We introduce a new class of ($2+1$) dimensional circle Pearcey beams (CPBs) with the abruptly autofocusing (AAF) characteristics. Compared with circular Airy beams, CPBs can increase the peak intensity contrast, shorten the focus distance and, especially, eliminate the oscillation after the focal point. Furthermore, we discuss the influence of the optical vortices (including on-axis, off-axis, and vortex pairs) on the light intensity distribution of the CPBs during propagating.

We show the tight-focusing properties of a linearly polarized circular Airy Gaussian vortex beam (CAiGVB) with a high-numerical-aperture objective lens; the light intensity distributions exhibit diversity with different positions of the vortex pairs (on-axis or off-axis vortex pairs). By choosing different optical distribution factors, the CAiGVB turns into a circular Airy vortex beam or Gaussian vortex beam, and the depth of focus can also be controlled. It is known that the vortex beam possesses both orbital and spin angular momentum. The spin density vector changes its direction in three-dimensional space during beam propagation, as long as it is not purely transverse or longitudinal, which would cause 3D polarization ellipse topologies. In contrast, the polarization topologies degenerate into 2D when the spin density vector is purely transverse or longitudinal. Furthermore, the direction of the spin density vector is closely related to the Gouy phase difference between longitudinal and transverse electric field components of the vectorial beam.

The evolution of the chirped ring Pearcey Gaussian vortex (CRPGV) beams in free space is numerically investigated. The optical bottle beam and two-consecutive optical bottle beam can be directly generated from a ring Pearcey Gaussian beam with central vortices and the second-order chirp in free space. In particular, the focusing intensity will decrease when the vortices’ locations are far away from the center of the beam or the topological charges are smaller. Our results show that the beam shapes, the normalized focusing intensity, and the focal length of the CRPGV beams can be controlled by choosing different beam parameters.

We address the propagation of vortex beams with the circular Airy–Gaussian shape in a ( 2 + 1 )-dimensional optical waveguide modeled by the fractional nonlinear Schrödinger equation. Systematic analysis of autofocusing of the beams reveals a strongly non-monotonous dependence of peak intensity in the focal plane on the corresponding Lévy index α , with a strong maximum at α ∼ 1.4 . Effects of the nonlinearity strength, the ratio of widths of the Airy and Gaussian factors in the input, as well as the beams’ vorticity on the autofocusing dynamics are explored. In particular, multiple autofocusing events occur if the nonlinearity is strong enough. Under the action of azimuthal modulational instability, an axisymmetric beam may split into a set of separating bright spots. In the case of strong fractality (for α close to one), the nonlinear beams self-trap, after the first instance of autofocusing, into a breathing vortical quasi-soliton. Radiation forces induced by the beams’ field are considered too, and a capture position for a probe nanoparticle is thus identified.

We introduce the propagation properties of the circular Pearcey Gaussian (CPG) waves in Kerr medium for the first time. The breathers-like structure and breathers-like groups structure (filamentions) of CPG waves will form due to the interaction between the linear waves and the nonlinear medium. The focusing characteristics in Kerr medium can be adjusted by the deviation factor and the initial input power of the CPG waves. By choosing appropriate input power, the imaginary part of the CPG waves can split into some wavelets during the propagation. It is worth noting that the distinctive stepwise focusing of the imaginary part of the CPG waves which can be applied to wave modulation. Furthermore, the numerical experiment results show good agreement with the numerical simulation results.

We simulate the propagation of the abruptly autofocusing chirped ring Pearcey Gaussian vortex (CRPGV) beams with caustics by modulating the phase of a circularly symmetric optical wavefront appropriately. The propagation characteristics of the CRPGV beams are explored in the Kerr medium. Different caustic surfaces of revolution which can be used as optical bottles are formed during the propagation. We also introduce the influence of the initial input power, the chirp factor and the stochastic type perturbations for the CRPGV beams during the propagation. Furthermore, the dynamics of the optical bottle and the breathers-like structures are explored in detail.