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

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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 subleading-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.

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Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile

    Giuseppe De Nittis

  2. Advanced Institute of Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan

    Max Lein

Authors
  1. Giuseppe De Nittis
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  2. Max Lein
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Correspondence to Max Lein.

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Communicated by Jean Bellissard.

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De Nittis, G., Lein, M. Derivation of Ray Optics Equations in Photonic Crystals via a Semiclassical Limit. Ann. Henri Poincaré 18, 1789–1831 (2017). https://doi.org/10.1007/s00023-017-0552-7

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  • DOI: https://doi.org/10.1007/s00023-017-0552-7

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