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The effect of the photoinduced absorption of terahertz (THz) radiation in a semi-insulating GaAs crystal is studied by pulsed THz transmission spectroscopy. We found that a broad-band modulation of THz radiation may be induced by a low-power optical excitation in the spectral range of the impurity absorption band in GaAs. The measured modulation factor achieves 80%. The amplitude and frequency characteristics of the resulting THz modulator are critically dependent on the carrier density and relaxation dynamics in the conduction band of GaAs. In semi-insulating GaAs crystals, the carrier density created by the impurity excitation is controlled by the rate of their relaxation to the impurity centers. The relaxation rate and, consequently, the frequency characteristics of the modulator can be optimized by an appropriate choice of the impurities and their concentrations. The modulation parameters can be also controlled by the crystal temperature and by the power and photon energy of the optical excitation. These experiments pave the way to the low-power fast optically-controlled THz modulation, imaging, and beam steering.