| Beams diffracted by planar surfaces having a two-dimensional (2-D) periodicity may exhibit strong spatial modifications that cannot be explained in terms of conventional geometric arguments. These effects include 2-D longitudinal and transverse lateral displacements, focal shifts, beam waist modifications, and angular deflections for each of the diffracted beams. We perform a rigorous analysis of these phenomena by first developing a formalism for plane-wave scattering by 2-D periodic impedance surfaces, and applying it to investigate the diffraction of Gaussian beams in the paraxial regime. Some of the beam modification effects due to the 2-D grating action are analogous to those for uniform or 1-D periodic surfaces. However, these effects acquire a complexity that involves novel features generated by the 2-D aspect of the periodic scattering surface.; For plane-wave scattering, we satisfy the periodic boundary conditions at the surface and obtain an infinite set of coupled recurrence equations whose solution yields the scattered field expansion coefficients. We present numerical results that compare favorably with those previously published for copolarized returns. We also show that no depolarization occurs for the diffracted orders that propagate in the same plane of the incident wave and incidence is in a symmetry plane of 2-D periodic structure. Using the plane-wave formalism, the geometrical properties of Gaussian beams diffracted by a 2-D periodic impedance plane are first analyzed by neglecting variations in the scattering function. We thus show that, in general, waist values are unequal in the two transverse directions across the beam and they appear along different locations along the beam axis, i.e., the diffracted beams are astigmatic. Spatial modification effects are derived by accounting for the actual scattering function describing the spectral contents of each diffracted field. The formal solution for those fields is a coherent superposition of Gaussian beams arising from the principal and cross polarized returns. Each one of these beams may undergo anomalous spatial modifications which are expressed in terms of derivatives of the scattering function. We obtain analytic expressions for these effects by developing a Padé approximant scattering model and demonstrate its surprising accuracy. The 2-D effects are shown to exhibit a greater complexity and include additional features that are absent in the simpler 1-D grating situation. |
