Power-dependent Raman scattering in focused geometry

Diffraction and dispersion effects in Raman scattering are investigated, with focused beams and pump depletion. Power broadening and Stark shifting of the molecular energy levels are included in the equations up to second order in intensity (an intermediate-power approximation). The steady-state paraxial wave equation is used to propagate the electric fields which are expanded in a series of rotationally symmetric Gauss-Laguerre functions. These functions allow diffraction and power-dependent effects to be included quite easily with the introduction of a recursive technique used to evaluate the nonlinear coupling coefficients. The resulting equations are solved analytically for a single transverse mode and numerically for the multimode case. Several examples are given for conversion in ${\mathrm{H}}_2$. In order for the intermediate-power approximation to remain valid, the pump power must be less than 0.015 MW for a beam confocal parameter of 10 cm. Under these conditions we show that power-dependent focusing reduces the Stokes conversion efficiency by a factor of 5 as compared to power-independent focusing.