Paraxial theory of electromagnetic waves in plane inhomogeneous media

The problem of electromagnetic waves propagating in inhomogeneous media is formulated within the paraxial approximation. The analysis is restricted to a medium with a plane and smooth inhomogeneity. The general form of a Gaussian beam is obtained in terms of the permittivity and permeability of the medium. Particular attention is paid to the case of internal reflection, where a short-wavelength approximation breaks down; a possible treatment of the problem is proposed. A simple model is worked out to illustrate the formalism.