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SPIE PRESS Bellingham, Washington USA
George Asimellis
LECTURES IN OPTICS
Volume 2
GeometricalOptics
Library of Congress Cataloging-in-Publication Data
Names: Asimellis, George, 1966- author.
Title: Geometrical optics / George Asimellis.
Description: Bellingham, Washington, USA : SPIE Press, [2020] | Series:
Lectures in optics ; vol. 2 | Includes index.
Identifiers: LCCN 2019001142| ISBN 9781510619456 (softcover) | ISBN
1510619453 (softcover) | ISBN 9781510619463 (pdf) | ISBN 1510619461 (pdf)
Subjects: LCSH: Geometrical optics. | Refraction. | Reflection (Optics)
Classification: LCC QC381 .A85 2019 | DDC 535/.32--dc23 LC record available at https://lccn.loc.gov/2019001142
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COVER IMAGE: ‘UMBRELLAS’ – MODERN ART CREATION BY GEORGE ZOGGOPOULOS, IN THESSALONIKI, GREECE,
IMAGED VIA A BALL LENS.
IMAGE CREATION: EFSTRATIOS I. KAPETANAS. FACEBOOK.COM/PHOTOSTRATOSKAPETANAS/
“Μηδείς ἁγεωμέτρητος εἱσίτω”
Πλάτων.
300 BC
Greek mathematician Euclid (Εὐκλείδης), the Father of Geometry, authored Optica (Οπτική). Euclid
asserted that light travels in straight lines and proposed mathematical formulae for reflection and
refraction. Shown above is one of the oldest preserved papyrus sheets with his writings.
GEOMETRICAL OPTICS
i
TABLE OF CONTENTS
Table of Contents .............................................................................................................................................................................. i
Foreword ............................................................................................................................................................................................ vii
Preface ................................................................................................................................................................................................. ix
About this Series .................................................................................................................................................................. ix
About this Book ..................................................................................................................................................................... x
Acknowledgments ............................................................................................................................................................. xiii
1 REFRACTION AT A SPHERICAL INTERFACE .................................................................................... 1-1
1.1 Reflection and Refraction ................................................................................................................................................ 1-1
1.1.1 The Laws of Reflection ................................................................................................................................... 1-1
1.1.2 The Refractive Index ........................................................................................................................................ 1-2
1.1.3 The Laws of Refraction ................................................................................................................................... 1-4
1.1.4 Critical Angle and Total Internal Reflection .......................................................................................... 1-6
1.2 The Single Spherical Refracting Interface ................................................................................................................. 1-9
1.2.1 Curvature and Radius of Curvature .......................................................................................................... 1-9
1.2.2 The Convex and Concave SSRI ................................................................................................................ 1-11
1.2.3 Refraction by an SSRI .................................................................................................................................. 1-11
1.2.4 Optical Power and Focal Length ............................................................................................................. 1-13
1.2.5 The Flat SSRI .................................................................................................................................................... 1-19
1.2.6 The Nodal Point in an SSRI ....................................................................................................................... 1-20
1.2.7 The Listing Eye Model as an SSRI ........................................................................................................... 1-21
1.3 Advanced Practice Examples ...................................................................................................................................... 1-22
1.4 Refraction Quiz ................................................................................................................................................................. 1-25
1.5 Refraction Summary ....................................................................................................................................................... 1-34
2 LENS REFRACTION AND POWER ............................................................................................... 2-37
2.1 What is a Lens? ................................................................................................................................................................. 2-37
2.2 Principles of Lens Operation ....................................................................................................................................... 2-40
2.2.1 Refraction by Two Surfaces ....................................................................................................................... 2-40
2.2.2 Two Prisms ....................................................................................................................................................... 2-40
2.2.3 Principle of Least Time ................................................................................................................................ 2-42
2.2.4 Wavefront Transformation ........................................................................................................................ 2-44
2.2.5 The Gravitational Lens ................................................................................................................................. 2-45
2.3 The Thin Lens ..................................................................................................................................................................... 2-48
LECTURES IN OPTICS
ii
2.3.1 Radii of Curvature and Material .............................................................................................................. 2-48
2.3.2 Primary and Secondary Focal Points ..................................................................................................... 2-49
2.3.3 Focal Planes and Optical Axis................................................................................................................... 2-51
2.3.4 Lens Shape ....................................................................................................................................................... 2-53
2.4 Lens Optical Power ......................................................................................................................................................... 2-56
2.4.1 Lens-Maker’s Formula ................................................................................................................................. 2-56
2.4.2 Dependence on Orientation, Media, and Geometry ...................................................................... 2-58
2.5 Advanced Practice Examples ...................................................................................................................................... 2-63
2.6 Lens Power Quiz ............................................................................................................................................................... 2-65
2.7 Lens Power Summary ..................................................................................................................................................... 2-68
3 IMAGING DEFINITIONS ............................................................................................................ 3-71
3.1 Object and Image ............................................................................................................................................................ 3-71
3.1.1 Real and Virtual Object; Real and Virtual Image ............................................................................. 3-73
3.1.2 Object and Image in a Plane Mirror ...................................................................................................... 3-75
3.2 Sign Conventions ............................................................................................................................................................. 3-77
3.2.1 Object / Image Height and Angle Sign Conventions..................................................................... 3-78
3.3 Magnification .................................................................................................................................................................... 3-79
3.3.1 Angular Magnification ................................................................................................................................ 3-80
3.4 Vergence ............................................................................................................................................................................. 3-82
3.4.1 Wavefront Vergence .................................................................................................................................... 3-82
3.4.2 Vergence and Propagation ....................................................................................................................... 3-84
3.4.3 Vergence and Optical Interfaces............................................................................................................. 3-87
3.5 Vergence in Imaging ...................................................................................................................................................... 3-89
3.5.1 Vergence of a Real and a Virtual Object ............................................................................................. 3-89
3.5.2 Vergence of a Real and a Virtual Image .............................................................................................. 3-90
3.5.3 Upstream and Downstream Vergence in Lens Imaging ............................................................... 3-91
3.5.4 Vergence and a Flat Refracting Interface ............................................................................................ 3-94
3.5.5 Vergence and the SSRI Power ................................................................................................................. 3-97
3.6 Advanced Vergence Examples ................................................................................................................................... 3-98
3.7 Vergence and Imaging Concepts Quiz .................................................................................................................3-103
3.8 Vergence and Imaging Concepts Summary .......................................................................................................3-106
4 IMAGING WITH LENSES ......................................................................................................... 4-109
4.1 Lens Imaging Relationship .........................................................................................................................................4-109
4.1.1 Imaging: Left to Right or Right to Left? ............................................................................................4-113
4.1.2 Image Magnification ..................................................................................................................................4-114
4.1.3 Image Reversal .............................................................................................................................................4-116
GEOMETRICAL OPTICS
iii
4.1.4 Newton’s Imaging Relationship ............................................................................................................4-117
4.2 Imaging by a Plus-Powered Lens ............................................................................................................................4-118
4.2.1 Summary .........................................................................................................................................................4-122
4.3 Ray Diagrams...................................................................................................................................................................4-124
4.3.1 Ray Diagrams for a Plus-Powered Lens .............................................................................................4-124
4.3.2 Some Considerations About Construction Rays ............................................................................4-125
4.3.3 Ray-Tracing Examples with Plus Lens Imaging ...............................................................................4-128
4.4 Imaging by a Minus-Powered Lens........................................................................................................................4-130
4.4.1 Ray Diagrams for a Minus-Powered Lens .........................................................................................4-131
4.5 Imaging by an SSRI .......................................................................................................................................................4-134
4.6 Notes on Imaging ..........................................................................................................................................................4-137
4.6.1 The Optical Invariant ..................................................................................................................................4-137
4.6.2 The Wild Ray .................................................................................................................................................4-140
4.6.3 Optical Infinity ..............................................................................................................................................4-141
4.7 Virtual Object Imaging ................................................................................................................................................4-143
4.7.1 Virtual Object Imaging with a Plus Lens ............................................................................................4-144
4.7.2 Virtual Object Imaging with a Minus Lens........................................................................................4-147
4.8 Advanced Lens-Imaging Examples ........................................................................................................................4-153
4.9 Lens Imaging Quiz .........................................................................................................................................................4-156
4.10 Lens Imaging Summary...............................................................................................................................................4-161
5 IMAGING WITH MIRRORS ...................................................................................................... 5-165
5.1 Plane Mirror Principles ................................................................................................................................................5-165
5.1.1 The Cartesian Convention in Mirrors ..................................................................................................5-167
5.1.2 Multiple Plane Mirror Surfaces ..............................................................................................................5-168
5.2 Spherical Mirrors ............................................................................................................................................................5-170
5.2.1 Geometry of a Spherical Mirror ............................................................................................................5-170
5.2.2 Focal Points and Focal Lengths in a Spherical Mirror ..................................................................5-174
5.2.3 Nodal Point in a Spherical Mirror .........................................................................................................5-175
5.2.4 Optical Power in a Spherical Mirror.....................................................................................................5-176
5.2.5 Vergence and Spherical Mirror Power ...............................................................................................5-178
5.3 Imaging with a Spherical Mirror ..............................................................................................................................5-180
5.3.1 Spherical Mirror Imaging Relationship ..............................................................................................5-180
5.3.2 Convex Mirror Imaging Examples ........................................................................................................5-182
5.3.3 Ray Diagrams for Convex Mirrors ........................................................................................................5-183
5.3.4 Concave Mirror Imaging Ray Diagrams and Examples ...............................................................5-188
5.3.5 The Virtual Object in Mirror Imaging ..................................................................................................5-196
LECTURES IN OPTICS
iv
5.4 Advanced Mirror Imaging Examples .....................................................................................................................5-202
5.5 Mirror Imaging Quiz .....................................................................................................................................................5-205
5.6 Mirror Imaging Summary ...........................................................................................................................................5-210
6 THICK LENSES AND LENS SYSTEMS ......................................................................................... 6-215
6.1 The Thick Lens .................................................................................................................................................................6-215
6.1.1 Equivalent Optical Power and Focal Length ....................................................................................6-215
6.1.2 Specialty Lenses ...........................................................................................................................................6-221
6.1.3 The Cornea Equivalent Power ................................................................................................................6-221
6.2 Cardinal Points: Concept and Applications.........................................................................................................6-223
6.2.1 Principal Points and Principal Planes ..................................................................................................6-223
6.2.2 Nodal Points ..................................................................................................................................................6-229
6.2.3 Cardinal Points in an SSRI and a Mirror .............................................................................................6-231
6.3 Vertex Powers in a Thick Lens ..................................................................................................................................6-233
6.3.1 The Concept of Front and Back Vertex Power and Focal Length ...........................................6-233
6.3.2 The Measure of Vertex Powers ..............................................................................................................6-234
6.4 Imaging with a Thick Lens ..........................................................................................................................................6-240
6.4.1 Ray Propagation in a Thick Lens ...........................................................................................................6-240
6.4.2 Ray Diagrams in a Thick Lens .................................................................................................................6-242
6.4.3 The Wild Ray in a Thick Lens ..................................................................................................................6-244
6.4.4 Thick Lens Imaging Relationship ..........................................................................................................6-245
6.5 Lens Systems....................................................................................................................................................................6-249
6.5.1 Optical Power in a Lens System ............................................................................................................6-249
6.5.2 Cardinal Points in a Thin Lens System ................................................................................................6-252
6.5.3 The Afocal System.......................................................................................................................................6-254
6.5.4 Cardinal Points in a Thick Lens System ..............................................................................................6-257
6.5.5 Intermediate Image Technique in Two or More Lenses .............................................................6-258
6.5.6 The Thick Lens as a Two-Element System ........................................................................................6-263
6.6 Advanced Practice Examples ....................................................................................................................................6-265
6.7 Thick Lens and Cardinal Points Quiz .....................................................................................................................6-278
6.8 Thick Lens and Cardinal Points Summary ...........................................................................................................6-289
7 FINITE TRANSVERSE OPTICS ................................................................................................... 7-295
7.1 Aperture Stop and Pupils ...........................................................................................................................................7-296
7.1.1 The Aperture Stop.......................................................................................................................................7-296
7.1.2 Significance of the Aperture Stop ........................................................................................................7-304
7.1.3 Entrance and Exit Pupil .............................................................................................................................7-306
7.1.4 Numerical Aperture and F-number .....................................................................................................7-315
GEOMETRICAL OPTICS
v
7.2 Principal / Chief and Marginal Rays ........................................................................................................................7-317
7.2.1 The Principal / Chief Ray ..........................................................................................................................7-317
7.2.2 The Marginal Rays.......................................................................................................................................7-318
7.3 Fields, Stops, and Related Effects............................................................................................................................7-325
7.3.1 Field of View ..................................................................................................................................................7-325
7.3.2 The Field Stop ...............................................................................................................................................7-327
7.3.3 Entrance and Exit Windows / Ports ......................................................................................................7-331
7.3.4 Size of the Field of View ...........................................................................................................................7-334
7.3.5 Fields of Half and Full Illumination ......................................................................................................7-342
7.3.6 Vignetting and Glare ..................................................................................................................................7-344
7.4 Depth of Field and Depth of Focus ........................................................................................................................7-347
7.5 Brightness, Contrast, and Resolution ....................................................................................................................7-353
7.6 Geometrical Image Blur ..............................................................................................................................................7-358
7.7 Advanced Practice Examples ....................................................................................................................................7-363
7.8 Transverse Optics Quiz ................................................................................................................................................7-377
7.9 Transverse Optics Summary ......................................................................................................................................7-390
8 OPTICAL ABERRATIONS ........................................................................................................ 8-395
8.1 Imaging to an Idealized Point ..................................................................................................................................8-395
8.1.1 The Origin of Optical Aberrations ........................................................................................................8-397
8.1.2 The Paraxial Approximation ....................................................................................................................8-398
8.1.3 Classification of Optical Aberrations ...................................................................................................8-400
8.2 Chromatic Aberration ..................................................................................................................................................8-402
8.2.1 Management of Chromatic Aberration .............................................................................................8-405
8.3 Monochromatic Aberrations .....................................................................................................................................8-407
8.3.1 Spherical Aberration ..................................................................................................................................8-407
8.3.2 Coma ................................................................................................................................................................8-415
8.3.3 Oblique / Radial Astigmatism .................................................................................................................8-420
8.3.4 Field Curvature and Distortion ..............................................................................................................8-423
8.4 Aberrations Quiz ............................................................................................................................................................8-428
8.5 Aberrations Summary ..................................................................................................................................................8-431
APPENDIX ...................................................................................................................................... 433
Conventions and Notation ...................................................................................................................................................... 433
Conventions ....................................................................................................................................................................... 433
Object-Space versus Image-Space Notation ....................................................................................................... 433
The Cartesian Sign Convention ................................................................................................................................. 434
Frequently used Notation ............................................................................................................................................ 435
LECTURES IN OPTICS
vi
Useful Notes ...................................................................................................................................................................... 436
Geometrical Optics Formulation ........................................................................................................................................... 437
Refraction ............................................................................................................................................................................ 437
Vergence, Optical Power and Focal Lengths, and Imaging ........................................................................... 439
Imaging Relationships ................................................................................................................................................... 443
Optics of the Human Eye ............................................................................................................................................. 445
Answers To Quiz Questions .................................................................................................................................................... 447
Index ................................................................................................................................................................................................. 449
GEOMETRICAL OPTICS
vii
FOREWORD
At the very beginning of my optics classes, I promise my students that I will probably turn a number of
them into optics geeks. At the very least, I hope that I am able to make them more aware that optics is
everywhere. By sending them out into the world with their cameras and the goal of finding and capturing
optical phenomena, they do indeed start to develop an optics awareness and perhaps a bit of optics
geekiness. It is, after all, the core of what we do as Optometrists.
When Dr. Asimellis asked me to provide feedback on his Geometric Optics, I took it on thinking that it
wouldn't hurt to see another approach to the subject. I’d already amassed a sizable collection of optics
texts spanning at least 50 years of approaches, always looking for different perspectives. Once I started
reviewing the lectures, I realized that George’s take on the subject was more of a journey through the
concepts with a narrative that draws you in, rather than a series of derivations and formulas. It turns out I
had found a kindred spirit. The interactions, discussions and, yes, even sometimes disagreements on the
topic were enjoyable for us and I believe translated into a valuable resource for the educator and the
student of optics.
The material is presented in a manner that provides a way to visualize the concepts, while still
following well-established notation and formulation. The real value of this book is its didactic approach:
emphasis is given to understanding the effects; the mathematical formulation then follows naturally and is
easier to comprehend. Step by step, students are guided from the simple effects of refraction and
reflection to the refractive effects of lenses and prisms, then on to the more complex concepts of thick
lenses and lens systems, and the optics of lateral restriction. Educators will be pleased to see the
complete coverage of the material prescribed in most optics curriculae in optometric education.
Corina van de Pol, OD, PhD, FAAO
Assistant Professor, Southern California College of Optometry, Marshall B. Ketchum University
Fullerton, California
June, 2020
GEOMETRICAL OPTICS
ix
PREFACE
“Geometrical Optics is either very simple or else it is very complicated.”
Richard Feynman, The Feynman Lectures on Physics, Volume I, 27-1
About this Series
Optics is fundamentally simple. At first glance, optics can be, indeed, formidably complicated. Sign
conventions are difficult to memorize, the reciprocal of (meter-converted) distances involved in imaging
equations are hard to rationalize, and focal distances are impossible to add. These are just a few of the
hurdles encountered in the ostensibly easier part of optics, that of geometrical optics. Throw into the mix
the wave nature of light, the complicated integrals involved in the description of light propagation
through a small aperture, or some aspects of interference and polarization, and you have the perfect
recipe for confusion.
This perspective is permeated by the fact that optics instruction is fragmented, most often as part
of a Physics 102 course or sometimes as part of a classic electromagnetism curriculum. The presentation
of optics as a whole is rare.
As a graduate student, I enrolled in two courses, one in Fourier Optics and another in Teaching
Methodology. The recommended books were Introduction to Fourier Optics by Joseph W. Goodman and
The Feynman Lectures on Physics by Richard Feynman. Albeit uncorrelated, these two courses changed
my view of optics forever. I appreciated how certain phenomena can be explained in a straightforward
manner, for example, through a simple Fourier transform, or by the connection between quantum physics,
phase diagrams, and interference.
Geometrical optics can be vastly simplified if we adhere to the Cartesian convention and the
vergence method. Breaking away from the traditional approach, this formulation provides a much simpler
and unified tool to address imaging in geometrical optics, and, to a substantial extent, in visual optics.
The philosophy that optics is simple permeates this book series. Once the reader appreciates this
essential simplicity, it is a lot easier to build the foundation of fundamental knowledge, from the basics all
the way to the more esoteric topics. I feel that without an understanding of this basic simplicity on which
to develop, the structure of accumulated knowledge is unsteady at best, and at worst, will crumble under
its own weight.
I hope that this second volume of the Lectures in Optics series will be appreciated by those
seeking a bottom-up textbook, fitting the needs for any college-level optics or optometry optics course.
George Asimellis, PhD
Pikeville, Kentucky
June 2020
LECTURES IN OPTICS
x
About this Book
Geometrical optics is perhaps among the most challenging courses in many programs, including
optometric education. The material comes, unfairly in my opinion, with a reputation of being hard and
challenging.
I side with the ‘challenging,’ topping it with ‘rewarding.’ Optics is the foundation of how the eye works,
how we image the eye for diagnosis, and, progressively, how we use many laser-based therapeutic and
cosmetic applications. Optics helps to develop critical thinking skills that are necessary in a successful
diagnostic career such as that of an Optometrist or an Optical Engineer.
This book builds on the previous volume 1, Introduction to Optics, mainly on the topics of refraction and
its applications to prism deviation and simple optical instruments. The present volume 2, Geometrical
Optics, completely develops the instructional requirements pertaining to the foundation of the topic,
including: refraction at a spherical surface (SSRI), lens refraction, and imaging by lenses, SSRIs, and
mirrors; thick lenses and optics of stops and pupils; and optical aberrations. The material is presented at a
level applicable to medical students with a limited optical science background and covers the rubric
presented by the National Board of Examiners in Optometry. Emphasis is placed on conceptual
development, with ample examples ranging from very simple to advanced practice exercises.
Using the two books, Introduction to Optics (ItO) and Geometrical Optics (GO), the following brief
curriculum structure can be used as a general guideline in order to deliver an introductory and
foundational 4-credit optics course (50 + 2 lecture hours).
Unit 1 (4 hours): Light, Rays, Wavefronts, Vergence, Reflection and Refraction
ItO, Chapter 1 (§ 1.3 Propagation of Light, § 1.4 Index of Refraction, § 1.5 Light-Matter Interactions)
ItO, Chapter 2 (§ 2.1 Angle Measurement)
ItO, Chapter 3 (§ 3.1 Reflection, § 3.2 Refraction, § 3.4 Refraction Applications, § 3.5.1 Refractive Atmospheric
Phenomena)
Unit 2 (4 hours): Prisms and Color Dispersion
ItO, Chapter 3 (§ 3.3 Prisms, § 3.5.2 Prismatic Atmospheric Phenomena)
Unit 3 (4 hours): The Single Refracting Spherical Interface
GO, Chapter 1: Refraction in a Spherical Interface
Unit 4 (4 hours): Lenses and Lens Power
GO, Chapter 2: Lens Refraction and Power
GO, Chapter 6: Thick Lenses and Lens Systems (§ 6.1 The Thick Lens)
Unit 5 (4 hours): Imaging Definitions and Vergence
GO, Chapter 3: Imaging Definitions and Vergence
Unit 6 (8 hours): Lens Imaging
GO, Chapter 4: Imaging with Lenses
GEOMETRICAL OPTICS
xi
Unit 7 (4 hours): Mirror Imaging
GO, Chapter 5: Imaging with Mirrors
Unit 8 (6 hours): Imaging with Thick Lens and Lens Systems
GO, Chapter 6: Thick Lenses and Lens Systems (§ 6.2 Cardinal Points: Concept and Applications, § 6.3 Vertex
Powers in Thick Lens, § 6.4 Imaging with a Thick Lens, § 6.5 Lens Systems)
Unit 9 (10 hours): Pupils, Stops, and Related Effects
GO, Chapter 7: Finite Transverse Optics (§ 7.1 Aperture Stop and Pupils, § 7.2 Principal / Chief and Marginal Rays,
§ 7.3 Fields, Stops, and Related Effects)
ItO, Chapter 5 (§ 5.1.2 Microscope Principle of Operation, § 5.2.2 Telescope Principle of Operation)
Unit 10 (4 hours): Simple Optical Instruments
ItO, Chapter 4 (§ 4.1 Camera Obscura, § 4.2 The Human Eye, § 4.3 The Magnifying Lens)
Unit 11 (8 hours): Image Quality and Optical Aberrations
GO, Chapter 7: Finite Transverse Optics (§ 7.4 Depth of Field and Depth of Focus, § 7.5 Brightness, Contrast, and
Resolution, § 7.46 Geometrical Image Blur)
GO, Chapter 8: Optical Aberrations
REFRACTION IN A SPHERICAL INTERFACE
1-11
1.2.2 The Convex and Concave SSRI
The shape of the spherical surface comprising an SSRI can be convex [Figure 1-11 (left)] or
concave [Figure 1-11 (right)]. By definition, a convex interface wraps around a medium of
higher refractive index (center of curvature situated in the higher-index medium, e.g., glass),
while a concave interface wraps around a medium of lower refractive index (center of curvature
situated in the lower-index medium, e.g., air).
Figure 1-11: (left) A convex SSRI. (right) A concave SSRI.
1.2.3 Refraction by an SSRI
As in the case for any refracting surface, a ray incident on an SSRI is refracted. To properly draw
the refracted ray from a spherical surface, we apply the law of refraction. We assume the simple
case of a convex SSRI that separates air from glass and a ray that propagates parallel to the
optical axis, striking the SSRI from the air side (Figure 1-12). The first step is to identify the
center of curvature, which is the center of the hypothetical spherical surface that defines the
SSRI. We then identify the normal to the surface at the point of incidence; this line draws along
the spoke that connects the point of incidence to the center of curvature.
Figure 1-12: Identification of the center of curvature and the normal to the surface at the point of incidence.
GEOMETRICAL OPTICS
1-36
Ray-Tracing Rules in an SSRI
1. The ray parallel to the optical axis refracts to the secondary focal point.
2. The ray originating from (or crossing through) the primary focal point refracts to become
parallel to the optical axis.
3. The ray targeting the center of curvature refracts without any ray deviation (it crosses the
center of curvature, which is the nodal point).
The topic of ray-tracing rules in an SSRI is further discussed in § 4.5.
Figure 1-27: Summary of ray tracing in a convex SSRI.
Figure 1-28: Summary of ray tracing in a concave SSRI.
GEOMETRICAL OPTICS
2-56
2.4 LENS OPTICAL POWER
2.4.1 Lens-Maker’s Formula
A lens is a combination of two refracting surfaces with radii of curvature r1 and r2 that separate a
medium with refractive index n lens from a surrounding medium with index next.
Figure 2-35: Positive (left) and negative (right) radius of curvature in a lens.
To calculate the lens optical power, each surface that defines the lens may be
considered as an independent surface. Then, we just add the individual optical powers F1 and F2
of each refracting surface, using Eq. (1.8), which expresses the SSRI optical power. The SSRI
optical power is proportional to the difference of the refractive indices and is inversely
proportional to the radius of curvature. Consider a lens made of glass with refractive index n lens
surrounded by a medium with next:
2lens ext ext lens
2
1
1
and n n
Fr
n nF
r
−=
−= (2.2)
( )llen
elens e
1
ext ns2
2 2
sxt
e1
1
xt
1 1F
n nF
r
n
r
nF
rF
rn n
= + = + = −
−−
− (2.3)
If the lens is surrounded by air, then, simply, next = 1.0. In addition, the optical power
(reported in diopters, D) is simply the reciprocal of the lens focal length f (expressed in meters,
m), which is the distance at which the lens focuses a collimated beam to a single point. Then, if
the lens refractive index is denoted by n, the lens power F in air is expressed as
Lens Power in Air: ( )l
2
e s
1
n
11
1 1.0F
f rn
r
= = − −
(2.4)
This relationship is known as the lens-maker’s formula, which is an approximate
relationship, based on the following assumptions:
GEOMETRICAL OPTICS
2-68
2.7 LENS POWER SUMMARY
Focal Points
A lens has two focal points. The secondary, or image-space focal point F΄ is situated after a
positive lens or before a negative lens. The primary, or object-space focal point F is situated
before a positive lens or after a negative lens.
For a positive (converging) lens:
The secondary focal point is the (real) image point if a collimated, parallel-to-the-optical-axis
(plane wave) ray bundle enters the lens.
The primary focal point is the (real) object point that produces a collimated, parallel-to-the-
optical-axis (plane wave) ray bundle leaving the lens.
Figure 2-41: Secondary (left) and primary (right) focal points in a plus lens.
For a negative (diverging) lens:
The secondary focal point is the (virtual) image point from which an originally collimated pencil
of rays, when refracted by the lens, appears to originate as a diverging beam leaving the lens.
The primary focal point is the (virtual) object point to which a ray bundle appears to converge,
prior to being refracted by the lens as a collimated beam (parallel plane wave).
Figure 2-42: Secondary (left) and primary (right) focal points in a minus lens.
IMAGING DEFINITIONS
3-73
3.1.1 Real and Virtual Object; Real and Virtual Image
The concept of object is directly tied to rays ‘leaving a point.’ Naturally, these rays (and, by
association, the wavefronts) are diverging. An object is placed in front of (before) the optical
element, and the rays reach the optical element (a positive lens, a negative lens, a mirror, or a
single refracting interface) in a divergent configuration. This is a real object.
Figure 3-2: A real object can be placed in front of a positive lens (left) or in front of a negative lens (right).
A real object is the ‘common sense’ physical object, which is how ‘object’ has been
described so far. We can conceptually extend the notion of an object to associate it with any
light formation incident on the optical system. In optics, therefore, the object is associated with
light incident on the optical system. However, light formation is not necessarily diverging. It is
possible that the wavefront incident on the optical system is converging, although this does not
occur naturally.13,14
Assume, for a moment, that we remove the optical element (lens or a mirror). Light
would converge to a point after (to the right of) that element—the location of the object, which,
in essence, exists. The object is not physically formed due to the presence of a lens (or a mirror)
in the way. This light formation incident on the optical element is a virtual object.
Figure 3-3: A virtual object can be formed after a positive lens (left) or after a negative lens (right).
13 Virtual objects are formed in multi-lens, or, in general, multi-element, imaging systems. A converging lens or a concave mirror will
form a converging beam, forming a real image, which in turn can be incident on another optical element, forming a virtual image.
14 If the incident wavefront is flat, the incident vergence is zero. This object is located at optical infinity (see § 4.6.3).
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Figure 4-47: Summary of the image formation configurations in lens imaging. [Left column: Positive lenses, top (1) to bottom (6)] 1, 2, and 3: Real object, real image. 4: (Real) object at the primary focal point,
image at optical infinity. 5: Real object, virtual image. 6: Virtual object, real image. [Right column: Negative lenses, top (1) to bottom (6)] 1, 2, and 3: Virtual object, virtual image. 4: (Virtual) object at the primary focal point, image at optical infinity. 5: Virtual object, real image. 6: Real object, virtual image.
(Compare to Figure 5-63.) Note: For simplicity, the rays shown correspond to two of the three construction rays (rules 1 and 3).
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5.3.3 Ray Diagrams for Convex Mirrors
Alternatively, to find the image location and size, we can apply three simple ray diagram rules
similarly to the way we applied lens ray diagram rules (presented in § 4.3). For a convex
reflecting surface, the ray-tracing rules are as follows:
Figure 5-29: Ray-tracing diagrams for a convex mirror.
A ray parallel to the principal optical axis (parallel ray) is reflected as if it originated
from the focal point.
A ray directed at (targeting) the focal point (focal ray) becomes parallel to the
principal optical axis upon reflection.
A ray directed at the center of curvature of the mirror (radial or nodal ray) is retro-
reflected (simply reverses direction).
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5.6 MIRROR IMAGING SUMMARY
Mind the Sign!
In mirror imaging, all of the relationships are the same as their counterparts in lens imaging, and
the algebraic signs follow the Cartesian sign convention (§ 3.2). The notation for object location is
identical to that applied to lenses: An object location to the left of the mirror has a negative sign
and to the right of the mirror has a positive sign.
Figure 5-59: The Cartesian sign convention in mirrors for object location.
The directional distances that apply to image space are associated with a reflected wave that has
a reversed direction of propagation with respect to the initial, incident-to-the-mirror wave. Thus,
image location, radius of curvature, and focal length values to the left of the mirror have a
positive sign and to the right of the mirror have a negative sign.
Figure 5-60: The Cartesian sign convention in mirrors for image location, radius of curvature, and focal length.
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Figure 5-63: Summary of image formation configurations. [Left column: Concave mirrors, top (1) to bottom (6)] 1, 2, and 3: Real object, real image. 4: (Real) object at the primary focal point, image at optical infinity. 5: Real object, virtual image. 6: Virtual object, real image. [Right column: Convex mirrors, top (1)
to bottom (6)] 1, 2, and 3: Virtual object, virtual image. 4: (Virtual) object at the primary focal point, image at optical infinity. 5: Virtual object, real image. 6: Real object, virtual image. (Compare with Figure 4-47.)
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6.2.2 Nodal Points
A thick lens has two nodal points. Their role is equivalent to that of the center of curvature in
an SSRI (§ 1.2.6) or a mirror (§ 5.2.3), or the center of a thin lens. The undeviating ray (§ 4.3.1)
crosses the center of a thin lens, maintaining its inclination with the optical axis. We note that
there is no nodal plane; there are only nodal points. However, we use the notion of ‘nodal ray,’
which is a ray directed toward, or appearing to originate from, a nodal point.
The nodal points function as follows: A ray directed at the object-space nodal point N
emerges from the lens as if it originated from the image-space nodal point N΄ without a change
in the angle formed with the optical axis (parallel to its original direction).
Figure 6-18: Nodal points in a thick lens and ray tilt preservation: Angle ϑ, which expresses the ray tilt, is equal along both sides of the lens.
The two focal points, the two principal points, and the two nodal points are the six
cardinal points. These points are all situated on the optical axis.
If both sides of the lens are surrounded by the same medium (n = n΄), the nodal points
coincide with the corresponding principal points: PN = P΄N΄ = 0. If not (n ≠ n΄), the nodal points (N,
N΄) and the principal points (P, P΄) are separated by
Principal-to-Nodal Point Displacement: e e e
n΄ n n΄ nPN P΄N΄ f΄ f
F F F
−= = + = − = (6.7)
Note : There is no direct formula for determining the nodal point locations; they are, essentially,
referenced to their corresponding principal points. The good news is that Eq. (6.7) is not restricted to
thick lenses but also applies to single refracting interfaces and reflecting surfaces.
Cardinal
Points
• A thick lens or lens system has six (6) cardinal points:
• These are the two focal points, two principal points, and two nodal points.
• They are all situated on the optical axis.
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The reciprocal of the back vertex power (in air) is the back focal length BFL or fBFL
measured from the back vertex point V΄; likewise, the reciprocal of the front vertex power is the
front focal length FFL or fFFL measured from the front vertex point V.
Figure 6-79: Summary of power concepts and formulas in thick lenses.
• Is the beam vergence that leaves the front (first) lens surface if a collimated beam (object at
infinity) is incident on the lens.
• Is calculated using the SSRI power formula.
• Is referenced at the front (object-space) vertex point V.
Front Surface Power F1
• Is the beam vergence that leaves the image-space principal plane H΄ of a thick lens if a collimated
beam (object at infinity) is incident on the lens.
• Is the sum of the front surface power F1 , the back surface power F2, and the third term introduced
by Gullstrand's formula.
• Is referenced at the back (image-space) principal plane H΄.
Equivalent Power Fe
• Is the beam vergence leaving the object-space front surface of a thick lens if a collimated beam is
incident on the lens from the back side.
• Is the sum of the front surface power F1 and the downstream-adjusted back surface power F΄2.
• Is referenced at the front (object-space) vertex point V.
Front Vertex Power FFVP (also known as the neutralizing power)
• Is the beam vergence leaving the image-space back surface of a thick lens if a collimated beam
(object at infinity) is incident on the lens.
• Is the sum of the downstream-adjusted front surface power F΄1 and the back surface power F2.
• Is referenced at the back (image-space) vertex point V΄.
Back Vertex Power FBVP (also known as the prescription power)
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7.1.3.4 Locating the Pupils if the Aperture Stop is Unknown
If the aperture stop is unknown, to find the pupils (and the aperture stop), we follow these steps:
7.1.3.5 The Entrance Pupil in the Human Eye
The aperture stop of the eye is the anatomical iris. The entrance pupil is the object-space image
of the iris, formed by the cornea, which is the ‘lens’ that is preceding it. For this imaging, the
positive direction is from right to left. The object of the imaging is the iris opening, situated in
the aqueous, which is the ad hoc object space with naqueous = 1.336.
To determine the entrance pupil of the human eye, we assume the following values:
average corneal power +42 D; anatomical iris situated 3.6 mm to the right of the cornea;31 eye
filled with aqueous with refractive index naqueous = 1.336, and air with refractive index nair=1.0.
Figure 7-28: Entrance pupil of the human eye. This is the apparent pupil, which is situated slightly closer to the cornea and is about 12.7% larger in diameter than the anatomical iris, which is the aperture stop.
31 This is the anterior chamber depth. See Visual Optics § 2.4.2 Iris and Pupil.
En
tran
ce P
up
il ☞ is the image of the aperture stop
formed by the optical elements
preceding it.
☞ determines the angular breadth of
the rays that enter the system.
☞ is associated with object space.
Ex
it P
up
il ☞ is the image of the aperture stop
formed by the optical elements
succeding it.
☞ determines the angular breadth of
the rays that exit the system.
☞ is associated with image space.
1. Form the
images of any
possible
aperture stop
(AS) in object
space.
2. Identify the
angle subtended
from the on-axis
object point to the
edge of each AS
image.
3. The image
of the element
with the
smallest angle
is the entrance
pupil.
4. The element
producing this
image is the
aperture stop.
5. The image
of the
aperture stop
in image
space is the
exit pupil.
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We can compute the image-space aFoV considering the angular subtense of the exit
port from the exit pupil. In the case where the object is at infinity, the image is formed at the
focal plane of the lens. Now, the separation of the exit pupil from the exit port (distance dp)
equals the focal length of the lens f; therefore,
Field of View (lens focused at infinity): 1
aFoV 2 tanh
f−
=
(7.4)
Figure 7-61: The aFoV when the lens is focused at infinity and the field stop is at the sensor (image) plane.
In many devices such as the photography camera, the sensor (field stop) has a fixed size,
so the aFoV is inversely proportional to the focal length: The shorter the focal length, the larger
the aFoV. This is why short-focal-length lenses (e.g., 35 mm or less) are considered to be wide
field, while long-focal-length lenses (e.g., 125 mm or more) are considered to be narrow field.
Figure 7-62: Two photographs with a large and a small field of view, taken from the same spot (Molyvos, Lesvos Island, Greece) with different focal length lenses: (left) short-focal-length (35 mm) wide-angle lens with a large field of view and (right) long-focal-length (200 mm) telephoto lens with a small field of view.
When a lens is used as a collimating magnifier, the FoV that matters is the linear field of
view, and specifically, just the linear extent of the viewable object. Consider a lens of focal
length f (power F = 1/f) with a semi-diameter h. The lens is held at a distance d. We seek the
size of the linear field through this lens, which is simply the linear length of the object.
The object is situated at the primary focal point of the lens; therefore, the rays leave the
lens collimated. If the lens diameter 2h is sufficiently larger than the eye’s pupil diameter, the
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Principal Ray and Marginal Rays in a Three-Lens System
Example ☞: The system presented in Figure 7-110 comprises three lenses and one aperture stop (AS). The
object and image locations are indicated, as well as the entrance pupil and exit pupil. Draw the marginal
ray and the principal rays.
Figure 7-110: Three-lens system. Note the two intermediate images and the entrance and exit pupil.
To draw the marginal ray and the principal ray, we follow the strategy outlined in § 7.2.2.1. The marginal
ray originates at the on-axis object point and is initially directed toward the edge of the entrance pupil. It
bends (refracts) at each lens, aiming at the on-axis image point of that lens upon refraction. On its way, it
passes by the edge of the aperture stop and leaves the system by the edge of the exit pupil.
We note in Figure 7-111 that the marginal ray does not actually cross the edges of either the entrance
pupil or the exit pupil; it is the extrapolation of the marginal ray that does so. This is because both the
entrance pupil and the exit pupil are virtual images of the aperture stop.
Figure 7-111: The marginal ray in the three-lens system.
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8.3.3 Oblique / Radial Astigmatism
If the rays are tilted when they encounter the lens, or if the lens has a tilt with respect to the optical
axis, in addition to the difference between the peripheral and paraxial rays (that causes coma), there
is one more difference: The ray bundle may lie on a tilted plane, called the sagittal plane, or on a
plane with no tilt with respect to the lens, called the tangential or meridional plane.
Figure 8-35: Tangential focus and sagittal focus in oblique astigmatism.
These two planes intersect the lens interfaces quite differently. As a result, the projected
(perceived) lens optical thickness is different in each plane. Specifically, the ray pencil along the
sagittal plane interacts with an increased lens thickness compared to the ray pencil along the
meridional plane. Therefore, the lens optical power appears to be different along these planes.
Thus, there exist two different focal lengths, depending on whether the rays are sagittal or
meridional. This is oblique (or radial) astigmatism.
Figure 8-36: Tangential and sagittal dependence of the projected lens radius of curvature. Rays along the sagittal plane intersect a lens with a smaller radius of curvature (r2), while rays along the tangential plane
(optical axis) intersect a lens with a larger radius of curvature (r1).
George Asimellis, PhD, serves as Associate Professor of Optics and
Research Director at the Kentucky College of Optometry, Pikeville,
Kentucky, which he joined in 2015 as Founding Faculty. He oversees
development and coordination of the Geometric Optics and Vision
Science courses and development of the Laser Surgical Procedures
course.
In the past, he served as head of Research at LaserVision.gr Institute,
Athens, Greece, and as faculty in: the Physics Department, Aristotle
University, Greece; Medical School, Democritus University, Greece; and
the Electrical Engineering Department, George Mason University, Virginia.
His doctorate research involved advanced optical signal processing and pattern recognition techniques
(PhD, Tufts University, Massachusetts), and optical coherence tomography (Fellowship, Harvard University,
Massachusetts). He then worked on research and development of optoelectronic devices in a number in
research centers in the USA. He has authored more than 75 peer-reviewed research publications, 8
scholarly books on optics and optical imaging, and a large number of presentations at international
conferences and meetings.
He is on the Editorial Board of eight peer-reviewed journals, including the Journal of Refractive Surgery, for
which he serves as Associate Editor. He received the 2017 Emerging Vision Scientist Award by the
National Alliance for Eye and Vision Research (NAEVR).
His research interests include optoelectronic devices, anterior-segment (corneal and epithelial) imaging,
keratoconus screening, ocular optics, and ophthalmological lasers. His recent contributions involve
publications in clinical in vivo epithelial imaging and corneal cross-linking interventions.
