Circuit Analogs for Wave Propagation in Stratified Structures

Abstract

To simulate or design a reasonably large system, fast and simple models are necessary. To verify the design versus the specifications, more detailed (and costly) calculations can be performed and final adjustments made. In wave propagation problems, circuit analogs provide a powerful, yet simple, means of computing the desired response of the system, such as reflection or transmission coefficients. The reason circuit analog models are good for wave propagation problems, is that they are exact for one-dimensional wave propagation, regardless of whether we consider acoustic or electromagnetic waves. Typically, wave propagation through homogeneous media is modeled as a transmission line with propagation constant β and characteristic impedance Z, whereas obstacles such as thin sheets are modeled as lumped elements. If the sheets are lossless, the circuit models contain only reactive elements such as capacitors and inductors. Modeling complex wave propagation problems with circuit analogs was to a large extent developed in conjunction with the development of radar technology during the Second World War. Many of the results from this very productive era are collected in the Radiation Laboratory Series and related literature, in particular (Collin, 1991; 1992; Marcuvitz, 1951; Schwinger & Saxon, 1968). Further development has been provided by research on frequency selective structures (Munk, 2000; 2003). In recent years, the circuit analogs have even been used in an inverse fashion: by observing that wave propagation through a material with negative refractive index could be modeled as a transmission line with distributed series capacitance and shunt inductance, i.e., the dual of the standard transmission line, the most successful realization of negative refractive index material is actually made by synthesizing this kind of transmission line using lumped elements (Caloz & Itoh, 2004; Eleftheriades et al., 2002). This chapter is organized as follows. In Section 2 we show that propagation of electromagnetic waves in any material, regardless how complicated, boils down to an eigenvalue problem which can be solved analytically for isotropic media, and numerically for arbitrary media. From this eigenvalue problem, the propagation constant and characteristic impedance can be derived, which generates a transmission line model. In Section 3, we show how sheets with or without periodic patterns can be modeled as lumped elements connected by transmission lines representing propagation in the surrounding medium. The lumped elements can be given a firm definition and physical interpretation in the low frequency limit, and in Section 4 we show how these low frequency properties