The interplay of symmetry and scattering phase in second harmonic generation from gold nanoantennas

Nonlinear phenomena are central to modern photonics but, being inherently weak, typically require gradually accumulation over several millimetres. For example, second harmonic generation (SHG) is typically achieved in thick transparent nonlinear crystals by phase-matching energy exchange between light at initial, ω, and final, 2ω, frequencies. Recently, metamaterials imbued with artificial nonlinearity from their constituent nano-antennas have generated excitement by opening the possibility of wavelength-scale nonlinear optics. However, the selection rules of SHG typically prevent dipole emission from simple nano-antennas, which has led to much discussion concerning the best geometries; for example those breaking centro-symmetry or incorporating resonances at multiple harmonics. In this work we explore the use of both nano-antenna symmetry and multiple harmonics to control the strength, polarisation and radiation pattern of SHG from a variety of antenna configurations incorporating simple resonant elements tuned to light at both ω and 2ω. We use a microscopic description of the scattering strength and phases of these constituent particles, determined by their relative positions, to accurately predict the SHG radiation observed in our experiments. We find that the 2ω particles radiate dipolar SHG by near-field coupling to the ω particle, which radiates SHG as a quadrupole. Consequently, strong linearly polarised dipolar SHG is only possible for non-centro-symmetric antennas that also minimize interference between their dipolar and quadrupolar responses. Metamaterials with such intra-antenna phase and polarisation control could enable compact nonlinear photonic nanotechnologies.