Introduction

1. Transformation Optics based Local Mesh Refinement for solving Maxwell's Equations Jinjie Liu1, Moysey Brio2, Jerome V. Moloney2 1Department of Mathematical Sciences, Delaware State University, Dover, DE 19901 2Arizona Center for Mathematical Sciences, The University of Arizona, Tucson, AZ 85721 Numerical Examples Introduction TO Based Maxwell Solver One of the major difficulties of the explicit finite difference method (such as the FDTD [1,2]) is the computational cost of resolving small structures. Current subgridding and adaptive mesh refinement FDTD methods often suffer from the late time instability problems [3]. Recently, transformation optics (TO) [4,5] has been applied to design the metamaterials, with many interesting applications, such as the invisibility cloak [5,6]. In this work, we propose a novel local mesh refinement algorithm based on the use of the transformation optics. The new local mesh refinement algorithm applies transformation optics to enlarge a small region and then numerically solves the new anisotropic Maxwell's equations as in [6,7]. The newly introduced TO-FDTD method (that uses invariance of the Maxwell's equations and transforms both dependent and independent variables), in comparison to the subgridding and AMR methods (that only transform the dependent variables), is proven to be stable. In addition, our method does not have coarse-fine mesh interfaces, so that other errors such as the dispersion error due to the coarse-fine mesh interface is avoided. Step 2:We apply a numerical method to solve the transformed Maxwell's equations in the new computational domain. For example, the anisotropic FDTD method proposed in [6,7]. If some eigenvalues of the material matrices and are less than one, we can map them to dispersive media as follows. For example, if, and , we have where and . As a result, the new material matrix has all eigenvalues greater than 1. Step 3:The solution is transformed back to the original coordinate system to obtain the solution of the original problem: and The TO based algorithm can be directly extended to dispersive media. • Example 3: Metallic Bowtie with small gap = 1 nm • TO-FDTD result (160x160 mesh) • If use FDTD with Δ = 0.1 nm, mesh size is 8000 x 8000, so the computational cost is unattainable on a regular laptop computer. • For TO-FDTD simulation, a small region enclosing the gap can be enlarged 100 times, so that the 1 nm gap can be resolved by a coarse mesh with Δ= 10 nm. • CPU time for TO-FDTD simulation (160x160 mesh): 10 seconds. TO Based Maxwell Solver Conclusion The transformation optics based local mesh refinement algorithm: Step 1: We enlarge a small region using transformation optics [4,5]. After the transformation, new Maxwell's equations with anisotropic permittivity and permeability are obtained in the new computational domain. where ,, and is the Jacobian matrix from the original coordinates to the new coordinates . and . For example, to transform an annulus to another annulus, we can try the following three transformations: , where is the coordinate stretching function. Combining the three transformations, we get the Jacobianmatrix from to : The resulting inner circular region is enlarged so it can be resolved by larger grid cells. In this work, we propose a novel local mesh refinement algorithm based on the use of transformation optics. The new algorithm is an alternative way to achieve local mesh refinement. It applies transformation optics to enlarge a small region and then we numerically solve the new anisotropic Maxwell's equations in the transformed space by an anisotropic FDTD method. In comparison to the subgridding and AMR methods, one of the major advantages of our method is the proven stability property of the numerical methods applied to the anisotropic Maxwell equations.  Our method does not have coarse-fine mesh interfaces, so that other error such as dispersion error due to the coarse-fine mesh interface is avoided.  The TO method is a natural stable, efficient and robust alternative to traditional subgridding methods that suffer from late-time instabilities. Numerical Examples Example 1: Cylinder Scattering problem (TM plane wave λ = 800 nm, R = 50 nm, ϵ = 12) FDTD (Δ = 20 nm) FDTD (Δ= 10 nm) TO-FDTD (Δ= 20 nm) Example 2: Metallic Bowtie FDTD (400x400 mesh) FDTD (800x800 mesh) * Bowtie gap = 10 nm * Material: gold modeled by Drude dispersive medium * A small region enclosing the gap is enlarged 10 times TO-FDTD (80x80 mesh) Line plots of (b) and (c) CPU time comparison: References [1] K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media”. IEEE T. Antenn. Propag., 1966, 14, 302-307 (1966). [2] A. Taflove and S. Hagness, “Computational Electrodynamics: The Finite-Difference Time-Domain Method”, 3rd  ed.  Norwood, MA: Artech House (2005). [3] J. P. Berenger, The Huygens subgridding for the numerical solution of the Maxwell equations, J. Comput. Phys. 230, 5635-5659 (2011) . [4] U. Leonhardt, T. G. Philbin, Transformation optics and the geometry of light, Prog. Opt. 53, 69-152 (2009). [5] J. B. Pendry, D. Schurig, D. R. Smith, Controlling electromagnetic fields, Science312, 1780-1782 (2006). [6] U. Leonhardt, Optical conformal mapping, Science 312, 1777-1779 (2006). [7] G. R. Werner, J. R. Cary, A stable FDTD algorithm for non-diagonal, anisotropic dielectrics, J. Comput. Phys. 226, 1085-1101 (2007). [8] G. R. Werner, C. A. Bauer, J. R. Cary, A more accurate, stable, FDTD algorithm for electromagnetics in anisotropic dielectrics, J. Comput. Phys. In press. arXiv:1212.4857. Acknowledgements This work is supported by the AFOSR under Grant numbers FA9550-10-1-0127 and FA9550-10-1-0064. JL was also supported in part by NSF Grant HRD-1242067 and US ARO Grant W911NF-11-2-0046. MB was also supported in part by US AFOSR BRI Grant FA9550-12-1-0482.