Surface-enhanced second-harmonic generation at a metallic grating

The theory of surface-enhanced second-harmonic generation at a metallic grating is developed. Using the form of the nonlinear source polarization given by Bloembergen et al. [Phys. Rev. 174, 813 (1968)], we solve Maxwell's equations to obtain the fields at the second-harmonic frequency. The calculations are done up to second order in the surface-roughness parameter. These perturbation expressions are used to evaluate numerically the second-harmonic intensity, in various directions, produced by a plane wave incident on a metallic grating. The resonant enhancement in the second-harmonic intensity due to surface-plasmon excitation at fundamental frequency $\omega$ is discussed and the results compared with some recent experimental observations. The second-harmonic fields are also shown to get enhancement due to excitation of surface plasmons at $2\omega$; these, however, correspond to local-field enhancements at $2\omega$ and are evanescent in nature.