Second-harmonic generation with arbitrary pump-beam profiles

We consider optical second-harmonic generation for a pump beam of arbitrary transverse profile. The analysis is carried out in the parametric approximation and the phase-matching condition is assumed to be satisfied along a principal axis of the nonlinear crystal. The diffraction effects associated with propagation of the pump beam and the second-harmonic beam are fully incorporated. An explicit expression for the harmonic field is obtained and it requires a twofold surface integration over the pump-beam cross section. The general formalism is shown to be readily extended to incorporate the case of partially coherent pump beams.