Photon transport in binary photonic lattices

TL;DR

This paper reviews mathematical methods for analyzing classical and quantum photon transport in binary photonic lattices, highlighting operator approaches, Floquet-Bloch theory, and connections to Fibonacci polynomials.

Contribution

It provides a comprehensive overview of classical and quantum analysis techniques applied to binary photonic lattices, including novel insights into finite lattice dynamics using Fibonacci polynomials.

Findings

Classical propagation analyzed via coupled mode theory and Floquet-Bloch operator approach.

Quantum photon transport studied using coupled mode theory and orthogonal polynomials.

Finite lattice dynamics expressed through roots and functions of Fibonacci polynomials.

Abstract

We present a review on the mathematical methods used to theoretically study classical propagation and quantum transport in arrays of coupled photonic waveguides. We focus on analysing two types of binary photonic lattices where self-energies or couplings are alternated. For didactic reasons, we split the analysis in classical propagation and quantum transport but all methods can be implemented, mutatis mutandis, in any given case. On the classical side, we use coupled mode theory and present an operator approach to Floquet-Bloch theory in order to study the propagation of a classical electromagnetic field in two particular infinite binary lattices. On the quantum side, we study the transport of photons in equivalent finite and infinite binary lattices by couple mode theory and linear algebra methods involving orthogonal polynomials. Curiously the dynamics of finite size binary lattices can be expressed as roots and functions of Fibonacci polynomials.