Theory of Optical Parametric Amplification from a Focused Gaussian Beam

The theory of parametric amplification of focused Gaussian beams in a uniaxial medium in the approximation of no pump depletion is studied using the mode theory developed in previous papers. We consider a Gaussian pump beam interacting with a Gaussian signal beam both propagating perpendicular to the optic axis of the medium. We show how the gain varies with the focusing parameter of the pump, the relative focusing of the pump and signal, phase matching, and a degeneracy parameter defined as $\mu =\frac{(k_3-2\mathrm{}{k}_1)}{k_3}$, where $k_3$ is the wave number of the pump beam and $k_1$ that of the signal beam. The conditions under which the resultant idler field may be approximated by a Gaussian beam are then studied. It is shown that when the interaction takes place in the near field of both beams, the idler field may be assumed to be Gaussian with spot size given approximately by $\frac{1}{w_{02}^2}=\frac{1}{w_{03}^2}+\frac{1}{w_{01}^2}$, and when the pump and signal are focused for maximum gain the idler field is approximately Gaussian with spot size given approximately by $\frac{1}{w_{03}^2}=\frac{1}{w_{02}^2}+\frac{1}{w_{02}^2}$.