Ultra Wideband (UWB) Pulse Reflection from a Dispersive Medium Half Space

Abstract

Electromagnetic wave reflection from dispersive media has been a subject of interest to researchers for many years. The advent of ultra wideband (UWB) short pulse sources has recently attracted renewed interest in this aspect. Accurate modeling and improved physical understanding of pulse reflection from dispersive media is crucial in a number of applications, including optical waveguides, UWB radar, ground penetrating radar, UWB biological effects, stealth technology and remote sensing. Numerous researchers have demonstrated that Lorentz, Debye and Cole-Cole models can be used to accurately predict dispersive properties of many media. In the seminal work of Sommerfeld (Sommerfeld, 1914) and in the subsequent refinements of Oughstun and Sherman (Oughstun & Sherman, 1988) (Oughstun & Sherman, 1989) (Oughstun & Sherman, 1990), the investigations have focused on the Lorentz material, which is a good model for many materials encountered in optics and engineering. The reflection of a short pulse by a Lorentz medium has been considered for TE (transverse electric) polarization by Gray (Gray, 1980) and for TM (transverse magnetic) polarization by Stanic et al (Stanic et al., 1991). In each of these studies, the authors find the impulse response of the reflected field by calculating the inverse transform of the frequency domain reflection coefficient as an infinite series of fractional order Bessel functions. Although this gives a convenient analytical result, the series form provides little insight into the behavior of the reflected field waveform. Cossman et al have presented a compact form for both TE (Cossman et al., 2006) and TM (Cossman et al., 2007) reflection coefficients, which provide useful intuition about the response of a Lorentz medium half space. However, the mathematical derivations are lengthy and the solutions involve exponential and modified Bessel functions and require convolution operation to evaluate. Particularly, because of the greater complexity of the TM frequency domain reflection coefficient, a more involved process is needed, including the introduction of a term that does not appear in the TE case. Under some conditions this term is noncausal, although the final expression for the impulse response is causal. Moreover, when the incident angle is equal to 450, the general expressions cannot be used directly due to singularities, and a special form of TM time domain reflection coefficient is separately achieved. The use of short pulses to probe materials has prompted the study of the reflection of transient waves from material half space of other types. The Debye model (Debye, 1945) is