Derivation of Ray Optics Equations in Photonic Crystals Via a Semiclassical Limit

In this work we present a novel approach to the ray optics limit: we rewrite the dynamical Maxwell equations in Schr\"odinger form and prove Egorov-type theorems, a robust semiclassical technique. We implement this scheme for periodic light conductors, photonic crystals, thereby making the quantum-light analogy between semiclassics for the Bloch electron and ray optics in photonic crystals rigorous. Our main results, Theorems 3.3 and 4.1, give a ray optics limit for quadratic observables and, among others, apply to local averages of energy density, the Poynting vector and the Maxwell stress tensor. Ours is the first rigorous derivation of ray optics equations which include all sub-leading order terms, some of which are also new to the physics literature. While the ray optics limit we prove initially (Theorem 3.3) applies to photonic crystals of any topological class, we also consider the ray optics limit for real electromagnetic fields propagating in non-gyrotropic photonic crystals. Such an extension is non-trivial, because the ray optic limit for real fields is necessarily a multiband problem.