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.

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The self-accelerating Airy Ince–Gaussian (AiIG) and Airy helical Ince–Gaussian (AihIG) wave packets in strongly nonlocal nonlinear media (SNNM) are obtained by solving the strongly nonlocal nonlinear Schrödinger equation. For the first time, the propagation properties of three dimensional localized AiIG and AihIG breathers and solitons in the SNNM are demonstrated, these spatiotemporal wave packets maintain the self-accelerating and approximately non-dispersion properties in temporal dimension, periodically oscillating (breather state) or steady (soliton state) in spatial dimension. In particular, their numerical experiments of spatial intensity distribution, numerical simulations of spatiotemporal distribution, as well as the transverse energy flow and the angular momentum in SNNM are presented. Typical examples of the obtained solutions are based on the ratio between the input power and the critical power, the ellipticity and the strong nonlocality parameter. The comparisons of analytical solutions with numerical simulations and numerical experiments of the AiIG and AihIG optical solitons show that the numerical results agree well with the analytical solutions in the case of strong nonlocality.

A type of chirped Airy Gaussian vortex (CAiGV) localized wave packets in a quadratic index medium are studied by solving the paraxial differential equation. For the first time, the propagation properties of spatiotemporal CAiGV light bullets in the quadratic index medium are demonstrated. Some typical examples of the obtained solutions are based on the temporal and spatial chirp parameters, the initial velocity, the distribution factor, and the topological charge. The radiation force of the spatial CAiGV wave packet on a Rayleigh dielectric particle has the periodically reversion and recovery abilities due to the quadratic potential. What we report here can obtain different radiation force trajectory and may have potential application in optical tweezing and bio-medical field.