Path Integrals for Photonic Crystals

Abstract

We develop a path integrals approach for analyzing stationary light propagation appropriate for photonic crystals. The hermitian form of the stationary Maxwell equations is transformed into a quantum mechanical problem of a spin 1 particle with spin-orbit coupling and position dependent mass. After appropriate ordering several path integral representations of a solution are constructed. One leaves the propagation of polarization degrees of freedom in an operator form integrated over paths in coordinate space. The use of spin 1 coherent states allows to represent this part as a path integral over such states. Finally a path integral in transversal momentum space with explicit transversality enforced at every time slice is also given. As an example the geometrical optics limit is discussed and the ray equation is recovered together with the Rytov rotation of the polarization vector.