Phase matching of electromagnetic wave on moving interface

Abstract

Due to the boundary conditions of electromagnetic fields and phase matching of electro magnetic waves on interface being the basis to drive the Snell’s laws and Fresnel’s laws, they are also crucial for the analysis of electromagnetic wave propagation in a moving medium. There are mainly two methods to derive the boundary conditions of electromagnetic fields on moving interface. One of them is to use the kinematic integral form, yet this method is based on the classical time-space, and the other is based on the relativistic transformation, the boundary conditions are derived from the scaling effect with geometric method, or from the principle of relativity directly. However, the first one has a form the same as the form obtained by using the kinematic integral form, while the second one obtains a different form. At the same time, the phase matching of electromagnetic wave on moving interface is only discussed by Galileo transformation, however this is unreasonable, because of the relativistic effect cannot be ignored here. Therefore, it is necessary to reexamine the boundary conditions of electromagnetic fields and phase matching of electromagnetic wave on moving interface. Herein, firstly, the relativistic transformation formula of the unit normal vector of moving surface is derived from the surface equation and principle of relativity. Secondly, the boundary conditions of electromagnetic fields on moving interface are given based on the relativistic transformation formula and the non-relativistic transformation formula of the unit normal vector and electromagnetic fields, which show that the boundary conditions of electromagnetic fields on moving interface under the relativistic case and the non-relativistic case have the same form. This is not accidental but definite, because the change of flux of electromagnetic fields, like the change of magnetic flux, from the induction of electromagnetic filed is the same as that from the variation of surface element. Thirdly, the phase matching of electromagnetic wave on moving interface is given based on the relativistic transformation formula of the unit normal vector and the phase matching of electromagnetic wave on resting interface. In the problem of light incident on a homogeneous medium moving at a constant velocity in vacuum or air, using the phase matching of electromagnetic wave on moving interface, the same results can be easily obtained through other methods. The discussion in this study belongs to classical electrodynamics with no quantum effects considered, but the results will provide some conveniences for theoretically analyzing electromagnetic communication, remote sensing and telemetering.