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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.

We introduce a new class of cosine complex variable function Airy–Gaussian (Cos-CVFAiG) beams and investigate their propagation through the gradient-index medium. Deriving the analytical expression, we show the intensity and phase distributions, the propagation trajectory and the Poynting vector. We carry out the numerical experiment to verify the rationality of the model and find that the light intensity distribution is parallel to the y axis in the transverse plane. With the distribution factor increasing, the axial intensity maximum and the number of the side lobes increase as well. Additionally, the distribution factor can change the bend extent of the ballistic trajectory and the intensity distribution region in the x–z plane. We also compare the influences of the gradient refractive index coefficient with the distribution factor on the propagation properties.

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.