Photons in the presence of parabolic mirrors

TL;DR

This paper provides a detailed vectorial analysis of electromagnetic fields near parabolic mirrors, emphasizing symmetry use, polarization, and phase angular momenta, and quantizes the field in terms of elementary modes, including the case of ideal parabolic mirrors.

Contribution

It introduces a novel vectorial framework for electromagnetic fields in parabolic geometries, explicitly quantizes the modes, and clarifies the role of symmetries and angular momenta in this context.

Findings

Explicit expressions for electromagnetic modes near parabolic mirrors.

Quantization of the electromagnetic field in parabolic geometries.

Modes in this work generalize previously reported modes.

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.