Conservative Difference Scheme for Maxwell Equations Describing a Nonstationary Nonlinear Response of Matter in a Semi-Classical Approximation

Abstract

A conservative difference scheme is constructed for the system of equations that describe, in the semi-classical approximation, resonance and nonresonance interaction of an electromagnetic pulse with matter. The conservativity of the scheme is proved. A cascade mechanism for the excitation of high energy levels in the medium under the impact of a narrow-spectrum electromagnetic pulse is demonstrated. In the cascade mechanism, if the spectrum does not contain a pulse of sufficiently high frequencies to trigger direct transitions from the ground state to high excited states, the molecules of the medium may still reach the high states through a series of successive transitions between nearer lying energy levels. This may lead to the appearance in the transmitted pulse of higher frequencies than those observed in the incident pulse. For a medium whose length is comparable with the spatial extent of the pulse, distortion of the transmitted pulse spectrum is observed. The absolute phase of the small-period pulse substantially affects the transformation of the pulse spectrum during the transmission through the matter. If the matter contains several close energy levels, the corresponding transition rates will depend on the absolute phase of the field. We demonstrate and explain the generation of the third harmonic during the interaction of the electromagnetic pulse with the medium due to the nonstationary and nonlinear character of the response.