Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space

The problem of transmission and reflection of TE electromagnetic X-waves normally incident on a lossy dispersive half-space is investigated. Such X-waves are represented as a superposition of pulsed plane waves in all azimuth directions and tilted by a constant angle with respect to the X-wave propagation direction. The problem of transmission and reflection of pulsed TE electromagnetic plane waves in the presence of a lossy dispersive half-space is formulated in the frequency domain and is converted to the time domain via Fourier transformation. The transmission coefficient, the reflection coefficient, and the longitudinal propagation function inside the lossy dispersive medium are frequency dependent. A new efficient and accurate method is used to evaluate the inverse Fourier transform where the asymptotic forms of the frequency dependent parameters are obtained, and the corresponding fields are evaluated analytically. The remaining part of the integrand is a fast decaying function of frequency that can be approximated by a series of exponential functions via Prony's method. The Fourier transform of such exponential terms is evaluated analytically in closed form.