We show that the refractive index change induced by a focused incident beam with an additional incoherent uniform illumination in photovoltaic-photorefractive crystals under open-circuit condition has a saturable nonlinearity form. The incoherent uniform background illumination can be used to increase the effective dark irradiance. The formation time of the photovoltaic soliton can be decreased by keeping the intensity of the soliton at a higher value without over-saturation by use of the background illumination.