Photons in the presence of parabolic mirrors

Abstract

We present a vectorial analysis of the behavior of the electromagnetic field in the presence of boundaries with parabolic geometry. The relevance of the use of symmetries to find explicit closed expressions for the electromagnetic fields is emphasized. Polarization and phase related angular momenta of light have an essential role in the proper definition of the generator $\mathfrak{A}_3$ of a symmetry transformation that distinguishes the parabolic geometry. Quantization of the electromagnetic field in terms of the resulting elementary modes is performed. The important case of a boundary defined by an ideal parabolic mirror is explicitly worked out. The presence of the mirror restricts the eigenvalues of $\mathfrak{A}_3$ available to the electric and magnetic fields of a given mode via compact expressions. Modes previously reported in the literature are particular cases of those described in this work.