Asymptotics and energy estimates for electromagnetic pulses in dispersive media

Abstract

We examine electromagnetic pulse propagation in anomalously dispersive media, using the Debye model as an example. Short-pulse, long-pulse, short-time, and long-time approximations and amplitude rate-of-decay estimates are derived with asymptotic methods. We also study the following problem: If we know only the peak amplitude and the energy density of an incident pulse, what can be said about the amplitude of the propagated pulse? We provide tight upper and lower bounds for the propagated amplitude, which may be useful in controlling the electromagnetic interference or the damage produced in dispersive media. We explain a factor-of-nine effect in the speed of pulses in a Debye model for water, which seems to have been previously unnoticed, and we also explain some observations from experimental studies of pulse propagation in biological materials. Finally, we propose some guidelines for sample size in transmission time-domain spectroscopy studies of dielectrics. Most of our results are easily extended to multiple space dimensions.