A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides

TL;DR

This paper introduces an exact Dirichlet-to-Neumann method for computing guided modes in photonic crystal waveguides, simplifying the eigenvalue problem to a nonlinear fixed point problem localized near the defect.

Contribution

It presents a novel, exact DtN approach that reduces the eigenvalue problem to a nonlinear fixed point problem on a small domain, improving accuracy over existing methods.

Findings

Exact reduction to a nonlinear eigenvalue problem

Localized computation near the defect

Improved accuracy over traditional methods

Abstract

This works deals with one dimensional infinite perturbation - namely line defects - in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and the computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann (DtN) approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. On contrary to existing methods, this one is exact but there is a price to be paid : the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature.