# American Institute of Mathematical Sciences

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

 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.
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
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##### 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.  doi: 10.1137/S0036142901384162.  Google Scholar [2] G. A. Baker, Finite element methods for elliptic equations using nonconforming elements,, Math. Comput., 31 (1977), 45.  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.   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.   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.   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., 6 (2007).   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.  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.  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.  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.  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., 5 (2006).   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.  doi: 10.1109/66.475184.  Google Scholar
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