Influence of nonuniform amplitude on the optical transfer function

Influence of nonuniform amplitude on the opticaltransfer function

Chang S. Chung and Harold H. Hopkins

The purpose of this work was to develop formulas and accurate numerical techniques for computation of theoptical transfer function (OTF) in the general case of unrestricted aberration and with the following differentforms of nonuniform real amplitude: (1) when the real amplitude is described by a polynomial, (2) a Gaussiandistribution of real amplitude, and (3) a pupil with a central obstruction. The resulting computer programhas been carefully tested and used to study the influence of nonuniform amplitude on the OTF in typicalcases, for which detailed numerical results are given.

1. Introduction

The optical transfer function can be calculated fromthe design data, but its calculation has so far beenmainly restricted to cases where the image formingwavefront is of uniform amplitude. This is not alwaysthe case: the real amplitude across the pupil can vary,and in recent years a good deal of work has beenpublished about such cases.'-5 For example, annularapodizers,' radial Walsh filters,2 and Gaussian laserbeams 3 4 have been investigated in terms of both thepoint spread function and the optical transfer func-tion; diffraction by an aperture with a central obstruc-tion 5 has also been studied. However, the mathemati-cal difficulties of evaluating the OTF in the generalcase of an aberrated optical system with any kind oftransmission function over the exit pupil have severelylimited the analytical results obtained in this field.

The present work develops the formulas and nu-merical techniques for the general case of unrestrictedaberration with the following forms of nonuniform realamplitude: (1) when the real amplitude is describedby a polynomial which is of similar form to the wave-front aberration polynomial, (2) a Gaussian distribu-tion of real amplitude, and (3) the case of a centralobstruction which is a characteristic of catoptric andcatadioptric systems. In each case the computer pro-

C. S. Chung is with Chonnam University, Physics Department,Kwangju 505, Korea, and H. H. Hopkins is with University of Read-ing, J. J. Thomson Physical Laboratory, Whiteknights, ReadingRG6 2AF, U.K.

Received 12 February 1988.0003-6935/89/061244-07$02.00/0.© 1989 Optical Society of America.

gram calculates the OTF of the optical system from aknowledge of its wavefront aberration.

Typical cases of the influence of the above nonuni-form amplitudes on the OTF are then illustrated bynumerical results.

1. Theory

The position and shape of the pupil must be firstdetermined, because the shape of the pupil determinesin turn the area of integration in OTF calculations.The shape and position of the pupil are obtained fromthe constructional data of the system in question bymeans of a pupil exploration scheme which employsfinite ray tracing. The basic ray tracing consists ofthree parts; the opening formula, the transfer proce-dure, and the closing calculations in which the re-quired aberration information is obtained.

All the computations are carried out using Hopkinscanonical coordinates.6 7

A. OTF Formulas

The OTF for the frequency pair (so,to) of a 2-D objectis given by

Do(so,to) = (1/A) J f o (Xo + soYo + t to)

X f* (xo - 1/2s,yo - 12to)dxodyo, (1)

where fo(xo,yo) is the pupil function, S is the region ofoverlap between two pupils centered on the points (xoL 1/2so,yo d 1/2to) and

A JA Ifo(xoyo)I2 dxodyo,

with the region of integration being the whole of thepupil as the normalizing factor to make Do(O,O) = 1.

1244 APPLIED OPTICS / Vol. 28, No. 6 / 15 March 1989

A

X

S

Fig. 1. Region of integration for the OTF.

B. Cases of Nonuniform Amplitude

The variation of real amplitude over the image form-ing wave is expressed in terms of the unrotated pupilcoordinates (xoyo). For any value of 4A, the values ofi-(x,y) are then obtained from To(x0,y0) using Eq. (4c).

The three different types of nonuniform amplitudeconsidered in this paper are:

Case (i) When the real amplitude varies accordingto a polynomial which is similar in form to that for thewavefront aberration polynomial; that is,

To(xoyo) = a0o + allyo + a20(xO + yo) + a2 2Yo

+ a31(x0 + YO)YO + a33yg + a4 0(X +y

The spatial frequency pair (so,to) corresponds to a 1-D Fourier component of reduced spatial frequency s =+V/(s2 + t2) in the azimuth A = arctan(to/so). We thusrotate the coordinate axes (xo,yo) to give rotated axes(x,y), with they axis lying in the azimuth 4', as shown inFig.-1. Do(so,to) isnowdenotedbyD(s,4' andEq. (1) isrewritten as

D(s,@ = (1/A) J A f (x + sy) f* -2 sy) dxdy, (2)

in which f(x,y) derives from f(xo,yo) by the substitu-tions

x = x cos4/ - y sin4',

y = cos + x sin4 (3)

A = J J If(xy)I 2 dxdy

is the normalizing factor. With fo(xo,yo) for the pupilfunction referring to the unrotated axes, the pupilfunctions in Eq. (2) referring to the rotated axis will begiven by

f(x ± 1/2 s,y) = f{(x 1/2s) cost - y sincy coso + (x i '/2 s) sinqI

(4)

after using Eq. (3) for (x0,y0).The pupil function fo(xo,yo), which refers to the un-

rotated coordinates, is defined by

fo(xoyo) = ro(xo,y,) epti2irW 0(x 0y 0 )} (xOyO) E A

=0 (xo,yo) $ A, (4a)

where A is the domain of the exit pupil, r0 (x0,y0 ) is thereal amplitude of the wave at (x0,y0 ), and Wo(xoyo) isthe wavefront aberration at this point, using in thelatter wavelength X as the unit of length. Referring tothe rotated axes, we shall have

f(xy) = r(x,y) expji2irW(x,y)j (x,y) e A

= (xy) s A, (4b)

where the values of r(x,y) and W(x,y) are found, forany values of (xy), using

+ a 42(xo + Y0)Yo + a 4 Y0, (5)

where (x0,y0) are exit pupil coordinates and a00, all, a2o,a22, a31 , a33, a40, a4 2, and a44 are coefficients, the sub-scripts indicating the dependence on polar coordinates(ro,00), with a typical term being amprm cosPo0.

Case (ii) A Gaussian distribution of real ampli-tude, which has

r0 (x,y) = expf-(xo + y0)/o2b (6)

where is a constant.Case (iii) The case of a central obstruction, which

is one of the characteristics of catadioptric systems.For such systems, the outer periphery of the exit pupilis a circle for the axial image point, and for extra-axialimage points we assume that it can be represented withsufficient accuracy by an ellipse. In both cases the exitpupil coordinates are scaled to make the outer periph-ery the unit circle x2 + y2 = 1. For the axial image, thecentral obstruction is a circle concentric with the outerperiphery of the pupil; for extra-axial image points thecentral obstruction will in general be neither a circlenor concentric with the outer periphery of the pupil.We assume that its shape can be adequately approxi-mated by an ellipse which is displaced from the centerof the pupil. Account is then taken of the centralobstruction simply by putting the real amplitude'0(x0,y0) = 0 in its interior. We thus define

To(XO,yO) = 0 [Xo/a]2 + [(yo - c)/b]2 < 1

= 1 1 S [x0 /a]2 + [(Yo Jc)/b]2 < [X +

=0 [X2+yO]>1, (7)

where (a,b) are the semiaxes of the elliptically shapedcentral obstruction, and (o,c) are the coordinates of itscenter. The values of a, b, and c are determined bypupil exploration for each image point.

C. Wavefront Aberration

The wavefront aberration will be specified initiallyby the coefficients Wmp in the polynomial:

-r(x,y) = To(x cost - y sin4',y cosik + x sin),

W(x,y) = W0 (x cosV, - y sini/,y cos4' + x sin4')

in accordance with Eq. (3).

(4c) (8)W0(X0,Y0 ) = E Wmp(X + m p

where, in polar coordinates, the terms are Wmpr'

15 March 1989 / Vol. 28, No. 6 / APPLIED OPTICS 1245

X.

cosPOG. The coefficients Wmp are determined from theresults of ray tracing.

Ill. Numerical Method

In all cases, the outer periphery of the pupil is takento be a circle or an ellipse; and, in all cases, the pupilcoordinates are scaled to make the pupil bounded bythe unit circle x2 + y2 = 1. Referring to the rotatedaxes, this becomes x2 + y2 = 1.

To evaluate Eq. (2), the integration over x is effectedfirst, using the Hopkins method6 because the inte-grand will often be highly oscillatory. For subsequentintegration over y Gauss integrations is used. Thus, inEq. (2) we take fixed values y = k, where yk (with k =1,2,... n) are the Gauss points; and then, for eachvalue of y, we integrate along the line P1P2 in Fig. 2,with the limits

x = -[/(1 y2 ) - 1/2s] tox = +[V(1-y 2) - 2S],

and Eq. (2) then becomes

/1 \ +[-(/ 2S)21 r+I/(j-Y2)-'/0I x 1'A J\/[l..(/1s)21 JdIV(ljy2)1/2S k 2 '

Xf*(x- 1/2s,y)dx (9)

since (CQ1 ) = /[1 - (1/2s)2].

To treat the integration over x in Eq. (9) for y = yk,

we write it as

+[|(I-k/0)- 2 s ( 2 ) ( 2

Y

X

Fig. 2. Integration along the line y = yk.

Xaj = 4( - 12)ck (j = 1,2,* ,Jk),

Y =k, (13)

and the mesh element j will extend from (xj - /2ek) to(Xj + 1/20)

Since the real amplitude will vary only slowly with x,we assume r(X,y) to be constant over each of the abovesmall intervals and then write Eq. (10) as

I(s P;yk) = Xj + 2 SYk) (j - A-

U'S0)

X expji2vrW(xj,yk;s)jdx, (14)

where

X expji2r[W(x + 1 /2s,yk) - W(x - / 2s,yk)fldx, (10)

where the pupil functions are expressed in terms of r(xi

1/2S,Yh) and W(x t 1 /2S,Yk), which represent, respec-tively, the real amplitude distribution across the pupiland the wavefront aberration polynomial, expressed asfunctions of the rotated coordinates. The numericalvalues of these quantities are obtained by substituting(x i 1/2s,yk) for (x,y) in Eq. (4c) and using the appropri-ate forms for To(xo,yo) and Wo(xoyo).

We determine first the range of x variation for eachGauss point Yk for the spatial frequency s of interest,which will lie in the range from 0 to 2. We choose anumber J such that the size of the interval over thepupil radius will be eo = 1/J and then for line Yk, wetake the interval eh to be nearly equal to eo but whichdivides (PoP2 ) in Fig. 2, into an exact number of inter-vals Jk. Thus, since (PoP2) = [/(i - y) - 1/2s], wedetermine an integer Jk such that

(1/co) [(1-yA) - /2S] = Jk - (11)

where X is a positive fraction, and the size of the inter-val given by

= [/(1- y) - 12S]/Jk (12)

is then used for the given value of Yk. The midpointsof the Wth interval along the line y = k will havecoordinates

W(xy;s) = W(x + /2 s,y) - W(x - /2s,y) (15)

is the value of the aberration difference function at(x,y), and the real amplitudes are given their values atthe centers of the corresponding mesh elements. Thenumerical evaluation of Eq. (14) is described fully inRef. 6. Following this method, Eq. (14) is integratedto give

+Jk

(S,;Yk) = k (j + 2 Sayk) r (x- 2 SsYk) expji2rWj'k

o 0)

X inctw/2[Wj+ 1, - Wj-lk], (16)

where

Wjk = W(Xj + 1 /2 S;S) - W(Xj - 1/2S;S) (16a)

is the value of the aberration difference function at thecenter of the mesh element (j,k), and the approxima-tion

ax (Wj+l k Wj-lk)/ek (16b)

is used for the value of the x derivative of fV(x,y;s) atthe point (Xj,Yk). Equation (16) is the final expressionused for the value of integral (9).

Using Eq. (16) for the value of integral (9) over x, andGauss integration to integrate this over y, formula (8)for the OTF becomes


D(x,V,) = (1A)[ - s)h] Z Wk (xj + SY)k=1 j=_J

(j #-0)

X T(X- 1 /2SYk) expji27rWkjk sinc 1/2(TWj+lk - jj k)}>

(17)

with (k = 1,2,. ,n), where ck are the weighting fac-tors for the n Gauss points yk, and the factor /[1-(1/2s)2] is present because the limits of integration aredifferent for different values of s.

For the final calculation of the OTF, we write

D(xV = R(s,4/) expjiO(s,4)}, (18)

and then need to find

T(sq) = +\[R(s,'p)]2 + [I(sq,)]2, (19)

KsM = argjR(x,q,) + iI(s,})j (20)

for the contrast transfer function T(s,4) and the phasetransfer function (s,).

Test case (4): when -0(xo,y0 ) = a00 + a2 0(X4 + y)+ a 22y0 + a40(xO + y0) + a42(x + yo)

yo + a44yO + ally 0 + a 3 (X4 + y0)yO

+ a33y8, and W0(xoyo) = 0.

Test case (5): when r0(x0,y0) = expl-(xo + y0)/o2I,

and W0(xoyo) = 0.

In the above test cases, (1) is the in-focus OTF of adiffraction-limited system having uniform amplitudeand a circular aperture, for which the result is wellknown. Case (2) is merely case (1) with an annularaperture, for which formulas are given in Ref. 9.

Case (3) is a diffraction-limited system with a para-bolic variation of real amplitude and simple focus er-ror, of which both terms are rotationally symmetric.The OTF is thus independent of the azimuth 4'; and,for any value of 4', we shall have f(x,y) = fo(xy). For-mula (9) for the OTF then takes the form

1 [I-(I/ )2] I+[V(-y2

)-1 2 s

D(si) = l 2J-V[1-(_)

2] J-V(1Y2)

{a00 + a2x2+xs + S2 + 2]}

IV. Numerical Results and Comparisons with AnalyticalResults

A. Comparisons with Analytical Results

The computer program for evaluating Eq. (17) waswritten in FORTRAN for the general case of opticalsystems with aberrations and nonuniform real ampli-tude. Computations were carried out using an HP-3000 computer whose word length is 16 bit.

To find suitable values for the numbers of intervalsalong the x and y axes in the computer program, aninitial choice of J = 50 was made for the number ofintervals used to find the value of E0 = 1/J. Differentnumbers of Gauss points, from 10 to 32, were then usedto calculate the OTF. It was found that the differ-ences between the computed values and analytical andsemianalytical values were <8 X 10-4 even if only 10Gauss points were used. We, nevertheless, chose 16Gauss points to ensure good accuracy in all cases. Thenumber of intervals along the u axis was thereforefixed at 16, and the OTF was calculated using differentnumbers of intervals J along the x axis. With J = 20 asthe number of intervals, the above differences did notexceed 5.0 X 10-3 in all cases. The subsequent calcula-tions of the OTF were thus all carried out using 16Gauss points along the y axis and J = 20.

The program was tested using five different cases ofnonuniform amplitude for which the OTF can be ob-tained analytically, semianalytically, or using straight-forward Gauss integration. The five cases were:

Test case (1): T0(x0,y0) = 1 and W0(x0,y0) = 0.

Test case (2): when the system has a concentric cir-cular central obstruction andW0 (x 0,Y0 ) = 0.

X {a00 + a 20 [x2 -Xs + 1/4S2]I exp(i47rW2 0 sx)dxdy,

in which the integration over x was evaluated analyti-cally, followed by Gauss integration over y.

Case (4) is the in-focus OTF for a diffraction-limitedsystem having nonuniform amplitude represented by apolynomial in (x0 ,y0). In this case W(x,y) = Wo(xo,yo)= 0, and the formula (9) now merely requires theintegration of a polynomial of degree 8 in (x,y). Thiswas evaluated using Gauss integration over x and thenover y.

Case (5) again has W(x,y) = Wo(xo,yo) = 0; and, since(X2 + y2) = (x + y2), it gives r(x,y) = expl- (x2 + y2)/a 2 i.Formula (9) now requires the evaluation of

D(x,, = (-) exp (- 22)ram(1;2s exp ( d2) dy

J+I/(1-y2)-'/2s] / 2x2

X exp - dx,J-[V/(1-y2)-'12s] \ a /

in which the integral over x can be expressed in termsof the error function followed by Gauss integrationover y. Since, however, both integrands are smoothlyvarying and nonoscillatory, Gauss integration wasused both for the integration over x and then over y.

B. Grouping of the Nonuniform Amplitudes Used inExamples

To compare the effects of nonuniform amplitude onthe OTF, the variations of amplitude are grouped asfollows:Group (I) for the amplitude polynomial [Eq. (5)]:

Test case (3): when r0 (x0,y0 ) = a00 + a2 0(Xo + 0)

and W0 (xoyo) = W20 (x2 + 2).I(a) a 00= 1, a2 = all = a22 = a31 = a33 = a 4 = a42 = a44 =

(uniform amplitude),


,-, I.

\ /I \/ \I

0.5.

(I-a)~~~~~~~~~~~~~~~~~~~~ (H-a) /

" /

(I-c)'x/ 'N

/fib) \-I 0 +1

r(X2+Y2) 2

Fig. 3. Amplitude variation, group I.

I(b) a 00= a 20 = 1, all = a 22 = a3 l = a33 = a4 = a4 2 = a44 =0

(center amplitude = 0),

I(c) a0 = 1, a 2 0=-1, all = a22 = a3l = a 33 = a 4 = a42 = a44 0

(edge amplitude = 0).

Group (II) for the Gaussian amplitude distribution[Eq. (6)]:

II(a) a (uniform amplitude),

II(b) a = 1.0 (edge amplitude = 0.37),

II(c) a = 0.5 (edge amplitude = 0.02).

Group (III) for the central obstruction [Eq. (7)]:

III(a) a = 0, b = 0, c = 0 (uniform amplitude),

III(b) a = 0.5, b = 0.3, c = 0.2 (central obstruction).

1.0

OTF

0.5

Fig.

OTF

1.0

i.

, I ~~~~~wso Wi

\ -(I-c)

I-Cl-a)

(I-b)-4--

OTF

05

,/, /

, , / 0.5I/

/

" \ui-b)

\ "

\ "b

I 0 +12 2

r=(X +Y ) 2

Fig. 4. Amplitude variation, group II.

+y

+1

-I

Fig. 5. Pupil with a central obstruction, group III(b).

1 + Iss W-o (ii)

U -C \ (fl-)

OTF w=o (iii)

.0 s 2.0 0 1.0 s 20 0 1.0 s 20

Optical transfer functions of diffraction-limited systems with nonuniform amplitudes in (i) group I, (ii) group II, (iii) group III.

.0 1.0

i) W.a=3.OAW0o=-24A (i) Wo-30AWao-30A (ii) W-3.0AW2=-3.6AOTF OTF

'(r-d (I ~~~~~~ \ --0. 1 ~~~~~~~~~~~~0. (1-b)

(-a)(1-b) (fl-b)

(.0s

Fig. 7. Optical transfer functions in the presence of spherical aberration W4 o = 3X in different focal planes with nonuniform amplitudes in

group I: (i) W20 = -2.4X, (ii) W20 ='-3.OX, (iii) W2 0 -3.6X


v - - al - -~~~~~~~~~~~~~~~~~~~~~

- l -

' I O ,,-_ .. i.

(I- a) (E-a)

-

-11

Cii) W40=3WAVho=-3DA OTF

-(l-a)

U(f-b)

S

U(-a)0.

(-a)

,(-b)

(U - c)

Fig. 8. Optical transfer functions in the presence of spherical aberration W40 = 3X in different focal planes with nonuniform amplitudes ingroup II: (i) W20 = -2.4X, (ii) W20 = -3.OX, (iii) W20 = -3.6X.

Ci)W4030A.W2=-24AOTF

Cii) W4O30AWa-3.0A OTF

-a)

(2-b)

(iii) W4=3.0AW2-3.6A

(D-b)

-(m-a)

2.0

-O. 2L -O. 2 L -0.2L

Fig. 9. Optical transfer functions in the presence of spherical aberration W40 = 3X in different focal planes with nonuniform amplitudes ingroup III: (i) W20 = -2.4X, (ii) W 20 = -3.OX, (iii) W20 = -3.6X.

(i) W =30AW2.OP=0 OTF OTF(iii) W=30A\W.0O,>PO

Fig. 10. Optical transfer functions in the presence of coma W31 = 3X in the azimuth A/ = 0 with nonuniform amplitudes in (i) group I, (ii) groupII, (iii) group III.

Figure 3 shows the variations of amplitude acrossthe pupil for each of the above groups. I(a) is the caseof uniform amplitude; 1(b) and I(c) are, respectively,the case of the amplitude increasing from the center tothe edge and the case of the amplitude decreasing fromthe center to the edge. Group (II), shown in Fig. 4, isthe case of a Gaussian amplitude distribution, with theedge amplitude of cases II(b) and 11(c) being 0.37 and0.02, respectively. Figure 5 shows the central obstruc-tion of case 111(b).

C. Numerical Results and Discussions

The influence of nonuniformity of amplitude on theOTF, in the absence of aberration and no focus error, isshown in Figs. 6. The curves in these figures showgood agreement with the results of previous workers.10

Figures 7-9 show comparisons of the OTFs in focalplanes at and on each side of the optimum focus W20 =

- W40, when the system has nonuniform amplitudegiven, respectively, by groups (I), (II), and (III), andspherical aberration W40 = +3X.

Figures 10 show comparisons of the OTFs in thefocal plane W20 = 0 for systems having coma W31 = 3Mand nonuniform amplitudes given by groups (I), (II),and (I).

One may observe the general tendency of the abovecurves. If we have the amplitude decreasing from thecenter to the edge of the pupil, and OTF values arehigher in the low spatial frequency region and lower inthe high spatial frequency region than for uniformamplitude. But for amplitude increasing from thecenter to the edge of the pupil, the OTF values arelower in the region of low spatial frequency and higherin the region of high spatial frequency than for a uni-form amplitude. This general tendency can be ob-served whether aberration is present or not.


OTF

(E-b)

OTF

(i W4=30AW2=-24A (ii) W.c=3.0A2=-36AOTF OTF

(iD WS=3oA\%kof=O

The authors would like to thank J. Macdonald foruseful discussions and are grateful to KOSEF for thefinancial support.

References1. J. Ojeda-Castaneda, L. R. Berriel-Valdos, and E. L. Montes,

"Bessel Annular Apodizers: Imaging Characteristics," Appl.Opt. 26, 2770 (1987).

2. L. N. Hazra and A. Guha, "Far-Field Diffraction Properties ofRadial Walsh Filters," J. Opt. Soc. Am. A 3, 843 (1986).

3. K. Tanaka and 0. Kanzaki, "Focus of a Diffracted GaussianBeam Through a Finite Aperture Lens: Experimental and Nu-merical Investigation," Appl. Opt. 26, 390 (1987).

4. P. Kuttner, "Image Quality of Optical Systems for TruncatedGaussian Laser Beams," Opt. Eng. 25, 189 (1986).

5. R. Barakat, "Diffracted Electromagnetic Fields in the Neigh-borhood of the Focus of Paraboloidal Mirror Having a CentralObscuration," Appl. Opt. 26, 3790 (1987).

6. H. H. Hopkins, "Calculation of the Aberrations and Image As-sessment for a General Optical System," Opt. Acta 28, 667(1981).

7. H. H. Hopkins and M. J. Yzuel, "The Computation of Diffrac-tion Patterns in the Presence of Aberrations," Opt. Acta 17, 157(1970).

8. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathemat-ical Functions (Dover, New York, 1972).

9. E. L. O'Neil, "Transfer Function for an Annular Aperture," J.Opt. Soc. Am. 46, 285 (1956).

10. M. Mino and Y. Okano, "Improvement in the OTF of a Defo-cused Optical System Through the Use of Shaded Apertures,"Appl. Opt. 10, 2219 (1971).

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Influence of nonuniform amplitude on the optical transfer function