CARL W. HELSTROMV

Fl(r1 ,r2 ; W; vs)-(22c/9h)vf d 2r 3 R(ri,r 3 ; W; s)

Xfl(r 3,r2; ; v,s) =R(ri,r 2 ; W; s), (A23)

TrH(7,,s) = (2McT/l) jd2rf H(r,r; w; v,s)dco/2wr. (A24)

We now define the eigenvalues vk and the eigenfunc-tions llk(r) of the kernel

(2 Qc TI ) sO(rir2)

as in Eq. (5.6) and write the kernel itself and the func-tions P, R, and H in terms of them,

(2 QcT/A') sc(r1 ,r2) = IkZk(n)k*(r2),

P(ri, r2; W; s) =Z Pk (C; s) 1k(rl>),k-*(r2),k (A25)

R(rr2; W; s) =_ Rk(W; s)j 1k(r 1)71 k*(r 2),k

fl(ri, r2; Wo; v, s) = TH h(111; V S) qk (rl) 7k* (r2).

In this way we get from Eqs. (A21)-(A24)

9(01+1)0Pk(W; s)IaS

= v13T1 X(W)Pk(W; s) + (h/2QcT)vkX(W),

Rk(&; s) = [EX+vkTT'X(w)]Pk(o; s),

Hk(f ; vs)[1-(2Qc/h)vRk(W; s)]=Rk(w; s),

TrH(v,s)dv

=- T E n~ -ln (20c/h)Rk(w; s)]dwl2r-

(A26)

(A27)

(A28)

(A29)

Finally, from Eqs. (A26-A28),

(2Qc X {)Rk (W; s) = [01+,XVk7 iY(+) )X { exp[vk7llX(-1)s/91(91+ 1)]- 1)} (A30)

which with Eqs. (A6) and (A29) yields the logarithmof the mgf ,u. (s; f) given in Eq. (5.7).

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 59, NUMBER 8 AUGUST 1969, PART 1

Image of a General Periodic Bar Pattern through anAberration-Free Annular Aperture

K. SINGH AND A. K. KAVATHEKAR

Department of Physics, Indian Institute of Technology, New Del hi-29, India(Received 4 September 1968)

The image of a periodic bar pattern formed by a diffraction-limited annular aperture with incoherentincident light has been investigated theoretically. Results for the irradiance distribution in the image andthe rectangular-wave response have been obtained. Three typical values of the central obscuration ratioare considered and the value of bar width that gives a rectangular wave response nearly equal to that ofsine wave is identified.

INDEX HEADINGS: Image formation; Diffraction; Aberrations; Coherence; Modulation transfer.

As a result of the studies by Duffieux, 1 Schade,2

Marechal,3' 4 and Hopkins,5 it is now standard practice'--to regard image formation in optics as a filtering ofspatial frequencies and to treat image-forming optical

I P. M. Duffieux, L'Integrale de Fourier et ses Applicationsa' I'Optique (Besanson, privately printed, 1946).

2 0. H. Schade, RCA Rev. 9, 245, 490, 653 (1948).3 A. Mar6chal, Rev. Opt. 26, 257 (1947).4 A. Mar6chal, Rev. Opt. 27, 73, 269 (1948).5 H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).6 H. H. Hopkins, Proc. Phys. Soc. (London) 79, 889 (1962).7K. Murata in Progress in Optics, V, E. Wolf, Ed. (North-

Holland Publ. Co., Amsterdam, 1966), p. 201.8 K. Rosenhauer in Advanced Optical Techniques, A. C. S. van

Heel, Ed. (North-Holland Publ. Co., Amsterdam, 1967), p. 634.9 K. Rosenhauer and K. J. Rosenbruch, Rep. Progr. Phys. 30,

1 (1967).10 E. L. O'Neill, Introduction to Statistical Optics (Addison-

Wesley Publ. Co., Inc., Reading, Mass., 1963), Chs. 5, 6.

systems as low-pass filters. Various studies1 l-2 0 of theperformance of optical systems with slit and circularapertures have been based on this concept. Systemswith annular apertures have been considered by Steel,21

O'Neill,2 2 and Barakat and Houston. 23 All of these

11 H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).12 W. H. Steel, Opt. Acta 3, 67 (1956).13 M. De, Proc. Roy. Soc. (London) A233, 91 (1955).14 G. Black and E. H. Linfoot, Proc. Roy. Soc. (London) A239,

552 (1957).15 H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).16 See Ref. 10, Chs. 5, 6.17 R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).18 R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992

(1962).19 R. Barakat, J. Opt. Soc. Am. 54, 920 (1964).20 R. Barakat and A. Hougton, J. Opt. Soc. Am. 55, 1142 (1965).21 W. H. Steel, Rev. Opt. 32, 4 (1953).22 E. L. O'Neill, J. Opt. Soc. Am. 46, 285 (1956); see also Ref.

10, p. 99.23 R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538 (1965).

936 Vol. 59

Augustl969,Partl BAR PATTERN THROUGH

studies"l-2 3 are, however, concerned with the evaluationof optical transfer functions when the object is anincoherently illuminated sinusoidal grating.

To avoid the difficulties in making sinusoidal gratingswith correct wave form, use of rectangular wave gratingshas been suggested by various workers, such as Washerand Rosberry,'4 Hutto," Rosenhauer and Rosenbruch,2 6

Murata and Matsui,2 7 and Emara.2 5 However, notheoretical studies have been made by these workers forthe irradiance distribution and transfer functions (imagecontrast) for this type of target.

At present, considerable attention is being paid tothe investigations of the use of square-wave objects forstudies of the effect of diffraction on optical imaging.This is evident from the investigations of Markchal andFrangon,25 Barakat and Houston,3 0 and Newman,Barakat and Humphreys." Recently Barakat andLerman3 2 have investigated the imagery of truncatedsquare-wave targets. Kottler and Perrin" have made anextensive study of the irradiance distribution and con-trast in the diffraction images of a general rectangular-wave object. Their investigations are limited to theaberration-free slit and circular apertures operating inincoherent incident light.

Like the slit and circular apertures, the annularaperture is also of considerable practical importance.Annular apertures are encountered during attempts touse a central stop in optical systems with circularapertures, which causes the central maximum of theAiry pattern to become narrowed and increases thedepth of focus of the system; hence it is obvious thatthe various aspects of the images formed by annularapertures need extensive investigation. Considerablework in this direction has been done by Steel,2 ' O'Neill,"B arakat,3 Lansraux,34 Roizen-Dossier, 3 ' Toraldo diFrancia,'6 Linfoot and Wolf,37 Taylor and Thompson,"Welford,' 5 Thompson, 4 0 Cornacchio, 4 ' and Dhillon. 4 1 We

24 F. E. Washer and F. W. Rosberry, J. Opt. Soc. Am. 41, 597(1951).

25 E. Hutto, J. Soc. Motion Picture Television Engrs. 64, 133(1955).

28 K. Rosenhauer and K. J. Rosenbruch, Opt. Acta 4, 21 (1957).27 K. Murata and H. Matsui, Oyo Butsuri 32, 223 (1956).28 S. H. Emara, J. Res. Natl. Bur. Stds. 65A, 465 (1961).29 A. Marechal and M. Frangon, Diffraction Structure des Images

(Editions de La Revue d'Optique Paris, 1960), pp. 38, 50, 169.20 R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1371 (1963).3A. Newman, R. Barakat and R. Humphreys, Appl. Opt. 5,

670 (1966).22 R. Barakat and S. Lerman, Appl. Opt. 6, 545 (1967).

F. Kottler and F. H. Perrin, J. Opt. Soc. Am. 56, 377 (1966).34 G. Lansraux, Rev. Opt. 32, 475 (1953).35 B. Roizen-Dossier, Rev. Opt. 33, 57, 147, 267 (1954).36 G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).37 E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66,

145 (1953).28 C. A. Taylor and B. J. Thompson, J. Opt. Soc. Am. 48, 844

(1958).29 W. T. Welford, J. Opt. Soc. Am. 50, 749 (1960).40 B. J. Thompson, J. Opt. Soc. Am. 55, 145 (1965).41 J. V. Cornacchio, J. Opt. Soc. Am. 57, 1325 (1967).4' H. S. Dhillon, "Diffraction of Partially Coherent Light by an

Annular Aperture"; thesis, Indian Institute of Technology, NewDelhi (1968).

have undertaken theoretical studies of the diffractionimagery of a general bar pattern through an opticalsystem using aberration-free annular apertures andaberration-free incoherent incident light. The presentpaper deals with the irradiance distributions in theimage and the generalized rectangular-wave response ofsuch systems.

THEORY

Let the one-dimensional object function be denotedby G(Z). Since the process of image formation in in-coherent illumination can be written as a convolutionintegral, the irradiance distribution in the image is givenby the convolution theorem,4 3

H(Z') = F(Z'-Z)G(Z)dZ, (1)

where F(Z') is the spread function appropriate to thesystem and Z' is the distance in the image plane. Toeliminate the factor of magnification, the ideal image ofthe object is considered as the object. In our case, thespread function appropriate to the system is the linespread function (distribution of irradiance in the imageof an infinitesimally thin incoherent line source). Thisfunction can be obtained by starting with the irradiancepoint-spread function4 4

I= ~ ° | ()- r72[ ( 2 (2)(I z'iI2)2(2)~

where j is the obstruction ratio, i.e., the ratio of theradii of obstructed part to the total aperture of thesystem and Z' = 7rD sinO/X; X, D, and 0 are, respectively,the wavelength of light used, diameter of the aperture,and the angle of diffraction. We assume that this dis-tribution (point-spread function) does not change as weexplore the working field of the instrument, i.e., thesystem is isoplanatic. Following Steel,2 ' we can writethe line-spread function of (also known as the impulseresponse) an annular aperture as,

37r Hl(2Z') H1(2,qZ')F(Z')=-I ~ +rq3

8 L Z'2 ?2Z12

4-q2 rTsin(Z'[1-2,qcosq5+,q1])-- nsin2,OdO], (3)7rZ' 1 cos- +,j' i

where H, (Z') denotes the Struve function4 ' and q is theangular coordinate of the variable point in the pupilplane. For the object function G(Z) we take a general

43 R. M. Bracewell, Fourier Transform and Its Application(McGraw-Hill Book Co., New York, 1965), p. 27.

44 M. Born and E. Wolf, Principles of Optics (Pergamon Press,Inc., New York, 1964), p. 416.

45 A. Gray, G. B. Mathews and M. M. MacRobert, A Treatiseon Bessel Functions (Dover Publications, Inc., New York, 1966),p. 214.

937ANNULAR APERTURE

938 K. SINGH AND A. K. KAVATHEKAR Vol.59

10 In Eq. (6)

.2 nw nw- [,d 2-A=-jcosg--- 0•w/2<1

2 208 =0 wl/2> 1

B= cos'---- 1i-(- f I<1Lr 2X7 27 1\28 -J-

0 Iac .10

0.6 -

C= -2X12 _<co/2< (1-7)/2

N 277=-2-q2+- sinO+E(l+q72)17r]O

0.4 2 1+--(1-72) tan-l -tan-l7r - 1 I 1 2J

(I -- q)/2 <co/2 < (1 +X7)/2

= I 0. w/2> (1+X1)/20.2 \

Here----t pe A=cos-l(j+,2_n2X2,,2) ,

w0o1s705\ 1.0\ For n= 1, the function F(n) is the modulation-transfer

0 0.5 1.0 1.5 2.0 2.5 3.0 1.WZ 7r-

FIG. 1. Irradiance distribution in the images of periodic barpatterns in the case of annular apertures. Bar width a= O.10; barfrequencies are c= 0.10, 0.50 and 1.0. Solid and dotted curves arefor X7 = 0.25 and 77 = 0.50 respectively. The vertical line (--)represents the geometrical image.

0.8

periodic bar pattern (general rectangular wave). Thewidths of bright and dark bars are ap and (1-a)p, -\

respectively. The Fourier-series representation of such a \wave can easily be shown to be given by --... \ O.5l

0.6 - a 0 504b X sinnira \ '\

G(Z) =a-b+2ab+-- T cosncoZ, (4)i7.n=l 1 flh

where n= 1, 2, 3 ... and a and b are the mean irradiance N;

and the modulation, respectively; co is the angular fre- 0.4 :1 \ -

quency and is given by wc= 27rv, where v is the number \of repetitive elements per unit distance, so that the woi .5\i.operiod is p= 1/v.

To obtain the irradiance distribution H(Z') in theimage, Eqs. (3) and (4) have to be substituted in Eq.(1). If this is done and the integral is solved by the 0.2 -method of Steel 2' and O'Neill2 ' we get,

4b n' sinn7rcaH(Z')=a-b+2ab+-- 5 F(n) cosnczZ', (5)

7r U-i 1I I3 I I.00 0.5 1.0 1.5 2.0 25 3.0where F (it) is given by eoz 7 7 ,

FIG. 2. Same as Fig. 1, bar width, a=0.50.F(n)= (1-,nl)-'[A+B+Cj. (6)

Augustl969,Partl BAR PATTERN THRO;UGH ANNULAR APERTURE

function or the aperture autocorrelation of the annularaperture." 22' 46 In Eq. (5), the upper limit of n is n'instead of x as in Eq. (4). This is because F(n) becomeszero for values of nco 2.

When ao-- 0, the object consists of infinitely narrowlines. In such a case, b should tend to infinity if a finiteamount of light is to be transmitted. Further, if thenormalized irradiance in the line is taken as unity,2ab 1-> and therefore the object function for this case(and for a= b) becomes

G(Z) -- 1+2 E cosncWZ

4.5

4.0

3.5

3.0

(7)

2.5

Iand the irradiance distribution in the image can beshown to be

n,H(Z') =1+2 E F(n) cosnwZ'.

n=1(8)

In addition to the irradiance distributions given byEqs. (5) and (8), the ratio of the contrast in the image

1.0

0.8

N

46 L. Levi, Applied Optics; A Guide to Modern Optical SystemnDesign (John Wiley & Sons, Inc., New York, 1968), p. 505.

2.0

1.5

1.0

0.5

N -31 3-l --COlin

0 0n

f \x

W =0. I O.5 I .a '.

0 0.5 1.0 1.5 2.0 2.5 3.01WZ -

FIG. 4. Same as Fig. 1, bar width, a=0.

to that in the object is also of interest. For simplicity,the object contrast (Gmax-Gmin)/(Gmax+Gmin) may beassumed to be unity, so that the generalized rectangular-wave response is given by

Hmax-Hmin H(Z')z,=o-H(Z') =,z=H= =. (9)

Hmax+Hmin H(Z').z= 0+H(Z').z',

RESULTS AND DISCUSSION

The irradiance distribution in the image has beencalculated for different bar widths a (a= 0.10, 0.50, and0.75) using Eq. (5). These computations have beenmade for three different bar frequencies co(w= 0.10, 0.50,and 1.0). Parameters a and b have been taken to be suchthat a= b= 0.50. Irradiance H(Z') is shown as a functionof coZ' in Figs. 1-3. For the special case of discrete lines(a= 0), the irradiance distribution (Fig. 4) was calcu-lated by use of Eq. (8). In all of these cases the solidand the dotted curves represent the results for centralobscuration ratio of 0.25 and 0.50, respectively. Verticallines represent the edges of the geometrical images. Tobring out salient features of the irradiance distributionin these images formed by annular aperture, Fig. 5

WZ Tr

FIG. 3. Same as Fig. 1, bar width, a=0. 75 .

939

K. SINGH AND A. K. KAVATHEKAR

.9

.5 1*

I, aaZ 7

Ii

I,/ 0.2.

/-. a w=15

Ir - at G7

FIG. 5. Comparison of normalized irradiance distribution in theimages formed by a circular and an annular aperture (X7=0.50).(a) W = 1.0 and (b) co=1.5, a =0. 10, 0.50, and 0.90 in each case.Dotted curves are for annular and solid curves are for circularapertures.

shows the comparison of normalized irradiance dis-tributions in the image formed by a circular and anannular aperture with -q=0.50. For co=1.0 (Fig. 5a)there is a broadening of the diffraction figure for allvalues of a. For co=1.5 (Fig. 5b) however, the half-widths of the diffraction figures formed by an annularaperture, for all the values of a, are smaller than thosefor the circular aperture. This is in agreement with theexperimental and theoretical results that, for highspatial frequencies, the image contrast in the case of anannular aperture is better, although at low spatial fre-quencies it is not as glood as is the case with a circularaperture.

The contrast was calculated by use of Eqs. (5) and(9) for ca= 0. 10 through 0.90 and by use of Eqs. (8) and(9) for ax=O. The results are shown in Figs. 6-8 forq= 0.25, 0.50, and 0.75, respectively. The curveslabelled C and A in each case represent the sine-waveresponse for a circular and an annular aperture respec-tively. The curves labelled A in each case are the same

as the results of O'Neill.22 For a given a, the contrastimproves a little for the higher spatial frequencies atthe expense of lower spatial frequencies. Figures 6-8show the extent to which the rectangular-wave responseis degraded for lower spatial frequencies and improvedfor higher spatial frequencies when a circular aperturewith central obstruction is employed as the image-forming system.

If the effects of higher harmonics in the object on theimage, are eliminated by opto-electric Fourier analysis,7

the sine-wave response is obtained. The method there-fore can be used for testing annular apertures such asmirror systems. Alternately, the sine-wave responsemay be calculated by using Coltman's formula,47 whichgives the relationship between the sine-wave responseand the square-wave response. Also of interest is thevalue of a for which rectangular-wave response is nearlythe same as the sine-wave response. This is approxi-

0.0

7=25

0.8 "

aa=aaa a

aIa =av 2a rs6i a=

a \ ~ C

40.2

a90 0.5~ a . . .a

FIG.6. ectnguar-av repns o q025 a5wdharea=0, . 0,0.2, .50 07a an090CuvslbleAadCrepeset th sie-av repnefraannlradcruaapetuesrepeaivly

47 J. W. Coltman, J. Opt. Soc. Am. 44, 468 (1954).

940 Vol. 59

August 1969, Part 1 BAR PATTERN THROUGH ANNULAR APERTURE

0.8

10.6

1

0004

0.2

77 .75

a=0

0.5 1.0 1.5 2.0 0.5 .0 1.5

FIG. 7. Same as Fig. 6, for ?1=0.50. FIG. 8. Same as Fig. 6, for =0.75.

mately 0.60 for all values of n. Thus for an appropriatebar width, a periodic bar pattern may be designed andused to measure the sine-wave response of the systemunder test.

ACKNOWLEDGMENTS

The authors wish to record their gratitude to Pro-fessor P. K. Katti for helpful guidance and Dr. M. Defor encouragement.

Karl G. Kessler (President, OSA; NBS) talking to R. R.Shannon (Univ. of Arizona, Optical Sciences Center) at SanDiego meeting. Photo by B. Sherman.

941