Mode mapping in paraxial lossless optics

Under the Collins transformation, the orthonormal set of Hermite-Gaussian modes maps into an orthonormal set of beams with a Gaussian envelope. Among them are the Laguerre-Gaussian beams and the recently introduced Hermite-Laguerre-Gaussian beams. Compact expressions for the complex field amplitudes of these modes are derived. The obtained results are useful for the description of light propagation through first-order optical systems, for the solution of the phase retrieval problem by non-interferometric techniques, and for the design of mode converters and information processing systems.