Theory of the photorefractive effect for Bi_12SiO_20 and BaTiO_3 with shallow traps

We analytically solve charge-transport equations for a photorefractive crystal with shallow and deep traps. We predict that, if shallow traps can accumulate a high density of charge, the photorefractive trap density and space-charge field will be strong functions of light intensity and the photoconductivity will scale sublinearly with intensity. We show that, depending on light intensity and grating spacing, shallow-trap charge gratings form either in phase or out of phase with the light pattern. As shallow traps thermally depopulate in the dark, the space-charge field either partially decays or partially develops for a few seconds. The amount of decay increases as grating spacing increases in Bi12SiO20 and as grating spacing decreases in BaTiO3.