Diffraction at a Curved Grating: Approximation by an Infinite Plane Grating

We study the influence of the curvature of a circular cylindrical grating on the diffraction of a TE or TM polarized electromagnetic plane wave. For every ratio of the wave length l to the period d, we prove the local approximation in the planar field by a plane infinite grating, when the radius R of the circular grating goes to infinity. The error estimates are optimal and of order O drR.outside some transition regions corresponding to a tangent Bragg’s mode or to transition between the region of direct illumination and the region of shadow, where the error is like O dr l2r3R1r3... In each region, we calculate the first corrector, solution of an infinite plane grating diffraction problem.