Image construction: optimum amplitude and phase masksin photolithography

Bahaa E. A. Saleh and Karen M. Nashold

It is desired to preprocess an input image so that when it is distorted by an imaging system a prescribed out-put image is produced. The system of interest is a linear shift-invariant bandlimited system followed byan absolute value squaring operation and a hardlimiter. This is the system that represents coherent imageformation using a high-contrast printer. A complex-valued solution is found using an iterative methodwhich employs alternating projections, first onto the set of bandlimited functions, then onto the set of func-tions whose magnitudes are above and below the desired theshold. Enlarging the class of allowed inputs toinclude complex-valued signals, instead of real-valued signals, widens the domain of possible solutions toinclude solutions for problems with smaller -bandwidth products.

I. Introduction

A new image processing problem, which may be calledimage construction, image design, or image synthesis,has been investigated recently.1-4 In an image recon-struction problem an unknown image is distorted by aknown imaging system; the distorted and noise-con-taminated image is used to reconstruct (recover) theunknown image. The goal of image construction, on theother hand, is to determine an image which when fed toa known distorting imaging system produces a desiredknown output image.

In both problems the output and distorting systemare known, and the input is to be determined. Thedifference between the two problems is mainly one ofmotive. The image reconstruction problem is a prob-lem of signal estimation; the measured output resultsfrom an actual, albeit unknown, input; therefore, at leastone solution must exist, and uniqueness is a problem ofconcern. The image construction problem is a controlproblem. Existence of an input that generates a desiredoutput is the issue of concern.

The image construction problem is formulatedmathematically by writing the imaging equation

g = Sf, (1)

where S is an operator representing the imaging systemwhose input and output are the signals f (x) and g(x).The system S is known. A prescribed image g (a realnon-negative function) is to be generated at the outputof the system by selecting an input image f satisfying

The authors are with University of Wisconsin, Department ofElectrical & Computer Engineering, Madison, Wisconsin 53706.

Received 19 November 1984.0003-6935/85/101432-06$01.00/0.© 1985 Optical Society of America.

given constraints f e F. The nature of the system S,constraints F, and dimensionality of x determinewhether a solution exists and how difficult it is to find.If an exact solution does not exist, one may be satisfiedwith an approximation , that generates an output asclose as possible (according to some criterion) to thedesired image g.

Previous efforts in solving this problem have beenlimited to a system S which is composed of a linearbandlimited system (such as that of a diffraction-lim-ited camera) followed by a hardlimiter (representingvery high-contrast photographic film). The output gis thus forced to have binary values (representingblack/white prints). The input f was constrained tobelong to the class of real-valued functions.

In one investigation' the input was also forced to bebinary. A numerical iterative algorithm was developedto solve this binary programming problem. In a secondinvestigation it was shown analytically2 that when x isone dimensional, g takes the form of two pulses of equalheights but arbitrary widths and separations (a problemcorresponding to printing parallel lines), and the inputf is allowed to take arbitrary gray values, a solution al-ways exists regardless of the separations between thelines. The resolution is limitless. This is not the casein the 2-D problem; analytical solutions are not alwaysfeasible.

In subsequent investigations in which the input wasallowed to take arbitrary gray levels (real-valued), it wasshown that approximate solutions may be obtained byusing linear programming techniques3 and by using aniterative approach based on the Gerchberg-Papoulis-Youla algorithm. 4

In this paper we wish to consider the image con-struction problem when the input f is allowed to becomplex-valued and when the imaging system is a cas-cade of a bandlimited system followed by an absolute-value-squaring operation, and a hardlimiter as illus-trated in Fig. 1. Such a system represents a coherent

1432 APPLIED OPTICS / Vol. 24, No. 10 / 15 May 1985

imaging diffraction-limited optical system employingvery-high-contrast detection. The complex-valuedinput signal f represents the optical field which may begenerated by use of an amplitude and a phase-shiftingmask. There are no constraints on the values the am-plitude and phase may take. Our approach is based onan ad hoc iterative algorithm resembling the Gerchberg,Papoulis, Youla, and Fienup algorithms.5 -9 The ideaof improving resolution in optical lithography by use ofcoherent illumination and phase-shifting masks hasbeen previously suggested and tested.1 0"11 The am-plitude was not manipulated, and phase changes werelimited to a 180° shift between neighboring lines. Al-though no attempts at optimization of the phase shiftwas made, substantial improvement in resolution wasachieved. This paper aims at providing a theoreticalbasis for mask optimization in this new technology.Both amplitude and phase are allowed to change.

11. Algorithm

A. Problem Formulation

In our case the imaging system S of Eq. (1) is a cas-cade of a bandlimiting operation B followed by an op-eration L of hardlimiting the squared-absolute-value

g = Sf, S = LB, (2)

where f = f (x) is a complex-valued signal; gb = Bf isdefined by

gb(X) = Bf(X) = F-'[H(w)F(w)], (3)

where

H(w)=1, forwcQ= 0, elsewhere,

and Q is a region in the w space representing the supportof the bandlimited system; and

g(x) = 1, for Igb(x)12 > T, (4)

=0, otherwise,

where T is the threshold of the hardlimiter. Theimaging system is completely characterized by thethreshold T and the support region Q. (An example inthe 1-D case is that for which is the interval[-27rvo,27rvO], where vo is the cutoff frequency in lines/mm.)

We wish to determine a complex-valued input imagef(x) that produces a prescribed binary image g(x) =

{1,01. That image is completely described by the regionsA+ and A in which g(x) = 1, or 0, respectively.

g(x) 1, xeA+ (5)

=0, XeA_.Ideally we would solve the problem if we could de-

termine the inverse operator S-1. However, becauseof the nonlinearity of the hardlimiting operation, thisappears to be an impossible task.

B. Illustration

Consider the 1-D problem of printing two thin linesseparated by a distance d through a diffraction-limited

f H- = I - 12 g

-V.0 _ _ _ T _

Fig. 1. System.

imaging system of numerical aperture (N.A.). If oneuses coherent illumination at a wavelength X, theimaging system is equivalent to that described in thissection with the bandwidth vo = N.A./X (lines/mm).When the product dvo is large, the problem is easy; onecan come up with numerous input images f(x) thatgenerate the desired output. An example is to choosea real f (x) that has the form of two impulses of ampli-tude AO and separation d:

f(x) =Aob x-2) + A0 +

The bandlimited function fb (X) is then1 2

fb(x) = A 0 2vo sinc -) 2v]

+ A0 2vo sinc + ) 2voJ

where sinc(x) = sin(rx)/(7rx). Its squared absolutevalue I fb (X) 12 has a double-peaked shape as long as d Po> 0.66. The amplitude AO may then be adjusted tomake the points at which Ifb(x)12 crosses thresholdcorrespond to the desired thickness of the desired imageg(x). The distance do = 0.66/vo = 0.66X/N.A. may beregarded as the two-line Sparrow's resolution limit ofthe imaging system under coherent illumination(compare this to the value 0.415X/N.A. in the incoherentcase).

When d Po < 0.66, a double-peaked I fb (X) 12 may notbe produced by use of two impulses of real values sep-arated a distance d. Two alternatives are available.

In the first alternative, f (x) remains real-valued, butforms other than the two-impulse form are allowed.This approach has been shown2 to always yield a solu-tion for arbitrary values of dvo. As dO decreases,however, the solution becomes unstable in the sense thatslight deviations from the exact computed values for theinput will produce large deviations in the output. Thesecond alternative is to use two impulses having a phasedifference So:

f (x) =Aoe (x -- ) + Ao exp(lo)b (x +-)

corresponding to

Jfb(x)I2 I= Ao2 ,I 2 sinC2 [(x -- ) 2v0] + sinc2 +) 2v0]

+ 2 sinc x ) 2-O] sinc + ) 2vo] cos(00)}

For a given 00, we ask for the minimum product dvo forwhich lfb(x)12 is double-peaked. As is well known fromstudies of resolution of coherent imaging systems, when00 = 1800, fb (0) = 0. The function I fb (x) is forced tohave more than one peak regardless of dvo. Resolutioncan obviously be improved.12

15 May 1985 / Vol. 24, No. 10 / APPLIED OPTICS 1433

C. Iterative Algorithm

In this section an iterative algorithm is proposed forcomputing an input complex signal f that produces adesired output signal g, defined by Eq. (1), when passedthrough the system S defined in Eqs. (2)-(5) and illus-trated in Fig. 1. The problem is equivalent to findinga complex signal f (x) that satisfies the following con-ditions:

(1) it is bandlimited to the support of the system;(2) its absolute value If(x) I satisfies some inequali-

ties, namely,

If(x)I>V2T forxeA+,

If(x)I • VT for x .

(6a)

(6b)

Because the found signal f (x) is bandlimited, it passesthrough the bandlimiting system unchanged. Becauseit satisfies Eq. (6), it generates the correct outputg(x).

The question is now how to find a complex-valuedfunction that satisfies the above two conditions.

The following iterative algorithm motivated by theGerchberg-Papoulis-Youla-Fienup 5 -9 algorithm issuggested:

(a) We begin with some initial guess for the desiredcomplex-valued function f (x).

(b) We force it to satisfy condition (1) by passing itthrough a bandlimiting system.

(c) We force the result to satisfy condition (2) bytesting the inequality in Eq. (6) at each point x andconstructing a new function according to the followingrule: If the inequality (6a) or (6b), depending onwhether x A+ or x E A-., is satisfied, keep If(x) I un-changed; if not, force it to the value of the thresholdV/T. The phase is not changed. The newly con-structed function now satisfies condition (2) but notcondition (1).

(d) The function obtained in (c) is used as a newguess, steps (b) and (c) are repeated, and so on.

The algorithm is illustrated in Fig. 2.

D. Convergence

Although the algorithm converged in almost all thetests that we attempted, it converged to a different so-lution depending on the initial guess. We cannot,however, prove that it should in general converge.

Although this algorithm is motivated by the methodof convex projections,8 it actually does not satisfy thatmethod's requirements.

The method of convex projections is an iterativemethod for finding a function f in a Hilbert space H thatlies in the intersection Co of a number of closed convexsubsets {CJ} of H:

feCo= n ci.i=1

The m sets {CiI correspond to m constraints or proper-ties that the function f must satisfy. To find a functionf which has all the m properties, all that is needed is aprojection operation Po that will project members of Honto C0 . Since Co is generally more complex in naturethan any of the CQ sets and a direct realization of Po is

f (x)

fn +,()or I

Fig. 2. Algorithm.

usually not feasible, an alternate approach is considered.If the projection operator Pi onto the subset Ci can bedetermined for each i, a composition operator T can bedefined: T = PmPm- , . . . ,Pi, which when repeatedlyapplied on an arbitrary f E H will result in a vector in Co.The iterative scheme generates fixed points of T bytaking f = Tnfo, n being the number of the iteration.Every point in Co is a fixed point of every Pi, and,therefore, of T because if f e Co, then f E Ci and Pif = f,i = 1, . . . m. If Pif = f, then Tf = f and f remains fixedafter each iteration.

In the present problem we wish to find a complex-valued function that satisfies three properties definingthree subsets of Hilbert space: C is the subset offunctions that are bandlimited (according to a pre-scribed support), and C2 and C3 are the subsets offunctions whose absolute values are below or above aprescribed threshold in prescribed regions, respectively[as in Eq. 6(a) and 6(b)].

The projection operators P, P2, and P3 which wesuggest for use in our algorithm are as follows:

Plf = F-[H(w)F(w)]

as defined in Eq. (3),

P2f=f-(x) X EA-,= f(x) otherwise,

and

P3f = f+(x) X A= f(x) otherwise,

where

f-(x) = f(x) if If(x) I < T,= dIT explJOWx] if W~x > /T,

and

f+( = (X) if I(x) > iT,= -VT expJO(x)] if If(x) I <I7i.

In each case 0(x) is equal to the phase of f (x) = If(x) IexpL7O(x)].

C1 and C2 are indeed convex sets. The problem isthat set C3 (complex functions whose magnitudes areabove a prescribed value in prescribed regions) is nota convex set. Although the operator P3 that we suggestusing is a projection operator, convergence of the algo-rithm is not assured because of the nonconvexity of setC3 .

1434 APPLIED OPTICS / Vol. 24, No. 10 / 15 May 1985

fbX )

Fig. 3. Desired output.

The fact that convergence did occur in the exampleswe attempted is probably due to the local convexity ofthe set because the initial guess was close to a solution.This situation has been experienced in other nonconvexapplications. 13

Ill. Example

As an example we undertook the task of producingthe 1-D signal g(x) shown in Fig. 3 at the output of asystem of bandwidth v0. The widths and separationsof the pulses in g(x) decrease with increasing values ofx; hence the resolution of the system required to pro-duce the pulses increases with x. The distances be-tween the nine pulses {dj, j = 1,8} correspond to space-bandwidth products dj=1,8} = 11.75,1.09,0.875,0.875,0.547,0.438,0.438,0.438}. This provides uswith a scale which can be used to compare the resolu-tions achieved by different methods. The system isshown in Fig. 1. The threshold of the hard-limiter wasset at T = 0.5. Convergence was checked by computingthe ratio

a. Fn(kQ)l EF.(kg)k 5-vo ,

after each iteration, where IF,} represents the discreteFourier transform of the sequence 1fV,}. If a,, = 1, itimplies that the resultant {t., after being restricted tobe above and below threshold in the desired regions, isstill bandlimited. Further iterations thus do not resultin any change.

Our results are summarized in the following:(1) Figure 4 shows the effect of simply inputting the

desired output into the system. Figure 4(a) shows theoutput from the bandlimiter, and Fig. 4(b) shows theresulting output from the hard-limiter. The thresholdis adjusted to recover the first pulse perfectly. This isequivalent to adjusting the heights of the input pulsesby a scaling factor. As seen, six pulses are resolved, butonly four are reasonable reproductions of the desiredpulses. (The other two are not of the proper width orlocation.) The resolution of the system corresponds todvo = 1.75, meaning that the last place where the pulsesare resolved exactly as desired is in the region where dyo= 1.75.

(2) Resolution can be improved by introducing a180° phase shift between consecutive pulses. This isshown in Fig. 5. When the function Ifo(x) I exp[.1 0 (x)]

1.0

0.5 _

0.0-0 100 200 x

(a)

glx )g()1.0-

0.6-

0.2

0 100 200 x(b)

Fig. 4. (a) Magnitude of output of bandlimiter when desired outputis used as input. (b) Resulting output from hardlimiter when input

to hardlimiter is as in (a).

1.0

0.5

0.0

g(x)1.0-

0.6-

0.2-

0

x

(a)

100

_ ,, In

111,,11 l I

Illl I l

x200

(b)

Fig. 5. (a) Magnitude of output of bandlimiter when input of Fig.

6 is used; (b) resulting output from hardlimiter.

with magnitude and phase as shown in Fig. 6 is input tothe system, the function in Fig. 5(a) results after thebandlimiter. This produces the output shown in Fig.5(b) after the hardlimiter. Here five pulses are resolvedat the output. The resolution of the system is nowimproved to dvo = 1.09.

15 May 1985 / Vol. 24, No. 10 / APPLIED OPTICS 1435

--. ... .. .1. .. . .. . . . . .

I Y~x )l

I

III

I

I

III

I

I

II

I

I

S(X

Fig. 6. Magnitude and phase of input as a function of distance x.

x

Fig. 7. Constructed input found after 100 iterations when the initialinput used in the algorithm was the real valued function in Fig. 3.

The effect of introducing a phase other than 180°between consecutive pulses was then checked. Inputswith phase differences of 90, 60, 45, and 30° betweeneach consecutive pulse were tried. Also an input withrandom phase on each pulse was tried. The achievedresolution varied but never exceeded that achieved withthe 180° phase shift.

(3) Improvement can be made beyond this point,however, by using our iterative method. When thedesired output (Fig. 3) is used as the initial input, withthe initial phase being zero at all points, the function inFig. 7 results after bandlimiting in the 100th iteration.It also has zero phase. This is expected because of thenature of the algorithm. If it is used as the designedinput, the output of the hardlimiter is the function inFig. 8. There are now seven pulses present. All threepulses in the region were d = 0.875 have been re-trieved. The two pulses that are produced in the regionwhere d = 0.44 are still, however, too large and notpositioned properly.

(4) We then tried initial guesses that include phases,e.g., a phase shift of 0o on alternate pulses and a randomphase. The iterative algorithm converged within 100iterations to results that are better than the 0 = 0 case.The best results were found when the algorithm wasused with an initial phase difference of 90° betweenpulses as in Fig. 9. After 100 iterations the results inFig. 10 were found. When the function with the mag-nitude as shown in Fig. 10(a) and phase as shown in10(b) is input to the system, the output is as shown inFig. 11. All the pulses are retrieved including the onesin the region where d = 0.44, the first and third ofthese only slighlty misplaced. With each of the othervalues of phase difference tried, inputs were foundwhich produced the pulses in both the regions where dvo= 1.75 and dvo = 0.875. Pulses were also produced inthe region where do = 0.44, but they were not alwayslocated properly.

IV. Summary and Conclusions

We addressed the problem of finding the complexamplitude transmittance of a mask, which when illu-minated coherently, imaged through a diffraction lim-ited system, detected, and printed on a very-high-con-trast film, produces a desired binary pattern. Thesimplest approach is to use the desired pattern itself

g(x)1.0

0.6

0.2-

100

11111

II I V

I II

I II

III

III I

200 x

Fig. 8. Output obtained using the constructed input of Fig. 7.

I f(X)l1.0-

0.6-

0.2-

0 100

7

200

eO(X)

_ft

- 7/2

- C

- IT/2

~_ -r

X

Fig. 9. Magnitude and phase of initial input which resulted in thebest constructed input.

(with zero phase) as the input mask. This works downto resolutions corresponding to a space-bandwidthproduct of approximately dyo = 1.75. Improvementcan be obtained by assigning to every other pulse of thepattern a 180° phase shift while keeping the amplitudesunchanged. This is the method used in Refs. 10 and 11.Resolution is then improved to dvo = 1.09.

A more sophisticted method was investigated. Itallows modification of both the amplitude and phase ofthe input pattern by use of an iterative algorithm. Inall the cases we tried, the algorithm converged but to adifferent solution depending on the initial guess. Thisis not surprising in view of the fact that the solution ofthe posed problem is not supposed to be unique. It was

1436 APPLIED OPTICS / Vol. 24, No. 10 / 15 May 1985

I....lo | Elz s all hil l | |b I b | l I @

I r I .I .I 1'' Il -

I I s | . l l. . . . . . . . . . . .l . l . . . l . . .

I,I,I,

I

I

I

I

I

I

1.0

0.5

0.0

3(x)

Tr/ 2

0'

- M

0 100 200

(d)

111,111

I g I I

I I I

I I I

I i i ii

I I IlIlII.

I II I

I I !IIIII,,II III

0 100 200 x(b)

Fig. 10. (a) Magnitude of constructed input found after 100 itera-tions when initial input had a 90° phase difference between pulses;

(b) phase of constructed input.

g(x)1.0-

0.6-

0.2-

100 200S

Fig. 11. Output obtained using the constructed input of Fig. 10.

found that the complex inputs determined by thismethod are capable of producing details correspondingto a space-bandwidth product of dyo = 0.44, less thanone-third of that obtained by using a real replica of thedesired pattern as an input (with no phase) and lessthan one-half of that obtained by adding a 180° phaseshift between consecutive pulses.

Improvement of resolution beyond the limit dyo =0.44 is, in principle, possible but, due to practical dif-ficulties, not meaningful. The problem is that as dvobecomes too small, the bandlimited function undergoesvery small excursions around the threshold value. Inpractice, the contrast of the detecting device is not in-finite, and very small deviations from threshold will notbe sufficient to initiate a transition to the 1 or 0levels.

The optimum complex-valued function that was ob-tained as a result of the iterative procedure is band-limited. It, therefore, has a magnitude that variessmoothly in gray level rather than discretely in blackand white bands. The phase also varies continuouslyand is typically a smoothed version of the phase of theinitial guess as shown in Fig. 10(b).

While the example presented in this paper is a 1-Dexample, generalization to the 2-D case is straightfor-ward. Masks that introduce a prescribed complexamplitude transmittance can be constructed by usingtwo layers-one for the amplitude and another for thephase as discussed in Ref. 10. A continuous variationof gray level may be obtained by use of a halftoneliketechnique.

The authors thank S. Sayegh for helpful comments.This work was supported by the National ScienceFoundation.

References1. B. E. A. Saleh and S. I. Sayegh, "Reduction of Errors of Micro-

photographic Reproductions by Optimal Corrections of OriginalMasks," Opt. Eng. 20, 781 (1981).

2. S. I. Sayegh and B. E. A. Saleh, "Image Design: Generation of

a Prescribed Image at the Output of a Band-Limited System,"IEEE Trans. Pattern Recognition Mach. Intell. 5,441 (1983).

3. S. 1. Sayegh, B. E. A. Saleh, and K. M. Nashold, "Image Design:Generation of a Prescribed Image Through a Diffraction-LimitedSystem with High-Contrast Recording," IEEE Trans. Acoust.Speech Signal Process (33 Apr. 1985).

4. K. M. Nashold and B. E. A. Saleh, "Image Construction Through

Diffraction-Limited Systems: An Iterative Approach," J. Opt.Soc. Am. A 2 (May 1985).

5. A. Papoulis, "A New Algorithm in Spectral Analysis and Ban-

dlimited Extrapolation," IEEE Trans. Circuits Syst. CAS-22,735 (1975).

6. R. W. Gerchberg, "Super-Resolution Through Error Energy

Reduction," Opt. Acta 21, 709 (1974).7. D. C. Youla, "Generalized Image Restoration by the Method of

Alternative Orthogonal Projections," IEEE Trans. Circuits Syst.CAS-25, 695 (1978).

8. D. C. Youla and H. Webb, "Image Restoration by the Method of

Convex Projections: Part 1-Theory," IEEE Trans. Med.Imaging MI-1, 81 (1982).

9. J. R. Fienup, "Phase Retrieval Algorithms: A Comparison,"

Appl. Opt. 21, 2758 (1982).10. M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Im-

proving Resolution in Photolithography with a Phase-ShiftMask," IEEE Trans. Electron Devices ED-29, 1828 (1982).

11. D. S. Goodman, M. D. Levenson, H. Santini, and V. Viswanathan,

"Improved Photolithographic Resolution with a Phase-ShiftingMask," in Proceedings, Microcircuit '82, the 1982 GrenoblePhotolithography Conference.

12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1968), p. 131.

13. D. C. Youla, Department of Electrical Engineering, PolytechnicInstitute of New York; private communications.

14. Z. Opial, "Weak Convergence of the Sequence of Successive Ap-proximations for Nonexpansive Mappings," Bull. Am. Math. Soc.

73, 591 (1967).

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.,. - - - . .... .. .