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.
