Integral transform for gradient-index media with linear dependence of dielectric permittivity on transverse cartesian coordinates

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