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The full wave surface integral equation computation of the second harmonic generation (SHG) dynamics for metal spheres and nanorods - presented as multimedia files - is performed to reveal the dynamics of the modes supported by the nanostructure. We demonstrate that the interplay between different modes controls the nonlinear response and that the size-induced redshift of the eigenmodes can be manipulated by adjusting the nanostructure geometry, so that the SHG signal can be boosted at specified frequencies. We show that the SHG radiation is not necessarily quadrupolar in spherical nanoparticles, as it is often assumed. Finally, we introduce an efficient way to reduce the SHG calculation time.

Optical second harmonic generation (SHG) from nanostructured graphene has been studied in the framework of classical electromagnetism using a surface integral equation method. Single disks and dimers are considered, demonstrating that the nonlinear conversion is enhanced when a localized surface plasmon resonance is excited at either the fundamental or second harmonic frequency. The proposed approach, beyond the electric dipole approximation used in the quantum description, reveals that SHG from graphene nanostructures with centrosymmetric shapes is possible when retardation effects and the excitation of high plasmonic modes at the second harmonic frequency are taken into account. Several SHG effects similar to those arising in metallic nanostructures, such as the silencing of the nonlinear emission and the design of double resonant nanostructures, are also reported. Finally, it is shown that the SHG from graphene disk dimers is very sensitive to a relative vertical displacement of the disks, opening new possibilities for the design of nonlinear plasmonic nanorulers.

We numerically investigate the second harmonic generation from different plasmonic systems and evidence the key role played in their nonlinear response by the phase at the fundamental wavelength. In the case of a single plasmonic nanorod, the interference between the second harmonic dipolar and quadrupolar emission modes depends on their relative phase, which is deeply related to the excitation wavelength. The knowledge obtained in this simple case is then used to describe and understand the nonlinear response from a more complex structure, namely a gold nanodolmen. The complex phase evolution associated with a Fano resonance arising at the fundamental wavelength enables dramatically modifying the second harmonic emission patterns from plasmonic metamolecules within minute wavelength shifts. These results emphasize the importance of the phase in the nonlinear optical processes arising in plasmonic nanostructures, in addition to the increase in conversion yield associated with the excitation of localized surface plasmon resonances.