  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.
