The electromagnetic theory of optical second-harmonic generation from small spherical particles comprised of centrosymmetric material is presented. The interfacial region where the inversion symmetry is broken provides a source of the nonlinearity. This response is described by a general surface nonlinear susceptibility tensor for an isotropic interface. In addition, the appropriate weak bulk terms for an isotropic centrosymmetric medium are introduced. The linear optical response of the sphere and the surrounding region is assumed to be isotropic, but otherwise arbitrary. The analysis is carried out to leading order in the ratio of the particle radius to the wavelength of the incident light, and can be considered as the Rayleigh limit for second-harmonic generation from a sphere. Emission from the sphere arises from both induced electric dipole and electric quadrupole moments at the second-harmonic frequency. The former requires a nonlocal excitation mechanism in which the phase variation of the pump beam across the sphere is considered, while the latter is present for a local-excitation mechanism. The locally excited electric dipole term, analogous to the source for linear Rayleigh scattering, is absent for the nonlinear case because of the overall inversion symmetry of the problem. The second-harmonic field is found to scale as and to be completely determined by two effective nonlinear susceptibility coefficients formed as a prescribed combination of the surface and bulk nonlinearities. Characteristic angular and polarization selection rules resulting from the mechanism of the radiation process are presented. Various experimental aspects of the problem are examined, including the expected signal strengths and methods of determining the nonlinear susceptibilities. The spectral characteristics associated with the geometry of a small sphere are also discussed, and distinctive localized plasmon resonances are identified.
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