After the diagonalization, the FDTD method for diagonal cases can be applied. Since the FDTD stability depends on the eigenvalues of ? ? ? and ? ? ?, to analyze nondiagonal cases, we must first find the eigenvalues and diagonalize ? ? ? and ? ? ?. Where Δ ? is the time step, ℎ is the grid spacing, and ? = 1, 2, and 3 dimensions. The phase velocity of light in a material is given by ? = ? 0 / ? ( ? 0 = vacuum light speed), and the Courant-Friedrichs-Lewy (CFL) stability limit becomes Δ ? ≤ ? ℎ ? 0 √ ?, ( 3 ) Consequently, ? ? ?, ? ? ? have real eigenvalues with orthogonal eigenvectors and are thus diagonalizable. Because ? ? ?, ? ? ? are constructed from the symmetric metric tensor ? ? ?, they are symmetric. Where ? ? ? is the relative permittivity, ? ? ? is the relative permeability, ? ? ? is the metric tensor, and ? = d e t ? ? ?. Under a coordinate transformation for a cloak, material parameters can be expressed as 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. Numerical Stability for Nondiagonal Permittivity Tensor
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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. We apply our method to simulate light propagation in the vicinity of an elliptic-cylindrical cloak. In this paper, we analyze the numerical stability for a cloak with a nondiagonal permittivity tensor and derive the FDTD formulation. However, we found that mapping the nondiagonal elements to dispersion models causes the computation to diverge. The diagonal case can be stably calculated by mapping material parameters having values less than one to a dispersion model. FDTD modelings of cloaks with a diagonal (uniaxial) permittivity tensor have been demonstrated, but a cloak with a nondiagonal permittivity tensor has never been calculated by the FDTD method. 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. In this paper, we present a finite-difference time-domain (FDTD) analysis of a cloak. Numerical simulations are useful to analyze complicated metamaterial structures. However, not many metamaterials have been realized in the optical region, because material parameters given by coordinate transformations have complicated anisotropies. For example, concentrators, rotation coatings, polarization controllers, waveguides, wave shape conversion, object illusions, and optical black holes have been designed. This approach enables one to design not only cloaks but also other metamaterials that can manipulate light flow. Material parameters (permittivity and permeability) can be obtained in the transformed coordinate system and put into Maxwell’s equations. The coordinate transformation is such that light is guided around the cloak region.
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developed a method to design cloaks via coordinate transformations. IntroductionĪn optical cloak enables objects to be concealed from electromagnetic detection. Numerical simulations demonstrated the stable calculation and cloaking performance of the elliptic-cylindrical cloak. Our approach is implemented for anĮlliptic-cylindrical cloak in two dimensions. Numerical instability due to material anisotropies is avoided by mapping the eigenvalues of the material parameters to a dispersion model. We demonstrate a finite-difference time-domain (FDTD) modeling of a cloak with a nondiagonal permittivity tensor.