Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams

We calculated the second-harmonic wave pattern induced by focused cylindrically symmetric, polarized vector beams. The second-order nonlinear polarization was expressed for fundamental electric field components passed through a dielectric interface based on vector diffraction theory. Furthermore, the second-harmonic wave pattern was represented on the basis of the far-field approximate expression derived from the formulation of higher-order harmonic generation including a Green's function. For a (110) zinc selenide crystal, the calculated forward emission patterns of the second-harmonic wave were eight-figure shaped as observed in experiment.