Gaussian beam in linear gradient-index medium

We have obtained an integral transform describing paraxial propagation of a light beam in gradient-index medium with linear dependence of dielectric permittivity on transverse Cartesian coordinates. We have shown that propagation of light in such medium is equivalent to passing through the prism, propagating in homogeneous medium and again passing through the same prism. We have also shown that for the Gaussian beam, propagating in such medium, its center is being shifted along a parabola, its radius is coinciding with radius of the Gaussian beam in homogeneous medium.