An alternating direction method of multipliers for inverse lithography problem. (English) Zbl 07814755
Summary: We propose an alternating direction method of multipliers (ADMM) to solve an optimization problem stemming from inverse lithography. The objective functional of the optimization problem includes three terms: the misfit between the imaging on wafer and the target pattern, the penalty term which ensures the mask is binary and the total variation regularization term. By variable splitting, we introduce an augmented Lagrangian for the original objective functional. In the framework of ADMM method, the optimization problem is divided into several subproblems. Each of the subproblems can be solved efficiently. We give the convergence analysis of the proposed method. Specially, instead of solving the subproblem concerning sigmoid, we solve directly the threshold truncation imaging function which can be solved analytically. We also provide many numerical examples to illustrate the effectiveness of the method.
MSC:
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
| 78M50 | Optimization problems in optics and electromagnetic theory |
References:
| [1] | R. A. ADAMS AND J. F. FOURNIER, Sobolev Spaces, Elsevier, 2003. |
| [2] | A. CHAMBOLLE, An algorithm for total variation minimization and applications, J. Math. Imaging Vision 20 (2004), 89-97. · Zbl 1366.94048 |
| [3] | S. H. CHAN AND E. Y. LAM, Inverse image problem of designing phase shifting masks in optical lithography, in: Proceedings of the 15th IEEE International Conference on Image Processing, IEEE, (2008), 1832-1835. |
| [4] | S. H. CHAN, A. K. WONG, AND E. Y. LAM, Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography, Opt. Express 16 (2008), 14746-14760. |
| [5] | T. F. CHAN, G. H. GOLUB, AND P. MULET, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput. 20 (1999), 1964-1977. · Zbl 0929.68118 |
| [6] | T. F. CHAN AND J. SHEN, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, 2005. · Zbl 1095.68127 |
| [7] | R. CHARTRAND AND B. WOHLBERG, A nonconvex ADMM algorithm for group sparsity with sparse groups, in: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, (2013), 6009-6013. |
| [8] | Z. CHEN AND J. ZOU, An augmented Lagrangian method for identifying discontinuous pa-rameters in elliptic systems, SIAM J. Control Optim. 37(3) (1999), 892-910. · Zbl 0940.65117 |
| [9] | S. K. CHOY, N. JIA, C. S. TONG, M. L. TANG, AND E. Y. LAM, A robust computational al-gorithm for inverse photomask synthesis in optical projection lithography, SIAM J. Imaging Sci. 5 (2012), 625-651. · Zbl 1250.65081 |
| [10] | H. W. ENGL, M. HANKE, AND A. NEUBAUER, Regularization of Inverse Problems, Kluwer Academic Publishers, 1996. · Zbl 0859.65054 |
| [11] | A. ERDMANN, T. FUEHNER, T. SCHNATTINGER, AND B. TOLLKUEHN, Toward automatic mask and source optimization for optical lithography, in: Proceedings of the SPIE Optical Microlithography XVII, Vol. 5377 (2004), 646-657. |
| [12] | D. GABAY AND B. MERCIER, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. with Appl. 2(1) (1976), 17-40. · Zbl 0352.65034 |
| [13] | T. GOLDSTEIN AND S. OSHER, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci. 2(2) (2009), 323-343. · Zbl 1177.65088 |
| [14] | J. W. GOODMAN, Introduction to Fourier Optics, Roberts & Co. Publishers, 2005. |
| [15] | Y. GRANIK, Fast pixel-based mask optimization for inverse lithography, J. Micro Nano-lithogr. MEMS MOEMS 5(4) (2006), 043002. |
| [16] | R. LAI AND S. OSHER, A splitting method for orthogonality constrained problems, J. Sci. Comput. 58 (2014), 431-449. · Zbl 1296.65087 |
| [17] | J. LI, S. LIU, AND E. Y. LAM, Efficient source and mask optimization with augmented Lagrangian methods in optical lithography, Opt. Express, 21(7) (2013), 8076-8090. |
| [18] | X. MA AND G. R. ARCE, Generalized inverse lithography methods for phase-shifting mask design, Opt. Express 15 (2007), 15066-15079. |
| [19] | X. MA AND G. R. ARCE, Computational Lithography, John Wiley & Sons, 2010. |
| [20] | X. MA AND G. R. ARCE, Pixel-based OPC optimization based on conjugate gradients, Opt. Express 19 (2011), 2165-2180. |
| [21] | X. MA, Z. WANG, J. ZHU, S. ZHANG, G. R. ARCE, AND S. ZHAO, Nonlinear compressive inverse lithography aided by low-rank regularization, Opt. Express 27 (2019), 29992-30008. |
| [22] | Y. NESTEROV, Lectures on Convex Optimization, in: Springer Optimization and Its Appli-cations, Vol. 137, Springer, 2018. · Zbl 1427.90003 |
| [23] | J. NOCEDAL AND S. J. WRIGHT, Numerical Optimization, in: Springer Series in Operations Research and Financial Engineering, Spinger, 2006. · Zbl 1104.65059 |
| [24] | S. OSHER, M. BURGER, D. GOLDFARB, J. XU, AND W. YIN, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul. 4 (2005), 460-489. · Zbl 1090.94003 |
| [25] | L. PANG, Inverse lithography technology: 30 years from concept to practical, full-chip reality, Journal of Micro/Nanopatterning, Materials, and Metrology 20 (2021), 030901. |
| [26] | L. PANG, Y. LIU, AND D. ABRAMS, Inverse lithography technology (ILT): What is the impact to the photomask industry?, in: Proceedings of SPIE Photomask and Next-Generation Lithography Mask Technology XIII, Vol. 6283, 2006. |
| [27] | Y. PENG, J. ZHANG, Y. WANG, AND Z. YU, Gradient-based source and mask optimization in optical lithography, Trans. Image Process. 20 (2011), 2856-2864. · Zbl 1372.94207 |
| [28] | A. POONAWALA AND P. MILANFAR, OPC and PSM design using inverse lithography: A non-linear optimization approach, Optical Microlithography XIX, 6154 (2006), 1159-1172. |
| [29] | A. POONAWALA AND P. MILANFAR, Double-exposure mask synthesis using inverse lithogra-phy, J. Micro Nanolithogr. MEMS MOEMS 6 (2007), 043001. |
| [30] | A. POONAWALA AND P. MILANFAR, Mask design for optical microlithography -an inverse imaging problem, IEEE Trans. Image Process. 16(3) (2007), 774-788. |
| [31] | A. POONAWALA, B. PAINTER, AND J. MAYHEW, Model-based assist feature placement: An in-verse imaging approach, in: Proceedings of SPIE Photomask Technology 2008, Vol. 7122 (2008), 277-286. |
| [32] | Y. SHEN, F. PENG, AND Z. ZHANG, Semi-implicit level set formulation for lithographic source and mask optimization, Opt. Express 27 (2019), 29659-29668. |
| [33] | Y. SHEN, N. WONG, AND E. Y. LAM, Level-set-based inverse lithography for photomask synthesis, Opt. Express 17 (2009), 23690-23701. |
| [34] | D. STRONG AND T. CHAN, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems 19 (2003), S165. · Zbl 1043.94512 |
| [35] | H. SUN, J. DU, C. JIN, J. FENG, J. WANG, S. HU, AND J. LIU, Global source optimisation based on adaptive nonlinear particle swarm optimisation algorithm for inverse lithography, IEEE Photon. J. 13 (2021), 1-7. |
| [36] | Y. WANG, W. YIN, AND J. ZENG, Global convergence of ADMM in nonconvex nonsmooth optimization, J. Sci. Comput. 78 (2019), 29-63. · Zbl 1462.65072 |
| [37] | Z. WEN, C. YANG, X. LIU, AND S. MARCHESINI, Alternating direction methods for classical and ptychographic phase retrieval, Inverse Problems, 28 (2012), 115010. · Zbl 1254.78037 |
| [38] | A. K.-K. WONG, Resolution enhancement techniques in optical lithography, in: SPIE Tuto-rial Texts in Optical Engineering, Vol. 47, SPIE Press, 2001. |
| [39] | A. K.-K. WONG, Optical imaging in projection microlithography, in: SPIE Tutorial Texts in Optical Engineering, Vol. 66, SPIE Press, 2005. |
| [40] | H. YANG, S. LI, Z. DENG, Y. MA, B. YU, AND E. F. YOUNG, GAN-OPC: Mask optimiza-tion with lithography-guided generative adversarial nets, IEEE T. Comput. Aid. D. 39(10) (2020), 2822-2834. |
| [41] | H. YANG, Z. LI, K. SASTRY, S. MUKHOPADHYAY, M. KILGARD, A. ANANDKUMAR, B. KHAILANY, V. SINGH, AND H. REN, Generic lithography modeling with dual-band optics-inspired neural networks, arXiv:2203.08616 (2022). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
