-/1-diagram . . Isocandela diagram !so-luminance diagram Isolux diagram

Plane- ..... . Polar- ..... .

XII-4 XII-7

Vl-2, IX-I XII-2 .X-IO . V-2 .X-IO

in quasi central projection

XII-7 VII-8

VII-IO

Lambert ........ . I-4 Photometric system of- . . I-4, Ch. XII

VIII-7 XIV-2 XIII-2 . IX-3

Lambert (unit of luminance) Lambert photometer Lambert's Jaw . . Law of reciprocity Light

Cold- .... Light distribution .

Polar - - curve . Solid of-- . - - apparatus

Light ether . . . . Lighting fittings

Classification of- - . Efficiency of - -

Light quanta . . Light source

Circular-­Linear-- . Non-point - -Point-­Standard - - .

Light-watt . . . . Linear light source Line spectrum . Lumen Lumen method . Lumen-hour . Lumen-second Luminance ..

Average-. Equivalent -

I-2 Ch. IV

. IV-I . IV-I XVI-3

I-I I-5

IV-11 . IV-4

I-I

IX-3, 4, 8 IX-9, IO . Ch. IX .IX-I, 6 . XIII-2 . XII-6 IX-9, IO

1-6 . III-2 VII-2 . III-6 . III-6

Ch. VIII, IX-I . VIII-I

.... XII-8 Ch. XIX XIX-3

Measurement of­Measurement of- distribution . Measurement of - from illumi-

nation measurement . . . . . XIX-2 Measurement of - of road sur-

faces . . . . . . . . . . . XIX-3

428

Luminance -of images - of light beams -of optics ..

. X-2,XI-3

.... IX-5

. X-2, XI-3 Luminance criterion . Luminance factor . .

.. XII-2 .. X-5, 7, 8

XX-4 Measurement of- - . Measurement of - - of road

surfaces . . . . . . Luminance indicatrix . . Luminance meters (portable) Luminosity . . . . . .

International - factors Luminous efficiency. . .

- - of radiation . . . Relative - - of radiation .

Luminous emittance . . .

XX-4 . X-5 XIX-I XII-I XII-4

0 111-5 XII-6 XII-4 Vlll-5

Measurement of- - . Luminous flux . . . . .

Measurement of- - .

.XVII-7 0 0 III-I Ch. XVII

- - from projection systems - - of optical systems . . . - - of uniform diffuse cylinder - - of uniform diffusers

XI-18 .XI-8 VIII-4 VIII-3

Luminous intensity Average-- ..... Horizontal - - . . . . Mean hemispherical-­Spherical - - . . . . .

111-1, IX-I 111-3, IV-7

Polar curve of- - . . . Luminous influsity indicatrix Lummer-Brodhun prism . Lux .. Luxon ....

Macula lutea. Martens polarisation photometer Maxwellian view . . . . . . . McAllister (Equilux spheres of-) . Mean lower and upper hemispheric-

al candle-power . Measurement of

0 111-4 .111-3 0 111-4 . IV-1 . X-5 XIV-2 . VI-I XII-2

XII-7 XIV-4 XVI-5 . IX-4

0 111-3

absorption . XX-3 density XX-2 emittance . XVII-7 exit pupil . XIX-4 gloss . XX-4 illumination . Ch. XVIII luminance . Ch. XIX luminance distribution . XIX-3 luminance factors. . . . XX-4 luminance of road surfaces . . XIX-3 luminous emittance . . . . . XVII-8 luminous flux . . . . . . Ch. XVII luminous flux of fluorescent lamps

and their fittings . XVII-5 luminous intensity . Ch. XVI photo-current XV-2, 3 projectors . . . . XVI-4 quantity of light . XVII-7

INDEX

Measurement of reflection factor . . . . . . transmission factor . . . .

Mechanical equivalent of light MgO-coated standard plate of re-

flection . Millilambert . . . . . . . . . . Millilux . . . . . . . . . . . . Min. perceptible luminance differ-

ence ..... Mirrors

Cylindrical -Facetted­Luminance of-

Mixed reflection . Mixed transmission Moll thermo-pile . Monochromatic radiation Multiple reflection Munsell scale of luminances

Natural light . New candle . Neutral filter . Neutral wedge Nicol ... . Nit .... . Non-point sources

Objective .... Objective gloss . .

--meters .. Optical pyrometer Optical system . . Optical systems for image-projec­

tion. Optics ...... .

Parallel beam of light Perfect diffuser Periodic time . Phase .... Phot Photo-cathode Photo-cell

Barrier-layer - . Caesium-Correction factors for - . Front-wall -Gasfilled. . . . Photo-voltaic -Potassium - . Rear-wall-Selenium-Vacuum-

Photo-effect Barrier-layer - . External­Internal- ...

XX-I XX-2 XII-6

XX-I VIII-7 . Vl-5

XII-9

Xl-11 XI-I2 .XI-3 . X-5 .X-I4 XV-4

1-2 .X-14 XII-9

1-3 . IV-2 XIV-3 XIV-3 XIV-3 VIII-I Ch. IX

.XI-7 XX-4 XX-4

XIX-I . XI-I

XI-I3 .XI-I

.XI-6 X-6 1-2 1-2

. VI-5 XV-2

XV-3 XV-2 XV-6 XV-3 XV-2 XV-3 XV-2 XV-3 XV-3 XV-2

XV-3 XV-2 XV-3

INDEX

Photometer Bechstein flicker XIV -4 Bouguer- XIV-2 Bunsen- . . XIV-2 Gonio. . . . XX-4 Grease-spot- XIV-2 Illumination - . XVIII-3 Lambert- XIV-2 Martens polarisation- XIV-4 Rumford- . . . . XIV-2 Weber- . . . . . XIV-4 with Maxwellian view XVI-4

Photometer bench XIV-4, XVI-2 Photometer field . . XIV-I, 2 Photometer head . . XIV-I Photometric balance .... XIV-I Photometric system (of Lambert)

. . . . . 1-4, Ch. XII Photometry

Flicker­Heterochromatic - . Homochromatic -Photographic -Physical-. Visual-

Photon ..... Photo-multiplier cell Photopic vision . . . Photo-voltaic cell . .

... XII-3 XII-3, XIV-5 XII-3, XIV-5

. XV-5

. Ch.XV

.Ch. XIV 1-1

XV-2 XII-7 XV-3

Physical illumination photometer Physical photometry

XVIII-4 Ch.XV XII-6 VII-7

Planck's formula . . Plane isolux diagram Plane of polarisation Point-by-point method Point source . . . . .

1-3

Polar curve of luminous intensity Polarisation . . . . . Polarisation photometer Polarisation gloss meter

VII-3 . IX-1, 6

. IV-I I-3

XIV-4 XX-4 VII-8 XV-2

Polar isolux diagram . Potassium cell . . . . Preferred numbers

in isocandela diagrams in isolux diagrams

Primary standard . . . . - - of luminous intensity .

Projection Azimuthal - . Diascopic - . Episcopic - . Quasi central -Sinusoidal - . . Slide- .... Standard-film -Sub-standard film -

Projectors, measurement of­Purkinje effect . . . . . . .

. V-9 Vll-8

XIII-2 . III-2

. V-6 .XI-13, 14 . Xl-13, 19

. V-8

. V-4 XI-15 Xl-16 XI-17 XVI-4 XII-7

Quantity of light . . . . . Measurement of- - -

Quantum theory . . . Quasi central projection

Rear-wall cell Reciprocity, law of- . Red limit ..... Reflection . . . . .

Critical angle of­Diffuse- . Mixed- . Multiple­Regular-. from Road surfaces Specular­Spread- .... Uniform diffuse-

Regular reflection . . Regular reflection factor . Regular transmission . . Relative !urn. eff. of radiation Retina ....... . Reflection factor . . .

- - of Conductors Diffuse-- .... Measurement of- - . -- of Non-conductors. Regular--

Ritchie wedge Rod vision ... Room index .. Rotating sector disc . Rousseau diagram Rumford photometer Russel-Bloch angles .

429

. III-6 .XVII-7

1-l . V-8

XV-3 . IX-3 XV-2

X-1 X-3 X-5 X-5

.X-14

. X-2

.X-10 X-2 X-5 X-6 X-2 X-5

.X-ll XII-4 XII-2

X-1 . X-4 . X-5 XX-1 . X-3 . X-5 XIV-2 XII-7 XII-2

XIV-3 Ch. IV

XIV-2 . IV-8

Scotopic vision . . . XII-7 Sea-mile candle . VI-5 Secondary-emission cells . XV-2 Secondary standards XIII-2 Selenium cell. . . . . . . XV-3 Sensitivity to luminance difference. Xll-9 Simultaneous method . Xlll-3 Sine condition . . . . XI-3 Sinusoidal projection . V-4 Slide projection. Xl-15 Snell's law . . . . . . X-3 Solid angle . . . . . . Ch. II

Table of solid angles page 407 Solid of illumination VII-6 Solid of light distribution . IV-1 Spectrum

Continuous - . Line- ....

Specular reflection Sphere paints . . Sphere window . . Spherical candle . Spherical co-ordinates .

1-6 1-6 X-2

.XVII-4

.XVII-I . III-3, 4 . . V-3

430

Spherical luminous intensity Spread reflection . . . . Spread transmission . . . Standard-film projection . Standard light sources. Standard observer Step-by-step method Steradian ..... Stilb . . . . . . . Stiles-Crawford effect Stilling's colour tables . Stray light . . . . . .

. III-4

. X-5

. X-11 XI-I6 XIII-2 XIV-6 XII-3 . II-I VIII-I XII-2

XIV-6 XIII-4

XII-I, 2 XIII-2 XI-I7 XIII-3

Subjective brightness . Sub-standard . . . . . Sub-standard film projection Substitution method . . . . Summation law . . . . . . Systems of obtaining variation in

XII-5

luminance in photometers ... · XIV-3

Talbot's law . . . . . . . . . . XIV-3 Taylor's method of measurement of

reflection factor . . . . . . Technique of light production Temperature radiator . Thermo-couple . . . . Totally reflecting prism Transmission .

Diffuse- . Mixed- . Regular-. Spread- . Uniform diffuse-

Transmission factor . Measurement of- - .

XX-I I-5 I-2

XV-4 . X-3

X-I, II .X-11 .X-11 .X-11 .X-11 . X-11 . X-I XX-2

INDEX

Transmissive exponent . Tungsten lamps . . . . Tungsten ribbon lamp .

.X-I5 I-6

XIII-2

Ulbricht sphere. . . . . . . . XVII-I Ultra-violet radiation . . I-2 Uniform diffuser . . . . VIII-2, X-5 Uniform diffuse reflection . . . X-6 Uniform diffuse transmission. . . . X-1I

Vacuum cell . . . . Variable diaphragm . Velocity of radiation Vertical illumination Vignetting . . . . . . . . . . Visual illumination photometer . Visual photometry Visual purple . .

Wavelength . . Wave-mechanics Wave-motion. . Wave theory . . Weber photometer Weber's law ... Weber-Fechner's law Wollaston prism . Working standards

Yellow-blue ratio Yellow spot . . .

Zonal luminous flux diagram

. XV-2 XIV-3

I-2 . VI-4 .XI-7

XVIII-3 .Ch. XIV

XII-7

I-2 I-I I-2 I-1

XIV-4 Xll-9 XII-9

XIV-3 XIII-2

XIV-6 XII-7

IV-5