  The scattering of quasiperiodic waves for a two-dimensional Helmholtz
equation with a constant refractive index perturbed by a function which is
periodic in one direction and of finite support in the other is considered. The
scattering problem is uniquely solvable for almost all frequencies and formulas
of Breit-Wigner and Fano type for the reflection and transmission coefficients
are obtained in a neighborhood of the resonance (a pole of the reflection
coefficient). We indicate also the values of the parameters involved which
provide total transmission and reflection. For some exceptional frequencies and
perturbations (when the imaginary part of the resonance vanishes) the
scattering problem is not uniquely solvable and in the latter case there exist
embedded Rayleigh-Bloch modes whose frequencies are explicitly calculated in
terms of infinite convergent series in powers of the small parameter
characterizing the magnitude of the perturbation.
