method for the calculation of partially coherent imagery

Method for the calculation of partially coherent imagery

Eric C. Kintner

The tedious numerical computations associated with the calculation of partially coherent imagery are alle-viated by a method which uses dimensionless coordinates and takes advantage of the properties of the Fouri-er transform. A 1-D periodic object function can model many objects of practical interest, including nonper-iodic objects. The properties of a given optical system are described in terms of the transmission cross coef-ficient. For aberration-free systems with circular pupils, including annular sources (dark-field illumina-tion), the cross coefficient can be calculated analytically. For aberrated or apodized systems, a 1-D approxi-mation can be used. The effect of a convolving slit in the image plane of a scanning microscope can also beincluded.

1. Introduction

Although it has long been recognized that imagingin a microscope is partially coherent and nonlinear, thetheory of partially coherent imaging has made littleimpact in practical optical calculations. This is largelydue to the greatly increased complexity of imagingcalculations using a nonlinear theory. Nevertheless,a qualitative understanding of the effects of partiallycoherent imaging has been obtained from the varietyof numerical investigations, which have been reportedin the literature.

However, in certain applications such a qualitativeapproach is definitely inadequate. In optical mi-crometrology, for example, the imaging characteristicsof a microscope must be predicted to high accuracy.Moreover, a large number of arduous computationsmust be performed in order to investigate a wide varietyof imaging situations. Nyyssonenl has recently de-scribed such computations for the imagery of inte-grated-circuit photomasks. This investigation clearlyindicated the need for an efficient method to calculatethe imaging performance of a microscope.

The method to be described here simplifies the cal-culation of partially coherent imagery by using a di-mensionless coordinate system and by taking advantageof the properties of the Fourier transform. Computa-tional speed and numerical accuracy are gained byemploying 1-D periodic objects. The properties of agiven optical system are described in terms of the

When this work was done the author was with U.S. National Bureauof Standards, Washington, D.C. 20234; he is now with Charles StarkDraper Laboratory, Inc., Cambridge, Massachusetts 02139.

Received 13 March 1978.

transmission cross coefficient. An analytic algorithmcan be used to model the performance of an unaberrated2-D optical system with circular apertures. Alterna-tively, numerical integration can be used in some casesto model the performance of a 1-D optical system withaberrations or apodization. The method also incor-porates a simple technique for including the effect of asampling slit in the image plane of a scanning micro-scope.

Many features of the method have been discussedpreviously.2 3 Hopkins4 employed dimensionlesscoordinates and periodic objects in his early discussionsof partially coherent imaging. Ichioka and Suzuki5used the concept of the transmission cross coefficientin their investigation, although they did not stress itsspecial convenience for practical computations.

In this paper, the theory of the computational methodwill be emphasized. Another paper describing resultsobtained with this method is in preparations

II. Computational Method

Let positions in the object plane of a microscope bedescribed by the dimensionless Cartesian coordinatesx and y, where

x= [(PO)/XIY, (la)

y= [(Po)/X]Y. (lb)

The geometrical coordinates are given by x and Y; po isthe numerical aperture of the objective, and X is themean wavelength of the (quasi-monochromatic) illu-mination. Points in the image plane conjugate to (x,y)in the object plane are described by the same coordi-nates (x,y). Similarly, points in the pupil plane can bedescribed by the coordinates

1 September 1978 / Vol. 17, No. 17 / APPLIED OPTICS 2747

= t/po, (2a)

= Vpo- (2b)

Again, t and X are the geometrical coordinates, and pois the numerical aperture. Where the Fraunhofer dif-fraction theory is valid, the coordinate system ()corresponds to the Fourier-transform plane of thecoordinate system (x,y). The imaging properties of theobjective lens system may be described by the ampli-tude point spread function K(xy), whose Fouriertransform is given by

KQt,-) = 3s K(x,y) exp[27ri(xt + yn)]dxdy. (3)

In Fourier optics, the function K(Q,-) may be identifiedwith the pupil function of the optical system. The di-mensionless radius p ( 2 + 72) = 1 corresponds to theedge of the optical pupil and simultaneously corre-sponds to the limiting spatial frequency detected by theobjective lens system with coherent illumination.

The image spatial frequency spectrum I,) is de-fined to be the Fourier transform of the image intensitydistribution I(x,y) through the equations

I0= 3 X I(x,y) exp[27ri(xt + y-)]dxdy, (4a)

I(xy) = 3 f I(%-) exp[-2ri(xt + yn)]dtdr7. (4b)

Similarly, if the complex amplitude transmittance (orreflectance) of the object is given by the function F(xy),its spatial frequency spectrum is given by the Fouriertransform

FQ = 3' ' F(x,y) exp[27ri(xt + yn)]dxdy. (5)

Finally, when the partially coherent illumination isstationary (which in effect implies wide-field illumi-nation), the mutual intensity of the illumination in theobject plane may be described by the function J(x,y).Its Fourier transform

JQn) = 3' J(x,y) exp[27ri(xt + y)dxdy (6)

is called the effective source4 ; physically, it correspondsto the intensity distribution in the aperture of thecondensing-lens system. Thus, for example, in a mi-croscope system with matched numerical aperturesbetween the objective and the condenser, the effectivesource is a uniform distribution, which cuts off at thelimiting radius Ps = (Q2 + 712) = 1.

With these definitions, the image spatial-frequencyspectrum produced by a specified optical system is givenby the equation

I,r) = Am L TQ + ', + lt,)x F( + (', + ')F*(Q',r')d'd1 ', (7)

where

TQ1,nl;tMD 2= f ' J(Q",,)KQ(1+ t ,1l + Xn)

* K*(Q2 + CMX + n)d#"dXn (8)

In Eq. (7), the asterisk denotes complex conjugation.The function T(Q1,71 2;42,12), called the transmission crosscoefficient (Ref. 7, Sec. 10.5), characterizes the completeoptical system including condenser and objective. (Theterm transmission cross coefficient may be misleading,since this function could equally well be used to char-acterize a system imaging light reflected from theobject.) When the illumination is incoherent, it can beshown that the function T becomes the optical transferfunction; similarly, for coherent illumination, thefunction becomes the product of the two pupil functionsK and K*. Therefore, Eq. (7) combines the attributesof the optical system with the attributes of the objectin a manner which is both intuitively meaningful andpractically useful.2 3

Imaging computations may be greatly simplified bymaking the reasonable assumption that the functionsrepresenting the optical system, K(,) and J(, 7 ), aresymmetric about the origin (i.e., the optical axis).

If the object function varies in one dimension only(e.g., a straight edge), Eqs. (7) and (8) may be simplifiedinto a 1-D form:

I(t)= F 3' T( + ',,')F(Q + ')F*(Q')dS', (9)

where

T(1,6) = f 3 J(Q",,?1)K(Q1 + t"',n")

X K*(Q2 + t",n")dS,"dn,". (10)

Equation (10) still allows for a 2-D optical system.Equation (9) is further simplified if the 1-D object is

periodic (e.g., a grating). Then the object may be rep-resented by a Fourier series, whose transform is a dis-tribution of delta functions. If the periodic functionis even, the Fourier series consists of cosine terms only.Thus, an even periodic function with period P = 1/tmay be represented by the Fourier series

F(x) = E a,, cos(27rntpx) a = an, (11)

whose Fourier transform is simply

F(Q) = E anb(nt - ). (12)n--

The coefficients an} are complex when the objectfunction includes phase variations.

Equation (12) may be inserted into Eq. (9), whichafter simplification becomes

I(M) = anaT(ntp;O) + E an+,,,,an,T[(n+ n)p;n (pl

+ an-n,'a 'T[(n - n')#;-n'P] )(np -

= E Cn(n p- ) (Cn = Cn)n=-X

(13)

Therefore, the intensity in the image plane is givenby

I(x) = E cn cos(27rntpx).n=-- (14)

2748 APPLIED OPTICS / Vol. 17, No. 17 / 1 September 1978

Because the functions describing the object and theoptical system are assumed to be symmetric, and be-cause the intensity distribution in the image is neces-sarily real, the coefficients {c,,} are also real. When theseries in Eq. (14) is evaluated, only one cosine need becalculated directly; the remainder of the harmonics maybe calculated by using a simple recurrence relation forthe cosine function:

cosnx = 2 cosx cos(n - 1)x - cos(n -2)x.8 (15)

The use of a periodic object greatly simplifies thecalculation of the intensity distribution in partiallycoherent imaging. In general, the calculation of theimage intensity requires three separate integrations:(1) the integration over the source as given by Eq. (8)to determine the performance of the optical system; (2)the integration over the object spectrum as given by Eq.(7) to determine the intensity spectrum; and (3) theintegration over the intensity spectrum as given by Eq.(3b) (the inverse Fourier transform) to obtain the in-tensity distribution itself. In practical calculations,these integrations can be intolerably tedious. Theabove analysis shows that with a 1-D periodic object,two of the three integrations, Eqs. (3b) and (7), reduceto discrete summations, Eqs. (13) and (14), thus elimi-nating the problems of convergence associated withnumerical integration. Although infinite limits arespecified for the summations in Eqs. (13) and (14), thefunction T(Q1;42 ) vanishes for large values of the argu-ments due to the finite extent of the pupil function andthe source function. Therefore, the summations in Eqs.(13) and (14) are finite and may be truncated at prede-termined limits without any loss of accuracy.

So great are the advantages of using a periodic objectthat it is worthwhile to express even a nonperiodicobject in terms of a periodic model. This is readilyaccomplished if a large period is chosen so that theobject is effectively isolated and does not interfere withits periodic neighbors. The periodic model may becompared directly with the nonperiodic object to verifythe accuracy of the approximation. It should be em-phasized that this periodic approximation, which is soeasily checked, is in many cases the only approximationin the imaging calculation. Except possibly for thedetermination of the cross coefficient T, all the othersteps of the calculation are exact.

In a scanning microscope system, the image plane issampled by a scanning slit, which must be convolvedwith the intensity distribution to obtain the distributionrecorded by the scanning system. Since convolutionin the image plane corresponds to multiplication in thepupil plane, it is convenient to multiply the intensityspectrum in Eq. (13) by the Fourier transform of the slit.Thus, the intensity spectrum actually observed is

Is t) = S(4) .-( ) = sin(rw) ),7rw4

equation

Is(x) = co + 2 E c,,s,, cos(27rn4px),n=1

(17)

where

Cn= aaOT(nap;a) + a,,+,,aT[(n + n')4p;n'p]n'o 1

+ an-nanTI(n - n')tp;-n'tp]}, (18)

sin(n7rw4p)sn = nirwtp

(19)

I The Fourier coefficients of the symmetric objectfunction F(x) may be obtained from the equation

a, = 2 F(x) cos(27rnx/P)dx,P fo

(20)

where P is the period of the repeating function.Among the variety of periodic objects which may be

expressed in terms of Fourier series, two have beenparticularly important to our investigation. The firstis a periodic bar chart [Fig. 1(a)], whose Fourier coeffi-cients are

a. = tb + (2wt/P) (tt - tb),

an = sin(27rnwt/P)/rn.

(21a)

(21b)

The amplitudes tb and tt may, of course, be complex tomodel an object with phase variations. When wt = 0.5,the object is a square wave with bars and spaces of equalwidth. Alternatively, when wt < 0.2, each bar is effec-tively isolated and behaves as a single bar.

The second object is the familiar tri-bar resolutionchart [Fig. 1(b)], which can simultaneously illustrate the

F(x)

P 1we H

x

(a)

(16)

where w is the full width of the scanning slit in dimen-sionless coordinates.

Therefore, the complete calculation for the partiallycoherent imaging process is characterized by the

x

(b)

Fig. 1. (a) Periodic chart. (b) Periodic tri-bar chart.


I

III

imaging of both isolated and closely spaced bars. Forthis object, the Fourier coefficients are

a. = tb + (6wt/P) (tt - tb), (22a)

a,, [sin(2irnwt/P) + sin(27rn(3wt + Wb)/P)

-sin(27rn(wt + Wb)/P)I (t - tb)/7rn. (22b)

Although the procedures described above simplifythe calculation of the image intensity distribution inpartially coherent imaging, the integration for thefunction T describing the optical system, Eq. (8) or (10),remains a formidable problem. It will be noted thatthis integration represents a generalized convolution ofthe source function J with the objective pupil functionK. Where the source and objective apertures are cir-cular and the objective pupil includes aberrations, thisconvolution has no simple solution and implies an ex-tremely tedious numerical integration. To avoid this,two alternative compromises are available.

First, where the source aperture is small (smaller thanthe objective aperture), it is possible to reduce the in-tegration to one dimension rather than two, in whichcase numerical integration becomes feasible. Thus thefunction becomes

T(QI;42) J(Q")K(41 + 0")K*(42 + 4")dt",

where

K(t) = {exp[ 2 ri'()] 141 • <1° > 141>1,

and (t) is an arbitrary phase variation in the pupilfunction. With this model, it is possible to test the ef-fects of aberrations such as defocusing and sphericalaberration. In the limit of coherent imaging, where theeffective source is a point, J(Q) = constant; 5(Q), sucha model would exactly describe a 2-D imaging system.As the source size becomes appreciable, this model be-comes less accurate, although Nyyssonen reports thatacceptable results can be obtained for source sizes upto Ps = 1.

Second, if all variations in the source aperture andobjective pupil are suppressed (i.e., no aberrations), the2-D integration corresponds to the generalized convo-lution of three uniform circles, that is, the problem re-duces to finding the area of the mutual intersection ofthe source aperture and the shifted objective pupils (seeFig. 2). This is a fairly strenuous exercise in logic andgeometry, but it yields an efficient algorithm for thefunction T (see Appendix).

This algorithm may also be applied to systems withannular sources, such as systems with dark-field illu-mination. First, the function T is computed for a sys-tem including a source with radius equal to the outerradius of the annulus. Second, the function T is re-computed for a source with radius equal to the innerradius of the annulus. Then the difference betweenthese two values of T yields the value of T correspond-ing to the annular source (see Fig. 3). This is entirelyanalogous to the method used to calculate the diffrac-tion pattern of an annular aperture (Ref. 7, Sec. 8.6).

With the techniques described in this paper, a com-

Fig. 2. Computation of the cross-coefficient function T(Q1;42).

ANNULUS

/ (C, 1)

[I'l

OUTER CIRCLE

[T ]

INNER CIRCLE

[/I ]

Fig. 3. Computation of the cross coefficient for an annular source.

plete image profile can be calculated in several secondson a typical computer. This represents a considerablesavings over previous computational methods andpermits the assessment of a wide variety of imagingsituations.

Appendix: Unaberrated 2-D Optical System

As noted in the text, the transmission cross coefficientT(Q1; 2) for a 2-D aberration-free optical system corre-sponds to the area of mutual intersection of three uni-form circles. Figure 2 shows that this area depends onone parameter Ps, which specifies the radius of the ef-fective source centered on the origin and on two explicitvariables, 0, and 42, which specify the displacement ofthe objective pupil along the t axis. The algorithmdescribed below classifies the problem into one of sev-eral geometrical possibilities on the basis of certainmathematical inequalities. In general, each classifi-cation may be further reduced into combinations ofareas, each common to two circles (rather thanthree).

Thus the basic tool for the calculation of T is a for-mula for the area common to two intersecting circles,A(rl,r 2 ,d), where r, and r2 are the radii of the two cir-cles, and d is the separation between their centers.Examination of Fig. 4 shows that the required area isseparated into two regions by the common chord joiningthe two points of intersection of the circles. The in-


Fig. 4. Computing the area of intersection of two circles.

tersection points determine a wedge-shaped section ofeach circle, and the radii of this section together withthe common chord determine a triangle within thissection. The area of each of the two common regionsis then the difference between the area of the sectionand the area of the triangle; thus

A(r1 ,r2 ,d) = A1 + A2= (ri - cdi) + (r202 - cd2)= (r?0, + r2°2) - cd.

The quantities 01, 02, and c may be found from thespecified variables by using the law of cosines:

°1 = arccos(d 2 + r - r2)/2drl;

02 = arccos(d 2 + r2 - r')/2dr 2 ;

then

c = r sino1 or c = r2 sin02 .

(Note that when r = r2 the formula corresponds toHopkins'9 expression for the optical transfer functionof an unaberrated optical system.)

The algorithm for calculating the transmission crosscoefficient can be described step by step using words,diagrams, and mathematical expressions. The algo-rithm classifies the problem geometrically on the basisof several successive tests. When the cross coefficientT can be calculated explicitly, the algorithm termi-nates.

Algorithm for T 1 ,4 2 )

It is assumed that 4, 2 2-

Test 1: Is the source effectively coherent?

Ps < .O ?

If so, go to Step 10 and use a simple routine to calculateT.

Test 2: Do the two unit pupils (representing theobjective pupil) intersect?

(11-42) 2 ?

If not, T vanishes identically.

Step 1: Determine the maximum and minimumlimits of the region of overlap between the unit pu-pils.

max = 62 + 1,

(min = - 1.

Test 3: Does the source intersect this region ofoverlap?

6min p and 6max -Ps ?

If the source does not intersect the region of overlapbetween the two unit pupils, T vanishes identically.

Test 4: Is the source smaller than the unit pupil(objective aperture underfilled)?

Ps<1 ?

If so, go to Test 7.Test 5: Does the source entirely envelope the area

common to both unit pupils (Fig. 5)?

Ps 2 max and -s S min ?

If not, go to Test 6.

Fig. 5. Large source envelopes intersection of unit pupils.

Step 2: If the source encloses the area of overlapbetween the two unit pupils, T is simply equal to thisarea (Fig. 5).

T(Q1;U = A[1,1,( 1 - 2)] (finish).

Test 6: Does the source include the two points ofintersection between the two unit pupils (Figs. 6 and 7)?(The distance of the points of intersection from theorigin may be determined using the law of cosines.)

PS 2 (Q16 + 1) ?

If not, go to Step 4.Step 3: The required area is bounded by both unit

circles and the source (Fig. 6). This area is equal to thearea of overlap between the two unit circles, plus thearea of overlap between the source and the innermostunit circle, minus the area of the unit circle, that is,


Fig. 6. Area determined by both unit pupils and large source.

T(0 = A[1,1,(t - 2)l + AIt8,lmin(kIIt2I~l - ir (finish).

Step 4: The required area is bounded by the sourceand one unit circle (Fig. 7). Then

TQtl#2) = A(.,1, max(It1,1t2I) (finish).

Test 7: When the source is smaller than the unitcircle, does the source overlap the boundary of the areacommon to the two unit circles?

Condition (a) Condition (b)

tmin > Ps 6max < Ps

If the source overlaps either pupil but not both (Fig. 8),go to Test 8. If the source overlaps both pupils (Fig. 9),go to Test 9.

Step 5: The source is entirely enveloped by the areacommon to both unit circles, so the required area ismerely the area of the source.

T(Q1;02)= irp. (finish).

17

T

Fig. 7. Area bounded by one unit pupil and large source.

Test 8: Which unit circle does the source intersect(Fig. 8)? If the unit circle corresponds to 0j [Condition(a) in Test 7], go to Step 6. If the unit circle correspondsto 02 [Condition (b) in Test 7], go to Step 7.

Step 6: The required area is bounded by the sourceand the unit circle corresponding to Al, so

TQl;U2) = A(1,p, l (ll) (finish).

Step 7: The required area is bounded by the sourceand the unit circle corresponding to 42, SO

TQl;U2) = A(1,p8,I 2I) (finish).

Test 9: Does the source entirely envelope the areacommon to both pupils? (This can be determined byfinding the distance of the points of intersection of thetwo unit circles from the origin, using the law of co-sines.)

Io1f t + 1) S o, 9

If so, go to Step 9.

I17

C.

k

Fig. 8. Small source overlaps one unit pupil.

C.

Fig. 9. Small source overlaps intersection of both units pupils.


I | r, .^,^.l l l l b

1

i C, 0/1' ic,

1<*

Step 8: The required area is bounded by the sourceand both unit circles. This area is equal to the sum ofthe areas of overlap between each of the unit circles andthe source, minus the area of the source, that is,

T(j;62) = A (l,p,Ij1) + A (l,p, 12 -1) rp. (finish).

Step 9: The required area is merely the area com-mon to both unit pupils.

T~j;62) = A[1,l,(Qj - 2)I (finish).

Step 10: If the source is very small, it may be ap-proximated by a coherent point source, so that the ef-fective function in Eq. (6) becomes

JTh) en 006(n).Then

T(Q1,) = 1 It11 < 1 and 121 1,0 elsewhere, (finish).

References1. D. Nyyssonen, Appl. Opt. 16, 2223 (1977).2. E. C. Kintner, "Investigations Relating to Optical Imaging on

Partially Coherent Light," Ph.D. Thesis, U. Edinburgh (1975).3. E. C. Kintner and R. M. Sillitto, Opt. Acta 24, 591 (1977).4. H. H. Hopkins, Proc. Roy. Soc. A 217,408 (1953); J. Opt. Soc. Am.

47, 508 (1957).5. Y. Ichioka and T. Suzuki, J. Opt. Soc. Am. 66, 921 (1976).6. D. Nyyssonen, in preparation (1978).7. M. Born and E. Wolf, Principles of Optics (Pergamon, London,

1959).8. F. S. Acton, Numerical Methods that Work (Harper and Row, New

York, 1970), p. 11.9. H. H. Hopkins, Proc. Phys. Soc. 79, 889 (1962).

Second Announcement

III

0 // 0 0at {t2§lL' At 76 I .,

-_ _

vS~~~~~~t0~~ ~ *9 ^ S - 3 SL * '0 'I,,.~~~~~N

0C%- -1-1v

_]x33_~~ ': --....

~~~~~~~ 1 a_-


method for the calculation of partially coherent imagery

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