The beam fanning effect in an electro-optic crystal has been usually explained by the photorefractivity of the crystal. We attempt to analyze the beam fanning effect in $BaTiO_3$ crystal from the improved viewpoint by considering not only the photorefractivity but also the self phase modulation. The beam fanning effect in a photorefractive $BaTiO_3$ crystal is investigated based on the pattern of the transmitted beam, which is recorded on a photographic plate before generation of phase conjugate waves takes place, and it is explained as the consequence of spatial variation in the refractive index due to photorefractivity and self-phase modulation. The convergent $Ar^+$ ion laser beam (20mW) which exhibits a beam fanning effect in the crystal is approximated by an asymmetric Gaussian beam. The shift of the center of the Gaussian beam increases with increasing incident laser beam diameter. From the equation of the boundary between the beam fanning region and non-beam fanning region in the cross section of the transmitted beam, we have obtained $n_2$ for an extra-ordinary wave to be $ (-1.1±0.1)×10^{-11}m^2/W$ at the wavelength of 514.5 nm.
We also carried out the self phase conjugation experiment by using the converging laser beam as an incident wave. In this experiment, we observe the patterns of the conjugate beam and the transmitted beam, the propagation of the incident beam in the crystal, and the output power of the conjugate beam, as the size of the input beam at the crystal surface is altered. It is found that the larger input beam is advantageous in generating the conjugate wave compared with the small input beam. The use of the large input beam results in the higher intensity reflectivity, and the small tilt of the crystal does not cause the serious degradation of the power of the conjugate wave.
The analytic solution of phase conjugate wave generated by degenerate four wave mixing(DFWM), considering both the transmission and reflection gratings, and absorption in photorefractive media is derived. Dependence of the phase conjugate reflectivity (PCR) on the incident angle of the $Ar^+$ ion laser beam, and the backward and forward pump power ratio($I_b/I_f$) have been determined. We obtained the maximum reflectivity to be about 10% in the condition of $I_b/I_f=1.64$ and the incident angle 12˚.
In our researches, the brief introduction of the mutual phase conjugation is also included. The mutual conjugation of the mixed type is proposed for the first time. We obtained theoretically in all types the intensity ratio of the mutual conjugate waves($R_2/R_1$) to be proportional to $I^2_1/I^2_2$ where $I_1$, $I_2$ are two probe intensities.
