0 RICHARD BARAKAT V

maximum value at the end of phase It, r= 2. Theintegrated field in phase III then undergoes a slow decayand actually becomes negative at r-5.9. In fact, I(r)stays negative for 5.9<'r< °°. This anomalous be-havior is entirely due to the wave-diffusion integral in(4.3). The interested reader should compare this graph

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

with the data for surface pressure in Ref. 4, wheresimilar behavior occurs.

ACKNOWLEDGMENT

I am indebted to Agnes Houston for her help withthe numerical work.

VOLUME 55, NUMBER 8 AUGUST 1965

Investigations of the Influence of the Degree of Coherence uponImages of Edge Objects*

ROBERT E. KiEZLY

Cornell Aeronautical Laboratory, Inc., Buffalo, New York 14221(Received 11 February 1965)

The theoretical irradiance is presented for the images of sharp-edged objects as a function of the coherenceof the illumination of the object. These distributions are presented for optical systems possessing bothsquare apertures and circular apertures. The irradiance for degraded edged objects is also determined foroptical systems with square apertures. The influence of the degree of coherence upon contrast and edgegradient is considered in detail. Experimental results are given which show good agreement with the theo-retical irradiance. In all cases the systems are assumed free from aberration and the quasimonochromaticassumption is employed.

1. INTRODUCTION

THE literature relating to partial coherence theoryTis well covered in standard texts.", 2 However, de-spite the mathematical development, only a few pa-pers3- 6 report investigations dealing with the influence ofthe coherence of the illumination in an optical systemupon the images of specific objects. This paper reportsthe details of an investigation employing edged objectsbecause they are common in photographic transpar-encies.

General coherence theory as developed by Wolf2 usesa statistical quantity called the mutual-coherence func-tion to investigate the properties of electromagneticfields. This function is written in Wolf's notation as:

F112(r) = (Vi(t1+'r) V2*(t)). (1)

\Vhen the spatial and temporal separations are allowedto vanish, the mutual-coherence function is equivalentto the intensity or irradiance. Consequently, we mayinvestigate the imaging properties of a system by "prop-agating" the mutual coherence through the system tothe image plane. This approach can be simplified underseveral reasonable assumptions about the optical sys-

* This work was supported by internal research funds ofCornell Aeronautical Laboratory, Inc.

I M. J. Beran and G. B. Parrent, Theory of Partial Coherence(Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

'M. Born and E. Wolf, Principles of Optics (Pergamon Press,Ic., Netv York, 1959), Chap. X.

3 D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).4 M. De and S. C. Som, Opt. Acta 4, 17 (1962).6 W. N. Charman, J. Opt. Soc. Am. 53, 410 (1963).6 W. T. Welford, Optics in Meteorology (Pergamon Press, Inc.,

New York, 1960), pp. 85-91.

tem. We assume, in the following analysis, that (1)quasimonochromatic illumination is employed, (2) theoptical systems are diffraction limited, and (3) the small-angle approximation is valid. These assumptions arereasonable in microscopic and spatial-filtering opticalsystems, for example. Hopkins 7 has developed expres-sions for the irradiance in the image plane of the opticalsystem which fulfills the assumptions and is shown inFig. 1. He employs, as a mathematical convenience, an"effective source" to describe the mutual-intensityfunction in the object plane.8 In an actual optical sys-tem, such a source need not exist. The mutual-intensitvfunction in the object plane J(ut,u2,vi,v2) is shown to begiven by the Fourier transform of the radiance of theeffective source.9 Hopkins expresses the irradiance in

7H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).8 The mutual intensity is found by setting r equal to zero in the

expression for the mutual-coherence function which is useful wheninvestigating the imaging process employing quasimonochromaticlight.

9 One of the restrictions imposed by Hopkins, namely that themutual-intensity function in the object plane is spatially sta-tionary, can be removed by employing other expressions for thepropagation of the mutual-intensity function. These are

J(XIX2)= KfJ~t,2 expci(X2- 02-Xl'k)]dtidt2

for the propagation of the mutual intensity between planes suffi-ciently separated to allow the far-field approximation and

J' (x1,x) = J (X1,X2) T (xl) T*(x2)for propagation through diffracting objects or apertures having atransmittance function T(x). We note in the first expression thatthe "far-field approximation may be valid" in the near field asshown by the case when J(EI,02) represents an incoherent sourceand the first expression reduces to the Van Cittert-Zerniketheorem, which Hopkins uses to describe the effective source.

1002 Vol. 55

August1965 INFLUENCE OF COHERENCE IN IMAGES OF EDGES

I NCOHERENTEFFECTIVE

SOURCE a

OBJECT-IMAGING SYSTEM

x

ENTRANCEPUPIL

FIG. 1. Optical system showing the entrance and exit pupils ofthe imaging system. The effective source need not exist in anactual optical system and the field lens images the effective sourceonto the entrance pupil of the imaging system. The space sur-rounding the system is assumed to be filled with air.

the image plane as a six-dimensional integral involvingthe amplitude spectrum of the object, the complextransmittance of the imaging system between the en-trance and exit pupils, and the radiance of the effectivesource. This integral can be separated into two more con-venient expressions. The magnitude squared of the in-tegral involving the product of the object amplitudespectrum 0(m,n) and the complex transmittance of thesystem f(x,y), which is given by

1 r{k(x,y,U',v') =- ffO(m,n)

27rJ

Xf(x+m, y+n)ei(mu'+n,')dmdn, (2)

can be interpreted as the irradiance in the image planedue to a point in the incoherent effective source. Thesubsequent irradiance in the image plane is determinedby the remaining integral over the effective source, or

I'(u',v') = 27rffTs(x,y) I V,(x,y,u',v') I 'dxdy. (3)

Expressions (2) and (3) are employed in the investiga-tions reported in this paper. We should point out thatthe u, v, u', and v' coordinates are not the geometricalcoordinates of the optical system. To determine thesecoordinates, referred to as the reduced coordinates,

we multiply the geometrical coordinate by the productof the wavenumber of the light k = 2 7r/X and the numeri-cal aperture of the entrance or exit pupil of the imagingsystem.

In Sec. 2, we consider images of rectangular orsharp-edged objects. We assume that the optical systempossesses square apertures in order to obtain mathe-matical separability of the integrations over x and yin (3).10 This section also deals with one-dimensionalrectangular objects where the optical system is assumedto have circular apertures. Section 3 considers the imageof a degraded edge whose amplitude-transmittance func-tion is trapezoidal and, finally, in Sec. 4, the theoreticalirradiance is compared to experimental measurements.

2. RECTANGULAR OR SHARP-EDGED OBJECTS

Square Apertures

Consider a one-dimensional object whose amplitudetransmittance is

T(uv)={A O<u<w, svl <c

0 otherwise.(4)

This "rectangular object" has a complex-amplitudeFourier spectrum given by

(5)

In order to simplify the resulting integrations to yielda one-dimensional analysis, the aperture stop of thesystem is assumed to be a square aperture with one sideparallel to the edged object under investigation; that is,

f(xy)= hers0 otherwise.

(6)

This assumption is relaxed in the next section.Substitution of the above expressions into Eq. (2)

yields an expression for I t(x,y,u') 12

[IY<1 (7)

where Si and Ci denote the sine and cosine integralfunctions, respectively. Consider the radiance of theeffective source to be given by

1/2ir IxI < ye, E (8)s8(Xxy) =(8

o otherwise.

The source size, which depends on the value of e, repre-sents a particular mutual-intensity function in the ob-

'OD. Canals-Frau and M. Rousseau, Ref. 3, report a similaranalysis employing the Fourier transforms of the functions whichwefuse. However, the normalization criteria are different.

ject plane." Substitution of the effective source areaand Eq. (7) into Eq. (3) yields the image irradiance.The evaluation of the resulting integral requires numeri-cal integration and was completed by using an IBM 704.

In order to reach the incoherent limit, the value of emust be allowed to approach so. However, the resultingintegration cannot be completed numerically. Instead,the incoherent image is found by the more familiarpractice of convolving the intensity distribution of the

'1 In this case the normalized mutual-intensity function or com-plex degree of coherence measured perpendicular to the edgedobject is given by ro(u, -u) = [sin("2-u1) J/E(u2- uI)e].

1003

0(mn) == 2A 3(n)[sin(mw/2)/mJe-i(mw/2).

14,(xyn')11= o/47r'(Esi(l+x)u'+Si(I-x)u'-si(l+x)(u'-w)-Si(l-x)(u'-w)]I0 for 1y>1 +Eci(l+x)u'-Ci(I-x)u'+Ci(I-x)(u'-w)- Ci(1_X)(U1_W)]2)

1004ROBERT E. KINZLY V 55

Circular Apertures

For circular apertures, the transmittance of thesystem is given by:

1.5 (2. 10)-0.9 (3. 50)1.0 (3. I4)

0.5

0.4-

0.3

0.2

0.1

0.0-5 -4 -3 -2 -I 0 1 2 3 4 5 6 7

u' UNITS

FIG. 2. Irradiance in an image of a sharp edge for varying degreesof coherence. A square entrance pupil is assumed. The dashedcurve is the irradiance in the image of an incoherently illuminatededge.

object with the incoherent point spread of the imagingsystem.

Although the normalization criterion of equal-energycontent within each image is desirable, it is not con-venient since this requires a knowledge of the total areaunder the irradiance curves, I'(u'). Instead, a nearlyequivalent criterion is employed in which the irradianceat t'= w/2 is set equal to 1.0. The resulting irradiancedistributions in the image for all values of object widthsgreater than 300 reduced-coordinate units are identical.Figure 2 contains the resulting intensity distributionsfor various degrees of coherence, including the incoher-ent limit. The number in brackets in the figure repre-sents the coherence interval" in reduced units corre-sponding to the effective source. The results presentedin the figure show a smooth transition between thelimits of complete coherence and incoherence, which is

to be expected. As e changes from 0.2 to 0.9, there is a

decrease in the amount of "ringing" of the image.Further increase of e beyond 0.9 shows an increase in

the energy content of the toe of the image. In addition,the value of the irradiance at the image coordinate con-jugate to the location of the edge object (i.e., u'=0)changes from 1? to ' in passing from the coherent to theincoherent limits, as would be expected. Also, the edgegradients (estimated primarily in the region of 0 to 1.0)

exhibit a monotonic decrease with decreasing coherence,

changing from 0.40 in the coherent case to 0.31 in theincoherent case.

12By coherence interval, we mean the minimum separationbetween points in the object plane (spatial separation) for whichthe mutual-intensity function vanishes.

I r<1f(r,O) =0 elsewhere.

1.2

I .11.0

0.9

0.8

0.7

0.6

Wo

-4

-t

C,

1.2

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0-5 -4 -3 -2 -I | 1 2 3 4 5 6 7

ul UNITS

FIG. 3. Same as Fig. 2, except a circular entrance pupiland effective sources are assumed.

0.2 (0.60.8

Similarly, the effective source is given by:

1/27r r< e

1 8(r,O) =0 otherwise.

(9)

(10)

'IE

0.2 -0.6.0.8-0.9s

I.0

Vol. 551004

The analysis proceeds in a fashion similar to that pre-sented above. Substitution of Eqs. (9) and (10) intoEq. (2) yields an expression for I q/(r,O,u',v') 2 and thesolution is completed in polar coordinates. This ex-pression is identical to Eq. (7) when a function of r and0, expressing the varying width of the entrance pupil,is substituted for the constant (one) in that equation.

The final integration over the effective source againmust be completed numerically and was solved on theIBM 704. The resulting irradiance curves are shown inFig. 3. The general properties discussed for the previouscase are still exhibited here. However, the value of theedge gradient for corresponding values of e is less forthe circular apertures than for the square apertures(see Fig. 7). This result is to be expected, since theequivalent square aperture "passes higher frequencies"than the circular aperture.

3. TRAPEZOIDAL OBJECTS OR DEGRADED EDGES

As our last application of coherence theory to theimaging process, consider a c]ass of trapezoidal objects

August 1965 INFLUENCE OF COHERENCE IN IMAGES OF EDGES E00

zI

IIII

I

I

RELATIVE POSITIONW + S W + 2S

FIG. 4. Intensity transmission for the trapezoidal object.

which allow for a finite resolution of the object. It hasan amplitude transmittance given by:

[0(A/S)u

T(u) = AI (A/S)(w+2S-u)LO

U<0O<u<SS<u<w+SW+S<u<w+2SW+2S<u.

.4x

z1

I. I

1.0

0.9

0.80.70.6

0.5

0.40.30.2

0. I

0.0v

(11)

The intensity transmittance of the object is shown inFig. 4. Its amplitude spectrum is

0 (m,n) = (2A IS) 5(n)X{[(cosIm-cosam)/m2]}e imz, (12)

where

O=w/2a(= (w/2)+S.

The transmittance of the optical system is given by ex-pression (6) and the area of the effective source by ex-pression (8). The integration over the effective sourcemust be done numerically as in the previous cases.

The resulting image irradiance for S= 1 and S= 10are shown in Figs. 5 and 6. The results observed pre-viously, i.e., decreasing edge gradient, decreased ring-ing, and decreased contrast with decreasing coherence,

1 .2

1. I

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.30.2

0. I

Is.0.20.60.80.9I.0

u' UNITS

FIG. 5. Irradiance in the image of a degraded edge for variousdegrees of coherence. The slant parameter s of trapezoidal objectis 1.0 reduced units.

0.80.91.0

2 3 4 5 6 7 8 9 10 If 12 13 14u' UNITS

FIG. 6. Same as Fig. 5, except s= 10.0.

15 16 17

are also evident in the figures. In addition, the edgegradient is noticeably decreased from the previous casesbecause of the finite resolution of the object. Figure 7summarizes the change in gradient with coherence forthe various objects considered. We see that the edgegradient varies most rapidly in the region of E =0.6 toe= 1.1 in all cases.

Finally, we observe a noticeable decrease in the ampli-tude changes involved in the ringing for correspondingdegrees of coherence between the trapezoidal objectsand the previous sharp-edged object (Fig. 2).

4. EXPERIMENTAL RESULTS

Several experimental measurements have been madeby Charman"3 which show reasonable agreement with

.3 - -B

.2 -- D

A 2 -

- ----- -- -- - ---- -- - -- --

.2 .4 .6 .8 1.0 1.2 1.6EFFECTIVE SOURCE SIZE

FIG. 7. The variation of the edge gradients with coherence forthe various objects. (A) sharp edge, system with square aper-tures; (B) degraded edge with s= 1.0; (C) sharp edge, system withcircular apertures; (D) degraded edge with s=4.25; and (E) de-graded edge with s= 10.0.

1" See Fig. 8 of Ref. 5. Private communication with W. N.Charman revealed that a typographical error resulted in the mis-labeling of the "s" value at the low-intensity region of the curvesin this figure.

1005

-. X

S

.4

-X

.-

I

ROBERT E. KINZLY

-4 -2 0 2u' UNITS

f

///

-- := =I

-4 -2 0 2u' UNITS

///

//

- II I I

1 6 8

(O

Li 6 8

1. I

1.0

.9o

..-

(C)

-4 -2 0 2 It 6 8u' UNITS

FIG.-S. Comparison between theoreticallirradiance and Char-man's experimental resultsl(see Ref. 13). (a)!e=0.6;J(b)Ie=O.S;and (c) 6=01.0.

EFFECT IVESOURCE FIELD LENS

TUNGSTEN LENS MASK SYSTEMSOURCE / \

GU ,GA O T PA

GROUND GLASS ORJECTIPl. AN E

I MAGI NGLENS

I-ENR, A

ENTRANCE

ANSCOIMAGE MODEL 4APLANE MICROIMAGE<, SCANNER

i

- f00., -)

OUTPUT TO ANALOGAND DIGITAL

APERTURE RECORDING DEVICES(IMAGE OF THE

EFFECTIVE SOURCE)

FIG. 9. Experimental optical arrangement.

.7

. 6

- .5

.I

.3

.2

the theoretical irradiance curves for the sharp edgeswith circular apertures previously presented in Fig. 3.The theoretical curves are compared to the experimen-tal results in Fig. 8. The:experimental curves have a

I.3,

/=

/

(a)

I I I I I

(a)1.2

1.1

1.0

.0

.7

.6

.5

.1

.6

41

(c)

1.0

.9

.8

.7

.6

.5

.3

.2

/. /

0 10 20 30 40 s00 10 20 30 '0 50

RELATIVE POSITIONI (HICRO12) RELATIVE POSITION (MICRONS)

FIG. 10. Experimental irradiance plotted against the theoreticaldistributions for a degraded edge with a slant s = 14 ,. (a) e=0.23;(b) e= 0.54; (c) e = 0.92; and (d) e = 1.03.

slightly decreased slope, indicating that the opaqueedge employed by Charman is slightly degraded. InCharman's paper, the ratio of the condenser numericalaperture to the objective numerical aperture (the ratio,

1006

I .2 -

1.0k-

1.2

1. I

1.0

.9

- .

. . 7

1

i .6

.5

Vol. 55

(b)

O 10 20 30 40 50RELATIVE POSITION (MICRONS)

(d)

.2

1.2,

1.0

L

2

.3 00 .. / .2

.2

.1 /I

.I

0 10 20 30 40 50

RELATIVE POSITION (MICROOS)

I.0

2, . 8 .I

.6

.2

-

I

I

o

0

August 1965 INFLUENCE OF COIHIERENCE IN IMAGES OF EDGES 100

is called "s"), is equivalent to e, the size of the effectivesource. Also, the z unit is identical to the it' unit em-ployed in this paper.

We conducted several experiments to verify thetheoretical irradiance curves for the degraded edge ortrapezoidal object. A diagram of the experimentalapparatus we employed is shown in Fig. 9. Note that itclosely resembles the system previously presentedin Fig. 1.

The degraded-edge object was made by contactprinting a strip of high-density film on to film which wasdeveloped to a high gamma. The effective source masksprovided square sources with relative widths 14 of 0.23,0.54, 0.92, and 1.03. The experimental procedure in-volves appropriate calibration of the output of the de-tector which, in this case, is an Ansco microimagescanner. The object and desired effective source arethen placed in the apparatus and the image is scanned.The output voltages are recorded on punched paper tapeby using an analog-to-digital converter, which permitsthe output data to be processed on an IBM 704 com-puter. Scans made of the test object showed changesfrom maximum to minimum transmittance over regionsof 10 to 20,u. A trapezoidal object with an S parameterin this range should agree with the experimental results.Unfortunately, the experimental irradiance curves couldnot be completely normalized, because of insensitivity

14 That is, the width of the image of the square effective sourcein the plane of the entrance aperture normalized by the width ofthe square entrance aperture.

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

of the experimental technique and noise in the outputelectronics. Consequently, the theoretical irradiancedistributions for all values of the S parameter in therange of 10 to 20 g fit almost equally well. Figure 10presents the experimental data points fitted to a trape-zoidal object with S= 14 M. We wish to emphasize theimportant conclusion that the experimental distribu-tions do, indeed, follow the theoretical distributionsand, consequently, the conclusions concerning the in-fluence of the degree of coherence of the illuminationupon images of edge objects are verified.

5. CONCLUSIONS

The following effects upon images of edges can be ex-pected when the degree of coherence of the illuminationin the object plane is decreased:(1) Slight loss in contrast through increasing energy in

the toe of the image.(2) Decreased ringing of the image.(3) Lower edge gradients (intensity vs position).

In addition we wish to point out that the results ofthis paper indicate that appropriate consideration mustbe given to coherence effects when using edge-traceanalysis for image evaluation.

ACKNOWLEDGMENT

The author wishes to thank Dr. Paul Roetling for hisaid through several helpful discussions on the subjectmatter of this paper.

VOLUME 55, NUMBER 8 AUGUST 1965

Space-Variant Image Formation*A. W. LOHmANN AND D. P. PARIS

IBM, Systems Development Division, Development Laboratory, San Jose, California 95114(Revision received 10 April 1965)

The application of optical transfer theory to the process of image formation requires that the image-form-ing system be linear and space invariant. In a space-invariant system, the point image retains its shape whilethe point source explores the object plane. The purpose of this paper is to investigate image-forming systemswhich are linear but space variant. Such systems may exceed performance limitations which are inherent inlinear space-invariant systems. A method for experimentally determining space variance is devised. Thedegree of space invariance is defined and evaluated for several examples of space-variant systems.

A. INTRODUCTION

IT is convenient to describe the process of image for-mation by means of the optical transfer theory. The

application of this theory to image-forming systems isbased on two significant systems properties: linearityand space invariance.1 In this paper, we investigate

* Presented in part at the 6th Conference of the InternationalCommission for Optics, Munich, Germany, 19-26 August 1962.

1 R. V. Pole (Thomas J. Watson Research Center, YorktownHeights, New York) proposed the term "space invariance" insteadof the formerly used "stationarity." The term "nonisoplanatism"which has been used in this context, refers traditionally to thespace variance caused by aberrations only.

what happens when the property of space invariance isabandoned.

Let u1 (r) and u2 (r) be two objects and vi(r) and v2(r)their respective images. In the case of coherent illumina-tion, i 1, u 2 , vl, and v2 represent complex amplitudes,while for incoherent illumination they represent intensi-ties. The image-forming system is linear if the objectAu1(r)+Bu 2 (r) yields the image Avv(r)+Bv2 (r) for anypair of constants A and B and object functions ui1 (r) andu2(r). The image-forming system is space invariant ifthe object u(r+C) yields the image v(r+C) for anyvector C and function u(r). In other words, in a linear

1007