We present a precise theoretical explanation and prediction of certain resonant peaks and dips in the electromagnetic transmission coefficient of periodically structured slabs in the presence of nonrobust guided slab modes. We also derive the leading asymptotic behavior of the related phenomenon of resonant enhancement near the guided mode. The theory applies to structures in which losses are negligible and to very general geometries of the unit cell. It is based on boundary-integral representations of the electromagnetic fields. These depend on the frequency and on the Bloch wave vector and provide a complex-analytic connection in these parameters between generalized scattering states and guided slab modes. The perturbation of three coincident zeros-those of the dispersion relation for slab modes, the reflection constant, and the transmission constant-is central to calculating transmission anomalies both for lossless dielectric materials and for perfect metals.

This chapter concerns the interaction between guided electromagnetic or acoustic modes of a penetrable periodic planar waveguide and plane waves originating from sources exterior to the waveguide. The interaction causes resonant enhancement of fields in the waveguide and anomalous transmission of energy across it. A guided mode is an eigenfunction of a member of the family of operators in the Floquet-Bloch decomposition of the periodic differential operator underlying the waveguide structure. The theory of existence or nonexistence of modes in ideal lossless waveguides is founded on variational principles. The mechanism for resonant scattering behavior is the dissolution of an embedded eigenvalue into the continuous spectrum, which corresponds to the destruction of a guided mode of a waveguide, upon perturbation of the wavevector or the material properties or geometry of the structure. Analytic perturbation of functions that unify the guided modes and the extended scattering states gives rise to asymptotic formulas for transmission anomalies.

Arrays of cylindrical metal micro-resonators embedded in a dielectric matrix were proposed by Pendry, et. al.,[17] as a means of creating a microscopic structure that exhibits strong bulk magnetic behavior at frequencies not realized in nature. This behavior arises for H-polarized fields in the quasi-static regime, in which the scale of the microstructure is much smaller than the free-space wavelength of the fields. We carry out both formal and rigorous two-scale homogenization analyses, paying special attention to the appropriate method of averaging, which does not involve the usual cell averages. We show that the effective magnetic and dielectric coefficients obtained by means of such averaging characterize a bulk medium that, to leading order, produces the same scattering data as the micro-structured composite.

For homogeneous lossless 3D periodic slabs of fixed arbitrary geometry, we characterize guided modes by means of the eigenvalues associated to a variational formulation. We treat robust modes, which exist for frequencies and wavevectors that admit no propagating Bragg harmonics and therefore persist under perturbations, as well as nonrobust modes, which can disappear under perturbations due to radiation loss. We prove the nonexistence of guided modes, both robust and nonrobust, in "inverse" structures, for which the celerity inside the slab is less than the celerity of the surrounding medium. The result is contingent upon a restriction on the width of the slab but is otherwise independent of its geometry. c S.

Resonant scattering of plane waves by a periodic slab under conditions close to those that support a guided mode is accompanied by sharp transmission anomalies. For two-dimensional structures, we establish sufficient conditions, involving structural symmetry, under which these anomalies attain total transmission and total reflection at frequencies separated by an arbitrarily small amount. The loci of total reflection and total transmission are real-analytic curves in frequency-wavenumber space that intersect quadratically at a single point corresponding to the guided mode. A single anomaly or multiple anomalies can be excited by the interaction with a single guided mode.