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We calculate the optical second harmonic ͑SH͒ radiation generated by small spheres made up of a homogeneous centrosymmetric material illuminated by inhomogeneous transverse and/or longitudinal electromagnetic fields. We obtain expressions for the hyperpolarizabilities of the particles in terms of the multipolar bulk susceptibilities and dipolar surface susceptibilities of their constitutive material. We employ the resulting response functions to obtain the nonlinear susceptibilities of a composite medium made up of an array of such particles and to calculate the radiation patterns and the efficiency of SH generation from the bulk and the edge of thin composite films illuminated by finite beams. Each sphere has comparable dipolar and quadrupolar contributions to the nonlinear radiation, and the composite has comparable bulk and edge contributions which interfere among themselves yielding nontrivial radiation and polarization patterns. We present numerical results for Si spheres and we compare our results with recent experiments.

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is the plasma frequency of the metal. In the case of Au, we employed the low frequency value ␤ = ͱ 1/3v F instead of the high frequency value ␤ = ͱ 3/5v F stated in the text. Equation ͑14͒ is wrong; the signs of the terms proportional to Q 2 in both numerator and denominator should be exchanged, so that it becomesFortunately, this was only a typographical error and the results were calculated with the correct equation. The corrections above do not alter the main conclusions of the paper, except for a reduction by a factor of 2 in the distances at which nonlocal corrections become important.

We present a very efficient recursive method to calculate the effective optical response of metamaterials made up of arbitrarily shaped inclusions arranged in periodic 3D arrays. We apply it to dielectric particles embedded in a metal matrix with a lattice constant much smaller than the wavelength of the incident field, so that we may neglect retardation and factor the geometrical properties from the properties of the materials. If the conducting phase is continuous the low frequency behavior is metallic, and if the conducting paths are thin, the high frequency behavior is dielectric. Thus, extraordinary-transparency bands may develop at intermediate frequencies, whose properties may be tuned by geometrical manipulation.