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Through investigating the sinh-Gaussian beam with complex beam parameters, it is the first time to find that such beams can carry vortices and exhibit dark hollow intensity distributions when the complex beam parameters are sufficiently small. A closed-form propagation equation for sinh-Gaussian beams through paraxial ABCD optical systems is derived based on the Collins formula and illustrated with numerical methods. It is shown that the perfect hollow configuration can retain a quite long propagating distance under small complex beam parameters. The analytical discussions affirm the numerical conclusions.

Propagation invariance and dark hollow structures of sinh-Gaussian beams with small complex parameters