Generalized light rays and relativistic front dynamics : theory and application to paraxial optics

Abstract

We recount here briefly the main advantages of using the front form for paraxial wave problems. It leads to a clean separation of the space-time variables into the combinations on each of which the amplitude has a characteristically different degree of dependence. Here it must be mentioned that while the usual terms monochromatic and quasimonochromatic denote harmonic or near-harmonic dependence on the physical time, in the front form the analogous terms henochromatic and quasihenochromatic refer to dependence on the “spatial” variable. Factoring away the exponential ei?te^{i\omega t}ei?t is replaced here by factoring away eikze^{ikz}eikz; in the instant form we then deal with the evolution of the residual wavefunction with respect to xxx, and this is governed by the radical. In the front form, on the other hand, the evolution of the residual wavefunction is with respect to ttt, and that is given exactly, free of radicals. But this must be tempered with the following remark, amplified by the analysis of Section IV.2: In order to obtain a physically cogent description of the action of systems of lenses on paraxial beams in the front language, we have to connect up the properties of henochromatic beams with ordinary monochromatic ones in some way. The identification of the transverse coordinates x?x_\perpx?? with those that appear in the Galilei algebra will be the basis of our treatment of vector waves in the succeeding chapter. This will mean that the metaplectic group retains its significance for the full Maxwell equations.