Propagation formulas and foci transformation of autofocusing beams

Autofocusing beams have attracted widespread attention due to their advantages in optical trapping, but their propagation behavior in complex optical systems is still unclear. Here, we obtain the analytical propagation formulas for autofocusing beams through optical systems described by ABCD matrices. Foci adjustment through a lens and oscillate behavior in a parabolic potential medium of the beams are discussed. Interestingly, besides the real focus, autofocusing beams possess a virtual focus, which can be observed with a lens or lens-like medium. Furthermore, we provide a method for predicting the focal position of autofocusing beams passing through a given optical system, which is beneficial for the design and parameter optimization of practical optical devices.