Abstract

With continuous shrinking of critical dimension (CD) and the application of immersion lithography system to technology nodes 22nm and beyond, the vector nature of electromagnetic fields propagating from mask to wafer plane cannot be ignored, rendering mask synthesis under scalar imaging model inadequate. In this paper, we develop a level-set based optimization framework for mask synthesis with a vector imaging model. The forward model of vector image formation is established, and then the photomask synthesis is addressed as an inverse imaging problem whose variational level-set reformulation is represented by a stable time-dependent model, which is solved by employing conjugate gradient methods of the cost function and readily available finite-difference schemes. Experimental results demonstrate pronounced performance in terms of pattern fidelity and edge placement error, together with notable computation acceleration and better convergence performance.

© 2017 Optical Society of America

1. Introduction

Driven toward printing ever smaller features with increasingly large integrated circuit (IC) designs, the semiconductor industry is exploring the untapped potential of the resolution capacity of lithography systems. Along the way, resolution enhancement techniques (RETs) [1, 2] are essential in optical lithography including modified illumination schemes and optical proximity correction (OPC) [3,4], where the latter predistorts the mask patterns such that printed patterns are as close to the desired shapes as possible, among which, inverse lithography technique (ILT) is becoming a strong candidate of 22nm node and beyond and low-k1 regime, enabling the continuation of current immersion lithography.

Generally, mask synthesis in ILT involves the mathematical modeling of the imaging formation process of the lithography systems and iteratively seeking the minimization of the properly designed cost functions to improve pattern fidelity. Consequently, the performance of ILT mainly depends on the accuracy of the imaging formation model and the efficiency of the inverse imaging calculations in the optimization framework. For lithography systems with NAs less than 0.4 where the scalar imaging models provide sufficient accuracy [5], various model-based optimization approaches including steepest-descent methods [6, 7], active set and conjugate gradient method [8,9], augmented Lagrangian method [10] and level-set methods [11–15], find their applications in iterated methods for phase-shifting masks (PSMs) [8, 9, 16, 17], manufacture enhancement [18, 19], pixelated mask [20], illumination source optimization [21] and source-mask optimization [10, 22–26], double-exposure lithography [27] and designs with robustness to variations [28–30]. These approaches are all based on the scalar imaging models, which consider only the amplitude of the electromagnetic field ignoring its vector nature; while they make engineering sense, the current performance still falls short where the scalar imaging model cannot track the whole process of polarization state as light propagate through the optical components of the projection instruments.

With the continuous shrink of critical dimension (CD) and the NA of immersion lithography reaching as high as 1.35, the imaging process must account for the vector nature of the electromagnetic field propagating the projection system. Over the years, Yeung proposed high-NA vector imaging models incorporating thin-film model into the Hopkins formulations [5], whose matrix format in coherent systems developed by Flagello [31] with extension to partially coherent imaging systems by Pistor using Abbe method [32] and the vector imaging model inside resist by Adam et al. [33]. Summarizing the above developments in high-NA projections optics, Peng et al. generalized a uniform and consistent formulation with further simplification [34], based on which Ma and his coworkers designed pixelated mask optimization and source-mask optimization [23, 35] and Shen proposed level-set based mask synthesis using gradient-based methods [36]. These methods, however, are very time consuming, which limits their applicability.

This paper focuses on the development of algorithms for mask synthesis using level-set-based inverse lithography with a vector imaging model. A complete set of variational level-set formulations of the optimization problem is provided which combines an internal energy term forcing the level set function close to a signed distance function and an extra energy term that drives the zero level set toward desired mask features minimizing the pattern difference between the printed wafer and the desired pattern, thereby optimizing the mask layout without the costly re-initialization procedure. We take advantage of Fast Fourier Transform and reusable data in the computation of the evolution velocity of the level surfaces in the partial differential equation defining the time-dependent model of the level-set formulation which greatly enhances the iterative speed by easing the computation load of convolution operation; we also apply Polak-Ribière-Polyak (PRP) conjugate gradient (CG) to updating the normal velocity of the level-set function which achieves accelerated convergence. Numerical simulations demonstrate improved performance of the proposed method in both pattern fidelity and computation efficiency.

2. Forward vector imaging model

The projection lithographic imaging process T{} can be divided into two function blocks, namely the aerial image formation and resist effects. A schematic of aerial image formation based on a vector imaging model is illustrated in Fig. 1. It is often found convenient to use the polarization systems when tracking the projection of light, while a spatial coordinate system (x, y, z) is preferred when computing the actual field or intensity. Consider a monochromatic wave propagating in the direction k, where k=(α,β,γ)T is the direction cosine with γ=1α2β2. Without loss of generality, the intrinsic local coordinate (e, e) which point to the TE- or s-direction and TM- or p-direction respectively, can be defined as [34]

e=z×k|z×k|=[βραρ0],e=k×e=[αγρβγρρ],
where ρ=α2+β2, such that [e;e;k] is right-handed. Since electromagnetic wave is transverse, an arbitrary E-field can be represented as
E=Ee+Ee=[e,e][EE],
in which the polarization vector [E, E]T with T being the transpose, E being the electric field in p-state and E in s-state, describes the polarization state of the plane wave. The mapping of the spatial coordinate system and the local intrinsic system is computed as
[ExEyEz]=[βραγραρβγρ0ρ][EE]=T[EE].

 figure: Fig. 1

Fig. 1 Projection optics in a vector imaging model.

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For a point source (αs, βs) emanating an electric field E0 = [E, E]T, the emitting electric field in spatial coordinate is computed as E0=TE0 Denoting the mask as the scalar matrix M(r) ∈ ℛN×N where N is the size of M(r) and the bold r denotes spatial coordinate (x, y), each entry of which represents the corresponding transmission coefficient on the mask, the mask near field in spatial coordinate can be calculated as

E(r;αs,βs)=E0BM(r),
in which ⊙ is the entry-by-entry multiplication and B is the mask diffraction matrix. In general, the mask near field has to be computed with a rigorous Maxwell-equation solver, such as FDTD and RCWA methods, which is literally impractical for every source point. In this paper, we apply a simple embodiment of the constant scattering coefficient assumption (CSCA) from [37], where the entry in B is defined as
B(k,l)=ej×2π×βs×kN×ej×2π×αs×lN,k,l=0,1,,N1.
The CSCA assumes that the various incident plane waves are scattered by the mask in the same way, i.e. with the same scattering coefficient. While rigorous Maxwell-equation solvers provide a much more accurate mask near field than the CSCA at the hyper-NA regime and with mask features smaller than the wavelength, extremely heavy computational load and lack of analytical format prohibit them to be incorporated into the inverse lithography process. Albeit simple and by no means giving the best approximation of the mask induced effects, B is indicative of how the CSCA applies to the approximation of the mask near field where more precise solutions can be achieved involving more rigorous mask models.

According to the Abbe method and assuming E0 has identical entries, the aerial image intensity under a partially coherent illumination can be described as [35]:

Iaerial(r)=1Nsαsβsp=x,y,zHpαsβs(BαsβsM(r))2,
where Hp, B are functions of (αs, βs) and ⊗ denotes convolution operation, Ns is the number of the source points. Hpαsβs, p = x, y, z are referred to as the equivalent filters of the x, y, z components and are computed as:
Hp=1{nmnγγh(α,β)Vp(α,β,γ)},p=x,y,z,
in which, 1 denotes inverse Fourier Transform operation, n′ is the refractive index at the wafer side, m denotes the transverse magnification, (α′, β′, γ′)T and (α, β, γ)T are the light propagation direction cosines in the wafer side and mask side, respectively, h denotes a low pass pupil function of the projection lens
h(α,β)={1α2+β2NA0elsewhere,
and V(α′, β′, γ′) characterizes the rotating factor in a hyper-NA system
V(α,β,γ)=[β2+α2γ1γ2αβ1+γααβ1+γα2+β2γ1γ2βαβγ].

The resist effect can be approximated using a logarithmic sigmoid function

sig(x)=11+ea(xtr),
with a being the steepness of the sigmoid function and tr being the threshold. Putting together, we can write the forward model T{} as
I(r)=T{M(r)}=sig(Iaerial(r))=sig(1Nsαsβsp=x,y,zHpαsβs(B)αsβsM(r)2).
In what follows, we will drop the argument r when there is no ambiguity.

Figure 2(a) shows the annular illumination source with σin = 0.6 and σout = 0.9, Fig. 2(b) depicts the target pattern and Figs. 2(c) and 2(d) depict the resist images based on the scalar imaging model [13,14] with pattern error (PE) 2210 and edge placement error (EPE) 1251 and the vector imaging model with PE 2517 and EPE 1277, respectively, where pattern error is defined as the square of the L2 norm of the difference between the target pattern and the resist image and edge placement error is defined as the CD error at one side to convey CD information, especially in the polygon-based OPC [15]. Severe distortion incurred by the low pass nature of the pupil function is observed in both Figs. 2(c) and 2(d), however, degrading pattern fidelity and EPE in Fig. 2(d), for example, the vanishing contacts, are noticed which is accountable by the degradation of p-polarized images compared to s-polarized images from a polarized-light emanating source [3].

 figure: Fig. 2

Fig. 2 (a) The annular illumination source with σin = 0.6 and σout = 0.9. (b) Target pattern I0 of size N = 257. (c) The resist image I based on scalar imaging model with pattern error (PE) 2210 and edge placement error (EPE) 1251. (d) The resist image I based on vector imaging model with PE 2517 and EPE 1277.

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3. Level-set based optimization framework

Given a target pattern I0 ∈ ℛN×N, the goal of mask synthesis is to find an optimal M^N×N which minimizes the distance between T{} and I0, namely

M^=argminMN×Nd{I0,T{M}},
with d(·,·) being the sum of the mismatches between the printed image and the desired one over all locations.

The level-set based optimization framework with vector imaging formation can be developed following the steps in [13,14,36]. We give M a level set description by introducing an unknown function ϕ which relates to M by defining

M={mintfor{r:ϕ(r)<0}mextfor{r:ϕ(r)>0},
and reformulate the inverse lithography problem to handle the level set function ϕ instead of the pixelated mask M, with the aim to solving Eq. (9) with a least squares fit leads to minimizing
F(M)=12T{M}I02,
where ║·║ stands for the l2 norm, subsequently arriving at the time-dependent model
ϕt=|ϕ|v(r,t),
in which t is the artificial time and v(r, t) is the velocity normal to the surface of ϕ defined as
v(r,t)=J{M}T(T{M}I0)=12M(II0)2=2aNsαsβsp=x,y,zReal[(Bαsβs)*((Hpαsβs)*{[Hpαsβs(BαsβsM)](I0I)I(1I)})],
in which * is the conjugate operation and ◦ flips the matrix in the argument in both up-down and right-left directions and 1 ∈ ℛN ×N is the all-ones matrix.

In previous level-set based methods of mask synthesis [13–15,36], re-initialization of level set function ϕ is usually not practiced, however, it has been observed that if ϕ is not smooth or much steeper on one side of the interface than the other, the zero level set of ϕ can move incorrectly from that of the original function [38], therefore, re-initialization has been extensively used as a numerical remedy for stable curve evolution and ensuring desirable results. The standard re-initialization method is to keep the evolving ϕ as a signed distance function during the evolution, especially in a neighborhood of the zero level set. With the signed distance function satisfying |∇ϕ| = 1, the closeness of ϕ to a signed distance function is characterized by

P(ϕ)=12|ϕ|12,
incorporating which in Eq. (11) with a multiplier μ gives the variational level set formulation
ϕt=|ϕ|v(r,t)μ|ϕ|[Δϕ(ϕ|ϕ|)]=|ϕ|g(r,t),
where Δ is the Laplacian operator and g(r,t)=v(r,t)+μ[Δϕ(ϕ|ϕ|)]. It should also be noted that the preserving ϕ close to a signed distance function in its evolution not only eliminated the need of re-initializing ϕ in every iteration, but also greatly facilitates the detection of neighboring pixels or narrow bands around the zero level set which is potentially beneficial for narrow-band level-set evolution.

4. Conjugate gradient based mask optimization

The computation of v(r, t) contains three convolution operations, two explicit in Eq. (13) and one implicit in the computation of the resist image I. Although the results of Epαsβs=|Hpαsβs(B)αsβsM(r)| in Eq. (8) can be repeatedly used in Eq. (13) therefore lightening the load of one convolution operation, the computation complexity of convolution is superlinear with respect to N which makes iteratively updating ϕ to convergence very time consuming. Applying Fast Fourier Transform (FFT) to remove the convolution operations will effectively reduce the computation intensity of Eq. (13), giving

v(r,t)=12M(II0)2=2aNsαsβsp=x,y,zReal[(Bαsβs)*1{[((Hpαsβs)*][(Epαsβs)(I0I)I(1I)])}].

In this paper, we update mask by employing the widely used Polak-Ribière-Polyak (PRP) conjugate gradient (CG) [39] which is believed most efficient, instead of the steepest descent (SD) practiced in [13,14,36] to achieve faster convergence and improved performance. It should be noted that, for convergence and stability when solving the partial differential equation Eq. (12), Courant-Friedrichs-Lewy (CFL) condition is often applied asserting that the numerical wave speed must be at least as fast as the physical wave which is described as

δtmax{|Vx|δx+|Vy|δy}=ϵ,
where δx and δy are the grid size of the discrete Cartesian grid, Vx and Vy are the components of V in the x and y direction. 0 < ϵ < 1 is the CFL number and δt is the Euler time step. In the PRP-CG method, the evolution velocity V (r, tk) in the kth iteration is defined as
V(r,tk)={g(r,tk)+ηkPRPV(r,tk1)ifk1g(r,tk)ifk=0,
with ηkPRP calculated as
ηkPRP=g(r,tk)2g(r,tk)g(r,tk1)g(r,tk1)2.
The updating procedure of level-set based PRP-CG method for mask synthesis is described in Algorithm (1).

Tables Icon

Algorithm 1. Mask synthesis with level-set based PRP-CG

5. Numerical results

The stable time-dependent model in Eq. (15) is a partial differential equation, which can be solved by readily available first-order temporal accurate and second-order accurate spatial finite-difference schemes [13, 14]. The imaging system parameters used in the numerical simulations are: λ = 193nm, NA = 1.35, resolution δx = 4nm/pixel, steepness of the sigmoid function a = 0.85, threshold tr = 0.3, annular illumination source with σin = 0.6 and σout = 0.9 which is shown in Fig. 2(a), and the target pattern is given in Fig. 2(b).

Figure 3 illustrates the OPC results using the scalar imaging formation, where the synthesized mask using the scalar imaging model [13, 14] is depicted in Fig. 3(a), the aerial image and the resist image with PE 392 of Fig. 3(a) using the scalar imaging model are given in Figs. 3(b) and 3(c), respectively. The same proposed level-based PRP conjugate gradient method is applied to the mask synthesis using the scalar imaging model to accelerate the optimization process. It is noted that while the pattern fidelity with PE 392 is greatly improved in the scalar imaging model, when the synthesized mask using the scalar imaging model in Fig. 3(a) as input enters the vector imaging model, the resist image PE reaches 1291, showing the incapability of handling large NA systems where the vector nature of the electromagnetic fields must be taken into account.

 figure: Fig. 3

Fig. 3 (a) The synthesized mask using the scalar imaging model. (b) The aerial image Iaerial of (a) using the scalar model. (c) The resist image I of (a) using the scalar model with PE 392. (d) The resist image I of (a) using the vector imaging model with PE 1291.

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The synthesized masks acquired using level-set methods based on steepest descent (SD) and proposed Polak-Ribière-Polyak conjugate gradient (PRP-CG) are given in Figs. 4(a) and 4(e), with corresponding aerial images in Figs. 4(b) and 4(f), and resist images in Figs. 4(c) and 4(f), respectively. Several trials of using different μ = 0.01, 0.1, 0.5, 1 in Eq. (15) show no significant distinction in terms of the synthesized mask, hence, μ = 0.1 is applied in the following simulations. Bearing almost the same pattern error around 400 and edge placement error around 150, the level-set based ILT with SD and proposed method both demonstrate great improvement in terms of pattern fidelity comparing with Fig. 2(d). Apart from a few scattered little blocks observed in Fig. 4(a), the similarity of the synthesized masks in Figs. 4(a) and 4(e) suggests that the SD and proposed techniques reach very close local minimum which is justified by the underlying gradient-based nature of both methods.

 figure: Fig. 4

Fig. 4 (a) Synthesized masks using the steepest descent level-set methods. (b) The aerial image of the mask in (a). (c) The resist image of the mask in (a) with PE 401 and EPE 152. (d) Synthesized mask using the proposed level-set methods. (e) The aerial image of the mask in (d). (f) The resist image of the mask in (d) with PE 401 and EPE 153.

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In Fig. 5, the computation time of the simulations with the SD method and the proposed method is depicted, illustrating computation linearity for both methods and much faster speed of the proposed method than that of the SD method. A more straight-forward comparison of the computation time of the two methods is given in Table 1 with the observation that it took over 230 hours for the SD method to complete 200 mask updates amounting to 1.16 hours each iteration in contrast to only 5.60 hours total time and 0.028 hours each iteration using the proposed method. While the strenuous SD method taxes the computation load in every iteration rendering the mask synthesis intolerably slow, the proposed method achieves over 40 times of acceleration in terms of computation speed owing to the application of Eq. (16) and the repeated usage of Epαsβs to the calculation of the evolution velocity. The computation time of mask synthesis using the scalar imaging model of up to 50 iterations is also given in Table 1, amounting to 0.0095 hours each iteration which is about 13 of that using the vector imaging model. This observation conforms to the involving of the equivalent filters of the x, y, z components in the computation of the resist image in Eq. (8) and the velocity function in Eq. (13) using the vector imaging model instead of only one filter in the scalar imaging model, hence tripling the computation.

 figure: Fig. 5

Fig. 5 Simulation time by the SD method and the proposed method, for the desired pattern in Fig. 2(b).

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Tables Icon

Table 1. Simulation time (hours) in Fig. 5 using the SD method and the proposed method.

The convergence drift of the pattern error and edge placement error which is popularly used in RETs to convey CD information, are illustrated in Figs. 6(a) and 6(b) respectively. It is duely noted from Fig. 6 that the edge placement error (EPE) has a positive relationship with the pattern error (PE) in general aside from small local fluctuations indicating the equivalency of minimizing PE and EPE in mask synthesis, however, explicit incorporation of EPE into the cost function is often too complicated, therefore the minimization of PE is often applied in ILT instead of EPE. As shown in Fig. 6(a), both the SD method and the proposed method do not need 200 iterations to reach the local minimum, although it takes the proposed method around 20 iterations compared to the 40 iterations by the SD method. Naturally, the PRP-CG method applied in the proposed method contributes to the 50% convergency performance improvement.

 figure: Fig. 6

Fig. 6 Convergence of pattern and edge placement error by the SD method and the proposed method, for the desired pattern in Fig. 2(b).

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6. Conclusion

We develop a complete variational level-set based optimization framework for inverse lithography with a vector imaging model. Detailed level-set formulation represented by a time-dependent model is extensively investigated by minimizing the sum of the mismatches between the printed image and the desired one over all locations. Moreover, a distance metric preserving the closeness of the evolution function ϕ to a signed distance function is incorporated into the cost function which eliminates the necessity of re-initialization in every iteration which helps stable curve evolution and ensure desirable results. Calculation of convolution in the frequency domain with Fast Fourier Transform (FFT) and repeated usage of Epαsβs in the computation of the evolution velocity V (r, t) expedite the iterative mask synthesis procedure. We also employ the Polak-Ribière-Polyak (PRP) conjugate gradient (CG) in the proposed level-set based ILT. Numerical results demonstrate a acceleration of the mask updating by over 40 times compared to the conventional steepest descent (SD) method and around 50% improved convergency performance. The theoretic and numerical analysis of the proposed ILT brings algorithmic insights into the exploitation of vector imaging model for mask optimization problems in next generation lithography with 22nm technology node and beyond.

Funding

Natural Science Foundation of Guangdong Province, China (2016A030313709, 2015A030310290). Guangzhou Science and Technology Project, China (201607010180).

Acknowledgments

The work in this paper is partially supported by Natural Science Foundation of Guangdong Province, China (2016A030313709, 2015A030310290) and Guangzhou Science and Technology Project, China (201607010180). The author greatly appreciates the suggestions from Professor Edmund Lam (Imaging System Lab, the University of Hong Kong).

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2. L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001). [CrossRef]  

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References

  • View by:

  1. A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, Bellingham, WA, 2001).
    [Crossref]
  2. L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
    [Crossref]
  3. A. K.-K. Wong, Optical Imaging in Projection Lithography (SPIE, Bellingham, WA, 2005).
    [Crossref]
  4. Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11(9), 2438–2452 (1994).
    [Crossref]
  5. M. S. Yeung, “Modeling High Numerical Aperture Optical Lithography,” in Proceedings of Microlithography Conferences, pp. 149–167 (1988).
  6. A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
    [Crossref] [PubMed]
  7. A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
    [Crossref]
  8. A. K. Wong, E. Y. Lam, and S. H. Chan, “Inverse Synthesis of Phase-Shifting Mask for Optical Lithography,” in Proceedings of OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).
  9. S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in Proceedings of IEEE International Conference on Image Processing, pp. 1832–1835 (2008).
  10. J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21(7), 8076–8090 (2013).
    [Crossref] [PubMed]
  11. L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
    [Crossref]
  12. V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
    [Crossref]
  13. Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
    [Crossref]
  14. Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
    [Crossref] [PubMed]
  15. W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
    [Crossref]
  16. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007).
    [Crossref] [PubMed]
  17. X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express 16(24), 20126–20141 (2008).
    [Crossref] [PubMed]
  18. X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
    [Crossref] [PubMed]
  19. X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).
  20. V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
    [Crossref]
  21. W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
    [Crossref]
  22. N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011).
    [Crossref] [PubMed]
  23. X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013).
    [Crossref]
  24. J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
    [Crossref] [PubMed]
  25. J. Li and E. Y. Lam, “Joint optimization of source, mask, and pupil in optical lithography,” Proc. SPIE 9052, 90520S (2014).
    [Crossref]
  26. X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
    [Crossref] [PubMed]
  27. A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
    [Crossref]
  28. N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
    [Crossref]
  29. N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).
  30. Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).
  31. D. G. Flagello, “High Numerical Aperture Imaging in Homogeneous Thin Films,” Ph.D. thesis, The University of Arizona (1993).
  32. T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography,” Ph.D. thesis, University of California, Berkeley (2001).
  33. K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
    [Crossref]
  34. D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
    [Crossref]
  35. X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. AM. A 29(7), 1300–1312 (2012).
    [Crossref]
  36. Y. Shen, “Level-set based ILT with a vector imaging model,” in Proceedings of 2017 China Semiconductor Technology International Conference (CSTIC), pp. 1–3 (2017).
  37. T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
    [Crossref]
  38. S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
    [Crossref]
  39. W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

2015 (1)

2014 (4)

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
[Crossref] [PubMed]

J. Li and E. Y. Lam, “Joint optimization of source, mask, and pupil in optical lithography,” Proc. SPIE 9052, 90520S (2014).
[Crossref]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

2013 (3)

2012 (1)

2011 (2)

2010 (2)

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).

2009 (4)

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

2008 (3)

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express 16(24), 20126–20141 (2008).
[Crossref] [PubMed]

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

2007 (3)

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007).
[Crossref] [PubMed]

A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
[Crossref] [PubMed]

2006 (1)

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

2005 (1)

A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
[Crossref]

2003 (1)

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

2001 (2)

S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
[Crossref]

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

2000 (1)

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

1994 (1)

Adam, K.

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

Arce, G. R.

Baik, K. H.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Cecil, T.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Chan, S. H.

A. K. Wong, E. Y. Lam, and S. H. Chan, “Inverse Synthesis of Phase-Shifting Mask for Optical Lithography,” in Proceedings of OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in Proceedings of IEEE International Conference on Image Processing, pp. 1832–1835 (2008).

Chen, D.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Cobb, N. B.

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

Dam, T.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Dong, L.

Dunham, T. G.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Erdmann, A.

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

Fedkiw, R. P.

S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
[Crossref]

Flagello, D. G.

D. G. Flagello, “High Numerical Aperture Imaging in Homogeneous Thin Films,” Ph.D. thesis, The University of Arizona (1993).

Gleason, B.

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Granik, Y.

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

Hager, W. W.

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

Han, C.

He, L.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Hu, P.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Jia, N.

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011).
[Crossref] [PubMed]

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

Kailath, T.

Lam, E. Y.

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
[Crossref] [PubMed]

J. Li and E. Y. Lam, “Joint optimization of source, mask, and pupil in optical lithography,” Proc. SPIE 9052, 90520S (2014).
[Crossref]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21(7), 8076–8090 (2013).
[Crossref] [PubMed]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011).
[Crossref] [PubMed]

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in Proceedings of IEEE International Conference on Image Processing, pp. 1832–1835 (2008).

A. K. Wong, E. Y. Lam, and S. H. Chan, “Inverse Synthesis of Phase-Shifting Mask for Optical Lithography,” in Proceedings of OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

Lavin, M. A.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Leipold, W. C.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Li, J.

Li, Y.

Liebmann, L. W.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Liu, S.

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21(7), 8076–8090 (2013).
[Crossref] [PubMed]

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

Lv, W.

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Ma, X.

Mansfield, S. M.

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

Milanfar, P.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
[Crossref] [PubMed]

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
[Crossref]

Neureuther, A. R.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

Osher, S.

S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
[Crossref]

Pang, L.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Pati, Y. C.

Peng, D.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Pistor, T. V.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography,” Ph.D. thesis, University of California, Berkeley (2001).

Poonawala, A.

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
[Crossref] [PubMed]

A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
[Crossref]

Shen, Y.

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

Y. Shen, “Level-set based ILT with a vector imaging model,” in Proceedings of 2017 China Semiconductor Technology International Conference (CSTIC), pp. 1–3 (2017).

Singh, V.

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

Sinn, R.

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

Slonaker, S.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Socha, R. J.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

Toh, K.

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

Tolani, V.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Torres, A.

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

Tyminski, J.

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Wong, A. K.

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

A. K. Wong, E. Y. Lam, and S. H. Chan, “Inverse Synthesis of Phase-Shifting Mask for Optical Lithography,” in Proceedings of OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

Wong, A. K.-K.

A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, Bellingham, WA, 2001).
[Crossref]

A. K.-K. Wong, Optical Imaging in Projection Lithography (SPIE, Bellingham, WA, 2005).
[Crossref]

Wong, N.

Wu, X.

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

W. Lv, S. Liu, X. Wu, and E. Y. Lam, “Illumination source optimization in optical lithography via derivative-free optimization,” J. Opt. Soc. Am. A 31(12), 19–26 (2014).
[Crossref]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

Xia, Q.

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Xiao, G.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

Yan, B.

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

Yeung, M. S.

M. S. Yeung, “Modeling High Numerical Aperture Optical Lithography,” in Proceedings of Microlithography Conferences, pp. 149–167 (1988).

Zhang, H.

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

IBM J. Res. Develop (1)

L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop 45(5), 651–665 (2001).
[Crossref]

IEEE Trans. Image Process. (1)

A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007).
[Crossref] [PubMed]

J. Comput. Phys. (1)

S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
[Crossref]

J. Opt. Soc. AM. A (4)

J. Vac. Sci. Technol. B (1)

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Opt. Express (9)

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007).
[Crossref] [PubMed]

X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express 16(24), 20126–20141 (2008).
[Crossref] [PubMed]

X. Wu, S. Liu, W. Lv, and E. Y. Lam, “Sparse nonlinear inverse imaging for shot count reduction in inverse lithography,” Opt. Express 23(21), 26919 (2015).
[Crossref] [PubMed]

J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21(7), 8076–8090 (2013).
[Crossref] [PubMed]

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011).
[Crossref] [PubMed]

X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014).
[Crossref] [PubMed]

J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014).
[Crossref] [PubMed]

Pac. J. Optim (1)

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim 2(1), 35–58 (2006).

Proc. SPIE (12)

J. Li and E. Y. Lam, “Joint optimization of source, mask, and pupil in optical lithography,” Proc. SPIE 9052, 90520S (2014).
[Crossref]

K. Adam, Y. Granik, A. Torres, and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78–91 (2003).
[Crossref]

D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

A. Poonawala, P. Milanfar, and B. Yan, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
[Crossref]

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
[Crossref]

N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 75200E (2009).

Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 1–8 (2010).

V. Singh, K. Toh, and B. Yan, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE 7520, 75200X (2009).
[Crossref]

V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE 7488, 74880Y (2009).
[Crossref]

A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
[Crossref]

T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE 4000, 228–237 (2000).
[Crossref]

Other (9)

A. K. Wong, E. Y. Lam, and S. H. Chan, “Inverse Synthesis of Phase-Shifting Mask for Optical Lithography,” in Proceedings of OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).

S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in Proceedings of IEEE International Conference on Image Processing, pp. 1832–1835 (2008).

M. S. Yeung, “Modeling High Numerical Aperture Optical Lithography,” in Proceedings of Microlithography Conferences, pp. 149–167 (1988).

A. K.-K. Wong, Optical Imaging in Projection Lithography (SPIE, Bellingham, WA, 2005).
[Crossref]

X. Wu, S. Liu, A. Erdmann, and E. Y. Lam, “Incorporating photomask shape uncertainty in computational lithography,” in Proceedings of SPIE Advanced Lithography, p. 97800Q (2016).

A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, Bellingham, WA, 2001).
[Crossref]

D. G. Flagello, “High Numerical Aperture Imaging in Homogeneous Thin Films,” Ph.D. thesis, The University of Arizona (1993).

T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography,” Ph.D. thesis, University of California, Berkeley (2001).

Y. Shen, “Level-set based ILT with a vector imaging model,” in Proceedings of 2017 China Semiconductor Technology International Conference (CSTIC), pp. 1–3 (2017).

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Figures (6)

Fig. 1
Fig. 1 Projection optics in a vector imaging model.
Fig. 2
Fig. 2 (a) The annular illumination source with σin = 0.6 and σout = 0.9. (b) Target pattern I0 of size N = 257. (c) The resist image I based on scalar imaging model with pattern error (PE) 2210 and edge placement error (EPE) 1251. (d) The resist image I based on vector imaging model with PE 2517 and EPE 1277.
Fig. 3
Fig. 3 (a) The synthesized mask using the scalar imaging model. (b) The aerial image Iaerial of (a) using the scalar model. (c) The resist image I of (a) using the scalar model with PE 392. (d) The resist image I of (a) using the vector imaging model with PE 1291.
Fig. 4
Fig. 4 (a) Synthesized masks using the steepest descent level-set methods. (b) The aerial image of the mask in (a). (c) The resist image of the mask in (a) with PE 401 and EPE 152. (d) Synthesized mask using the proposed level-set methods. (e) The aerial image of the mask in (d). (f) The resist image of the mask in (d) with PE 401 and EPE 153.
Fig. 5
Fig. 5 Simulation time by the SD method and the proposed method, for the desired pattern in Fig. 2(b).
Fig. 6
Fig. 6 Convergence of pattern and edge placement error by the SD method and the proposed method, for the desired pattern in Fig. 2(b).

Tables (2)

Tables Icon

Algorithm 1 Mask synthesis with level-set based PRP-CG

Tables Icon

Table 1 Simulation time (hours) in Fig. 5 using the SD method and the proposed method.

Equations (22)

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e = z × k | z × k | = [ β ρ α ρ 0 ] , e = k × e = [ α γ ρ β γ ρ ρ ] ,
E = E e + E e = [ e , e ] [ E E ] ,
[ E x E y E z ] = [ β ρ α γ ρ α ρ β γ ρ 0 ρ ] [ E E ] = T [ E E ] .
E ( r ; α s , β s ) = E 0 B M ( r ) ,
B ( k , l ) = e j × 2 π × β s × k N × e j × 2 π × α s × l N , k , l = 0 , 1 , , N 1 .
I aerial ( r ) = 1 N s α s β s p = x , y , z H p α s β s ( B α s β s M ( r ) ) 2 ,
H p = 1 { n m n γ γ h ( α , β ) V p ( α , β , γ ) } , p = x , y , z ,
h ( α , β ) = { 1 α 2 + β 2 N A 0 elsewhere ,
V ( α , β , γ ) = [ β 2 + α 2 γ 1 γ 2 α β 1 + γ α α β 1 + γ α 2 + β 2 γ 1 γ 2 β α β γ ] .
s i g ( x ) = 1 1 + e a ( x t r ) ,
I ( r ) = T { M ( r ) } = s i g ( I aerial ( r ) ) = s i g ( 1 N s α s β s p = x , y , z H p α s β s ( B ) α s β s M ( r ) 2 ) .
M ^ = arg min M N × N d { I 0 , T { M } } ,
M = { m int for { r : ϕ ( r ) < 0 } m ext for { r : ϕ ( r ) > 0 } ,
F ( M ) = 1 2 T { M } I 0 2 ,
ϕ t = | ϕ | v ( r , t ) ,
v ( r , t ) = J { M } T ( T { M } I 0 ) = 1 2 M ( I I 0 ) 2 = 2 a N s α s β s p = x , y , z Real [ ( B α s β s ) * ( ( H p α s β s ) * { [ H p α s β s ( B α s β s M ) ] ( I 0 I ) I ( 1 I ) } ) ] ,
P ( ϕ ) = 1 2 | ϕ | 1 2 ,
ϕ t = | ϕ | v ( r , t ) μ | ϕ | [ Δ ϕ ( ϕ | ϕ | ) ] = | ϕ | g ( r , t ) ,
v ( r , t ) = 1 2 M ( I I 0 ) 2 = 2 a N s α s β s p = x , y , z Real [ ( B α s β s ) * 1 { [ ( ( H p α s β s ) * ] [ ( E p α s β s ) ( I 0 I ) I ( 1 I ) ] ) } ] .
δ t max { | V x | δ x + | V y | δ y } = ϵ ,
V ( r , t k ) = { g ( r , t k ) + η k P R P V ( r , t k 1 ) if k 1 g ( r , t k ) if k = 0 ,
η k P R P = g ( r , t k ) 2 g ( r , t k ) g ( r , t k 1 ) g ( r , t k 1 ) 2 .

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