FDTD formulation for dispersive chiral media analysis using Z-transform

A finite difference time-domain (FDTD) scattered-field formulation for dispersive chiral media is developed and presented. The FDTD formulation uses the Z transform method to model the frequency dependent dispersive nature of permittivity, permeability and chirality as well. The permittivity and permeability are assumed to follow the Lorentz model whereas the chirality is assumed to follow the Condon model. The formulation is developed for three-dimensional electromagnetic applications. Results of this formulation are presented for the co-polarization and cross-polarization of the reflected and transmitted waves from a chiral slab and for the scattered field from a chiral sphere, a chiral cube and a finite length chiral cylinder, due to normal incidence of a plane wave. Validation of the formulation is performed by comparing the results with those based on the exact solutions and those obtained from method of moments solutions. Introduction The electromagnetic wave propagation in chiral and bi-isotropic media has been modeled by the FDTD technique recently in various studies. These studies are based on various assumptions of constitutive relations of bi-isotropic and chiral media. Akyurtlu et al. extensively studied modeling bianisotropic media and its subclasses (e.g. biisotropic, chiral, isotropic) using the FDTD method. In [1] and [2], they incorporated the dispersive nature of permittivity, permeability and chirality in the FDTD formulation, thus providing a full dispersive model; frequency dependence of permittivity and permeability follows the Lorentzian model, wheras chirality follows the Condon model. Their studies are based on decomposing the electric and magnetic fields in the medium into wavefields; with which they treated the chiral medium problem as the sum of two problems in associated isotropic media. They verified the validity of their formulations by providing results for only one-dimensional problems. In this study, a dispersive chiral FDTD formulation is developed based on a direct implementation of the coupled chiral constitutive relations incorporated into Maxwell’s equations, unlike the ones presented in [1] and [2], which use the decoupled equations. The scattering of electromagnetic plane wave from three-dimensional dispersive chiral scatterers as well as the reflection and transmission from one-dimensional slab are presented. The dispersion of permittivity and permeability follows the Lorentzian model, wheras chirality follows the Condon model. These models are incorporated into the FDTD formulation using the Z transform method. The FDTD formulation is used to calculate transient reflected and transmitted fields from a chiral slab due to the incidence of a Gaussian TEM field. Furthermore, the formulation is used to calculate co-polarization and cross-polarization of the reflected and transmitted waves from a chiral slab and a chiral sphere due to an incident plane wave. Very good agreements are observed while comparing the numerical results based on these new formulations and the corresponding values based on the exact solutions for these canonical problems. Furthermore, scattering from non-canonical objects such as a chiral cube and a finite chiral cylinder have been 0-7803-9544-1/05/$20.00 ©2005 IEEE 8 calculated and the results are compared with those based on method of moments solutions of the same problems. With this presented formulation, the analysis of dispersive composite structures made of combinations of dielectric, magnetic and chiral materials is possible. Dispersive chiral FDTD formulation using the Z transform method The constitutive equations for chiral media in frequency domain can be written as ( ) ( ) ( ) ( ) ( ) o o D E j H ω ε ω ω κ ω ε μ ω = − (1.a) ( ) ( ) ( ) ( ) ( ) o o B H j E ω μ ω ω κ ω ε μ ω = + (1.b) where ( ) ε ω , ( ) μ ω and ( ) κ ω are frequency dependent permittivity, permeability and chirality parameters, respectively. In most of the cases, the Lorentz model is used to describe the dispersive nature of permittivity and permeability, and the Condon model [3] is used for chirality [1-2,4]. These are given in the following forms 2