Second-order and fourth-order difference algorithms for the modeling of electromagnetic wave propagation in temporally dispersive media

Abstract

Second-order and fourth-order difference algorithms are presented for electromagnetic wave propagation in temporally dispersive media. Three media types are considered: the cold plasma, the Debye dielectric and the Lorentz dielectric. To achieve second-order temporal accuracy, the algorithms are couched in leap-frog form. Fourth-order temporal accuracy is achieved by means of the four-stage Runge-Kutta integrator. With respect to the spatial differencing, either second-order or fourth-order central differences are used in conjunction with the Yee stencil. Standard Fourier-type analysis is invoked to quantify the errors of each algorithm, and dispersion and dissipation plots are provided. In addition, reflection coefficient data, as obtained from numerical simulation, is provided to give credence to the algorithms' utility. To appreciate the scattering mechanisms of each media type, scattered field data are also provided. From these plots one can see multiple scattering events and/or ringing effects. INTRODUCTION There exists a fundamental need to understand, quantify and predict the propagation characteristics of an electromagnetic (EM) wave within a dispersive material. For example, in hyperthermia applications, antennas are used to focus electromagnetic energy on a biological tissue or organism in order to render it innocuous. The ability to achieve the desired focus depends upon, in part, a rigorous understanding of the dispersive nature of the biological materials that surround the unwanted organism [1]. In the previous example, the dispersive mechanisms are associated with the atomic structure of water or other aqueous solutions. For such materials where permanent dipole moments are present, there exists a tendency for the dipoles to rotate and vibrate under an applied EM field, thus heating the solution and consequently damping the wave. Macroscopically, this damping is observed between the displacement vector and the electric field vecCopyright ©by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. tor in terms of a permittivity model ascribed by the name of Debye [2]. In addition to water, materials such as plasmas and optical dielectrics are also temporally dispersive. With respect to the latter, a simplified model of an electron bound elastically to its nuclei is postulated (attributed to Lorentz) to create a mathematical model much akin to the one for the vibrating spring. That is, the model contains acceleration, restoring and damping forces as well as a forcing term. Not surprisingly then, this material has the capability to resonate, which is an important phenomena to consider in fiber optic technologies. When the restoring force is absent, such as in an isotropic cold plasma, the media can no longer resonate. Even so, acceleration and damping forces still exist to modify the wave's phase velocity as it passes through the material. This effect is well known to the radio scientists, who exploit the ionosphere for VLF communication [3]As can be inferred from the previous paragraphs, nature provides us with numerous materials that temporally disperse the EM wave. In order to make better use of these materials, accurate and robust numerical algorithms are needed that can predict the wave interaction with the material structure. Although one could employ a convolutionbased method [4]-[6], a direct integration approach [7] or a semi-implicit scheme [8]-[9], all in conjunction with the Yee algorithm [10], we summarize and extend methodologies that are simple, robust, and memory parsimonious [11]-[13]. Specifically, this paper summarizes and postulates new numerical algorithms associated with isotropic plasmas, Debye dielectrics and Lorentzian materials. Since the dispersive mechanism for all these materials appears as a polarization current in Ampere's law, we need only to consider in detail the temporal discretization of the auxiliary equations that describe the polarization current. Not only does this approach lead to a system of equations that is easy to discretize numerically, it also yields a system of equations that is conveniently switchable between a total field formulation and a scattered field formulation. Both secondand fourth-order algorithms are presented. When second-order accuracy is required, conventional leap-frog integrators are employed. With respect to fourth-order accuracy, we present, where possible, a leap-frog scheme and discuss the scheme's attributes both positive and negative. It is the negative attributes, however, that motivate us to contemplate other temporal advancement methods, namely the class of Runge-Kutta integrators. GOVERNING EQUATIONS In their most general form, Maxwell's equations describe the interrelationship between four field constituents: the magnetic flux density B, the electric flux density D, the electric intensity E and the magnetic intensity H. In source-free media, these constituents must satisfy [2],