Fractional Fourier transform of double-half inverse Gaussian hollow beams

In this paper, we study an appropriate mathematical model of dark-hollow beam by concerning double-half inverse Gaussian hollow (DHIGH) beams passing through fractional Fourier transform (FRFT) optical systems. Based on Collins integral formula and using the expression of the hard-edged aperture function into a finite sum of complex Gaussian functions, analytical expressions for the propagation properties of DHIGH beams through the FRFT optical systems without and with apertures have been derived in detail. Some numerical simulations are given to illustrate the normalized intensity distribution of DHIGH beams in a FRFT system by studying the important factors of both incident beams order and the FRFT order. Our results obtained by using an approximate analytical formula, are in a good agreement with those obtained by using numerical integral method. The results show that the FRFT optical system is a convenient way for DHIGH beams shaping.