Boundary-Integral Calculations of Two-Dimensional Electromagnetic Scattering in Infinite Photonic Crystal Slabs: Channel Defects and Resonances

We compute the transmission of two-dimensional (2D) electromagnetic waves through a square lattice of lossless dielectric rods with a channel defect. The lattice is finite in the direction of propagation of the incident wave and periodic in a transverse direction. We revisit a boundaryintegral formulation of 2D electromagnetic scattering [Venakides, Haider, and Papanicolaou, SIAM J. Appl. Math., 60 (2000), pp. 1686–1706] that is Fredholm of the first kind and develop a second-kind formulation. We refine the numerical implementation in the above paper by exploiting separability in the Green’s function to evaluate the far-field influence more efficiently. The resulting cost savings in computing and solving the discretized linear system leads to an accelerated method. We use it to analyze E-polarized electromagnetic scattering of normally incident waves on a structure with a periodic channel defect. We find three categories of resonances: waveguide modes in the channel, high-amplitude fields in the crystal at frequencies near the edge of the frequency bandgap, and very high-amplitude standing fields at frequencies in a transmission band that are normal to the direction of the incident wave. These features are captured essentially identically with the first-kind as with the second-kind formulation.