Based on the second-order and the higher-order moments, analytical expressions for the beam propagation factors of a Lorentz–Gauss vortex beam with have been derived, and analytical propagation expressions for the kurtosis parameters of a Lorentz–Gauss vortex beam with through a paraxial and real ABCD optical system have also been presented. The factor is determined by the parameters and and decreases with increasing the parameter or . The factor is validated to be larger than 2. The kurtosis parameters depend on the diffraction-free ranges of the Lorentz part, the parameters and , and the ratio . The kurtosis parameters of a Lorentz–Gauss vortex beam propagating in free space are demonstrated in different reference planes. In the far field, the kurtosis parameter decreases with increasing one of the parameters and . Upon propagation, the kurtosis parameter first decreases, then increases, and finally tends to a saturated value. In any case, the kurtosis parameter is larger than 2. This research is beneficial to optical trapping, guiding, and manipulation of microscopic particles and atoms using Lorentz–Gauss vortex beams.
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