A Method of Inferring Single-Path Band Transmission

Factors from Double-Path Measurements

M. A. McKiernan and H. W. Wessely

Assuming rectilinear propagation in a linearly stratified medium, a method of estimating single-pathband transmission factors from double-path measurements is described. Precise analytical bounds forthe uncertainty in the estimates are derived.

1. Introduction

In making transmission measurements over longpaths it is frequently desirable to use a double-pathrather than a single-path experimental arrangement.The outstanding advantage of the double-path arrange-ment, of course, is that the source and the spectrometerare in the same location and hence under close controlat all times.

The great shortcoming of the double-path type ofexperiment, however, is that for an inhomogeneousmedium there is apparently no way of exactly inferringthe single-path transmission from double-path data.Since it is often the single-path transmission which is ofinterest, the question naturally arises as to whether anyreasonably accurate estimate of the single-path trans-mission can be made from double-path data.

This paper describes a method of making such anestimate based on only two assumptions:

1. The medium is linearly stratified, i.e., it consistsof parallel plane layers.

2. The ray paths are essentially straight lines.

11. Basic Formulation

Consider a horizontally stratified medium whosespectral absorption coefficient at any altitude y is de-scribed by a function k(y,X), in which X is the wave-length (see Fig. 1). If a source of uniform spectralintensity Io is located at P, then for a path making anangle with the vertical, the incremental change in thespectral intensityAI(y,X) at the altitude y for an in-

The authors were both at the Laboratories for Applied Sciences,The University of Chicago, Chicago 37, Illinois. H. W. Wesselyis now with Aerospace Corporation, El Segundo, California.

Received 7 May 1962.This work was sponsored by the Air Force Cambridge Research

Laboratories with funds from the Advanced Research ProjectsAgency.

cremental path As is approximately

AI(y,X) = -k(y,X) I(y,X)As.

Since

Ay = As coso,

(1)

(2)

and

I(OX) = Io,

one easily obtains the well-known result for the spectralintensity at any height h,

I(h,X) = o exp [-seceJf k(y,X) dy]. (3)

The radiative power in a spectral band AX that wouldbe measured by a spectrometer placed at the height h issimply

f 1(h,X) dX = I, f exp [-seec h k(y,X) dyldX.AX Axo J (4)

The single-path band transmission factor, T(h,OAX),for the height h and the angle 0 is then

1 C rh 1T1 (h, 0, A) = J exp [-secOj k(y,x) dyjdX. (5)

Now suppose that the spectrometer at height h is re-placed with a retroreflector, and that the spectrometeris moved to P to measure the reflected radiation. It iseasy to see that the double-path band transmissionfactor, T 2(h, 0, AX), is simply

T,(h, 0,AX) = A exp [-2 seco et k(yX)dy d. (6)

111. Inferring Double-Path Values from Single-Path Data

If the problem were to infer double-path transmissionfactors from single-path data, the desired inference

May 1963 / Vol. 2, No. 5 / APPLIED OPTICS 503

It follows from the definition of the generalized trans-mission factor that

(12)T(h, 0, AX) = 1.

Since for double-path measurements,

p = 2 sec0, (13

Fig. 1. Slant path transmission in a linearly stratifiedmedium.

could be made directly. Suppose, for example, onewished to know the double-path transmission factor fora vertical path. The desired quantity, therefore, is

t2(h, 0,Xx) = - exp [-2 fJ k(y,X) dy]dX. (7)

But this is nothing else than the single-path transmis-sion factor for 0 = 600. Thus, since see 600 = 2,

7',(h, 600, AX) = exp AX2 J k(,) dydX

= 722(h, 0, AX). (8)

Hence, in order to obtain the double-path transmissionfactor for a vertical path, one simply measures thesingle-path transmission factor for an inclination angleof 600.

More generally, if one desires to know the double-path transmission factor for an inclination angle 02, onesimply measures the single-path transmission factor foran angle 01 such that

secO2 = 2 sec0. (9)

IV. Inferring Single-Path Values from Double-Path Data

It is clear from the preceding section that thesingle-path transmission factors for slant angles greaterthan or equal to 600 can be inferred directly from thedouble-path data. If one simply pairs the double-pathslant angles, 02, with the single-path slant angles, 01, inaccord with Eq. (9), then

Tj(h, 02, AX) = 7 2(h, 0, AX).

it is clear that the values of the generalized transmissionfactor for p) 2 are given directly by the double-pathdata. These values are shown as the solid portion ofthe curve in Fig. 2. Thus, the only unknown portionof the curve is that having the domain

0 < p < 2.

Now for single-path transmission,

p = sec0,

(14)

(15)

for which the domain of T is evidently p 1. Hence,the only unknown portion of the curve which is ofphysical interest is that having the domain

1 p <2. (16)

Since the generalized transmission factor is a smoothlyvarying function of p, an estimate of its behavior inthis interval can be obtained by extrapolating theknown curve for p , 2 to the unit point at p = 0.

V. Upper and Lower Bounds

The problem now is to put the extrapolation pro-cedure on a firm analytical foundation. Two facets tothis problem must be recognized. The first has to dowith establishing analytical upper and lower bounds forthe desired single-path values in terms of the assumedknown double-path values. The second has to do withthe inaccuracy of these bounds because of the experi-mental errors made in measuring the double-pathvalues. Of these two facets it is clear that the establish-ment of analytical bounds constitutes the real problem,since, once it is solved, a straightforward application ofthe calculus of errors is all that is needed to determinethe inaccuracy of the bounds.

(10)

The problem now reduces to finding the single-pathtransmission factors for slant angles less than 600. Forthese angles there is apparently no way of making directinterferences from the double-path data alone, butremarkably good estimates can be made by using asimple analytical extrapolation procedure.

To illustrate this procedure qualitatively, considerthe generalized transmission factor, T, defined by

T(h, p, AX) = -,fA exp [-p f0 k(y,X) dy d. (11)

T(h,p a)I

Extrapolated Single Path Vertical( =) Transmission Factor

Extrapolated Single Path TransmissionFactoro for 01<60°

Measured Double Path VerticalTransmission Factor (= Single PathTransmission Factor for a, = 60°)

Measured Double Path SlantTransmission Factors (= SinglePath Transmission Factors foreI, 60°)

Fig. 2. Graphical representation of the generalized transmissionfactor.

504 APPLIED OPTICS / Vol. 2, No. 5 / May 1963

This proves to be a very interesting analytical prob-lem. Although a variety of useful bounds have alreadybeen established, the problem of establishing optimumbounds, i.e., least upper and greatest lower bounds, stillresists a general solution. Nevertheless, the resultsthus far obtained indicate that from only a few, say,two or three, double-path measurements made atvarious slant angles, is the uncertainty in the estimatedsingle-path values based on the analytical boundsprobably less than the experimental error that would bemade if the single-path transmission were measureddirectly. In fact, for the limiting situation where thedouble-path transmission is precisely known for anarbitrarily large number of slant angles, it is shownthat the single-path values can be predicted with anarbitrarily small error.

In the following only the "better" bounds are pre-sented. It is not claimed that these bounds are optimal,but it is strongly suspected that further analysis willshow that this is indeed so.

In the following discussion the altitude, h, and thespectral interval, AX, are assumed constant. With thisin mind, the somewhat simpler notation of T(p) ishenceforth used to denote the generalized transmissionfactor defined by Eq. (11).

One-Point Bounds

Suppose that the double-path transmission is knownonly for the vertical direction, i.e., only T(2) is known.What bounds can be placed on the single-path verticaltransmission, i.e., T(1)?

An upper bound is immediately furnished by theSchwarz inequality, namely

T(0) = 1, but this does not cause any difficulty.Hence, if only T(2) is known, the best possible lowerbound for T(1) is T(2) itself. Furthermore, it is notdifficult to show that the best possible upper bound forT(1) is T(2). Thus, Ineq. (19) actually gives theoptimum bounds for T(1) when only T(2) is known.

It is interesting to note that the maximum uncer-tainty in T(1) occurs when T(2) 1/4. In this caseIneq. (19) reduced to

1/4 < T(1) < 1/2.

Thus, neglecting experimental errors, the maximumuncertainty in T(1) is 4 1/8.

The discussion thus far has been restricted to ob-taining bounds for T(1) in terms of T(2). To generalizethe discussion suppose now that the generalized trans-mission factor is known at a single arbitrary point p d

0. What general bounds can be prescribed for T(p) interms of the only available knowledge, viz., T(p,)?

There are at least two ways of treating this question:One is based on the Holder inequality,' and the other isbased on the idea of logarithmic convexity.2 Since forthe specific results to be presented the Holder inequalityoffers the more direct approach, only that approach isdescribed here.

The Holder inequality states that

(20)

where and are any pair of positive numbers such that'/ + /1, = 1, and f and g are any pair of functions forwhich the integrals in Ineq. (20) exist.

To apply this inequality let

T(1) < T 2(2). (17) f = exp [-p fh k(yX)dy]Since it is clear from Eq. (11) that the generalizedtransmission factor is a nonincreasing function of p, alower bound for the single-path transmission is evi-dently

T(1) > T(2). (18)

Thus, combining these two bounds,

T(2) < T(1) < T 11 '(2). (19)

At first sight it seems natural to question the equalitycondition of Ineq. (18), since it hardly seems possiblethat the double-path and single-path transmissionfactors could be identical. From an analytical stand-point, however, this possibility must be admitted.Suppose, for example, that for some fraction of thespectral interval AX, the medium is perfectly trans-parent, i.e., the spectral absorption coefficient is zero, andthat over the remainder of the interval the medium isperfectly opaque, i.e., the spectral absorption coefficientis infinite. Then T(p) = constant for all p id 0. Thisintroduces a discontinuity at p = 0 where, of course,

andg = 1.

With the understanding that the integration in Ineq.(20) covers the interval AX, substitution of Eqs. (21)into Ineq. (20) and division by AX yields

T(p) < 7 I'(p). (22)

Inequality (22) holds for any p and for any > 1.Now suppose that the generalized transmission factor

is known at p. If in Ineq. (22) p is replaced by p, andbpi is replaced by p, where now, of course, p > p, oneobtains a lower bound for the generalized transmissionfor all p > p. Thus,

( > Pt). (23)

The upper bound for the generalized transmission forp > pi follows from the nonincreasing character of thegeneralized transmission. Thus,

T(p) < T(p 1) (P > pi). (24)

To obtain bounds for p < p, let p in Ineq. (22) be

May 1963 / Vol. 2, No. 5 / APPLIED OPTICS 505

(21)

TP411( pl) < T( p)

la , 0Vfg� < VA, T [f V0]

replaced by pi. This yields the upper bound

T(p) < TP'P'(p1 ) (p < p).

The lower bound for p < p is correspondingly

T(p) 2 T(p,) (P < p).

Combining Ineqs. (23) through (26) yields

T(p1) < T(p) < T'(p 1)

and

T7P/PI(p) < T(p) < T(p1 )

(p < p),

( > PI).

Regrettably, at this writing the general solution to thisproblem is still unknown. Only a special result has thusfar been obtained.

Suppose that the generalized transmission is known at(26) p = 2 and at p = 3. Consider the Holder inequality

with 6 = / = 2, and with

f = exp [- fO k(yX)dX]1l - exp[- f0o k(y,X)dX I1/2(27) (32)

and

Two-Point Bounds

Suppose that in addition to knowing the transmissionat pi, the transmission is known at a second" point 2.

Because of this additional knowledge, it should bepossible to improve the previous bounds which werebased upon the knowledge of T(p,) alone.

To obtain these improved bounds, consider theHolder inequality with

and

f = exp [-i fo k(y)dX]

= [ fAS; k(yX)dX]$

(28)

where t and v are any pair of positive numbers. Sub-stituting Eqs. (28) into Ineq. (20) one obtains

T(u + v) < Ta (au)T"0 (0v). (29)

To apply this inequality, suppose, for example, thatj3v = 3, 8u = 1, and u + v = 2. For these choices it isclear that 8 = / 2 and, hence, Ineq. (29) reduces to

T(1) 2 T 2(2)T- 1(3). (30)

Since T(2)T-1(3) > 1, it is clear that Ineq. (30) con-stitutes a better lower bound for T(1) than that given byIneq. (18), namely, T(2).

The generalization of this result is straightforward.If the generalized transmission is known at a pair ofpoints pi and P2, where P2 > p' 5z 0, say, then it can beshown that

T(p) > 7(2-P)/(P2- Pl)(pI)T7(Pl)/(P2-Pl)(P2),(p < pi) or (p > P2),

and (31)

T(p) < T(P2-P)/(P2-P)(pI) 71(P-P')/(P2-P)(p,),

(pI < P < P2).

Thus, Ineq. (31) furnishes a lower bound for thegeneralized transmission for values of p outside theinterval (p', p2) and an upper bound for values of pinside the interval (p', p2).

The next step in the discussion, obviously, should bethe determination of an upper bound when p is outside(p, p2) and a lower bound when p is inside (, p2).

g = -1 -exp [- f k(mX,)dAj

Substituting Eqs. (32) into Ineq. (20), one obtains, aftera few simple manipulations,

[T(1) - '(2)] 2• [(2) - 7'(3)][1 - '(1)].

Solving for T(1), one obtains

T(1) <

T(2) + T(3) + V[T(2) + T(3)12 - 4[T2 (2) - T(2) + T(3)l2

(33)

An estimate of the accuracy of the estimates of T(1)based on measurements of T(2) and T(3) can now bemade. Numerical calculations using Ineqs. (30) and(33) show that the maximum uncertainty in T(1) occurswhen T(2) 0.22, and T(3) 0.18. Using theseresults, the maximum uncertainty in T(1) is approxi-mately +0.05.

Lower Bounds Based on Taylor's Theorem

It is interesting to note that Taylor's theorem fur-nishes a set of lower bounds which analytically can bemade to differ from the true value of the generalizedtransmission by an arbitrarily small amount.

From Taylor's theorem

T(p) - !1( - ) = + I) (P,)_(p - 2)N+ (34)N (p-2n -! 01 (N + 1)!

7z=0

in which p' lies between p and p - 2.It is clear from Eq. (11) that the derivatives of T with

respect to p alternate in sign. When n is odd (even),the nth derivative is negative (positive). The samealternation occurs for (p - 2)n for p < 2. Hence, everyterm in the Taylor expansion for p < 2, including inparticular the remainder term, is nonnegative. There-fore, for p < 2, and for every N,

N (n) (p - 2).(35)

Statement of the General N-Point Problem

It should be apparent that the preceding analysisrepresents a beginning, but by no means a complete

506 APPLIED OPTICS / Vol. 2, No. 5 / May'1963

treatment of the problem. The real problem is theestablishment of optimum bounds for any finite numberof double-path transmission measurements, whereas thebulk of the preceding analysis merely considers boundsbased on one and two measurements of the double-pathtransmission. Moreover, the question of the optimityof the bounds has been almost completely neglected.

In the hope that other analysts may become inter-ested in this problem, a restatement of the general prob-lem in purely analytical terms is given below. Thephysicist will recognize that no essential features havebeen lost in this restatement.

Problem: Consider the class of functions 0 < f < 1sectionally continuous on the closed interval [a, b].

Given a finite set of pairs of numbers (pi, Ai), i = 1,2,... .n, where pi > 2, and

Ai= ff.i(X)dX,

find the absolute maximum and minimum offbfi p(x)dx

for < p < 2.

References1. L. M. Graves, The Theory of Functions of a Real Variable, 2nd

ed. (McGraw-Hill, New York, 1956), p. 233.2. R. Courant, Differential and Integral Calculus (Interscience,

New York, 1948), Vol. II, pp. 325-330.

Meeting Reports continued from page 502connection with the determination of the masses of the geo-metrical structures constituting the scattering units. Finally,it was also shown, by treating the influence of elastic strains onthe components of scattered light of various polarizations withthe aid of Fourier transform methods, that the effect of suchstrains can be separated from the influence of size and shape ofthe scattering regions. Some possible generalizations of thistheory have been discussed.

The final afternoon session of the conference was devoted tomultiple scattering, and one of the authors of this report (C.C.G.)presented the introductory talk on recent progress in the develop-ment of a new approximate general theory of multiple scattering.After a brief review of his 1956 approximation, he outlined thefundamental ideas and formulas underlying his newest analyticalapproximate formalism endowed with a set of interesting proper-ties among which the applicability to problems involving anytype of nonisotropic single-scattering law with axial symmetryaround the direction of incidence deserves attention. This workwas followed by an elegant variable-order diffusion-type ap-proximation for treating isotropic multiple scattering in infinite,semi-infinite and finite geometries. An extension to anisotropicmultiple scattering, which would be important, was claimedpossible. A paper dealing with multiple scattering of waves indense distributions of large tenuous scatters came next, whichwas an extension of previously published papers in which prop-agation through "gas-like" statistical distributions of scattererswas considered for the purpose of discussing the "bulk" electro-magnetic properties of such media and their fluctuations, interms of elementary scattering processes. The generalizationconsisted in studying multiple scattering in a system whosepopulation is divided between a "gas-like phase" and a "crystal-like phase," both phases contributing to the coherent field whereasonly the gas phase contributes to the incoherent scattering.Fourth in line came an excellent contribution on multiple scat-tering in media with anisotropic scattering, putting greateremphasis on the polarization effects and constituting a substantialgeneralization of Chandrasekhar's classical treatment. Thesession closed with a report on some remarkable mathematicalstudies concerning the uniqueness of solutions to Chandrasekhar'sformulation of principles of invariance in the theory of radiativetransfer. It was not only shown that, in many cases, certainequations in this formulation have a definite multiplicity ofsolutions which had so far been only partially explored, butexplicit parametric representations of the families of solutionswere generally written down.

We cannot end this report without extending special con-gratulations to Dr. Kerker and his planning committee, for themagnificent job of organization and coordination which they did,a job that was as delicate as the focusing of light and the alignmentof lenses in a complicated scattering experiment.

1962 WESCON Meeting, Los Angeles, August21-24, 1962

Reported by J. Arndt, Space Technology Laboratories

The 1962 WESCON meeting was held August 21-24 in LosAngeles at the Statler Hilton Hotel and the Los Angeles SportsArena, the latter requiring a canvas-topped annex to accom-modate more than 850 exhibitors. Nearly 50,000 attended theshow, making it the largest technical exposition of the year.Some 80 papers were presented in 25 technical sessions, and mostof these have since been published in the eight-volume 1962WESCON Convention Record.* The WESCON convention wasobviously influenced, if not dominated, by the vast missiles andspace industry along the west coast from Seattle to San Diego.One paper of widespread interest dealt with interstellar com-munication on a realistic and quantitative level, correlatingprobability of contact (with an extrasolar civilization) withexpenditure of time and money. Laser demonstrations openedthe show, with a laser pulse from the Statler Hotel to the SportsArena triggering an electronic welcome message. Later, aneoprene balloon was sent aloft when its steel restraining wirewas severed by a focused laser beam.

Areas of interest were not constrained to electronics, and only afew papers dealt with any one particular field. Most sessionscovered complex systems and techniques encompassing severaldisciplines at a time. Examples of this are the five "special"sessions of general interest, including "Biological InformationTransfer on the Molecular Level," "Research in Nuclear TestDetection," and "Lunar Exploration." A few papers dealtwith optical devices in the field of pattern recognition includingone paper dealing with an optical decision filter capable ofclassifying pyramids, spheres, cubes, and ellipsoids, regardless oforientation in three-dimensional space. It is worthy of notesince the device represents a working machine based on the"perceptron" approach to pattern recognition developed atCornell University several years ago.

* Write WESCON, 1435 S. La Cienega Blvd., Los Angeles,California. for further information.

May 1963 / Vol. 2, No. 5 / APPLIED OPTICS 507

An Analysis of Radiation Transfer By Means ofElliptical Cylinder Reflectors

:S. B. Schuldt and R. L. Aagard

Expressions are derived giving the relative amount of pumping radiation transferred from a cylindricalsource to a cylindrical laser by means of a reflector in the shape of an elliptical cylinder. The generalexpression depends upon the radii of the source and laser, the eccentricity of the ellipse and the lengthof its semimajor axis, and an arbitrary angular distribution of source radiation. The effect of radiationreflected back into the source itself (source-blocking) is also considered. A very considerable simplifica-tion in the calculation results when the source distribution is invariant under a rotation of the sourceabout its axis, i.e., when the distribution depends only on the direction of the radiation with respect to thelocal normal, thus corresponding to virtually all practical cases. Moreover, the special case of a Lam-bertian source distribution yields efficiencies which may be evaluated directly. Especially simple andillustrative is the Lambertian source which is sufficiently small compared to the reflector that source-blocking can be ignored.

Introduction

Elliptical reflectors have often been employed totransfer pumping radiation from a cylindrical source tothe active medium in an infrared or optical maser. Thistype of reflector can be constructed with reasonableease,' and it has served equally well for ruby, 2- 4 metalvapor,', 6 and rare-earth doped laser materials. 7'-0

However, there is little or no information in the litera-ture concerning the focusing properties of an ellipticalreflector. In order to become more familiar with theelliptical reflector, we have studied the subject ana-lytically within the framework of the following threeconsiderations.

Considering an elliptical cylinder in cross section, wewill observe that an object at one focus is imaged at theother focus with the least magnification when the fociare closest together. If the image of a light source fallswithin the circular cross section of the laser rod, all ofthe rays leaving the source, except those blocked by thesource itself, are considered to be transferred to thelaser. We speak of an efficiency, then, which representsthe relative percentage of rays leaving the source whichenter the laser circle. Generally, the efficiency isgreatest when the source and laser are close togetherand the reflector is nearly circular. It is easy to see thatthis may lead to enormous reflectors, and one would like

The authors are with the Honeywell Research Center, Hopkins,Minnesota.

Received 21 November 1962.

to know how much the efficiency is compromised byreducing the dimensions of the ellipse.

The effect of the relative size of the source and re-ceiver leads to a second important consideration. Wewould also expect the efficiency to be greatest when thediameter of the source is small and the laser large. Justthe opposite condition pertains to the elements alone.That is, the laser material requires a high flux densityimplying small diameter, while the power availablefrom the pumping lamp increases with the diameter.Therefore, the designer would like to know if the addi-tional power obtained by increasing the lamp diametermore than makes up for the loss in efficiency.

Rays leaving normal to the surface of the source in thecross-sectional plane will, of course, reach the laser, butrays leaving at an angle will be imperfectly focused. Inthis consideration, we must satisfy ourselves that theproblem reduces to one of studying the distribution ofrays in the two dimensions of a cross section. If weassume an infinitely long cylindrical source and reflectorand a finite laser, it is obvious that the distribution oflight rays entering the laser is, except for end effects,the same for any cross section through the laser. Fur-thermore, the geometry of the focusing does not dependon the angle that a ray makes with the plane of the crosssection, since the surfaces of the source, reflector, andlaser are generated by parallel lines (perpendicular tothe cross section). The condition of infinitely longsource and reflector can be effectively met in practice byenclosing the ends of the elliptical cylinder with planereflectors of very high specular quality, located flush

May 1963 / Vol. 2, No. 5 / APPLIED OPTICS 509