Propagation properties of Hollow higher-order cosh-Gaussian beams in quadratic index medium and Fractional Fourier transform

Hollow higher-order cosh-Gaussian (HhCG) beam is a new mathematical model to describe the dark-hollow beam. An analytical expression for HhCG beams propagating through a paraxail ABCD optical system is derived and used to investigate its propagation properties in a quadratic index medium (QIM) and a Fractional Fourier transform (FRFT) optical system. By using the derived expressions, the properties of HhCG beams of both systems are graphically illustrated with numerical example. Several influence parameters on the evolution of HhCG in the both systems QIM and FRFT, respectively, are discussed in detail. Results show that the intensity distribution of HhCG evolves periodically during propagation in the QIM. In FRFT system, it is found that the normalized intensity distribution of HhCG beams plane depends not only on the fractional order but also on the beam order and the truncation parameter. The results obtained in this work are valuable to provide a convenient way for HhCG beams shaping, optical fibers, lenses, and other fields.