Wave Hierarchies for Propagation in Dispersive Electromagnetic Media

Abstract

In this talk we introduce the concept of a wave hierarchy whereas waves of different type (e.g., non-dispersive, dispersive, diffusive, higher-order dispersive and diffusive) coexist in a spatial domain and each order manifests itself in mutually exclusive regions by appearing as dominant over the others in a sequence which depends on the material properties. In this talk we will derive the wave hierarchy governing the propagation of arbitrary electromagnetic pulses in dispersive media whose dielectric properties are modeled by a conduction current mechanism, and by two types of polarization current mechanisms. A single partial differential equation will be shown to govern the evolution of the electric field. This single equation will exhibit 5 wave types, i.e., a hierarchy. Each wave type will be seen to be associated with a distinct speed and with a strength coefficient whose order of magnitude will determine when the associated wave order will dominate the response in the dielectric. Detailed results of the general procedure will be given in the Analysis section for a one relaxation Debye medium model. Our analysis will identify a “skin-depth” of length O(c ∞τmin) m for pulses incident on the air / dielectric interface, where τmin is the shortest relaxation time, and c ∞ is the infinite frequency phase velocity (the wavefront speed). In this short interval the pulse will be shown to travel with the wavefront speed, and to decay exponentially according to a telegrapher’s equation. Past this shallow depth (∼ O(10−3) m for a single relaxation Debye model of water) we will show that the main disturbance satisfies an advection-diffusion equation and that it travels with a subcharacteristic advection speed equal to the zero frequency phase velocity (∼ c ∞ /9 for the water model).