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

Abstract

In this work we present a novel approach to the ray optics limit: we rewrite the dynamical Maxwell equations in Schrödinger 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. One major conceptual difference between the two theories, though, is that electromagnetic fields are real, and hence, we need to add one step in the derivation to reduce it to a single-band problem. Our main results, Theorem 3.7 and Corollary 3.9, 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. The ray optics limit we prove applies to photonic crystals of any topological class.

Wave packet dynamics in slowly modulated photonic graphene

“…Moreover, the set of all Bloch modes {Φ b (x; k)} b≥1, k∈Ω * is complete in L 2 (R 2 ). That is for any f ∈ L 2 (R 2 ), 13) and the Parseval formula for the Bloch decomposition in L 2 (R 2 ) holds,…”

“…Amongst those, photonic graphene has attracted a lot of interest due to its potential applications and relatively simple experimental realizations [24,28,32,36]. To study electromagnetic waves in photonic graphene, we have to deal with Maxwell's equations in [12,13]. In a simple physical setting, for example, the propagation of transverse electronic fields can be described by the 2D wave equation (1.1).…”

Symmetry classification of topological photonic crystals

“…While the experimental confirmation [Wan+09] of Quantum Hall Effect for light settled the question that topological phenomena exist, very little effort was made to probe the quantitative validity of Haldane's Photonic Bulk-Edge Correspondence and derive it from first principles. Raghu and Haldane based their arguments on postulating ray optics equations which included an "anomalous velocity term"; however, the form of the subleading terms was a topic of discussion, one that was settled only recently with our rigorous work [DL17a] (see [DL17a, Section 5.2] for an in-depth discussion). Unfortunately, it is not possible to derive bulk-boundary correspondences purely on the basis of ray optics equations -not only because those govern the light dynamics in the bulk, but also because the semiclassical arguments with which one may show the quantization of the transverse conductivity (see e. g. [PST03a, Section 1]) do not generalize to electromagnetism due to the fundamental differences between both physical theories (cf.…”

The Schrödinger formalism of electromagnetism and other classical waves—How to make quantum-wave analogies rigorous

“…During our investigation of the ray optics limit [DL17a], we learnt that it was helpful to compare not just (i) states and (ii) the dynamical equation for the classical waves with the corresponding quantum system, but also include (iii) observables and (iv) additional information on the physics (such as typical states, regimes, length and time scales) in our considerations. This paper will initially focus on (i) and (ii) and revisit (iii) and (iv) in Sections 5.…”

“…We emphasize that the similarity to quantum expectation values is purely computational, and we are not ascribing some "quantum interpretation" to these electromagnetic observables. For such observables we were able to derive ray optics equations in photonic crystals where W is periodic using semiclassical techniques [DL17a]. Fortunately, the fundamental conserved quantities for Maxwell's equations are indeed of this form (see e. g. [BBN13, equations (3.40')-(3.45')] for a list).…”