We present detailed calculations of the temporal and spatial evolution of beam fanning in photorefractive crystals that is initiated by scattering from noise. We show that fanning starts from beam coupling between the incident radiation and part of the incident radiation scattered by noise at or near the input plane. We show that scattering within the volume of the crystal has negligible effect on fanning, that absorption affects the time response but not the spatial pattern of the fanning, and that the difference between calculations including only phase-matched terms and those including non-phase-matched terms is negligible.Beam fanning is the process of self-induced scattering observed in almost all photorefractive crystals. In its most common manifestation a single beam entering one face of a crystal fans out in the horizontal and/or vertical directions. It is now well established that fanning is an example of stimulated photorefractive scattering and is caused by energyexchange beam coupling between the incident beam and radiation scattered by inhomogeneities in the entrance face or within the bulk of the crystal.'-The alternative view, that beam fanning is caused by a whole-beam effect in which the trapped charge follows the beam profile while the space-charge field and refractive-index profiles follow a spatial integral of the trapped charge,"11 now seems inapplicable since it fails to predict fanning for an input wave that is a single plane wave and gives fanning that is too small relative to observations for collimated beams with diameters of the order of a millimeter.Previous calculations of beam fanning led to detailed steady-state results and to limited results of the temporal dependence. 5 Here we present detailed calculations of the temporal dependence of beam fanning obtained by numerical solutions of the paraxial wave equation for the optical beam and the equation for the time dependence of the refractive-index perturbation produced by interference between the incident beam and radiation scattered by imperfections in the crystal. Our calculations extend the results of Ref. 5 for steady-state fanning to temporal evolution and include the effects of absorption and Rayleigh scattering distributed throughout the crystal.We formulate the problem of beam propagation in photorefractive materials by using the planewave expansion technique that we have recently used for stimulated photorefractive backscattering' 2 and for counterpropagating beams.' 3 Similar expansions have been used previously for fanning by Kukhtarev' 4 and by Obukhovskii and Stoyanov. 2 Limitations inherent in this technique are discussed by Crosignani et al.' 5 To isolate the physics of beam fanning alone, we neglect interactions between orthogonal components (anisotropic scattering), assume a monochromatic incident beam and stationary scattering centers, and include only two spatial dimensions.The electric field E(x, z, t) of the optical beam propagating in the z direction may be written in the form( 2) where A(x, z, t) is the slo...