Temporal evolution of fanning in photorefractive materials

Abstract

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...

Temporal evolution of photorefractive double phase-conjugate mirrors

“…As found in Ref. 6, our calculations show that the non-phasematched terms are important only for propagation distances of a few optical wavelengths, and hence all our results are based on numerical solution of Eqs. (10) and (11).…”

“…(1)- (6) the field amplitudes, Ai, an, and bn, are given in units of the square root of the equivalent dark irradiance and time is in units of the dielectric relaxation time evaluated at the equivalent dark irradiance. 4 6 To make the calculations tractable, 6 we define Io as the spatial average of (1 + IAlI 2 + 1A 2 1 2 ) and substitute this into Eq. (6).…”

“…6 Angles are measured with respect to the +c axis, and 31 plane waves are used to describe the transverse dependence (angular separation of half a degree). We have solved the equations numerically by using the split-step methods, 4 and we have checked the calculations by doubling the number of steps in z and t and in the transverse resolution in x (namely, reducing the angular separation to 0.25°).…”

Reduction in the reconstruction error of computer-generated holograms by photorefractive volume holography

“…We assumed negligible absorption (which does not affect the qualitative process in the transmission geometry anyway), and all units are given in equivalent dark irradiance units. 8 The boundary conditions for recording process are Ai(0, t), AR(0,t), and A ni(O, t) (for i = 1 ... N), where the input fields alternate in time in the incremental recording procedures The initial conditions are Ani(z, 0) = 0 for all i, and one may vary the input amplitudes Ai(0, t) to optimize the recording process.…”

“…4 for N = 1, ... 30 holograms. The input imagebearing beams (A's) alternate periodically in time every At = 0.05 (in normalized time units, 8 chosen according to the criterion of Ref. 5) until all Ant's reach their steady state and the recording process is terminated.…”