Laser Beam Propagation in Nonlinear Optical Media

3.2.2      Reduction of d_eff to expressions in the literature

According to Ref. [2] (page 27), ‘the correct expressions for the $d_{\text{eff}}^{\text{ss}-\text{f}}$ and $d_{\text{eff}}^{\text{sf}-\text{f}}$.. were first obtained by Lavrovskaya et al.’ [3]). Since the explicit expressions (without walk-off) for the ssf and sff cases are given in Ref. [2], we first check the expressions above to see if we obtain the same results.

For crystals belonging to point group mm2, there are 5 non-zero d coefficients: d₁₅, d₂₄, d₃₁, d₃₂ and d₃₃. For a Type I (ssf) interaction, d_eff(ω₁) is obtained from Table 3.3 as:

$\begin{array}{l}{d}_{\text{eff}}\left({\omega }_3\right)=e_{fi}\left({\omega }_3\right)d_{il}{u}_l^{ss}\\ =d_{15}{e}_{fX}\left({\omega }_3\right)\left\{e_{sZ}\left({\omega }_1\right)e_{sX}\left({\omega }_2\right)+e_{sX}\left({\omega }_1\right)e_{sZ}\left({\omega }_2\right)\right\}\\ +d_{24}{e}_{fZ}\left({\omega }_3\right)\left\{e_{sY}\left({\omega }_1\right)e_{sZ}\left({\omega }_2\right)+e_{sZ}\left({\omega }_1\right)e_{sY}\left({\omega }_2\right)\right\}\\ +d_{31}{e}_{fZ}\left({\omega }_3\right)e_{sX}\left({\omega }_1\right)e_{sX}\left({\omega }_2\right)\\ +d_{32}{e}_{fZ}\left({\omega }_3\right)e_{sY}\left({\omega }_1\right)e_{sY}\left({\omega }_2\right)\\ +d_{33}{e}_{fZ}(\end{array}$