Volume electric dipole origin of second-harmonic generation from metallic membrane with none entro symmetric patterns

We analytically study second-harmonic (SH) generation from thin metallic films with subwavelength noncentrosymmetric patterns. Because the thickness of the film is much smaller than the SH wavelength, retardation effects are negligible. The far-field SH intensities are thus dominated by an effective electric dipole. It is further demonstrated that internal bulk SH polarization density is comparable with its surface counterpart. The electric dipole, consequently, originates from the entire volume of the metallic membrane.