Derivation of a matrix describing a rugate dielectric thin film

In optical coating calculations, a 2 x 2 matrix formalism is generally used to describe the effect of each layer on the electromagnetic field. In current applications, the layers are considered homogeneous, and, therefore, the matrix used is restricted to this particular case. In this paper, a 2 x 2 matrix describing an inhomogeneous dielectric thin film is derived from Maxwell's equations. The inhomogeneity is assumed to vary along the direction perpendicular to the film plane. No restriction is made on the amplitude of its variation. The matrix is illustrated in the case of a rugate filter designed according to Sossi's Fourier transform technique. An improved approximation for the Fourier transform technique is then introduced.