Propagation characteristics of elegant Laguerre–Gaussian beams in the fractional Hankel transform plane

The fractional Hankel transform (FRHT) system is applied to study the transformation properties of elegant Laguerre–Gaussian beams (ELGBs). An analytical formula is derived for the FRHT of ELGBs based on the definition of the FRHB. By using the derived formula and the definition of the kurtosis parameter, the analytical propagation expression for the kurtosis parameter of ELGBs in the FRHT plane is derived. Some detailed numerical examples are given to illustrate the analytical results. The results show that the intensity distribution and the kurtosis parameter of elegant Laguerre–Gaussian beams in the fractional Hankel transform plane not only depend on order p and index m of Laguerre polynomial, but also evolve periodically with fractional order p of fractional system.