PERFECTLY MATCHED LAYERS FOR DIFFRACTION GRATINGS IN INHOMOGENEOUS MEDIA. STABILITY AND ERROR ESTIMATES

We study the scattering of acoustic or electromagnetic waves at diffraction gratings in inhomogeneous media. The refractive index stabilizes to 1 as the distance to the grating increases. Outgoing solutions are characterized by means of the limiting absorption principle. We prove the unique solvability of the problem with perfectly matched layer of finite length. Further, we show that solutions of the latter problem approximate outgoing solutions of the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity. Contribution of eigenvalues and resonances to the error of approximation is clarified.