Scattering of a plane wave by an inhomogeneous 1D dielectric layer with gradient refractive index

We propose a new method for calculating reflection and transmission coefficients for an arbitrarily polarized electromagnetic plane wave incident on a one-dimensional dielectric medium of finite thickness and with dielectric permittivity being an arbitrary continuous function of the coordinate. We have shown that the problem of plane wave scattering by an inhomogeneous layer is reduced to a system of first order differential equations that contain the derivative of the refractive index or dielectric permittivity of the layer, which can be used, for example, when searching for an analytical solution. This method also makes it easy to obtain the distribution of the field strength within the layer. The reflection spectra and field distribution obtained using this method were compared with the analytical solution based on Mathieu functions.