Influence of the Bragg angle on the optimal energy exchange in the course of two-wave mixing in a $Bi_{12}SiO_{20}$ piezoelectric crystal

The problem of maximising the relative intensity of the signal optical wave in the course of two-wave mixing was solved taking into account the self-diffraction in a Bi$_{12}$SiO$_{20}$ piezoelectric crystal. The maximisation was carried out simultaneously with respect to four parameters: the polarisation ($\psi$) and orientation ($\theta$) angles, the crystal thickness $d$, and the Bragg angle $\varphi$. It was established that there are two sets of optimal parameters for a Bi$_{12}$SiO$_{20}$ crystal at the wavelength of 632.8 nm and for a typical acceptor concentration of $10^{22}$ m $^{-3}$: $\varphi \approx 11^{\circ}$, $\theta \approx 39.1^{\circ}$, $\psi \approx 98.85^{\circ}$, $d \approx 7.11$ mm (for the first maximum) and $\varphi \approx 11^{\circ}$, $\theta \approx 320.9^{\circ}$, $\psi \approx 54.15^{\circ}$, $d \approx 7.11$ mm (for the second maximum). The dependences of these optimal parameters on the acceptor concentration in the range $10^{21}-4 \cdot 10^{22}$ m $^{-3}$ were found. The model proposed for two-wave mixing in the special case of a fixed Bragg angle and the constant visibility approximation improved the agreement between the theoretical results and the known experimental data.