International Scholarly Research Network ISRN Optics Volume 2012, Article ID 536209, 7 pages doi:10.5402/2012/536209 Research Article FDTD Modeling of a Cloak with a Nondiagonal Permittivity Tensor Naoki Okada and James B. Cole Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan Correspondence should be addressed to Naoki Okada, okada@cavelab.cs.tsukuba.ac.jp Received 13 February 2012; Accepted 27 March 2012 Academic Editors: A. Danner, M. Midrio, and A. Tervonen Copyright © 2012 N. Okada and J. B. Cole. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We demonstrate a finite-difference time-domain (FDTD) modeling of a cloak with a nondiagonal permittivity tensor. Numerical instability due to material anisotropies is avoided by mapping the eigenvalues of the material parameters to a dispersion model. Our approach is implemented for an elliptic-cylindrical cloak in two dimensions. Numerical simulations demonstrated the stable calculation and cloaking performance of the elliptic-cylindrical cloak. 1. Introduction An optical cloak enables objects to be concealed from electromagnetic detection. Pendry et al. developed a method to design cloaks via coordinate transformations [1]. The coordinate transformation is such that light is guided around the cloak region. Material parameters (permittivity and permeability) can be obtained in the transformed coordinate system and put into Maxwell’s equations. This approach enables one to design not only cloaks but also other metamaterials that can manipulate light flow. For example, concentrators [2], rotation coatings [3], polarization con- trollers [4–6], waveguides [7–11], wave shape conversion [12], object illusions [13–15], and optical black holes [16, 17] have been designed. However, not many metamaterials have been realized in the optical region [18–25], because material parameters given by coordinate transformations have complicated anisotropies. Numerical simulations are useful to analyze complicated metamaterial structures. In this paper, we present a finite- difference time-domain (FDTD) analysis of a cloak. The FDTD method has gained popularity for several reasons: it is easy to implement, it works in the time domain, and its arbitrary shapes can be calculated [26–29]. FDTD modelings of cloaks with a diagonal (uniaxial) permittivity tensor have been demonstrated [30–38], but a cloak with a nondiagonal permittivity tensor has never been calculated by the FDTD method. The diagonal case can be stably calculated by mapping material parameters having values less than one to a dispersion model [31]. However, we found that mapping the nondiagonal elements to dispersion models causes the computation to diverge. In this paper, we analyze the numerical stability for a cloak with a nondiagonal permittivity tensor and derive the FDTD formulation. We apply our method to simulate light propagation in the vicinity of an elliptic-cylindrical cloak. To the best of authors’ knowledge, this is the first time that a cloak with a nondiagonal anisotropy has been calculated using the FDTD method. 2. Numerical Stability for Nondiagonal Permittivity Tensor In the stability analysis, we confirm that the FDTD method for a cloak with a diagonal permittivity tensor cannot directly be extended to the nondiagonal case. Under a coordinate transformation for a cloak [39], material parameters can be expressed as ε ij = μ ij =±  gg ij , (1) where ε ij is the relative permittivity, μ ij is the relative permea- bility, g ij is the metric tensor, and g = det g ij . Because ε ij , μ ij are constructed from the symmetric metric tensor g ij , they