Propagation properties of controllable dark-hollow beams through fractional Fourier,transform systems

Based on the definition of fractional Fourier transform （FrFT） in the cylindrical coordinate system, the propagation prop- erties of a controllable dark-hollow beam （CDHB） are investigated in detail. An analytical formula is derived for the FrFT of a CDHB. By using the derived formula, the properties of a CDHB in the FrFT plane are illustrated numerically. The results show that the properties of the intensity of the beam in the FrFT are closely related to not only the fractional order but also initial beam parameter, beam order and the lens focal length of the optical system for performing FrFT. The derived formula provides an effective and convenient way for analyzing and calculating the FrFT ofa CDHB.