Second-harmonic generation in nanostructured metamaterials

We conduct a theoretical and numerical study on the second-harmonic (SH) optical response of a nanostructured metamaterial composed of a periodic array of inclusions. Both the inclusions and their surrounding matrix are made of centrosymmetrical materials, for which SH is strongly suppressed, but, by appropriately choosing the shape of the inclusions, we may produce a geometrically noncentrosymmetric system which does allow efficient SH generation. Variations in the geometrical configuration allow tuning the linear and quadratic spectra of the optical response of the system. We develop a theory that allows the calculation of the nonlinear polarization from the geometry of the system and its linear dielectric function at the fundamental and second-harmonic frequencies, and we implement an efficient scheme for its numerical computation, extending a formalism for the calculation of the macroscopic dielectric function using Haydock's recursion method. We apply our formalism to an array of holes within an Ag matrix, but it can be readily applied to any metamaterial made of arbitrary materials and for inclusions of any geometry within the long-wavelength regime.