Pathological scattering by a defect in a slow-light periodic layered medium

TL;DR

This paper investigates the unique scattering phenomena caused by a defect in a slow-light layered medium, revealing how the frozen mode leads to pathological scattering behavior and different resonant cases.

Contribution

It provides a detailed analysis of electromagnetic scattering in slow-light periodic media, highlighting the role of Jordan blocks and the interaction of transfer matrices with mode spaces.

Findings

Pathological scattering occurs at the frozen mode frequency.

Two distinct scattering regimes are identified: resonant and non-resonant.

The analysis uses Laurent-Puiseux series to describe mode interactions.

Abstract

Scattering of electromagnetic fields by a defect layer embedded in a slow-light periodically layered ambient medium exhibits phenomena markedly different from typical scattering problems. In a slow-light periodic medium, constructed by Figotin and Vitebskiy, the energy velocity of a propagating mode in one direction slows to zero, creating a "frozen mode" at a single frequency within a pass band, where the dispersion relation possesses a flat inflection point. The slow-light regime is characterized by a $3\!\times\!3$ Jordan block of the log of the $4\!\times\!4$ monodromy matrix for EM fields in a periodic medium at special frequency and parallel wavevector. The scattering problem breaks down as the 2D rightward and leftward mode spaces intersect in the frozen mode and therefore span only a 3D subspace $\mathring{V}$ of the 4D space of EM fields. Analysis of pathological scattering near the slow-light frequency and wavevector is based on the interaction between the flux-unitary transfer matrix $T$ across the defect layer and the projections to the rightward and leftward spaces, which blow up as Laurent-Puiseux series. Two distinct cases emerge: the generic, non-resonant case when $T$ does not map $\mathring{V}$ to itself and the quadratically growing mode is excited; and the resonant case, when $\mathring{V}$ is invariant under $T$ and a guided frozen mode is resonantly excited.