Propagation properties of hollow sinh-Gaussian beams through fractional Fourier transform optical systems

• We study the propagation of a HsG beam through an ideal FRFT optical system.
• The properties of a HsG beam through an apertured FRFT system have also been studied.
• The intensity of the HsG beam in FRFT plane is closely related some parameters.
• The FRFT optical systems provide a convenient way for laser beam shaping.

A hollow sinh-Gaussian beam (HsG) is an appropriate model to describe the dark-hollow beam. Based on Collins integral formula and the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, the propagation properties of a HsG beam passing through fractional Fourier transform (FRFT) optical systems with and without apertures have been studied in detail by some typical numerical examples. The results obtained using the approximate analytical formula are in good agreement with those obtained using numerical integral calculation. Further, the studies indicate that the normalized intensity distribution of the HsG beam in FRFT plane is closely related with not only the fractional order but also the beam order and the truncation parameter. The FRFT optical systems provide a convenient way for laser beam shaping.