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We introduce Pearcey Gaussian beams with the astigmatic phase (APPGBs) for what we believe is the first time and study their propagation in the chiral medium. APPGBs are split into the left and the right circularly polarized beams (LCP-APPGBs and RCP-APPGBs), of which we derive the analytical expression under the paraxial approximation. The results show that RCP-APPGBs have a longer autofocusing distance, which we can alter by adjusting the rotating factor or the chiral parameter. Besides, the increase of the rotating factor leads the autofocusing distances of both LCP-APPGBs and RCP-APPGBs to increase, while the increase of the chiral parameter leads to opposite effects on LCP-APPGBs and RCP-APPGBs. During propagation, the rotating factor can rotate the transverse intensity distribution, changing the rotation angular velocity. In addition, the rotating factor has a great impact on the phase distributions. Last, the effects of the rotating factor and the chiral parameter on the gradient force and the scattering force are analyzed.

Optical geometrical transformation is a novel and powerful tool to switch orbital angular momentum (OAM) states in modern optics. We demonstrate a scheme to operate multiplication and division in OAM by Fermat’s spiral transformation. The characteristics of the output beams in the case of integer and fraction OAM operations are presented in detail. Additionally, the power weight of the output OAM modes and the interference patterns of the output beams are reported to confirm the expected ability of OAM mode conversion by Fermat’s spiral transformation. We further investigate the evolution of OAM beams in operations theoretically and experimentally. This work provides a practical way to perform an optical transformation mapping on OAM beams. It can find application in optical communications with larger OAM mode numbers as well as quantum information in high-dimensional systems.

Partially coherent dual nonparaxial accelerating beams are constructed from the perspective of a superposition of fully coherent modes and incoherent interference effects among these modes. It is found that divergence angle and partial coherence of such beams can be altered by a curvature parameter. At the beginning of the propagation, such beams focus in the case of constructive interference, but they leave a central dip in the destructive case. Additionally, the extension of a crossed‐shaped intensity distribution caused by interference is slower than that of the transverse beam frame evidently. In particular, moving along the nonparaxial accelerating trajectory on the micrometer scale, the main lobes of these beams can diverge to a nearly flat angle. As such, these beams are likely to have applications in microscopic particle manipulation and coherence image processing.