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

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.