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.

A virtual source that yields a family of a Pearcey wave is demonstrated. A closed-form expression is derived for the Pearcey wave that simplifies to the paraxial Pearcey beam (PB) in the appropriate limit. From the perturbative series representation of a complex-source-point spherical wave, an infinite series nonparaxial correction expression for a PB is obtained. The infinite series expression of a PB can give accuracy up to any order of the diffraction angle. By applying the integral representation of the Pearcey wave, the first three terms in the nonparaxial correction series to the paraxial PB are provided.

Controlling the focal length and the intensity of the optical focus in the media is an important task. Here we investigate the propagation properties of the sharply autofocused ring Airy Gaussian vortex beams numerically and some numerical experiments are performed. We introduce the distribution factor b into the initial beams, and discuss the influences for the beams. With controlling the factor b, the beams that tend to a ring Airy vortex beam with the smaller value, or a hollow Gaussian vortex beam with the larger one. By a choice of initial launch condition, we find that the number of topological charge of the incident beams, as well as its size, greatly affect the focal intensity and the focal length of the autofocused ring Airy Gaussian vortex beams. Furthermore, we show that the off-axis autofocused ring Airy Gaussian beams with vortex pairs can be implemented.

By using the method of moments, the critical power of the Airy–Gaussian (AiG) beam is given for different decay factors and different distribution factors numerically. The critical power Pcr of the AiG beam decreases as the distribution factor increases. Using the split-step Fourier method, the propagations of the AiG beam in the free space and in the Kerr medium are shown. It has been found that the self-acceleration effect becomes weaker when the distribution factor increases. As the initial input power increases, we can observe the quasi-breather finally. From the root mean square (rms) beam width and the peak intensity figures, one can see that the beam with large distribution factor is more sensitive to the change of the initial input power.

The evolution of the three-dimensional (3D) self-accelerating Airy-Ince-Gaussian (AiIG) and Airy-Helical-Ince-Gaussian (AiHIG) light bullets is investigated by solving the (3+1)D linear spatiotemporal evolution equation of an optical field analytically. As far as we know, the numerical experimental demonstrations of the Ince-Gaussian (IG) and Helical-Ince-Gaussian (HIG) beams in various modes are first developed to study the evolution characteristics of the different 3D spatiotemporal light bullets. A conclusion can be drawn that the different photoelastics, pulse stacked, boundary, elliptical ring and physically separated in-line vortices can be achieved by adjusting the ellipticity, the evolution distance and the mode-number of light bullets.

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