Dynamics of Magnetophotorefractive Wave Mixing
Authors: Johansen Per Michael , Jensen Arne Skov 
Published: February 2023

In resonant photorefractive experiments the charge-density distribution in the conduction band is moving and an intrinsic magnetic field proportional to the velocity of the distribution is produced. In such cases the free electron plasma can interact with an externally applied magnetic field. On top of that materials exposed to a magnetic field exhibit Faraday rotation (Voigt birefringence) of the state of polarization of the optical bean.*. The purpose of the present summary is to describe the effect of both phenomena in a standard wave mixing setup. The photorefractive band transport model in the case v here magnetic fields are present is only modified via a term in the current density equation and reads (1) where e is the numeric elementary charge, µ the free electron mobility, n the number density' of free electrons, E the total electric field which may consist of an externally applied term and the norihnear generated spacecharge field, D the diffusion constant, and B the magnetic field term which may consist of an external part and an internal part (stemming from electronic motion). The remaining equations describing the photorefractive effect are the conventional ones. Now, by inferring the small modulation approximation, a perturbational relation for the physical quantities and, by Fourie- transforming in space-time, a steady-state response function for the space-charge electric field are found. ThL photorefractive frequency response function, expressed as a function of the conventional photorefractive parameters, is given by where the function P is given by (3) The actual definition of the parameters in Eqs. (2) and (3) can be found in Ref. [1]. Equation (2) is taken in the case where an externally magnetic field is applied perpendicular to the grating wave vector K. If it is assumed that the constituents of the incident intensity are monochromatic plane waves, that may be frequency shifted a small amount, the Fourier transformation of the perturbed intensity is a sum of δ-functions peaking at k = ±K and at ω = ±Ω. The space-charge field is now given by (4) A typical response function for GaAs:Cr is shown in Fig. 1. The characteristic parameters used in the calculation are taken from Refs. [2, 3, 4], The externally applied electric field is Eθ = 10 V/m, the applied magnetic field is Bθ = 2 T, the angle between the applied electric field and the grating wave vector is 20°, and the angle between the applied magnetic field and the z-axis is 45°. Recently, experiments on photorefr?ctive wave mixing with Faraday rotation in a diluted magnetic semiconductor of Cd] MnTe have been demonstrated. The gain showed an oscillatory behaviour as a function of the' magnetic field, and it was demonstrated that the magnetic field controls the direction and magnitude of the energy transfer. In what follows the coupled wave equations including uie effect of optical activity, the Faraday effect, and the effect of linear absorption will be given. Mcreover, the magnetic effects in the space-charge field given above will be taken into account. Numeric solutions to the coupled differential equations will be given. The equations are given in the following form (5) (6) (7) (8) where α is the absorption constant, Γθ the birefringence coupling constant, p the rotatory power, V the Verdet constant, <5 the angle between the z-axis and the magnetic field vector, and Γ (K,Ω) the photorefractive coupling parameter. A(. and are the polarization components perpendicular and^ parallel to the plane of incidence, respectively. In Fig. 2 a typical numerical solution to the above equations is given for photorefractive GaAs:Cr. The applied magnetic field is Bθ = 2 T, The electric field is Eθ - 106 V/m, the grating spacing is A - 34 µm, and the frequency shift is Ω = 94 rad/s. In such a material the optical activity is zero so the oscillations are due to the Faraday effect. Figure 1. The space-charge response function for GaAs:Cr. Figure 2. Intensity of the probe beam in GaAs:Cr as a function of distance in the crystal for a 10 mm crystal.