Robust level-set-based inverse lithography. Citation Optics Express, 2011, v. 19 n. 6, p

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1 Title Robust level-set-based inverse lithography Author(s) Shen, Y; Jia, N; Wong, N; La, EY Citation Optics Express, 2011, v. 19 n. 6, p Issued Date 2011 URL Rights This work is licensed under a Creative Coons Attribution- NonCoercial-NoDerivatives 4.0 International License.; Optics Express. Copyright Optical Society of Aerica.; This paper was published in [Optics Express] and is ade available as an electronic reprint with the perission of OSA. The paper can be found at the following URL on the OSA website: [ Systeatic or ultiple reproduction or distribution to ultiple locations via electronic or other eans is prohibited and is subject to penalties under law.

2 Robust level-set-based inverse lithography Yijiang Shen, Ningning Jia, Ngai Wong and Edund Y. La Departent of Electrical and Electronic Engineering, The University of Hong Kong, Pokfula Road, Hong Kong Abstract: Level-set based inverse lithography technology (ILT) treats photoask design for icrolithography as an inverse atheatical proble, interpreted with a tie-dependent odel, and then solved as a partial differential equation with finite difference schees. This paper focuses on developing level-set based ILT for partially coherent systes, and upon that an expectation-orient optiization fraework weighting the cost function by rando process condition variables. These include defocus and aberration to enhance robustness of layout patterns against process variations. Results deonstrating the benefits of defocus-aberration-aware level-set based ILT are presented Optical Society of Aerica OCIS codes: ( ) Microlithography; ( ) Photolithography; ( ) Coputational iaging. References and links 1. A. K.-K. Wong, Resolution Enhanceent Techniques in Optical Lithography (SPIE Press, Bellingha, WA, 2001). 2. F. Schellenberg, Resolution enhanceent technology: the past, the present, and extensions for the future, Proc. SPIE 5377, 1 20 (2004). 3. O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, Autoated optical proxiity correction: a rules-based approach, Proc. SPIE 2197, (1994). 4. S. Shioiri and H. Tanabe, Fast optical proxiity correction: analytical ethod, Proc. SPIE 2440, (1995). 5. L. Pang, Y. Liu, and D. Abras, Inverse lithography technology (ILT): a natural solution for odel-based SRAF at 45n and 32n, Proc. SPIE 6607, (2007). 6. Y. Liu and A. Zakhor, Optial binary iage design for optical lithography, Proc. SPIE 1264, (1990). 7. Y. Liu and A. Zakhor, Binary and phase-shifting iage design for optical lithography, Proc. SPIE 1463, (1991). 8. S. Sherif, B. Saleh, and R. De Leone, Binary iage synthesis using ixed linear integer prograing, IEEE Trans. Iage Process. 4(9), (1995). 9. Y. C. Pati and T. Kailath, Phase-shifting asks for icrolithography: autoated design and ask requireents, J. Opt. Soc. A. A 11(9), (1994). 10. S. H. Chan, A. K. Wong, and E. Y. La, Initialization for robust inverse synthesis of phase-shifting asks in optical projection lithography, Opt. Express 16(19), (2008). 11. S. H. Chan and E. Y. La, Inverse iage proble of designing phase shifting asks in optical lithography, in Proceedings of IEEE International Conference on Iage Processing, pp (2008). 12. A. Poonawala and P. Milanfar, Prewarping techniques in iaging: applications in nanotechnology and biotechnology, Proc. SPIE 5674, (2005). 13. A. Poonawala and P. Milanfar, Mask design for optical icrolithography: an inverse iaging proble, IEEE Trans. Iage Process. 16(3), (2007). 14. S. H. Chan, A. K. Wong, and E. Y. La, Inverse synthesis of phase-shifting ask for optical lithography, in OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007). (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5511

3 15. V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, Making a trillion pixels dance, Proc. SPIE 6924, 69240S (2008). 16. A. Poonawala, Y. Borodovsky, and P. Milanfar, ILT for double exposure lithography with conventional and novel aterials, Proc. SPIE 6520, 65202Q (2007). 17. N. Jia, A. K. Wong, and E. Y. La, Regularization of inverse photoask synthesis to enhance anufacturability, Proc. SPIE 7520, (2009). 18. N. Jia, A. K. Wong, and E. Y. La, Robust photoask design with defocus variation using inverse synthesis, Proc. SPIE 7140, 71401W (2008). 19. S. Osher and R. P. Fedkiw, Level set ethods: an overview and soe recent results, J. Coput. Phys. 169(2), (2001). 20. J. A. Sethian and D. Adalsteinsson, An overview of level set ethods for etching, deposition, and lithography developent, IEEE Trans. Seicond. Manuf. 10, (1997). 21. S. Osher and N. Paragios, Geoetric Level Set Methods in Iaging, Vision, and Graphics (Springer Verlag New York, NJ, USA, 2003). 22. F. Santosa, A level-set approach for inverse probles involving obstacles, ESAIM Contröle Opti. Calc. Var. 1, (1996). 23. S. Osher and F. Santosa, Level set ethods for optiization probles involving geoetry and constraints I. Frequencies of a two-density inhoogeneous dru, J. Coput. Phys. 171(1), (2001). 24. A. Marquina and S. Osher, Explicit algoriths for a new tie dependent odel based on level set otion for nonlinear deblurring and noise reoval, SIAM J. Sci. Cop. 22, (2000). 25. L. Pang, G. Dai, T. Cecil, T. Da, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes, Proc. SPIE 6924, 69240T (2008). 26. Y. Shen, N. Wong, and E. Y. La, Level-set-based inverse lithography for photoask synthesis, Opt. Express 17(26), (2009). 27. E. Y. La and A. K. Wong, Coputation lithography: virtual reality and virtual virtuality, Opt. Express 17(15), (2009). 28. E. Y. La and A. K. Wong, Nebulous hotspot and algorith variability in coputation lithography, J. Micro/Nanolithogr. MEMS MOEMS 9(3), (2010). 29. N. Jia and E. Y. La, Machine learning for inverse lithography: Using stochastic gradient descent for robust photoask synthesis, J. Opt. 12(4), (2010). 30. Y. Shen, N. Wong, and E. Y. La, Aberration-aware robust ask design with level-set-based inverse lithography, Proc. of SPIE 7748, 77481U (2010). 31. M. Born and E. Wolf, Principles of Optics (Pergaon Press Oxford, 1980). 32. J. W. Goodan, Statistical Optics (Wiley-Interscience, New York, 2000). 33. H. H. Hopkins, On the diffraction theory of optical iages, Proc. of the Royal Soc. of London 217A(1130), (1953). 34. A. K.-K. Wong, Optical Iaging in Projection Microlithography (SPIE Press, Bellingha, WA, 2005). 35. B. E. A. Saleh and M. Rabbani, Siulation of partially coherent iagery in the space and frequency doains and by odal expansion, Appl. Opt. 21(15), (1982). 36. X. Ma and G. R. Arce, Pixel-based siultaneous source and ask optiization for resolution enhanceent in optical lithography, Opt. Express 17(7), (2009). 37. B. Nijboer, The diffraction theory of aberrations, Ph.D. thesis, Groningen University (1942). 38. R. J. Noll, Zernike polynoials and atospheric turbulence, J. Opt. Soc. A. A 66(3), (1976). 39. P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, Aberration retrieval for high-na optical systes using the Extended Nijboer-Zernike theory, Proc. SPIE 5754, 263 (2005). 1. Introduction Optical projection lithography reains the predoinant icrolithography technology until the foreseeable future as shrinkage of integrated circuit device diension outpaces introduction of shorter-exposure wavelengths and higher-nuerical-aperture lenses extending to sub-0.35 k 1 regie and iniu design pitches to sub-100n. Due to the wave nature of light, as diensions approach sizes coparable to or saller than the wavelength of the light used in the photolithography process, the bandliited iaging syste introduces undesirable distortions and artifacts. Along the way, resolution enhanceent techniques (RET) [1, 2] are essential in optical lithography, which include odified illuination schees and optical proxiity correction (OPC). The latter predistorts the ask patterns such that printed patterns are as close to the desired shapes as possible. Rule-based OPC [3, 4], in which various geoetries are treated by different epirical rules, and odel-based OPC, which is ore coplex and involves the (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5512

4 coputation of a weighted su of pre-siulated results for siple edges and corners that are stored in a library, are two ain approaches to OPC. Moving beyond odel-based OPC, inverse lithography technology (ILT) is becoing a strong candidate for 32n and below low-k 1 regie. Inverse odel-based techniques invert the iaging odel and attept to directly synthesize the optiized ask pattern, and are not constrained by the topology of the original design [5]. Early works include branch and bound algorith [6] and the bacteria algorith [7] with very liited applicability due to their high coputation coplexity. Since the id-1990s, iterative ethods have been applied to generate binary asks [8] and phase shifting asks (PSMs) [9, 10, 11], treating the proble as an inverse proble and solving it by an optiization process. Meanwhile, Poonawala and Milanfar designed a odel-based OPC syste and developed a faster optiization algorith using steepest-descent [12, 13], which has been further iproved by Chan et al. using an active set ethod and conjugate gradient [11, 14]. Applications of ILT are also reported in pixelated ask technology [15], double exposure lithography [16], anufacturability enhanceent [17] and designs specifically taking into account robustness to variations, such as focal length [18]. Level-set ethod [19] offers a feasible alternative to inverse lithography. It has been applied in forward optical lithography [20] and inverse iaging probles [21] involving obstacles [22, 23], and in nonlinear deblurring and noise reoval [24]. Level set approaches for ILT has been explored in [5, 25], and in [26], Shen et al. presented systeatic level-set forulations and developed finite-difference schees to solve the with high fidelity and robustness for a coherent iaging syste. It should also be noted while the above-entioned algoriths enrich the weaponry for solving inverse lithography probles, the current perforance still falls short, and is not yet fully applicable to real-world anufacturing [27, 28]. One such concern is that they are ostly applied to calculate solutions under noinal conditions, without taking process variation into account. Consequently, various efforts have been ade to incorporate process conditions such as defocus and aberration variations into OPC algoriths for robust design of printed patterns [29, 30]. This paper focuses on the developent of algoriths for photoask synthesis using levelset based inverse lithography proble in partially coherent systes. A statistical ethod is proposed to incorporate process variations into the optiization fraework, which treats defocus and aberrations as rando variables, taking their statistical properties into account, and develops an optiization fraework which ais to iniize the pattern difference between the printed wafer upon various defocus and aberration conditions and the target pattern. A coplete set of level-set forulations of the optiization proble is provided with coputationally efficient solutions. 2. The Constrained level set tie-dependent odel forulation 2.1. Matheatical odel for partially coherent systes The projection lithography iaging process can be described as a forward odel I(x)=T {U(x)}, (1) where the boldface x denotes spatial coordinates (x, y), and T { } aps the input intensity function U(x) to the output intensity function I(x). Due to the lowpass nature of the optical lithography iaging syste, I(x) is typically a blurred version of U(x). Suppose the desired circuit pattern is I 0 (x). The objective of inverse lithography is to find a predistorted input intensity function Û(x) which iniizes its distance with the desired output, i.e., Û(x)=arg in d(i 0 (x),t {U(x)}), (2) U(x) (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5513

5 (a) (b) (c) Fig. 1. Coon illuination sources: (a) conventional, (b) annular, and (c) dipole. in which d(, ) is an appropriately defined distance etric, such as the l 2 nor. The lithography process T { } in Eq. (1) can be divided into two functional blocks, naely the projection optics effects (aerial iage foration) and resist effects. The resist effects can be approxiated using a logarithic sigoid function [13] sig(u(x)) = e a(u(x) t r), (3) with a being the steepness of the sigoid and t r being the threshold. The nonlinear nature of aerial iage foration in partially coherent systes has been understood since the early developent of optical coherence theory [31, 32], but the coplexity of the calculations involved has liited use of the theory for actual optical design. In partially coherent iaging, the ask is illuinated by light traveling in various directions. In [33], the concept of effective source is developed to help understand the echanis of partially coherent iaging. Radiation of partially coherent light has been shown to be an expansion of coherent odes added incoherently in the iage plane. The expansion uses a basis of orthogonal functions which are eigenfunctions of the utual coherence function [34], however, coputation of the eigenfunctions is a difficult process which leads to coplicated functions and expensive coputation coplexity in ters of both speed and eory requireent. Alternatively, the Fourier expansion odel [35, 36] is applied to decopose the partially coherent iaging systes as the su of coherent systes. According to the Hopkins diffraction odel, the light intensity distribution exposed on the wafer in partially coherent iaging is bilinear and can be described as I aerial (x)= U (x 1 )U(x 2 )γ(x 1 x 2 )H (x x 1 )H(x x 2 ) dx 1 dx 2, (4) where x =(x,y), x 1 =(x 1,y 1 ) and x 2 =(x 2,y 2 ). U(x) is the ask pattern, γ(x 1 x 2 ) is the coplex degree of coherence and H(x) represents the aplitude ipulse response of the optical syste, naely point spread function (PSF). The ter γ(x 1 x 2 ) is generally a coplex nuber, whose agnitude represents the extent of optical interaction between two spatial locations x 1 =(x 1,y 1 ) and x 2 =(x 2,y 2 ) of the light source. It is the inverse Fourier transfor of the iage of the illuination shape Γ(x) in the lens pupil. Coon illuination sources are shown in Fig. 1, which include conventional circular, annular, and dipole, all introducing partial coherence into illuination. If the object intensity vanishes outside a square area A defined by x [ G 2, G 2 ], for coputations involved in (4), the only values of γ(x) needed are those inside the square area A γ defined by x [ G,G]. We can expand γ(x) as a 2-D Fourier series of periodicity 2G in both the x and y directions, and therefore, γ(x) can be rewritten as γ(x)= Γ e jω0 x, (5) (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5514

6 and also Γ = 1 G 2 A γ γ(x)e jω 0 x dx, (6) where j = 1, ω 0 = π/g, =( x, y ), x and y being integers within the range of [ D,D], and x = x x + y y. Substituting (5) into (4), the light intensity on the wafer is given by I aerial (x)= Γ U(x) H (x) 2, (7) where H (x)=h(x)e jω0 x. (8) Cobining the aerial iage foration in (7) and the logarithic sigoid function in (3) describing resist effects, we have the iage foration equation for a partially coherent iaging syste as I(x)=sig( Γ U(x) H (x) 2 ). (9) It is observed fro (7) that the partially coherent iage is equal to the superposition of coherent systes. In what follows, we will drop the arguents x whenever there is no abiguity Tie-dependent odel forulation in partially coherent iaging Developent of the tie-dependent schee in partially coherent iaging is in principle siilar to that in coherent iaging [26]. The inverse lithography proble can be treated as an obstacle reconstruction proble [22, 23], which is an inverse proble involving obstacles where the desired unknown is a region consisting of several subregions. We give U a level set description by introducing an unknown function φ(x), which is related to U by defining { Uint for {x : φ(x) < 0} U(x)= U ext for {x : φ(x) > 0}, (10) where U int = 1 and U ext = 0 if we are dealing with binary asks. Now we can define the inverse lithography proble as finding φ(x) such that T (U) I 0. Solving this with a least squares fit to the approxiation is equivalent to seeking the iniizer of F(U)= 1 2 T (U) I 0 2. (11) The boundary of the subregions in U where U = U int is governed by the zero level set of φ, naely, φ(x)=0. If φ depends on both x and tie t such that the evolution of the subregions in U is associated with φ(x,t), following the steps in [26], we arrive at the tie-dependent odel with α(x,t) being defined as φ t = φ α(x,t), (12) α(x,t)= J(U) T (T (U) I 0 ) = 1 2 U (I I 0) 2 = 1 ( ) 2 2 U sig( Γ U H 2 ) I 0 { } = a Γ H [(I 0 I) I (1 I) (H U)], (13) (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5515

7 where J(U) is the Jacobian of T (U) at U and denotes eleent-by-eleent ultiplication. Eq. (12) is a partial differential equation (PDE). Once φ and α are defined at every grid point on the Cartesian grid, Eq. (12) can be solved using finite-difference ethods with first-order teporal and second-order spatial accuracy as suggested in [26]. It should be noted the inverse lithography proble is non-convex with ultiple local inia. Since we use finite-difference schees to solve the PDE, there is no guarantee of reaching the global iniu. However, ILT is an ill-posed proble and it is often not necessary to arrive at the global iniu. Any good local iniu (where goodness is defined using data-fidelity and user-defined properties) can suffice as an acceptable solution Aberration-aware statistical odel We have thus far assued noinal conditions where the iage always lies on the focal plane and the iage wavefronts is spherical. However, in optical lithography, the eerging wavefronts fro the pupil are in general aspherical [34], even if the lens surfaces are spherical. Deviation of wavefronts, also known as aberration, is studied extensively in the Nijboer-Zernike theory [37, 38, 39]. Aberration function Φ can be represented as Zernike polynoials [39], Φ(ρ,θ)= c n R n (ρ)cosθ, (14) n, where ρ,θ are the polar coordinates in the exit pupil function, and c n R n (ρ)cosθ is a polynoial of the Zernike set with c n being aberration coefficient. For exaple, one of the ost widely studied aberrations, defocus [34], or the focus error f between the iage plane and focal plane, can be expressed as f = c 0 2 ρ2 where c 0 2 is the aberration coefficient for defocus. As a reasonable assuption, the coefficients are odeled as independent, norally distributed rando variables with zero ean and identical non-zero variance. PSF in an ideal case where there exists no defocus and aberrations is taken as the inverse Fourier transfor of a disc function [34], denoted as H 0 in this paper. Aberration ters are incorporated into the PSF by ultiplying an exponential ter with aberration function as power in the frequency doain, i.e. [37] F (H)=F (H 0 ) e jφ, (15) where F denotes Fourier transfor. To enable the coputation of optiu ask patterns iniizing deviation of iages fro their targets not only at noinal but also over a range of aberrations, the objective function takes expectations of the difference under various aberration conditions to optiize the average perforance of layouts. The optiization proble we ai to solve becoes U optial = in E { I I 0 2 } 2, (16) U where 2 2 denotes the square of the l 2 nor and E denotes expectation. One should note this expectation-orient iniizing proble practically weights the cost function I I by the statistical probability of aberration ters appearing over a range. Previous efforts in [18, 29, 30] developed various algoriths to solve the inverse lithography proble in Eq.(16). Yet in the fraework of level-set-based ILT in partially coherent systes, a stable explicit tie-dependent odel can be applied, φ = φ α(x,t), (17) t (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5516

8 in which α(x,t) is coputed as α(x,t)= 1 2 U E { (I I 0 ) 2} { ( = 1 ) } 2 2 U E sig( Γ U H 2 ) I 0 { } = a E Γ H [(I 0 I) I (1 I) (H U)]. (18) One should notice that Eq. (17) takes the sae for as Eq. (12) with a different coputation of α(x,t), therefore Eq. (17) is also a PDE which could be solved by the proposed finite difference schees in [26]. The coputation stops after a certain nuber of iterations or when the value of the cost function has decreased below a certain threshold value. 3. Nuerical results We apply the inverse lithography technique outlined above to designing various circuit patterns for partially coherent systes. We use the sae iaging syste paraeters: λ = 193n, NA = 1.35, resolution Δx = 10n/pixel, steepness of the sigoid function a = 85, threshold t r = 0.3, and therefore the sae PSF H 0 (x) with the sae size as that of the target pattern which is for various experients. The optiization stops after 50 iterations. In this paper, we apply the proposed algoriths on binary asks. However, it should be noted the sae fraework can be readily applied to phase-shifting asks (PSMs) by applying different levels sets to corresponding phases in the PSMs. Figure. 2(a) shows the target pattern, and (b), (c) and (d) are the output patterns under circular, annular, and dipole source illuinations in Fig. 1(a), (b) and (c), respectively. Respective pattern errors are also given. In Fig. 3, input pattern derived using the proposed level-set based ILT in partially coherent iaging under noinal condition with circular, annular and dipole illuination are presented. The first colun denotes the input pattern, the second colun aerial iage and the third colun the output pattern. Cobining the data in Fig. 2 and Fig. 3, we can see that the input pattern developed using the proposed level-set based ILT greatly iproves pattern quality in partially coherent iaging systes under coonly used illuination sources. Another set of siulation is given in Fig. 4 applying the proposed statistical ethod to target pattern in Fig. 2(a) with circular source in Fig. 1(a) as an exaple to test its effectiveness against aberrations. It should be noted that the optial input ask pattern is coputed by averaging the output patterns under different aberration conditions weighted by the statistical probability of the aberration coefficient. Consequently, for any specific aberration, PSF H(x) should be coputed by degrading H 0 (x) with aberrations as in Eq. (15). For defocus aberration, conventionally, the relationship between the defocus coefficient c 0 2 and the real iage space coordinate z by paraxial approxiation can be described as [39] c 0 2 = 2 π ) (1 λ z (1 NA 2 ) z πna2 λ, (19) therefore, defocus ter is incorporated into the aplitude spread function H(x) by ultiplying (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5517

9 (a) (b) (c) (d) Fig. 2. (a) Target pattern of size (b) Output pattern under circular source, resulting in a pattern error of 45 pixels. (c) Output pattern under annular source, resulting in a pattern error of 116 pixels. (d) Output pattern under dipole source, resulting in a pattern error of 144 pixels. U(x) I aerial (x) I(x) (a) (b) (c) Fig. 3. Siulation of lithographic iaging with different ask patterns coputed using level-set based ILT. The first colun denotes the input U(x), the second colun I aerial (x), and the third colun I(x). Rows (a), (b) and (c) use the derived pattern under circular illuination, annular illuination, and dipole illuination as input, resulting in pattern errors of 5, 24, and 45 pixels respectively. an exponential ter with power, jc 0 2 ρ2 jπz NA2 λ ρ2 = jπz NA2 λ [ ( u λ ) 2 ( + v λ ) ] 2 NA NA = jπλz(u 2 + v 2 ) [ ( = jπλz 1 ) 2 ( + n 1 ) ] 2 NΔx NΔx = jπλz 2 + n 2 (NΔx) 2, (20) in which N is the iage size and (u,v) and (,n) are the frequency coordinates and noral- (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5518

10 (a) (b) algorith under noinal condition statistical ethod against defocus 80 pixel error focus error in n (c) algorith under noinal condition statistical ethod against coa 60 pixel error Aberration coefficient for coa c 3 (d) Fig. 4. Perforances of the proposed level-set based statistical ethod with aberration variations. (a) focus-aware input ask pattern coputed using the statistical ethod. (b) coa-aware input ask pattern coputed using the statistical ethod. (c) Coparison of pixel errors under different focus errors. (d) Coparison of pixel errors under different coa. (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5519

11 ized frequency coordinates corresponding to spatial coordinates (x, y) respectively, in the frequency doain of the noinal PSF H 0 (x). We first construct the optial input ask pattern with defocus variation using the statistical ethod described in Section 2.3 and then copare the generated results with that of the optial input ask pattern constructed under noinal conditions presented in U(x) of Fig. 3(a). The optial input ask pattern with defocus variation is presented in Fig. 4(a). It should be noted that since we are not introducing large aberration paraeters and sall weights accopany large aberration paraeters, big difference in ask patterns obtained under noinal condition and using the statistical ethod in Fig. 4(a) is not expected. Figure. 4(c) plots the perforance of the statistical ethod versus optiization only under noinal conditions with a defocus range of (-90n, 90n). It is observed that while under noinal conditions, the optial input ask pattern under noinal condition outperfors the input ask coputed using the statistical ethod, which is not a surprise since the forer is intended specifically for noinal conditions and the latter is not, the input ask coputed with statistical variability accounted for iproves pattern fidelity with focus variation, obtaining fewer pattern errors than the optial input ask pattern coputed under the noinal condition. This is because the expectation operation tends to copensate the distortion brought by different defocus conditions on the input pattern ask. In Fig. 4(b), the input ask coputed with statistical identity of coa variation accounted for is presented. The Zernike polynoial for coa is given as c 1 3 (3ρ3 2ρ)cosθ, and coa ter can be incorporated into the PSF H(x) by ultiplying noinal PSF H 0 (x) with an exponential ter with power, jc 1 3(3ρ 3 2ρ)cosθ = jc 1 3 ( 3 u λ ) 2 ( + v λ ) 2 NA NA 3 ( 2 u λ NA ) 2 ( + v λ ) 2 cosθ NA = jc 1 3 [3 λ 3 ( u 2 NA 3 + v 2) λ ( u 2 + v 2) ] 1 2 cosθ NA [ = jc 1 λ 3 ( (NΔxNA) 3 + n 2) 3 2 λ ( n 2) ] 1 2 cosθ. (21) N xna Figure. 4(d) plots the perforance of the statistical ethod versus optiization only under noinal conditions with a coa range c 1 3 of (-0.08, 0.08). Likewise, we observe siilar results in Fig. 4(d) as that in Fig. 4(c). These observations justify the proposed statistical algorith in producing aberration-aware input ask pattern for critical structures. It should be noted that when two or ore aberrations are present, the root ean square deforation of the cobined aberration ΔΦ rs = n, c n 2 becoes a chi-square distributed rando variable with degree of freedo equal to the nuber of Zernike polynoials in the cobined aberration. Incorporating aberration ters in the ASF siilarly as in Eq. (20) and Eq. (21), the proposed algorith can produce aberration-aware input asks against this cobined aberration using the proposed level-set-based statistical ethod. 4. Conclusion In this paper, we investigate level-set forulations for photoask design in optical icrolithography for partially coherent systes with coputationally efficient solutions. The inverse lithography proble is described as a constrained tie-dependent odel with the for of a partial differential equation which is solved by available finite-difference schees. The proposed level-set-based ILT enables coputation of optiu ask patterns to iniize deviations of (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5520

12 iages fro their targets not only at noinal but also over a range of aberrations. The statistical optiization fraework proposed in this paper offers algorithic insights of how the cost functions are weighted by statistical probability of aberration coefficients to optiize average layout perforance which shows great robustness against aberrations. Acknowledgent This work was supported in part by the Research Grants Council of the Hong Kong Special Adinistrative Region, China, under Projects HKU 7139/06E, 7174/07E and 7134/08E, and by the UGC Areas of Excellence project Theory, Modeling, and Siulation of Eerging Electronics. (C) 2011 OSA 14 March 2011 / Vol. 19, No. 6 / OPTICS EXPRESS 5521