Ray and van Cittert-Zernike Characterization of Spatial Coherence

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Ray and van Cittert–Zernike characterization ofspatial coherence

Jose M. Sasian, Scott A. Lerner, Tony Y. Lin, and Lenny Laughlin

We discuss a ray and a van Cittert–Zernike characterization of spatial coherence in condensers forprojection systems. We present a rule of thumb with which to estimate the modulus of the coherencefunction at a given point of the illuminated object and a ray-tracing methodology with which to determinethis modulus. For uniform illumination of the pupil we relate the modulus of the coherence function andthe pupil-filling factor. We suggest that the rms of the angular ray spread at a given object point is anappropriate metric with which to characterize local coherence properties. © 2001 Optical Society ofAmerica

OCIS codes: 080.3620, 110.3960, 110.4980.

1. Introduction

The nature of the image produced by a projectionsystem depends on the coherence of the light that oneuses to illuminate the object or the mask to be im-aged. In the field of lithography, careful control ofthe coherence of the illumination produced by thecondenser optics is important in optimizing the ge-ometry and uniformity of the printed features.1–3

From the point of view of the optical design ofcondenser systems, theoretical and practical toolsto guide the condenser design are desirable. Forexample, some first-order design issues4 and aberra-tions of a condenser for extreme-ultraviolet lithogra-phy have been discussed.5,6 However, the topic of howto assess spatial coherence and its gradients duringthe lens design process needs to be treated in a prac-tical manner.

In this paper we provide a methodology that anoptical designer can follow to evaluate coherenceproperties of an illumination system. First we de-velop a rule of thumb that links spatial coherence tothe angular ray spread at a given point in the object.This theoretical link is necessary to support a ray-tracing methodology that is suggested for estimating

J. M. Sasian [email protected]!, T. Y. Lin, and L.aughlin are with the Optical Sciences Center, University of Ari-ona, Tucson, Arizona 85721. S. Lerner is with Lawrence Liver-ore National Laboratory, 7000 East Avenue, Livermore,alifornia 94550.Received 7 August 2000; revised manuscript received 12 Decem-

er 2000.0003-6935y01y071037-07$15.00y0© 2001 Optical Society of America

spatial coherence. The methodology that wepresent is a useful optical design tool. With it, onecan construct a coherence map by using standardray-tracing capabilities in an optical design package.

The ray characterization that we present provides afirst-order approach to assessing coherence and sug-gests an alternative and useful way to define the pupil-filling factor. The ray-tracing model can also be usedto implement the van Cittert–Zernike theorem with astandard optical design package. In this way we pro-vide a useful tool with which to characterize coherencein a plane of interest with both a simplified ray-tracingmethod and a wave-diffraction calculation.

In Section 2 we establish the link between the mod-ulus of the coherence function and the rms of theangular ray spread at a given object point. We iden-tify the rms of the angular ray spread as a metric tocharacterize coherence properties. We then discussour rule of thumb, establish a relationship betweenthe modulus of the coherence function and pupil-filling factor s, estimate the diameter of the circlethat is coherently illuminated, and discuss the con-ditions under which one can use pupil illumination toestimate the pupil-filling factor. In Section 3 wepresent a ray-tracing methodology for creating a co-herence map by using both the metric of Section 2and the van Cittert–Zernike theorem. In Section 4we summarize and conclude the paper.

2. Calculation of Modulus zg12z of the MutualCoherence Function in Terms of the Angular RaySpread

In this section we follow the partial coherence treat-ment of Hopkins7 to establish a link between themodulus of the coherence function and the angular

1 March 2001 y Vol. 40, No. 7 y APPLIED OPTICS 1037

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ray spread at a given object point. This link is nec-essary to justify the use of a ray-tracing methodologyto characterize coherence. We recall the fact thatspatial coherence at two points depends on the angu-lar subtence of a source. This is a result from thevan Cittert–Zernike theorem that one uses to esti-mate the spatial coherence of an incoherent and ex-tended source.8 More recently it was pointed9 out thathe coherence at a point in an observation plane iselated to the angular ray spread at that point.

In our calculation we assume an extended incoher-nt source and a Kohler illumination system, ashown in Fig. 1. We wish to determine the spatialoherence at two points, P1 and P2, on the observationlane. Following Hopkin’s notation, the mutual co-erence function g12 at those points is

g12 51

ÎI1I2*(

u1u2*ds, (1)

where

u1 5 A1 exp@2~2pyl!iW1~sx,sy,x1,y1!#,

u2 5 A2 exp@2~2pyl!iW2~sx,sy,x2,y2!# (2)

are the optical fields at points P1 and P2, respectively,on the observation plane, W is the wave aberrationfunction, l is the wavelength, the asterisk representsthe complex conjugate, and sx, sy, and x, y are thecoordinates on source S and on the observation plane,respectively. The irradiance at points P1 and P2 is

I1 5 *(

uA1uds, I2 5 *(

uA2uds. (3)

Substituting the optical fields into Eq. ~1!, we canwrite

g12 51

ÎI1I2*(

A1 exp@ 2 ~2pyl!iW1~sx,sy,x1,y1!#

A2* exp@~2pyl!iW2~sx,sy,x2,y2!dsxdsy# (4)

51

ÎI1I2*(

A1A2* exp@ 2 ~2pyl!iDW~sx,sy,P1,P2!#

3 dsxdsy, (5)

where

DW~sx,sy,P1,P2! 5 W1 2 W2 (6)

is the wave-front difference at points P1 and P2 fromsource point sx, sy.

As the points P1 and P2 are taken to be so close thatone can evaluate coherence in a small neighborhoodand because DW is small, we can expand the expo-nential function to second order and write

g12 <A1A2*

ÎI1I2*(

F1 1 ikDW 212

k2~DW!2Gdsxdsy, (7)

038 APPLIED OPTICS y Vol. 40, No. 7 y 1 March 2001

where we have taken the amplitudes to be constant.We call this a proximity approximation. The coher-ence at two points becomes the modulus of the mu-tual coherence function:

ug12u2 <A1A2*A1*A2

I1I2 *(

F1 1 ikDW 212

k2~DW!2Gds

3 *(

F1 2 ikDW 212

k2~DW!2Gds (8)

< 1 2

k2 *(

~DW!2ds

*(

ds

1

k2F*(

~DW!dsG2

S*(

dsD2

5 1 2 k2sDW2, (9)

where sDW2 represents the variance of the wave-front

difference with respect to the field ~source! coordi-ates at points P1 and P2, k is the wave number, and

we have dropped high-order terms. This result isvalid when the variance of the wave-front differenceis small. Then the modulus of the mutual coherencefunction is

ug12u < ~1 2 k2sDW2!1y2 (10)

Wave front W at P1 and P2 to second order is

W1 5 W000 1 W200s2 1 W111~sxx1 1 syy1! 1 W020r1

2,

(11)

W2 5 W000 1 W200s2 1 W111~sxx2 1 syy2! 1 W020r2

2,

(12)

where sx, sy, and x, y are source and pupil coordi-nates, respectively, and r 5 ~x2 1 y2!1y2 and s 5~sx

2 1 sy2!1y2.

Fig. 1. Optical layout showing an incoherent source, a condenserworking in Kohler illumination, the mask plane, two points P1 andP2, and the optical field from a point in the source arriving at thetwo points in the mask plane.

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wc~

The wave-front difference at points P1 and P2 is

DW 5 W111sx~x1 2 x2! 1 W111sy~y1 2 y2!

1 W020~r12 2 r2

2!, (13)

and the variance of the wave-front difference becomes

sDW2 5 ~x1 2 x2!

2ax2 1 ~y1 2 y2!

2ay2, (14)

where ax2 and ay

2 are the variances of the angularay spread in the x and y directions, respectively, as

follows after we note that the product W111sx,y rep-resents the angular direction of the wave-front andray directions in x and y, respectively. The focusterm W020 is constant with respect to the source co-ordinates, and therefore its variance is zero. In ad-dition, terms that are independent of fieldcoordinates can be factored out from Eq. ~5! and can-cel out after we multiply by the complex conjugate toobtain the modulus. Thus the coherence can bewritten as

ug12u < $1 2 k2@~Dx2!ax2 1 ~Dy!2ay

2#%1y2 (15a)

or, alternatively, as

ug12u < 1 2k2

2@~Dx!2ax

2 1 ~Dy!2ay2#. (15b)

This result states, as a rule of thumb, that the coher-ence at points P1 and P2 that are considered close toeach other is related to the variance of the angularray spread at those points and to the square of thepoints’ separation. The result is an approximationthat is valid for small angular ray spreads or pointseparations such that coherence ug12u is larger than.8. This approximation is reminiscent of the Strehlatio calculation, with the difference that the vari-nce is with respect to the source rather than to theupil coordinates. The derivation made in this pa-er is for the case of Kohler illumination. We notehat the essence of the calculation is to consider theave front at points P1 and P2 as plane; then the

treatment can be extended ~not done here! and wecan conclude that the derivation is valid in the rangeof Kohler to Abbe illumination.

The significance of expression ~15b! is that it re-lates the modulus of the coherence function to therms of the angular ray spread. This is a relationshipbetween a physical property and a geometrical prop-erty. The rms of the angular ray spread can bereadily computed with a standard ray-tracing pro-gram, as we discuss in Section 3 below. It is alsoappropriate to mention that in our calculations wehave assumed an incoherent source. However,some source-illumination systems call for a partiallycoherent source treatment. It is our purpose in thispaper to treat first the simpler case of incoherentillumination from a uniform source.

A. Relationship between the Modulus of MutualCoherence Function ug12u and Pupil-Filling Factor s Usedn Lithography

In lithography it is has become customary to charac-terize the spatial coherence of illumination by thefactor s. This pupil-filling factor is defined as theratio of the diameter of the illuminated area atthe camera entrance pupil to the diameter of theprojection camera entrance pupil. Pupil-filling fac-tor s is meant as a global factor to characterize theillumination coherence from well-behaved condens-ers. Like the mutual coherence function, pupil-filling factor s is a function of the angular ray spreadf the illumination provided by the condenser at theask. When the condenser provides Kohler or Abbe

llumination and a uniform circular fill of the projec-ion camera entrance pupil, the ray angles will beniformly distributed, with the maximum ray angleorresponding to the numerical aperture of the con-enser. By evaluating the rms of the transverse rayoordinates in a circular and uniformly illuminatedupil we can establish the relation

ax,y 5sNA

2, (16)

here s is the pupil-filling factor and NA is the nu-erical aperture of the projection camera in object

pace. Using expression ~15b!, we can relate theodulus of coherence function ug12u and pupil-filling

factor s as

ug12u < 1 2s2NA2k2r2

8, (17)

rom which we can solve for the pupil-filling factor:

s < S81 2 ug12uNA2k2r2D1/ 2

(18)

where r 5 ~Dx2 1 Dy2!1y2 is the distance betweenpoints P1 and P2.

These relationships are appropriate for the case ofuniform illumination at the pupil. If the condenserdoes not distribute the rays uniformly at the entrancepupil of the projection camera, expressions ~16!–~18!

ould not be correct. In that case, characterizingoherence properties would require using expression15b!.

B. Diameter of a Circle that is Coherently Illuminated

Following Hopkins, we regard 0.8802 as a tolerabledeparture from perfect coherence. From relation~17!, diameter d of a circle that is coherently illumi-nated by a source subtending an angle whose numer-ical aperture equals sNA is

d 50.1558l

sNA. (19)

The factor @~1 2 0.8802!8#1y2y2p ~or ;0.1558! in Eq.~19! differs in the third decimal place from the factorin the classic result, which is 1y2p ~or ;0.1592!.


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Rayleigh resolution R in object space associatedwith a projection camera is

R 50.61l

NA, (20)

and therefore the diameter of the circle that is coher-ently illuminated at the object plane in terms of theRayleigh resolution is

d >R4s

. (21)

Using a typical value of s 5 0.6, we can estimate thatdiameter d of the circle that is coherently illuminatedat the mask of a lithographic system is ;0.4 times theRayleigh resolution in the object space. A systemwith a pupil-filling factor of s 5 0.1 is considered to bea coherent one, and according to expression ~21! itwill coherently illuminate a circle that is ;2.5 timesthe Rayleigh resolution in the object space.

Thus in calculating the diameter of the circle thatis coherently illuminated we find that expression~15b! is in substantial agreement with the results ofclassic coherence theory. In addition, expression~15b! can be used only to estimate coherence at twopoints that are close to each other. This is a draw-back. However, the main value of expression ~15b!is that one can use it to establish a relationship be-tween the rms of the angular ray spread and themodulus of the coherence function.

C. Note on the Use of the Pupil-Filling Factor

Factor s is a measure of coherence as established byxpression ~17! whenever the condenser system pro-ides stationary illumination ~every point is illumi-ated in the same way!. However, when there is aariation of illumination at the mask, one can useactor s to estimate coherence near a mask point,rovided that we allow only light from that point toefine the illuminated area at the projection cameraupil. From the illuminated area at the pupil, onean calculate the pupil-filling factor. The ray distri-ution at the pupil is also assumed to be uniform.Thus, pupil-filling factor s is a global metric that

haracterizes coherence properties whenever everyoint in the mask or the object plane is illuminatedniformly and the pupil plane is also uniformly illu-inated ~i.e., the ray distribution is uniform!. Cau-

ion must be used whenever there are nonuniformariations in the planes of the object and the pupil.n this case the rms of the angular ray spread isheoretically a local metric with which to characterizehe coherence properties at a given object point.

3. Ray-Tracing Methodology for EstimatingPupil-Filling Factor s

One requirement in the design of a condenser systemis that every point at the mask plane be illuminatedin the same way as seen from the entrance pupil ofthe projection camera. It is therefore important toestimate the modulus of the coherence function, or


alternatively the pupil-filling factor, at a variety ofmask points during the optical design of a condenser-projection system. Section 2 provided a theoreticaljustification for the use of rays to estimate coherenceproperties. In this section we discuss the use of ray-tracing programs for this purpose. We provide amethod of estimating the pupil-filling factor accord-ing to the results of Section 2 and to practicallyimplement a wave calculation by use of the vanCittert–Zernike theorem.

In an actual condenser the illumination of the ob-ject and the pupil will not be uniform, and thereforethe global pupil-filling factor s will lose its value tosome extent. Therefore, to be more specific, we cancharacterize coherence by the rms of the angular rayspread at each point. Inasmuch as in the field oflithography the use of the angular ray spread or ofthe modulus of the coherence function is not custom-ary in describing coherence properties, we use Eq.~16! to define the pupil-filling factors sx and sy as

sx,y 52ax,y

NA, (22)

here NA is the numerical aperture of the projectionamera and ax

2 and ay2 are the variances of the an-

gular ray spread in the x and y directions, respec-ively. In the case of circular and uniform pupilllumination the filling factor will be equivalent totandard pupil factor s; that is,

s 5 sx 5 sy. (23)

To evaluate pupil factors sx and sy associated with amask point it is necessary to determine the angularray spread at that point. Two ways to do this withoptical design software are

~1! By use of the direct method, i.e., tracing a largenumber of rays from the radiation source, through thecondenser, to the camera pupil. A pinhole aperture isplaced at the mask around the point at which we wishto evaluate the coherence, such that only the raysthat pass through this pinhole aperture are traced tothe entrance pupil of the projection camera. Therays that reach the pupil plane create a spot diagramor an illuminated area that conveys in an insightfulway the nature of the angular ray distribution. Inusing this direct ray-tracing method there is a trade-off between the size of the pinhole aperture and thetime that it will take the computer to trace a suffi-cient number of rays to characterize the pupil fill.This problem can be solved to some extent by use ofray-aiming capabilities. Typically this method willbe implemented with an illumination optics programthat can model an extended source.

~2! By use of a symmetrical image of the condenserin a standard lens-design program. This methodologyis illustrated in Fig. 2. First, rays are traced from

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the mask point of interest through the condenser tothe source. An aperture placed at the source planerepresents the source geometry and limits the bundleof rays that can be propagated. Rays that passthrough this aperture both originate from the sourceand illuminate the original point on the mask. Thenthese rays are traced back through a mirror image ofthe condenser system to the camera entrance pupilafter they pass through the original point of themask. The setting of this configuration involvespropagating rays in a negative index of refraction ~orackward in negative thickness! from the mask to theource and then in a positive index from the sourcehrough the pupil. This procedure causes the raysfter the source to be mirror images of the rays beforehe source. An advantage of this method is that apot diagram can easily be created at the pupil andhat therefore one can obtain directly the rms of theransverse ray spread, which, divided by the object–upil distance, gives the rms of the angular raypread. A disadvantage is that this ray-tracingodel does not include source-radiance variations.In this methodology all the rays directed to the

upil pass through a single point, with no phase vari-tions in the geometrical wave front; that is, the waveront is spherical. Because the pupil fill is an imagef the source ~the effective source!, one can take theourier transform of the pupil and implement thean Cittert–Zernike theorem. One can readily dohis with an optical design package by calculating theoint-spread function that corresponds to the pupilll. In this case, including an apodizing mask at theupil could simulate source radiance variations.hus with this method pupil filling factors sx and sy

can be calculated and implemented with the vanCittert–Zernike theorem to assess spatial coherenceat a given mask point.

In this paper we discuss how spatial coherence canbe characterized in an illumination system; a detailedcomparison of the two methods outlined above is be-yond the scope of the paper. However, we point outthat in implementing the first method in Kohler andAbbe illumination systems we found the following:The Kohler system consisted of a parabolic mirrorwith a vertex radius of 2000 mm and an aperture of100 mm. A 1-mm-diameter source was located atthe focal point, and a 0.02-mm pinhole was located atthe illumination plane also at the focal plane. Tenmillion rays were traced in ;1 min from the source tothe mirror and then back to the illumination plane.Except for the optical axis ray, no rays passedthrough the pinhole, which clearly indicates the low

Fig. 2. Diagram of ray-tracing methodology for evaluating thepupil-filling factor of a point at the mask. A mirror image of thecondenser optics is created within the ray-tracing software.

efficiency of the direct method. Then we set the il-lumination system as Abbe by using a spherical mir-ror with a radius of 1000 mm and locating both thesource and the observation plane at the center ofcurvature; 1 million rays were traced, of which only370 were able to pass through a 0.02-mm pinhole.This number of rays could be sufficient to character-ize local coherence properties. However, in the illu-mination subroutine the program used did notcompute the rms position of the rays that passedthrough the pinhole. Thus the direct method maynot be so efficient or convenient as the method thatuses the symmetrical image of the condenser. Inaddition, the second method permits the implemen-tation of the van Cittert–Zernike theorem.

We illustrate the symmetrical-image ray-tracingmethodology by using it to evaluate pupil factors sxand sy for an astigmatic condenser. An astigmaticlens, as shown in Fig. 3, focuses light in the sagittalplane and collimates light in the tangential planefrom a small source. It provides illumination on aline-shaped region at a mask located at the sagittalfocus. The pupil of the projection camera would belocated at the medial focus, beyond the sagittal focus,and will receive nearly circular illumination pro-duced by rays passing through all the mask points.However, the pupil fill that is due to individual

Fig. 3. Cylindrical lens as an astigmatic condenser. At the sag-ittal focus a line illumination is formed that coincides with themask to be illuminated. At the medial focus an almost circularillumination is produced.

Fig. 4. The illumination at a mask located at the astigmatic lens’ssagittal focus is a narrow line, represented here by the rectangle.The letters represents points where coherence is evaluated.


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1

points, such as A–D in Fig. 4, will not be circular butwill be different, as in Fig. 5. Figure 5 was gener-ated with the symmetrical-image methodology de-scribed above and shows the use of spot diagrams toillustrate the pupil illumination that is due to indi-vidual points A–D in the mask. As we can see fromTable 1, the astigmatic lens provides a pupil-fillingfactor that varies across the mask and direction. Itis near zero in one direction where there is no signif-icant variation in the angular ray spread and is nearunity in the other direction, where there is angularray variation. Because the rays at the pupil fallalmost uniformly in a line, the pupil-filling factorbecomes slightly larger than one for mask positions Aand B. According to Eq. ~22!, the pupil-filling factorfor a uniformly filled pupil illuminated with a lineshould be sx 5 2~1⁄3!1/2, or sx > 1.15. This resultindicates that there is slightly more spatial incoher-ence in one direction when uniform line illuminationis used than for a circular pupil fill. This conclusionis supported by the fact that the width of the first zeroin the diffraction pattern of a circular aperture and ofthat in a square aperture of the same size are in theratio 1:22. That is, the diffraction pattern of asquare aperture in one direction is smaller than thatof a circular aperture, and therefore the illuminationthat the pattern characterizes according to the vanCittert–Zernike theorem is more spatially incoher-ent.

Figure 6 shows in a series of frames the squaredmodulus of the Fourier transform of the pupil fills ofFig. 5, including in frame E the transform of the idealcircularly filled pupil. These figures represent agraphic implementation of the van Cittert–Zernike

Fig. 5. Pupil fill that is due to points A–D in the mask, as shownby spot diagrams.

Table 1. Pupil-Filling Factors for Points along the Line Illumination atthe Mask

Pupil-Filling Factor

Point

A B C D

sy 0.039 0.041 0.044 0.017sx 1.15 1.09 0.85 0.22


theorem. In Fig. 6 two points that fall within thewhite regions where the Fourier transform is nonzerowill be significantly coherently illuminated. Notethat the pupil-filling factors in Table 1 are a predic-tion of the dimensional extent of the transforms inFig. 6. For example, a pupil-filling factor of 1 wouldbe represented by the extent of the transform inframe E. Frame D has associated pupil factors ofsx 5 0.22 and sy 5 0.017, indicating that point D athe mask will be coherently illuminated by approxi-ately 1ysx ; 4.5 more times along the x direction

than the ideally illuminated point E.The Fourier transforms in Fig. 6 were obtained

with the ray-tracing methodology that uses a sym-metrical image of the condenser. All that we didwas to calculate the point spread function of the pu-pil. Thus the model presented permits a hybridgeometrical–physical calculation of the pupil-fillingfactors and allows the van Cittert–Zernike theoremto be implemented. Note that this last implementa-tion requires a spherical wave front, as produced bythe ray-tracing methodology.

4. Conclusions

We have presented a ray-tracing methodology for es-timating coherence properties near a given point ofan illuminated plane. We derived a rule of thumbthat relates coherence at two close points in a maskplane to the angular rms ray spread and to the squareof the point separation. This rule of thumb is usefulin estimating coherence and coherence gradients atan observation plane as illuminated by an incoherentsource and condenser optics. Based in this rule ofthumb, we suggested the definition of two directionalpupil-filling factors, sx and sy, that one can use tocreate a coherence map at an illuminated plane.Characterizing coherence through the use of factorssx and sy has the limitations of any ray-tracing ap-proach to characterizing diffraction like properties.However, those factors are useful in quantifying spa-tial coherence, determining coherence gradients, and

Fig. 6. Fourier transform of the pupil-fill geometries of Fig. 5.Frame E corresponds to the Fourier transform of the ideal circularpupil fill.

offering a different and insightful perspectives of thecoherence properties of a system.

The ray-tracing methodology readily lends itself toimplementation of the van Cittert–Zernike theorem,and therefore the methodology provides both a phys-ical and a hybrid ray–wave estimate of coherence.Key to this implementation is the fact that the ray-tracing model uses throughout a spherical wavefront, as required in that theorem. The model alsoallows a designer to add an apodizing mask at thepupil to include source-radiation variations. Imple-mentation of the van Cittert–Zernike theorem withthis ray-tracing methodology is straightforward; allthat is needed is to compute the point-spread functionof the pupil, which is done easily within a ray-tracingprogram.

An example of an astigmatic condenser has beenpresented that illustrates and clarifies the use of thetheory described above. Thus we have developed aray-tracing methodology that permits a designer tocharacterize spatial coherence with a standard opti-cal design package.

This study was made with the support of the Semi-conductor Research Corporation.

References

1. Y. Borodovsky, “Partial coherence variations linked to line-width changes,” Solid State Technol. 38, 42 ~1995!.

2. J. P. Kirk and C. J. Progler, “Pupil illumination in situ mea-surement of partial coherence,” in Optical Microlithography, L.Van de Hove, ed., Proc. SPIE 3334, 281–288 ~1998!.

3. I. M. Grodnensky, E. Morita, K. Suwa, and S. Hirukawa, “Char-acterization of spatial coherence uniformity in exposure tools,”in Optical Microlithography, L. Van de Hove, ed., Proc. SPIE3334, 289–296 ~1998!.

4. M. Ceglio, A. M. Hawryluk, and G. E. Sommargren, “Front-enddesign issues in soft-x-ray projection lithography,” Appl. Opt.32, 7050–7056 ~1993!.

5. G. E. Sommargren and L. G. Seppala, “Condenser optics, partialcoherence, and imaging for soft-x-ray projection lithography,”Appl. Opt. 32, 6938–6944 ~1993!.

6. W. C. Sweatt, “Condenser for illuminating a ring field,” U.S.patent 5,361,292 ~1 November 1994!.

7. H. H. Hopkins, “The concept of partial coherence in optics,”Proc. R. Soc. London Ser. A 208, 263–277 ~1951!.

8. M. Born and E. Wolf, Principles of Optics ~Pergamon, New York,1959!, Chap. 10, Sec. 4.2.

9. D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations inKohler illumination,” in OpticalyLaser Microlithography, B. J.Lin, ed., Proc. SPIE 922, 108–126 ~1988!.


Ray and van Cittert-Zernike Characterization of Spatial Coherence

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