Convergent scheme for light scattering from an arbitrary deep metallic grating

The justification for continuing the Rayleigh expansion to the grating's surface (the Rayleigh hypothesis) and its convergence properties are considered. A class of gratings for which the Rayleigh hypothesis is exact is identified, a prime example of which is the sinusoidal grating. Based on the exposure of the origin for the Rayleigh expansion limited convergence, a modified expansion is introduced, dubbed the dressed Rayleigh expansion. This new expansion has presumably excellent convergence properties as explicitly demonstrated for the sinusoidal grating. The dimensionality N of the matrix which must be inverted for a sinusoidal grating of arbitrary depth g and periodicity d is found to be N{~}8{π}g/d.