Laser Beam Propagation in Nonlinear Optical Media

$\begin{array}{l}{\eta }_{\text{SHG}}^P=\frac{P_s\left(\mathit{\ell },t_1\right)}{P_p\left(0,t_1\right)}\\ =2{\displaystyle \underset{0}{\overset{\infty }{\int }}{r}_{1}d{r}_1{e}^{-r_1^2}{\tanh }^2\left(\sqrt{K_1^{\prime }{e}^{-r_1^2}}\right)}\end{array}$

(4.226)

using Eqn. 4.197 for the case of no phase mismatch.

For pulsed incident beams with Gaussian temporal and spatial shapes

$\begin{array}{l}{E}_s\left(\mathit{\ell }\right)=2\pi {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}dt{\displaystyle \underset{0}{\overset{\infty }{\int }}rdr{I}_s\left(\mathit{\ell }\right)}}\\ =2\pi r_{0p}^2{t}_{0p}{I}_{p0}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}d{t}_1}{\displaystyle \underset{0}{\overset{\infty }{\int }}{r}_{1}d{r}_1{e}^{-r_1^2}{e}^{-t_1^2}}\\ \times {\tanh }^2\left(\sqrt{K_1^{\prime }{e}^{-r_1^2}{e}^{-t_1^2}}\right)\end{array}$

(4.227)

so that the energy conversion efficiency for SHG can be written as

$\begin{array}{l}{\eta }_{\text{SHG}}^E=\frac{E_s\left(\mathit{\ell }\right)}{E_p\left(0\right)}\\ =\frac{2}{\sqrt{\pi }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}d{t}_1}{\displaystyle \underset{0}{\overset{\infty }{\int }}{r}_{1}d{r}_1{e}^{-r_1^2}{e}^{-t_1^2}}\\ \times {\tanh }^2\left(\sqrt{K_1^{\prime }{e}^{-r_1^2}{e}^{-t_1^2}}\right).\end{array}$

(4.228)

When $K_1^{\prime }$ ≪ 1, i.e., for the case of no pump depletion, using the relation tanh x ≈ x for small x, we have

$\begin{array}{l}{\eta }_{\text{SHG}}^P=2K_1^{\prime }{\displaystyle \underset{0}{\overset{\infty }{\int }}{r}_{1}d{r}_1{e}^{-2r_1^2}}\\ =\frac{K_1^{\prime }}{2}\\ =\frac{I_{p0}}{2I_a^{\prime }}\end{array}$

(4.229)

and

$\begin{array}{l}{\eta }_{\text{SHG}}^E=\frac{2K_1^{\prime }}{\sqrt{\pi }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}d{t}_1}{\displaystyle \underset{0}{\overset{\infty }{\int }}{r}_{1}d{r}_1{e}^{-2r_1^2}{e}^{-2t_1^2}}\\ =\end{array}$