Degenerate four-wave mixing by transmission holographic gratings in a Bi$_{12}$TiO$_{20}$ crystal of the (110)-cut

The regularities of the stationary degenerate four-wave mixing on transmission holographic gratings formed in the Bi$_{12}$TiO$_{20}$ crystal of the (110)-cut are analyzed. The system of differential equations has been obtained; this system that can be used to find the components of the vector amplitudes of linearly polarized light waves in the case of four-wave mixing on phase and phase-amplitude holographic gratings. The theoretical model take into account the linear electrooptic, the photoelastic and the inverse piezoelectric effects, as well as natural optical activity, circular dichroism and crystal absorption. The values of the orientation angle and the crystal thickness where the reflection coefficient can reach maximum values are determined. It was experimentally found that the reflection coefficient can reach 2.4 with an optimal choice of the orientation angle in the Bi$_{12}$TiO$_{20}$ crystal of (110)-cut with the thickness of 7.7 mm. It is shown that the best agreement between theoretical and experimental data is achieved if the phase-amplitude structure of transmission holographic gratings formed in the Bi$_{12}$TiO$_{20}$ crystal is taken into account in the mathematical model of diffraction.