Focusing properties and focal shift of vortex Hermite-cosh-Gaussian beams

In this work, we investigate the focusing properties of a vortex Hermite-cosh-Gaussian beam (vHChGB) passing through a converging lens system. The analytical propagation equation as well as the beam width expression of a focused vHChGB is derived based on the Huygens-Fresnel diffraction principle. From the obtained formulae, the effects of the Gaussian Fresnel number NF and the initial beam parameters, namely the beam order, the decentered parameter b, and the vortex charge number m, on the structure of the light intensity distribution and the beam spot size in the focal region are analyzed numerically. It is shown that the focal shift, which is determined from the minimum beam spot width criterion, is strongly dependent on the Fresnel number NF. And it is affected by the parameter b and the vortex charge m but is nearly insensitive to the beam order n. For a fixed value of NF, the focal shift is smaller when b or m is larger. The focal shift decreases monotonously with the increase of NF until it vanishes asymptotically for large values of NF. The obtained results may be useful for the applications of vHChGB in beam shaping and beam focusing.