Time-Frequency Integrals and the Stationary Phase Method in Problems of Waves Propagation from Moving Sources

TL;DR

This paper develops a unified time-frequency integral and stationary phase approach to analyze electromagnetic waves from moving sources in dispersive media, revealing nonlinear Doppler shifts and spectral behaviors in complex materials.

Contribution

It introduces a novel analytical method combining time-frequency integrals and stationary phase for dispersive media, providing explicit expressions and insights into wave effects and Doppler shifts.

Findings

Explicit expressions for field amplitudes and eigenfrequencies.

Nonlinear Doppler frequency shift in dispersive metamaterials.

Spectral analysis showing opposite directions of phase and group velocities.

Abstract

The time-frequency integrals and the two-dimensional stationary phase method are applied to study the electromagnetic waves radiated by moving modulated sources in dispersive media. We show that such unified approach leads to explicit expressions for the field amplitudes and simple relations for the field eigenfrequencies and the retardation time that become the coupled variables. The main features of the technique are illustrated by examples of the moving source fields in the plasma and the Cherenkov radiation. It is emphasized that the deeper insight to the wave effects in dispersive case already requires the explicit formulation of the dispersive material model. As the advanced application we have considered the Doppler frequency shift in a complex single-resonant dispersive metamaterial (Lorenz) model where in some frequency ranges the negativity of the real part of the refraction index can be reached. We have demonstrated that in dispersive case the Doppler frequency shift acquires a nonlinear dependence on the modulating frequency of the radiated particle. The detailed frequency dependence of such a shift and spectral behavior of phase and group velocities (that have the opposite directions) are studied numerically.