Excitation of the classical electromagnetic field in a cavity containing a thin slab with a time-dependent conductivity

Abstract

An exact infinite set of coupled ordinary differential equations, describing the evolution of the modes of the classical electromagnetic field inside an ideal cavity, containing a thin slab with the time-dependent conductivity $\sigma(t)$ and dielectric permittivity $\varepsilon(t)$, is derived for the dispersion-less media. This problem is analyzed in connection with the attempts to simulate the so called Dynamical Casimir Effect in three-dimensional electromagnetic cavities, containing a thin semiconductor slab, periodically illuminated by strong laser pulses. Therefore it is assumed that functions $\sigma(t)$ and $\delta\varepsilon(t)=\varepsilon(t)-\varepsilon(0)$ are different from zero during short time intervals (pulses) only. The main goal is to find the conditions, under which the initial nonzero classical field could be amplified after a single pulse (or a series of pulses). Approximate solutions to the dynamical equations are obtained in the cases of "small" and "big" maximal values of the functions $\sigma(t)$ and $\delta\varepsilon(t)$. It is shown, that the single-mode approximation, used in the previous studies, can be justified in the case of "small" perturbations. But the initially excited field mode cannot be amplified in this case, if the laser pulses generate free carriers inside the slab. The amplification could be possible, in principle, for extremely high maximal values of conductivity and the concentration of free carries (the model of "almost ideal conductor"), created inside the slab, under the crucial condition, that the function $\delta\varepsilon(t)$ is {\em negative}. This result follows from a simple approximate analytical solution, and it is confirmed by exact numerical calculations. However, the evaluations show, that the necessary energy of laser pulses must be, probably, unrealistically high.