  We conduct a theoretical and numerical study on the second-harmonic (SH)
optical response of a nano-structured 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 non-centrosymmetric system which does allow efficient SH
generation. Variations in the geometrical configuration allows tuning the
linear and quadratic spectra of the optical response of the system. We develop
an efficient scheme for calculating the nonlinear polarization, extending a
formalism for the calculation of the macroscopic dielectric function using
Haydock's recursion method. We apply the formalism developed here 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.
