The generation process of second harmonic (SH) radiation from holes periodically arranged on a metal surface is investigated. Three main modulating factors affecting the optical response are identified: the near-field distribution at the wavelength of the fundamental harmonic, how SH light couples to the diffraction orders of the lattice, and its propagation properties inside the holes. It is shown that light generated at the second harmonic can excite electromagnetic modes otherwise inaccessible in the linear regime under normal incidence illumination. It is demonstrated that the emission of SH radiation is only allowed along off-normal paths precisely due to that symmetry. Two different regimes are studied in the context of extraordinary optical transmission, where enhanced linear transmission either occurs through localized electromagnetic modes or is aided by surface plasmon polaritons (SPPs). While localized resonances in metallic hole arrays have been previously investigated, the role played by SPPs in SH generation has not been addressed so far. In general, good agreement is found between our calculations (based on the finite difference time domain method) and the experimental results on localized resonances, even though no free fitting parameters were used in describing the materials. It is found that SH emission is strongly modulated by enhanced fields at the fundamental wavelength (either localized or surface plasmon modes) on the glass metal interface. This is so in the transmission side but also in reflection, where emission can only be explained by an efficient tunneling of SH photons through the holes from the output to the input side. Finally, the existence of a dark SPP at the fundamental field is identified through a noninvasive method for the first time, by analyzing the efficiency and far-field pattern distribution in transmission at the second harmonic.
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