Certain particular solutions of the quasi-optics equation

The dependences of the refractive index on the coordinates, for which nontrivial symmetry groups of the two-dimensional quasi-optics equation exist, are considered. All cases where the Lie-symmetry algebra is two-dimensional are investigated. In such cases, the quasi-optics equation reduces to an ordinary differential equation. Several particular solutions of this equation are obtained. It is shown that all the particular solutions can be divided into three types: plain waves, source functions, and Gaussian beams. The distribution of the amplitude then obeys Gauss's law only for three cases: a transversely homogeneous medium, an optical wedge, and a distributed lens.