Internal transformations and internal symmetries in linear photonic systems

Abstract

Lorentz reciprocity, energy conservation, and time-reversal symmetry are three important global constraints of Maxwell's equations. Unlike time-reversal symmetry, Lorentz reciprocity and energy conservation usually are not considered as symmetries, i.e., they are not associated with operators. In this paper, we provide a unified treatment of these three global constraints from a perspective of internal symmetry. We define operators of transformations associated with each of these constraints, referred to as internal transformations. When Maxwell's equations are written as a linear system of equations, these internal transformations correspond to the operations of transpose, conjugate transpose, and conjugate of the system matrix, respectively. We show the three global constraints naturally follow from three fundamental identities of linear systems under the three matrix operations. We discuss the properties of electromagnetic fields and scattering matrices associated with these internal transformations. These internal transformations form the Klein four-group ${V}_{4}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$, and the internal symmetry group of any photonic structure corresponds to one of the five subgroups of ${V}_{4}$. Our paper provides a theoretical foundation for further exploration of symmetries in photonic systems.