Photonic band structure by direct second-kind free-space boundary integral equations

Abstract

We present a direct boundary integral equation (BIE) formulation of the Bloch eigenvalue problem for two-dimensional photonic crystals that uses only the free-space Helmholtz Green function. The unknowns are the Cauchy data of the Bloch mode on the inclusion boundary and on two cell walls. On the inclusion boundary, the transmission conditions are imposed via Müller's formulation. The wall equations, in turn, come from a stack formulation, in which Green's formula is applied on stacks of three cells, so that the wall self-interactions and the interactions between arcs sharing endpoints cancel exactly. The resulting system is of the second kind, Fredholm of index zero, and free of corner singularities, and the walls may be curved to keep the cell boundary away from the inclusions. Its operators are assembled once per frequency and reused for all Bloch vectors. We prove that every Bloch mode yields a nonzero null vector, and that a null vector comes from a Bloch mode exactly when its cell representation vanishes outside the cell, which gives an inexpensive test for spurious null vectors. The band frequencies are computed with a high-order Nyström discretization and the search of Bruno, Santana and Trefethen, which locates them as poles of adaptive Antoulas--Anderson (AAA) rational approximants to a randomly scalarized resolvent sampled on the real frequency axis. Numerical examples validate the method against exact and published band structures and include a challenging triangular lattice of air pores in a dielectric, whose cell requires curved walls.