JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Reciprocity Relations between the Transfer Function and Total Illuminance. IRICHARD BARAKAT AND AGNES HOUSTON

Optics Departmnent, Itek Corporation, Lexington, Massachusetts(Received 28 March 1963)

The functional relationship between the total illuminance and transfer function is obtained for systemshaving rotationally symmetric aberrations. It is shown that the behavior of the transfer function at zerospatial frequency determines the asymptotic behavior of the total illuminance. In addition, the moments ofthe transfer function determine the behavior of the total illuminance in the vicinity of the origin. Typicalnumerical results are presented.

1. INTRODUCTION

THE total illuminance (encircled energy) and theT transfer function are two quantities which char-acterize the performance of an optical system. Thetotal illuminance describes the average integrated be-havior of the point-source diffraction image, while thetransfer function is a measure of the contrast ratio ofan infinite sinusoidal object. We are, of course, onlyconcerned with incoherent light.

Both of the functions can be computed in principlefrom a knowledge of the wavefront emerging from theexit pupil. These calculations have been carried outfor actual systems by Barakat and Morello.'-3 Thesetwo functions are computed independently of eachother.

It seems reasonable to suppose that there is a func-tional relation between the total illuminance and thetransfer function in view of the fact that the point-spread function and the transfer function are Fouriertransform pairs. We have derived the necessary rela-tions in the next section. Given the transfer function(in analytical or numerical form), it is straightforwardto compute the total illuminance via our formulas.However, if the total illuminance is specified and wedesire to obtain the transfer function, then we areforced to solve a Fredholm integral equation of thefirst kind (a rather unpleasant task).

Part I (the present paper) deals with the first prob-lem: Given the transfer function, compute the totalilluminance. Part II (in preparation) deals with themore complicated problem: Given the total illuminance,compute the transfer function.

2. DERIVATION OF RECIPROCITY RELATIONS

The spread function (distribution of illuminance inimage plane due to point source) is related to thetransfer function by a Fourier transform'

I (v,v,) = KJ J T( w.,w,)ei(v- -+vy )ddxv, (2.1)

R I. Baralkat, J. Opt. Soc. Am. 52, 985 (1962).2 R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52,992 (1962).3 R. Barakat and M. V. Morello, "Computation of the Total II-

luminance of an Optical System from the Design Data for Rota-tionally Symmetric Aberrations" (to be published). 1

4 E. L. O'Neill, Selected Topics in Optics and Communication

where v, v are the reduced distances in image plane,c, w,, are the normalized spatial frequencies, and K isthe normalizing constant. By definition the total illumi-nance (encircled energy) is given by

V. 0

VY 0

L(vX,,Iv) = I (v.,,vy)dvxdv9,VX o -vvo(2.2)

where the center of the aperture is taken as the origin.Now integrate both sides of (2.1) as per (2.2) and

interchange the limits of integration of the fourfoldintegral on the right-hand side; then

L (v2, v) =KJ f T(wXwy)dwXxdw

v x0 f 0j-X -vy'

J~ , ei(v-X+vy-y)dzdv,-V,o -V0

rx {^00 sinv~w sinvly,4K | T(wcxswy) dc&ddoy.

I -. 0 J -. 0 co, coy ~ (2.3)

This is the general relation between the total illuni-nance and transfer function. The total illuminanceis a real function and the only way that the right-hand side can be real is for us to take the real partof the transfer function. This will happen in the off-axis case with an aberration such as coma. However,we restrict ourselves in the present paper to the casewhere the transfer function is real, which means thatonly rotationally symmetric aberrations are permitted.

In the special case of a slit aperture (very narrowrectangular aperture), (2.3) becomes

f'' sinvowL(vo)=Kj T (o) do.

0c

(2.4)

The constant K can be determined by recourse to the

Theory (Boston University Physical Research Laboratory, Boston,1959).

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NOVEM BER 1963VOLUME 53, NUMBER 11

November1963 TRANSFER FUNCTION AND TOTAL ILLUMINANCE. I

Dirichlet theorem5

sinvowlim f T(w) dco=irT(0)=-7r,"10- COc (2.5)

since the transfer function is unity at the origin. Also

lim L(vo) = 1, (2.6)

because all the illuminance must be contained in alarge enough line. Consequently

2 r 2 sinveowL(vo)=- T(o) dw. (2.7)

r co '

The infinite limits are replaced by finite ones; thetransfer function vanishes for w>2. The relation (2.7)is important because it allows us to determine the totalilluminance whose domain of definition is the infiniteregion (O<vo< oc) from a knowledge of the transferfunction whose effective domain is the infinite interval(0<w<2).

Now L(vo) is a monotone increasing positive func-tion. Therefore we place a constraint upon the formof the transfer function since the other function in theintegrand can assume positive and negative values.This question of constraint on T(w) is discussed inPart II as it naturally falls under the category of thesecond problem.

If we view (2.7) in the light of L(vo) as known andT(co) as unknown then (2.7) is a Fredholm integralequation of the first kind for T(co). The solution of thisequation is considered in Part II.

Before proceeding further let us consider the circularaperture. This is easily accomplished by converting(2.1) to polar coordinates

I (v) =K'f T(w)Jo(vw)wdo, (2.8)

and integrating

L (vo) J I (v)vdv = K'] T (c)codwj JO (vw)vdv

=K'voj T(w)Jj(voo)dw. (2.9)

Applying the polar form of Dirichlet's theorem, we have

lim vo T(w)Jj(vow)dw= T(0)= 1, (2.10)

I H. S. Carelaw, Introduction to the Theory of Fourier's Seriesand Integrals (Dover Publications, Inc., New York, 1956), 3rd ed.,p. 219.

so that K'= 1. Consequently (2.9) reads

2

L (o) = voJ T(co)J, (voco)dw. (2.11)

The same remarks apply to (2.11) as stated for (2.7).We are now ready to solve the first problem. We

merely effect the integrations of (2.7) and (2.11) byanalytical or numerical methods. As an example (ofa somewhat trivial nature) consider an aberration-freeslit aperture. The transfer function is

T(cw)= 2 (2-'w), (2.12)so that

2 sinvocw 1 f2

L (o) = - | -dw - - sinvodw7ro X 7r~

2 2= -Si (2vo) -- sin2vo,

7r Irvo(2.13)

where Si(x) is the sine integral. This is exactly whatwe would obtain by direct integration of the illuminance

to sin 2 F sin2 vo0

L (vo) K I dv =- Si (2vo)- I. (2.14).o v2 r voI

The constant K was determined by invoking the limitrelations (2.6) and noting that the sine integral has theasymptotic form

Sf (x)'- (7r/2)- (cosx/x). (2.15)

In general the transfer function is not given in simpleanalytic form but usually in tabular form and it isnecessary to resort to numerical integration schemesin order to obtain the total illuminance. The necessaryanalysis is considered in Sec. 4 along with numerousexamples.

3. ASYMPTOTIC RELATIONS

We can determine the behavior of L(vo) for large vo,given the behavior of T(w) at the origin. This is anexample of a Tauberian relation.' Willis7 has derivedthe following two formulas:

~~0 PO~~'() f"'(0)x f(s)Jj(xs)ds-f(0)+ -(°- (3.1)

Jo x 2x'

2 C sinxs 2 f'(0) 2f "' (0)- f(s) ds -f(0)+- -__ + -7r 0 s w x 37rx3

(3.2)

subject to some fairly mild restrictions on f (s).

6 B. van der Pol and H. Bremmer, Operational Calculus Based onthe Two-Sided Laplace Transform (Cambridge University Press,Cambridge, England, 1950), Chap. 7.

7 H. F. Willis, Phil. Mag. 39, 455 (1948). We had derived theserelations independently although in a somewhat less satisfactorymanner than Willis.

I1245

46. B RA KAT A N A A. - 0 IT N ol 5

TABLE I. Comparison of numerical values of L(vo) for aberra-tion-free circular aperture as determined by exact solution (3.9)and the Tauberian representation (3.8).

to L (exact) L (Tauberian)

2.0 0.6173 0.69163.0 0.8174 0.79074.0 0.8379 0.84215.0 0.8612 0.87336.0 0.9008 0.89437.0 0.9099 0.90938.0 0.9155 0.92069.0 0.9317 0.9294

10.0 0.9376 0.936412.0 0.9478 0.947014.0 0.9530 0.9.54616.0 0.9612 0.9602

Now it is known that the first derivative of thetransfer function evaluated at the origin is idependenttof the amount of aberration; furthermore

T'(0) -2/7r (circular aperture),T'(0) - (slit aperture).

Thus we have the following Tauberian representationsfor L(vo):

L (vo) 1- (2/7rvo)- T"' (0)/2vo3]+

(circular aperture), (3.3)

L (v0) - 1- (1/7rvo) - 2 T"' (0)/37rvo3]+ (slit aperture). (3.4)

These formulas show that ultimately the total illumi-nance of all systems (irrespective of their aberrations)must behave as

L(vo)-1-(27rv(o) (circular aperture), (3.5)

L (vo - 1- (1 /7nvo) (slit aperture). (3.6)

The exact solution was first obtained by Rayleigh, viadirect integration of the spread function

VO J 1 2(V)L (vo) = 21 vdv = 1-Jo2 (vo)-J2 (vo). (3.9)

Jov2

Numerical values of (3.8) and (3.9) are compared inTable I. The agreement is excellent even for values assmall as vo equal to 4. We must caution the reader that(3.8) is not the true asymptotic expansion of L(vo) butits Tauberian representation. In order to obtain thetrue asymptotic expansion it would be necessary tosubstitute the asymptotic formulas for J0 and J1 intothe right-hand side of (3.9). The results are quitecomplicated if terms in vo-1 are included.

As a second example, we treat the corresponding slitaperture whose transfer function has already beengiven in (2.12). Here all the higher derivatives vanishexcept the first so that

(3.10)

is the Tauberian representation. The true asymptoticexpansion can be derived from a consideration of thesine integral

r r 2! ]Si (X)_-cosX___+

2 _r x 2 _

1 3 ! - sinx __ .+ - . , (3.11)

X2 4 b

and after somne trigonometric manipulations becomes

1 sin2ivo cos2voL () -1- -- + +.7rty 27rz'o2 2 7rvo3

(3.12)

This result would be difficult to prove by direct ap-plication of asymptotic methods (stationary phase inparticular) to the double integral representation ofL(vo) previously derived by Barakat' and employedby Barakat and Morello.3

As our first example consider the transfer functionof a perfect circular aperture 4 :

T (w) = (2/7r) [cos-(/2) - (o4) (4 -2)-] (3.7)

Straightforward differentiation of (3.7) yields T()(0)= 1yr; hence (3.3) reads

(3.8)

8 R. Barakat, J. Opt. Soc. Am. 51, 152 (1961).L Lord Rayleigh (J. W. Strutt), Phil. Mag. 11, 214 (1881).

The first two terms of (3.12) agree with (3.10).One would expect that the behavior of T(co) for

large X will determine L(vo) for small v. This is truebut unfortunately very little information can be ob-tained from this inverse relation. For co> 2, the transferfunction is zero; consequently at w= co, it and all itsderivatives vanish. The only result that can be proved is

T(co)=L(0)=0 (3.13)

which is, of course, obvious on physical grounds.Nevertheless, we can still obtain the behavior of the

total illuminance in the vicinity of the origin in termsof the moments of the transfer function. Consider first

"T. J. Bromvich, An Introduction to the Theory of InfiniteSeries (Macmillan and Company Ltd., New York, 1955), 2nd ed.,p. 338.

IVol. 531 246

L (vo) - - (1 /7rvo)

L (vo) - - (2/rvo) - (1/47rvo3) +. ..

November1963 TRANSFER FUNCTION AND TOTAL ILLUMINANCE. I

the slit aperture (2.7). Expand the sine term in theintegrand in a power series and integrate termwise,then for small vo:

2 2 v03 r2

L(vo>)-voj T(w)dc---- T(w)(02dco+ (3.14)7r 3?r

The integrals are the moments of the slit-aperturetransfer function. The integral in the first term isnothing but the central illuminance or Strehl criterion,I(0). [Set v= 0 in the one-dimensional analog of (2.1)to prove this.] All integrals are 0(1) so that the domi-nating factor is vo; hence in the immediate vicinity ofthe origin the first term is the dominant one and wehave the important result

L(vo)- (2v/)7r)I(O) (slit aperture). (3.15)

1IG. 1. Total illuminancein various defocused receiv-ing planes for aberration-

- free slit aperture.

l.0

U .8zE .6

I- .

V.

The transfer function for a slitspherical aberration is given by thepupil function 4 and is

aperture sufferingconvolution of the

1 rAT(c)=- a eik[W(p)-w(-)dp,

X -1

(4.1)

Before commenting on the significance of this ex-pression, let us derive the corresponding formula forthe circular aperture. Expand the Bessel function in(2.11) into its power series and integrate termwise; forsmall v0 we have

v02

2 v04

,2

L(vo)-- I T(w)wdw-- T(W)o,4d+ - (3.16)2J 16 J

Upon setting v= 0 in 2.8, we see that the integral in thefirst term is I(0). Applying the same reasoning to(3.16) as we did to (3.14) yields

L (vo) - (vo2/2)I(0) (circular aperture). (3.17)

Note that for a slit aperture L(vo) is a linear functionof v, while for a circular aperture it is a quadraticfunction. The larger one makes the Strehl criterion,the faster the rise of the total illuminance at theorigin. Now the largest value that (0) can have isunity by Luneberg's first apodization theorem,"' 2 thevalue unity occurring only for an Airy system (uniformtransmission coefficient and aberration free). Conse-quently the total illuminance associated with an Airysystem will have the fastest rise of all possible systems.This result is closely related to a theorem that Barakat" 2

has proved, namely that the Airy system maximizesthe total illuminance in an infinitely small circle (orline).

4. NUMERICAL RESULTS

The purpose of this section is to present some typicalnumerical results using our procedure.

where

W (p) = W 2p2+ W 4p4 + W6P6+ * * , (4.2)

and W2 is the defocusing coefficient, W is the third-order spherical aberration coefficient, and W is thefifth-order spherical aberration coefficient. We are onlyconcerned with the first three terms of (4.2) in thispaper. The integral was evaluated by 24-point Gquadrature on the LGP-30 computer after making achange of variables to the standard interval (0,1).Incidentally only the real part of the integral is re-quired, it can be shown that the imaginary partvanishes; the transfer function is always real for ro-tationally symmetric aberrations. The calculations werealways performed at 21 points: w=0(0.1)2.0 for fairlygood systems, and for 41 points: w=0(0.5) 2.0 for othersystems.

In the case of a slit aperture, Barakat and Riseberg3

have shown that the optimum balancing of the aberra-tion coefficients yields

(A) W4, W2

W2=- (30/35)W4;

(B) W6, W 4, W2

W4=- (315/231)W6; W 2 = (105/231)W 6.

(4.3)

(4.4)

The equivalent circular-aperture results are14

W2= -W 4 , (4.5)

and

W 4 =-2W 6 ; W 2=5W 6. (4.6)

"t R. K. Luneberg, Mathemnatical Theory of Optics (BrownUniversity, Providence, Rhode Island, 1944).

12 R. Barakat, J. Opt. Soc. Am. 52, 264 (1962).

13 R. Barakat and L. Riseberg, "On the Theory of AberrationBalancing" (to be published).

14 M. Born and E. Wolf, Principles of Optics (Pergamon Press,Ltd., London, 1959), Chap. 9.

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SPATIAL FREQUENCY

FIG. 2. Slit-aperture transfer function for optimumbalanced third-order spherical aberration.

SPATIAL FREQUENCY

FIG. 4. Slit-aperture transfer function for optimumbalanced fifth-order spherical aberration.

SPATIAL FREQUENCY

1IG. 6. Circular-aperture transfer function for optimumbalanced fifth-order spherical aberration.

The integrals (2.7) and (2.11) were evaluated nu-merically using the transfer-function data

2 Nv sinvonL(v)=- HJT(cn) X (4.7)

7r n=° On

NL (vo) = vo E II0T(w.)J 1(vowo,), (4.8)

n=o

where N= 20 or 40 depending upon how widely thetransfer function oscillated. (In this paper only N = 20is employed.) The values of cot were 0(0.1)2.0 and theII,, are the usual Simpson qadrature weight coeffi-cients. All the computations were performed on theLGP-30 computer. The values of Its were usually chosento be vo=0(0.5)12.0.

AND A. HOUSTON

wUz4

2

--j

-j

f4

Vol. 53

FIG. 3. Total illuminance corresponding to transfer-function data of Fig. 2.

(.8z

ZM .6

4

.2

'0 2 6 8 10

V.

FIG. 5. Total illuminance corresponding to transfer-function data of Fig. 4.

0.824

.6

_- 4

..IfC-

FIG. 7. Total illuminance corresponding to transfer-function data of Fig. 6.

In Fig. 1 we show the total illuminance curve for aslit aperture with only defocusing present. Even assmall an amount of W2 as 0.5X has an appreciableinfluence. Note that all the curves vary linearly withvo in the neighborhood of the origin in accordance with(3.15).

The transfer function for a slit aperture having vari-ous amounts of third-order spherical aberration [op-timum balanced according to (4.3)] is shown in Fig. 2.The corresponding total illuminance curves are illus-trated in Fig. 3 and were computed using (4.7).Examination of Fig. 2 shows qualitatively that thethird derivative of the transfer function at the originis larger as W 4 increases. Hence by our Tauberianexpression (3.4) the total illuminance curves (for larger,) should be smaller numerically inversely as W. in-

R. BARAKAT1248

z

0

zF

I> .. . .U I., L4 1.6 LO 2.0

A.8.

.6-

SLIT APERTURE

4 10/ A W4 .80

B W4 120

.2 C W 1.40

o0 . . 6 . . 20 2 4 6 a e) C

z0

-)z

a:

z

a

A

SLIT APERTUREA W6 2.08 W3DOC W66-4.0D W. 0 5D

z4If)

CIRCULAR APERTURE.8

.2

0 .2 4 .6 .8 Lo L2 14 I . 1.

ISLIT APERTURE

.8 -A W, = 20

.6 - A B W, = 120C W = IAO

4

.2

0 - - . - . - .- .- .- --U .1

InI

I

November1963 TRANSFER FUNCTION AND TOTAL ILLUMINANCE. I

creases. This is exactly what happens in Fig. 3. Figures4 and 5 show the behavior of T(w) and Lvo for optimumbalanced fifth-order spherical aberration (see 4.5), againfor a slit aperture. The same general remarks hold asfor the previous case.

Finally we show T(w) (Fig. 6) and L(vo) (Fig. 7) fora circular aperture with optimum balanced fifth-orderspherical aberration (4.6). The transfer-function curveswere computed using a program already developed.', 2

Note that the circular-aperture transfer-function curvesare much smoother than the slit-aperture curves. Theconvolution integral is a smoothing operation, the in-fluence of the additional dimension enhances thesmoothing. The curves of total illuminance vary quad-ratically with v in the vicinity of the origin.

8

FIG. 8. Curves of con-stant total illuminance forslit aperture having W 4 =1as a function of defocusing.

wa:

0

0

5)

-1.4 -1.2 -1.0 -.8 -. 6

W2 IN X UNITS

-.4 -.2

The last curve (Fig. 8) shows how the lines of con-stant total illuminance behave as a function of de-focusing for a slit aperture having third-order sphericalaberration of amount W 4 = X.

1249