The problem of designing a periodic interface between two different materials. which gives rise to a specified far-field diffraction pattern for a given incoming plane wave, is considered. The time harmonic waves are assumed to be TM (transverse magnetic) polarized. The diffraction problem is modeled by a generalized Helmholtz equation with transparent boundary conditions. In this paper the design problem is relaxed to include highly oscillatory profiles. Existence of an optimal design is established. The principal method is based on the theory of homogenization for the model equation.