The factorization method in inverse scattering from periodic structures

The application of the factorization method, the refined version of the linear sampling method, to scattering by a periodic surface is considered. Central to this method is the near field operator N, mapping incident fields to the corresponding scattered fields on a horizontal line. A factorization of N forms the basis for the method. It is shown that the middle operator in this factorization, the adjoint of the single-layer operator, is coercive if the wavenumber is small enough. Thus this operator can, for general wavenumber, always be written as the sum of a coercive and a compact operator. We use this property to define an auxiliary positive operator N-# which can be constructed directly from N and which makes it possible to reconstruct the scattering surface directly using a simple numerical algorithm.