Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 096, 21 pages      arXiv:1212.1971      http://dx.doi.org/10.3842/SIGMA.2012.096
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

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

Gennadiy Burlak a and Vladimir Rabinovich b
a) Centro de Investigación en Ingeniería y Ciencias Aplicadas, Universidad Autónoma del Estado de Morelos, Cuernavaca, Mor. México
b) National Polytechnic Institute, ESIME Zacatenco, D.F. México

Received July 29, 2012, in final form December 02, 2012; Published online December 10, 2012

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.

Key words: dispersive media; two-dimensional stationary phase method; electromagnetic wave; moving modulated source.

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