Nonlinear combinations of gratings in Bi_12SiO_20: theory and experiments

Abstract

Multiple optical interference patterns generate phase gratings in Bi12SiO20 as a result of the photorefractive effect. Because of the nonlinear response of the space-charge field to the optical interference pattern, these primary gratings interact and cause the formation of additional gratings at frequencies that are combinations of the primary grating frequencies. Only the diffusion regime is considered in the present investigation. In our setup, two primary gratings are induced as the result of interference among one reference and two closely situated object beams. The interference between the two object beams leads to a negligible photorefractive grating, because the photorefractive Bi12SiO20 crystal cannot respond to the corresponding large fringe spacing. We show theoretically that nonlinear combinations of gratings are introduced through this third interference pattern even though the corresponding grating itself is absent. We obtain analytical expressions for the fundamental frequency components of the space-charge field in the special case of a weak reference beam. In the general case, the frequency components of the space-charge field are calculated numerically by the use of Fourier transforms and are incorporated into the coupled-wave equations. The optical beam propagation method is then used for solving these equations numerically. The effect is probed in a three-wave mixing experiment that uses a sinusoidal phase modulation on one of the object beams to control the grating strength. With this technique one of the gratings is specifically canceled out, and the relative change of the diffraction efficiency in the remaining grating is measured. It is shown that nonlinear combinations of gratings may cause relative changes of more than 35% in the diffraction efficiency. The theoretical predictions are in excellent agreement with our experimental results and the cross talk observed in dynamic optical interconnects.