Second- and third-harmonic generation with vector Gaussian beams

We consider second-harmonic generation (SHG) and third-harmonic generation (THG) in a nonlinear optical crystal illuminated by a vector Gaussian beam, i.e., a Gaussian beam in which the axial component of the excitation field is considered. This component exhibits twice the Gouy phase shift of the transverse component and vanishes at points on the beam axis. Harmonic generation stemming from this component exhibits a unique dependence on geometrical factors associated with the location and focusing of the beam relative to the location of the crystal. Using the first Born approximation (undepleted fundamental beam), we derive analytical formulas for the quantities that characterize these geometrical factors for a nonlinear optical crystal described by an arbitrary nonlinear susceptibility tenser, for both SHG and THG and for all polarization components. We also determine the efficiencies of these processes as functions of the geometry of the experimental arrangement for phase-matched crystals as well as for crystals of infinite length. (c) 2006 Optical Society of America.