Fresnel's formulas in time domain

A time-spatial representation of Fresnel's formulas is obtained that permits description of the interaction of complex electromagnetic signals with a flat interface between dielectric media to be made in the time domain. This description is made by means of integral equations and includes the case when the boundary is nonstationary, for example, if it is created at some instant in time. The solutions of these equations give the transmitted and reflected signals by virtue of resolvent integral operators that take into account the structure of the field polarization. These three-dimensional (3-D) operators are derived for the case when the medium in one of the two half-spaces changes its permittivity and acquires a conductivity. The weight factors in the kernels of these operators are simply the analogues of Fresnel's formulae. The oblique incidence of a plane, monochromatic, wave on the flat boundary of the medium whose permittivity changes abruptly in time is also considered. It is shown that the transient accompanies the transformed field in contrast to the case of the normal incidence, when the transformed field only consists of the monochromatic secondary waves.