Focusing properties and focal shift of a vortex cosine-hyperbolic Gaussian beam

In this paper, we investigate the focusing properties of a vortex-cosh-Gaussian (vChG) beam passing through converging thin lens. Based on Huygens–Fresnel diffraction integral, derived analytical propagation equation as well width expression focused vChGB. It is shown that including focal shift vChGB are crucially dependent incident parameters namely decentered parameter and topological charge in addition to Gaussian Fresnel number \(N_{F}\). From typical numerical examples, it found transformed into multi-lobes structure shape at real focus plane, principal maximum intensity located away from axis. The amount shift, which determined minimum spot size criterion, strongly number, vortex m. obtained results may be useful for applications vChGBs shaping focusing.