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Hyperbolic plasmonic metamaterials provide numerous opportunities for designing unusual linear and nonlinear optical properties. Here we report a full vectorial numerical model to study SHG in a plasmonic nanorod metamaterial slab. Our frequency-domain implementation of the hydrodynamic model of the metal permittivity for conduction electrons provided a full description of the nonlinear susceptibility in a broad spectral range. We show that the modal overlap of fundamental and second-harmonic light in the plasmonic metamaterial slab results in the frequency tuneable enhancement of radiated second-harmonic intensity by up to 2 orders of magnitudes for TM- and TE-polarized fundamental light, compared to a smooth Au film under TM-polarised illumination. A double-resonant condition with both the enhancement of fundamental field and the enhanced scattering of the second-harmonic field can be realised at multiple frequencies due to the mode structure of the metamaterial slab. The nanostructured geometry of the Au nanorod metamaterial provides a larger surface area compared to the centrosymmetric crystal lattice of gold, which is needed for exploiting the intrinsic surface nonlinearity of gold. The numerical model allows us to explain experimental investigations on the spectral behaviour and radiation diagram of the second harmonic signal. In the experiments SHG generated under femtosecond excitation with varying wavelength, polarization, and angle of incidence, was characterized in backward and forward directions. We show that the excitation of plasmonic modes in the array can remarkably enhance the nonlinear response of the system, as predicted by the model. The results open up wide ranging possibilities to design tuneable frequency-doubling metamaterial with the goal to overcome limitations associated with classical phase matching conditions in thick nonlinear crystals.
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