As a step toward analyzing second-harmonic generation (SHG) from crystalline Si nanospheres in glass, we develop an anisotropic bond model (ABM) that expresses SHG in terms of physically meaningful parameters and provide a detailed understanding of the basic physics of SHG on the atomic scale. Nonlinear-optical (NLO) responses are calculated classically via the four fundamental steps of optics: evaluate the local field at a given bond site, solve the force equation for the acceleration of the charge, calculate the resulting radiation, then superpose the radiation from all charges. Because the emerging NLO signals are orders of magnitude weaker and occur at wavelengths different from that of the pump beam, these steps are independent. Paradoxically, the treatment of NLO is therefore simpler than that of linear optics (LO), where these calculations must be done self-consistently. The ABM goes beyond previous bond models by including the complete set of underlying contributions: retardation (RD), spatial-dispersion (SD), and magnetic (MG) effects, in addition to the anharmonic restoring force acting on the bond charge. Transverse as well as longitudinal motion is also considered. We apply the ABM to obtain analytic expressions for SHG from amorphous materials under Gaussian-beam excitation. These materials represent an interesting test case not only because they are ubiquitous but also because the anharmonic-force contribution that dominates the SHG response of crystalline materials and ordered interfaces vanishes by symmetry. The remaining contributions, and hence the SHG signals, are entirely functions of the LO response and beam geometry, so the only new information available is the anisotropy of the LO response at the bond level. The RD, SD, and MG contributions are all of the same order of magnitude, so none can be ignored. Diffraction is important in determining not only the pattern of the emerging beam but also the phases and amplitudes of the different terms. The plane-wave expansion that gives rise to electric quadrupole magnetic dipole effects in LO appears here as retardation. Using the paraxial-ray approximation, we reduce the results to the isotropic case in two limits, that where the linear restoring force dominates (glasses) and that where it is absent (metals). Both forward- and backscattering geometries are discussed. Estimated signal strengths and conversion efficiencies for fused silica appear to be in general agreement with data where available. Predictions that allow additional critical tests of these results are made.
