Efficient construction of maximally localized photonic Wannier functions: locality criterion and initial conditions

TL;DR

This paper compares different criteria for constructing maximally localized photonic Wannier functions, introducing a new measure that simplifies the optimization process and proposing an analytical initial phase choice to enhance efficiency.

Contribution

It introduces the integrated modulus as a new locality measure and provides an analytical initial phase formula, improving the construction efficiency of Wannier functions.

Findings

IM criterion has a single extremum, enabling faster optimization.

The analytical initial phase formula often reaches the global maximum.

The new measure outperforms the second moment in optimization speed.

Abstract

Wannier function expansions are well suited for the description of photonic- crystal-based defect structures, but constructing maximally localized Wannier functions by optimizing the phase degree of freedom of the Bloch modes is crucial for the efficiency of the approach. We systematically analyze different locality criteria for maximally localized Wannier functions in two- dimensional square and triangular lattice photonic crystals, employing (local) conjugate-gradient as well as (global) genetic-algorithm-based, stochastic methods. Besides the commonly used second moment (SM) locality measure, we introduce a new locality measure, namely the integrated modulus (IM) of the Wannier function. We show numerically that, in contrast to the SM criterion, the IM criterion leads to an optimization problem with a single extremum, thus allowing for fast and efficient construction of maximally localized Wannier functions using local optimization techniques. We also present an analytical formula for the initial choice of Bloch phases, which under certain conditions represents the global maximum of the IM criterion and, thus, further increases the optimization efficiency in the general case.