  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 are characterized by
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.
