Perturbative analysis of the Wood anomalies of metallic diffraction gratings

Starting from the reduced Rayleigh equation and using elements of many-body perturbation theory, we present an expansion for the scattering amplitude of light reflected by a metallic sinusoidal grating. In light of our formalism, we show that the well-known Wood anomalies can be understood as a multiple scattering process. This process is generated by the interference, constructive or destructive, between a diffraction order that passes through a plasmonic state before being diffracted into the anomaly's order, and a simple scattering event propagating in the same direction, which is generally described by a lower order in the expansion. Furthermore, we demonstrate that the phase relationship between these two processes is independent of the grating depth, and the Fano nature of the anomalies. Under Gaussian beam illumination, this dependence generates a Goos-Hänchen-like shift and a spatial deformation of the diffracted beam at the SPP coupling condition.