Weak-localization effects in the generation of second harmonics of light at a randomly rough vacuum-metal grating

The diffuse component of second-harmonic light (frequency 2{ω}) generated at a weakly rough vacuum-metal grating through the interaction of incident (frequency {ω}) light with the optical nonlinearity of the surface is calculated. A multiple-scattering formulation is used, based on the Rayleigh hypothesis and the application of the electromagnetic boundary conditions at the vacuum-metal interface, which allows us to relate elements of the diffuse generation of 2{ω} radiation to resonant-scattering processes involving {ω} and 2{ω} surface polaritons. Specific attention is paid to the study of peaks in the angular intensity of diffusely generated 2{ω} light, about the normal to the mean surface and about the direction for 2{ω} light moving opposite to that of the incident {ω}-frequency light, arising from the Anderson localization of {ω} and 2{ω} surface polaritons, respectively. The peak about the normal to the mean surface is a type of weak-localization peak that occurs only at surfaces. For simplicity we assume that the roughness in our system is small enough that the general angle diffuse 2{ω} generation of light (away from the localization peaks) can be adequately described by lowest-order scattering in the surface roughness and then include the localization effects by summing the standard set of maximally crossed diagrams that gave rise to these effects in resonant processes involving both {ω} and 2{ω} surface polaritons.