Wave scattering by a periodic perturbation: embedded Rayleigh–Bloch modes and resonances

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.