JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Modulation of Square-Wave Objects in Incoherent Light. *RICHARD BARAKAT AND AGNES HOUSTON

Optics Department, Itek Corporation, Lexington, Massachusetts(Received 26 April 1963)

A study is made of the effect of diffraction on imaging a square-wave object (in intensity) in incoherentlight for slit and circular apertures suffering from spherical aberration. A modification of the usual Fourierseries representation of the square wave is required to eliminate the unwanted Gibbs phenomena. Themodulation of the image of the square wave is calculated by using transfer-function theory. Numerousexamples are presented.

1. INTRODUCTION

THE idea of employing a grating of parallel alter-nate bright and dark bars as a test object for

optical systems is evidently due to Foucault1 in hisfundamental paper of 1858 in which was also introducedthe (Foucault) knife-edge test. He utilized this pro-cedure to study the resolving power of telescope mirrors.Since then, the technique has remained as one of themost important test procedures for the practical opticianand systems engineer. The first really successful explana-tion of the Foucault grating test is due (as usual!) toLord Rayleigh2 in which the foundations of modernoptical systems theory were enunciated. Modern treat-ments of square objects are given by Frangon andMarechal.3

In spite of the unquestioned superiority of sine-waveobjects from a theoretical point of view, many peoplestill prefer (rightly or wrongly) to deal with square-wave objects. It is simple to make a square-wave tar-get; unfortunately the manufacture of a sine-wave tar-get is still a matter of some delicacy and luck.

The mere fact that people employ square-wave tar-gets behooves us to study the problem from a theoreticalpoint of view. The earlier and most elegant approach isthat of the transfer function. It is well known that

I (co)= ()O((X, 0(1.1)

2 7_

DISTANCE ACROSS OBJECT

FIG. 1. Square-wave and sine-wave objects ofthe same spatial frequency.

* Research supported in part by the Reconnaissance Labora-tory, Wright Air Development Center.

I L. Foucault, Ann. l'Observ. Imp. Paris 5, 197 (1858).2 Lord Rayleigh (J. W. Strutt), Phil. Mag. 42, 167 (1896).3 M. Frangon, article in Handbuch der Physik, edited by S.

Fltigge (Springer-Verlag, Berlin, 1956), Vol. XXIV, pp. 171, 342;A. Marechal and M. Francon, Diffraction-Structure des mages(Editions de la Revue d'Optique, Paris, 1960), pp. 38, 50, 169.

where O(co) is the spatial spectrum of the object in-tensity distribution, I(wo) is the corresponding spectrumof the image illuminance distribution, and T(w) is thetransfer function of the system. In this paper we con-sider only real transfer functions, so that our results onlyhold for simple defocusing and spherical aberration.Moreover, actual applications will only be made here todefocused and perfectly focused images. We are pre-paring a catalog of results for optimum balanced third-order spherical aberration and optimum balanced fifth-order spherical aberration for publication. A secondpaper in preparation will deal with complex transferfunctions such as are encountered in the off-axis case.The object spectrum can be expanded in a Fourierseries, so that in principle it is simple to compute I(w)given T(w). Now knowing I(X) is equivalent in thisspecial case to knowing the distribution of illuminancein the image.

Unfortunately, the usual approach to expanding asquare wave involves the Gibbs phenomena, which com-pletely distorts the object. We have employed an ap-proach due to Lanczos which circumvents this difficulty.Both slit and circular apertures are treated. The mainresult we wish to emphasize is that given the transferfunction we can calculate the shape and hence modula-tion of the image. We are not restricted to hypotheticalsystems but can employ (see Fig. 16) transfer functionscomputed from design data.4

FIG. 2. Fourierseries representationof square wave il-lustrating the Gibbsphenomena.

0 .5 -1.o

X/L

4 R. Barakat and M. V. Morello, J.(1962).

Opt. Soc. Am. 52, 992

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VOLUME 53, NUMBER 12 DECEMBER 1963

12. BARAKAT AND A. HOUSTON

X/L

FIG. 3. Fourier series representation of square waveemploying the Lanczos sigma factors.

2. LANCZOS SIGMA FACTORS ANDGIBBS PHENOMENA

The Fourier series representation of a square wave ofperiod 2L and unit amplitude centered at x= 0 (seeFig. 1) is

2 (-1)" F r_0 (x) 2+- ~ cos (2n+1)- , (2.1)

7 n-o (2n+1) L L_

as one can easily verify. Now a Fourier series providesus with a very poor representation of a square wave in

TABLE I. Numberwhich are allowed byfrequency range.

of harmonic components of square waveoptical system as a function of the spatial

the neighborhood of the edges (x=4zL/2) where O(x)has a discontinuity. This is the Gibbs phenomenon whichall Fourier series exhibit at their discontinuities and isdue to the failure of uniform convergence of the seriesat the point of discontinuity. Figure 2 shows how 15terms of (2.1) approximate the square wave. Note thelarge amount of oscillation in the vicinity of the dis-continuity. As the number of terms in the series is in-creased, the peak oscillations do not decrease in magni-tude but merely move closer to the discontinuity. Theamount of overshoot at the discontinuity is approxi-mately 18%, irrespective of the number of terms oneemploys. Consequently, (2.1) is not a sufficiently faith-ful representation of a square object. In order to avoidthe Gibbs phenomenon, we must therefore modify(2.1) using various procedures. For those readers whoare not acquainted with the Gibbs phenomenon, refer-ence is made to Carslaw5 for an extensive discussion, aswell as to Guillemin.6

We have chosen to employ the little-known method ofthe o factors due to Lanczos. 7' 8 It is not our intent to

'°0 describe the mitodus operandi of the method, for the in-terested reader can follow the lucid presentation ofLanczos himself. The essence of the method is an aver-aging of the finite Fourier series over one interval,

2N p+±12NON (x) =

2L JzL/2NON (x)dx,

where2 (-1)1 F r

ON (x) = 2 +- E cos (2n+ 1)-x .r ,=o (2n+1) L LI

Carrying out the integration yields

2 N (-1)nci 2,.+(&A)ON(X)= 2+ E

7r x1=0 (2c+ )

X cos (2n+ )-x ,

(2.2)

(2.3)

whereSpatial frequency range

0.6667 <coO<2.00.4000<co<0.6667

0.2857 <w<0.40000.2222 <coo<0.28570.1818<coo<0.22220.1538<w,<0.18180.1333 <wo<0.15380.1176<wo<0.13330.1053 <wo<0.11760.0952 <wo<0.10530.0870<wo <0.09520.0800 <wo<0.08700.0741 <wo<0.08000.0690 <wo<0.0 7 410.0646<wo<0.06900.0606 <o <0.06460.0572 <W<0.06060.0540 <w<0.05720.0512 <WO<0.05400.0488 <wo<0.0512

sin[(2n± 1)7r/2N]

[(2n+ 1)ir/2N]

(2.4)

(2.5)

As Lanczos very carefully shows, this procedure greatlyreduces the Gibbs phenomena. In Fig. 3 we show the 21-term approximation to the square wave. In our workwe used (2.4) with N= 21 as providing a highly faithfultrigonometric series representation of our square-waveobject.

There is one further method of eliminating the Gibbs

I H. S. Carslaw, Introduction to the Theory of Fourier's Series andIntegrals (Dover Publications, Inc., New York, 1956), 3rd ed.

0 E. A. Guillemin, Mlhatieiatics of Circuit Analysis (John Wiley& Sons, Inc., New York, 1951).

7 C. Lanczos, Linear Differential Operators (D. van NostrandInc., Princeton, New Jersey, 1962).

8 C. Lanczos, Applied Analysis (Prentice-Hall, Tnc., EnglewoodCliffs, New Jersey, 1956).

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December 1963

LO

.8.

W .6 -

zzZ_

(a) 3-i A _

MODULATION OF SQUARE-WAVE OBJECTS. I

1.0A~~~~~~~~~~~~~~~

CIRCULAR APERTURE CIRCUL

W2=.25 .8 W

A w. 05 B

B w. 2.0 B

W A.60zZ

REDUCED DISTANCE IN IMAGE PLANE - X/L

0

z

z

(c) D-j

REDUCED DISTANCE IN IMAGE PLANE - X/L

0zzn-j

REDUCED DISTANCE IN IMAGE PLANE - X/L REDUCED DISTANCE IN IMAGE PLANE - X/L

ID

t .. W A

zz 2~~~~~~~~~~~~~

2 .2 4 .6 .8 0 0 .2 4 .6 .8 ID

REDUCED DISTANCE IN IMAGE PLANE - X/L REDUCED DISTANCE IN IMAGE PLANE - X/L

FIG. 4. Distribution of illuminance in square-wave image for a circular aperture suffering a quarter wave of defocusing.

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(d)

(e) (f)

R. BIARAKAT AND A. HOUSTON

phenomena, the well-known Fejer theory of the arith-metic means of the partial sums.6 However, Fejer'stheory has a number of undesirable features, particularlythe slow rise time of the partial sums in the neighbor-hood of the discontinuity.

3. SOLUTION AND SOME NUMERICAL RESULTS

In terms of (1.1) we have the following expression forthe image of the square wave

2 N (-1)nI(x) T (O) +- 5 -7 ___

7,=o (2iit)

X T[(2n1 )wo] cos[(2n+ 1)owox], (3.1)

where wo= r/L is the "frequency" of the square wave.The actual transfer function involved will depend notonly on the geometrical contour of the aperture butalso on the nature and amounts of the aberrationspresent.

The method of computation was straightforward. Theexpression (3.1) was programmed for the LGP-30 com-puter, and all that is required is the manual insertionof the required values of the transfer function (slit orcircular aperture). The computer automatically evalu-ated (3.1) for a large number of values of x(O<x<L),and the final results were then plotted. The modulation,

modulation = (Imax ,x-I ~in )/ (Iinax+l'ini), (3.2)

10

z0

00

4

0L. _, . . _ , -0 4 8 1.2 1.6

NORMALIZED SPATIAL FREQUENCY

FIG. 5. Modulation of square and sine waves for anaberration-free slit aperture in focus.

was obtained directly from the graphs. As a typicalexample of the results, Figs. 4(a)-(f) illustrate how theimage behaves for various values of wo for a perfectsystem suffering a quarter wave of defocusing. Notehow rapidly the image rounds to a cosine term as w isincreased.

The maximum frequency that the lens pass is 2.0, sothat (2n+ 1)co= 2 represents the allowable harmoniccomponents of the square wave. When wo is fairly small(low-frequency square wave), then the optical systempasses a large number of harmonics. The number ofharmonics passed decreases rapidly as coL is increased(high-frequency square wave). In Table I we list themaximum number of harmonics passed as a function ofcoo. When wo> 2/3, then only the first harmonic of thesquare wave is passed by the optical system.

The results of computations for perfect slit and circu-lar apertures are shown in Figs. and 6. The smallcircles represent the computed points for the square-wave modulation. The solid lines represent the transferfunction for the corresponding sine wave. The first thingto note is that the modulation for the square wave isalways higher than that of the corresponding sine wave.The second is that, even for the two thirds of the allowedspatial frequency pass band in which only the funda-mental harmonic is passed, the two modulation curvesare exactly the same shape but have different numericalvalues. The reason why the square-wave response ishigher than the sine-wave response is simple. Since the

I.0SLIT APERTURE

8 < W2 .=25

z \\(B) A SINE WAVE

I .6 (ASU WAVE

o j4~

.8 1.2 1.6

NORMALIZED SPATIAL FREQUENCY

FIG. 7. Modulation of square and sine waves for an aberration-free slit aperture clefocused W2 =0.25 wavelength.

CIRCULAR APERTUREW2= o

A SINE WAVE

B SQUARE WAVE

z

0'S

00

8 1.2 1.6

NORMALIZED SPATIAL FREQUENCY

CIRCULAR APERTURE

.8 , = W2.25

B), A SINE WAVE

.6 ( a B SQUARE WAVE

N.2

0 4 8 1.2 1.6 2.0

NORMALIZED SPATIAL FREQUENCY

1G. 6. Modulation of square and sine waves for anaberration-free circular aperture in focus.

FIG. 8. Modulation of square and sine waves for an aberrationt-free circular aperture defocused W22=0.25 wavelength.

1.0

z0. 6

'S

0 40

0 4

1374 Vol. 53

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December 1963 MO D U LA T I0 N OF SQUARE-WAVE OBJECTS. I

I10

.8

z

0 6'10_J

0 40

SLIT APERTUREW =50

A SINE WAVE

B SQUARE WAVE(B)

.8 1.2

NORMALIZED SPATIAL FREQUENCY

FIG. 9. Modulation of square and sine waves for an aberration-free slit aperture defocused W2 =0.50 wavelength.

1 0

.8

.6g

_ 4

00

:F.2

-.2

Si IT APERTURE

W2 = 1.0

A SINE WAVE

B SQUARE WAVE

0 4 .8 1.2 1.6 2.0

NORMALIZED SPATIAL FREQUENCY

FIG. 13. Modulation of square and sine waves for an aberration-free slit aperture defocused W2 = 1.0 wavelength.

( A SINE WAVE0

04

0

o 4 8 12 1.6 20

NORMALIZED SPATIAL FREQUENCY

FIG. 10. Modulation of square and sine waves for an aberration-free circular aperture defocused W2 =0.50 wavelength.

lo > SLIT APERTURE

.8~~~~~~~~~W \W= 75

A SINE WAVE

z 8 X B SQUARE WAVE0

0

-.20 4 .8 1.2 1.6 2.0

NORMALIZED SPATIAL FREQUENCY

FIG. 11. Modulation of square and sine waves for an aberration-free slit aperture defocused W2 =0.75 wavelength.

1.0CIRCULAR APERTURE

.8 W2=.75

A SINE WAVE

Z.6 B SQUARE WAVE

4 1I1

0 I0

.2

-

z0

0

NORMALIZED SPATIAL FREQUENCY

FIG. 14. Modulation of square and sine waves for an aberration-free circular aperture defocused W2= 1.0 wavelength.

[. r Iw =.7

oh1.

-2L 2 -L 0 L~ L -L 21L

NORMALIZED SPATIAL FREQUENCY

FIG. 12. Modulation of square and sine waves for an aberration-free circular aperture defocused W2 =0.75 wavelength.

FIG. 15. Images of square-wave object for a circular aperturedefocused W2 =0.75 wavelength. Note the spurious resolution inlast two frames.

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R. BARAKAT AND A. HOUSTON

,0

.8

z01-

0 40

.2

CIRCULAR APERT

A SINE WAVEB SQUARE WAVI

0 4 .8 1.2

NORMALIZED SPATIAL FREQUENCY

FIG. 16. Modulation of square and sine wavesas determined theoretically from lens de

optical system is conservative, then thethe lens (whether perfect or suffering alalter the distribution of energy in the iltotal energy remaining constant. Fuoptical system is linear (in intensity); coifrequencies cannot be altered and onlyand phase can be varied. The amount o.square-wave object is larger than thatobject of the same frequency; consequenttion must be greater.

As an example of how defocusing canlation, we present in Figs. 7-14 the resulttations. As the amount of defocusing is

~ - 1 discrepancy between the square-wave and sine-waveURE curves decreases. Even when spurious resolution is

present (Figs. 13 and 14), the square-wave modulation-E is still igher than that of the sine wave. Figure 15

illustrates how the square-wave image is distorted ascOo is increased, finally culminating in spurious resolu-tion for the final two values of coo shown.

Finally we show (Fig. 16) the results for an actualoptical system (a Tessar lens in d light stopped down to

1.6 2,0 F/11). The transfer-function curve for the sine wavewas computed from the design data using the program

for a Tessar lens developed by Barakat and Morello. There is a sub-sign data. stantial difference between the two responses.

e only effect of)erration) is to 4. COMMENTSiage plane, the It is now possible to correlate the modulation of arthermore the square-wave target with its corresponding sine-wavensequently, the target for actual systems. From the theoretical point ofthe amplitude view there is nothing that the square-wave target canf energy in the tell us that cannot be obtained from the sine-wave tar-of a sine-wave get. Nevertheless, the practical dictates of lens testingly, the modula- require that a research effort (currently under way) ex-

plore the interrelationship of these two techniques. In aalter the modu- future paper in preparation we will consider how useful

of our compu- are the finite-bar targets (e.g., three-bar targets) inincreased, the determining image quality criteria.

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