American Institute of Mathematical Sciences

Advanced
March  2011, 15(2): 401-415. doi: 10.3934/dcdsb.2011.15.401

Accurate simulations of 2-D phase shift masks with a generalized discontinuous Galerkin (GDG) method

Xia Ji 1, and  Wei Cai 2,

1. 

LSEC, Institute of Computational Mathematics, Chinese Academy of Science, Beijing 100190, China
2. 

Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, United States

Received  November 2009 Revised  March 2010 Published  December 2010

In this paper, we apply a newly developed generalized discontinuous Galerkin (GDG) method for rigorous simulations of 2-D phase shift masks (PSM). The main advantage of the GDG method is its accurate treatment of jumps in solutions using the Dirac $\delta$ generalized functions as source terms of partial differential equations. The scattering problem of the PSM is cast with a total field/scattering field formulation while the GDG method is used to handle the inhomogeneous jump conditions between the total and scattering fields along the physical and perfectly matched layer (PML) interfaces. Numerical results demonstrate the high order accuracy of the GDG method and its capability of handling the non-periodic structures such as optical images near mask edges.
Keywords: Generalized discontinuous Galerkin method (GDG), discontinuous Galerkin method (DG), phase shift masks (PSM), perfectly matched layer (PML)..
Mathematics Subject Classification: Primary: 35Q60; Secondary: 35J0.
Citation: Xia Ji, Wei Cai. Accurate simulations of 2-D phase shift masks with a generalized discontinuous Galerkin (GDG) method. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 401-415. doi: 10.3934/dcdsb.2011.15.401
References:
[1]

D. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, United analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162.  Google Scholar

[2]

G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comput., 31 (1977), 45-59. doi: 10.1090/S0025-5718-1977-0431742-5.  Google Scholar

[3]

S. Burger and R. Kohle, Benchmark of FEM, waveguide and FDTD algorithms for rigorous mask simulation, Proc. SPIE, 5992 (2005), 368-379. Google Scholar

[4]

W. C. Chew, "Waves and Fields in Inhomogeneous Media," New York: Van Nostrand Reinhold, 1999. Google Scholar

[5]

W. C. Chew and W. H. Weedon, A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates, IEEE Microwave Guided Wave Lett., 7 (1994), 599-604. Google Scholar

[6]

B. Cockburn, S. Hou and C. W. Shu, Tvb Runge kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comput., 54 (1990), 545-581.  Google Scholar

[7]

A. Erdmann and P. Evanschitzky, Rigorous electromagnetic field mask modeling and related lithographic effcts in the low k1 and ultrahigh numerical aperture regime, J. Microlith., Microfab., Microsyst., 6 (2007), 031002. Google Scholar

[8]

K. Fan, W. Cai and X. Ji, A generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions, J. Comput. Phys., 227 (2008), 2387-2410. doi: 10.1016/j.jcp.2007.10.023.  Google Scholar

[9]

W. Lee and F. L. Degertekin, Rigorous coupled-wave analysis of multilayered grating structures, J. Lightwave Technol., 22 (2004), 2359-2363. doi: 10.1109/JLT.2004.833278.  Google Scholar

[10]

M. D. Levenson, N. S. Viswanathan and R. A. Simpson, Improving resolution in photolithography with a phase-shifting mask, IEEE Trans. on Electron Devices, 29 (1982), 1828-1836. doi: 10.1109/T-ED.1982.21037.  Google Scholar

[11]

K. D. Lucas, H. Tanabe and A. J. Strojwas, Efficient and rigorous three-dimensional model for optical lithography simulation, J. Opt. Soc. Am., 13 (1996), 2187-2199. doi: 10.1364/JOSAA.13.002187.  Google Scholar

[12]

T. Sato and A. Endo, Impact of polarization for an attenuated phase shift mask with ArF hyper-numerical aperture lithography, J. Microlith., Microfab., Microsyst., 5 (2006), 043001. Google Scholar

[13]

A. Taflove and S. C. Hagness, "Computational Electromagnetics: The Finite-Difference Time-Domain Method," 2nd, edition, ().   Google Scholar

[14]

A. K. Wong and A. R. Neureuther, Rigorous three-dimensional time-domain finite-difference electromagnetic simulation for photolithographic applications, IEEE Trans. on Semiconductor Manufacturing, 8 (1995), 419-431. doi: 10.1109/66.475184.  Google Scholar

show all references

References:
[1]

D. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, United analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162.  Google Scholar

[2]

G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comput., 31 (1977), 45-59. doi: 10.1090/S0025-5718-1977-0431742-5.  Google Scholar

[3]

S. Burger and R. Kohle, Benchmark of FEM, waveguide and FDTD algorithms for rigorous mask simulation, Proc. SPIE, 5992 (2005), 368-379. Google Scholar

[4]

W. C. Chew, "Waves and Fields in Inhomogeneous Media," New York: Van Nostrand Reinhold, 1999. Google Scholar

[5]

W. C. Chew and W. H. Weedon, A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates, IEEE Microwave Guided Wave Lett., 7 (1994), 599-604. Google Scholar

[6]

B. Cockburn, S. Hou and C. W. Shu, Tvb Runge kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comput., 54 (1990), 545-581.  Google Scholar

[7]

A. Erdmann and P. Evanschitzky, Rigorous electromagnetic field mask modeling and related lithographic effcts in the low k1 and ultrahigh numerical aperture regime, J. Microlith., Microfab., Microsyst., 6 (2007), 031002. Google Scholar

[8]

K. Fan, W. Cai and X. Ji, A generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions, J. Comput. Phys., 227 (2008), 2387-2410. doi: 10.1016/j.jcp.2007.10.023.  Google Scholar

[9]

W. Lee and F. L. Degertekin, Rigorous coupled-wave analysis of multilayered grating structures, J. Lightwave Technol., 22 (2004), 2359-2363. doi: 10.1109/JLT.2004.833278.  Google Scholar

[10]

M. D. Levenson, N. S. Viswanathan and R. A. Simpson, Improving resolution in photolithography with a phase-shifting mask, IEEE Trans. on Electron Devices, 29 (1982), 1828-1836. doi: 10.1109/T-ED.1982.21037.  Google Scholar

[11]

K. D. Lucas, H. Tanabe and A. J. Strojwas, Efficient and rigorous three-dimensional model for optical lithography simulation, J. Opt. Soc. Am., 13 (1996), 2187-2199. doi: 10.1364/JOSAA.13.002187.  Google Scholar

[12]

T. Sato and A. Endo, Impact of polarization for an attenuated phase shift mask with ArF hyper-numerical aperture lithography, J. Microlith., Microfab., Microsyst., 5 (2006), 043001. Google Scholar

[13]

A. Taflove and S. C. Hagness, "Computational Electromagnetics: The Finite-Difference Time-Domain Method," 2nd, edition, ().   Google Scholar

[14]

A. K. Wong and A. R. Neureuther, Rigorous three-dimensional time-domain finite-difference electromagnetic simulation for photolithographic applications, IEEE Trans. on Semiconductor Manufacturing, 8 (1995), 419-431. doi: 10.1109/66.475184.  Google Scholar

[1]

Zhiming Chen, Chao Liang, Xueshuang Xiang. An anisotropic perfectly matched layer method for Helmholtz scattering problems with discontinuous wave number. Inverse Problems & Imaging, 2013, 7 (3) : 663-678. doi: 10.3934/ipi.2013.7.663
[2]

Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817
[3]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319
[4]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[5]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139
[6]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078
[7]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29 (3) : 2375-2389. doi: 10.3934/era.2020120
[8]

Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104
[9]

ShinJa Jeong, Mi-Young Kim. Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems. Electronic Research Archive, 2021, 29 (2) : 1991-2006. doi: 10.3934/era.2020101
[10]

Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure & Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719
[11]

Qingjie Hu, Zhihao Ge, Yinnian He. Discontinuous Galerkin method for the Helmholtz transmission problem in two-level homogeneous media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2923-2948. doi: 10.3934/dcdsb.2020046
[12]

Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033
[13]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489
[14]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185
[15]

Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677
[16]

E. Fossas, J. M. Olm. Galerkin method and approximate tracking in a non-minimum phase bilinear system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 53-76. doi: 10.3934/dcdsb.2007.7.53
[17]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[18]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
[19]

Yulong Xing, Ching-Shan Chou, Chi-Wang Shu. Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Problems & Imaging, 2013, 7 (3) : 967-986. doi: 10.3934/ipi.2013.7.967
[20]

Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021077

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences