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We have numerically investigated the propagation properties of ring Airy beams modeled by a fractional Schrodinger equation (FSE). Beam splitting is exhibited and clarified from the view of group delay. Compared with the case of the condition in free space, the focal intensity is greatly enhanced and presents a trend from increasing to decreasing with the decrement of the Levy index, while the focal length largely increases; in addition, the parabolic trajectory is distorted with the Bessel pattern gradually disappearing. However, both the focal intensity and its length get decreased due to the absence of the initial phase of ring Airy beams; simultaneously, the size of the focal spot decreases for a larger Levy index. The changes of focusing properties are clearly clarified. Moreover, without initial phase, the ratio of focal intensity under the Levy index to the condition in free space is given to show the faster growth of focal intensity when the Levy index is lower than 1.25; otherwise, the trend gets reversed. The autofocusing properties of ring Airy beams modeled by FSE can find applications in optical trapping and manipulation.

The dynamics of finite energy Airy beams modeled by the fractional Schrodinger equation (FSE) with a linear potential are numerically investigated. Different from the propagation properties of Airy beams described by the standard Schrodinger equation, the splitting phenomenon, which is presented under the FSE without potential, is influenced by the quadratic chirp and the Levy index. As the linear potential is considered, the periodic evolution of Airy beams is shown, and the period is inversely proportional to the linear potential coefficient. The beam width can undergo an abrupt decrease or increase depending on the sign of potential coefficient. The beam deviation distance increases with the beam width greatly changed if the Levy index increases. Moreover, the quadratic chirp does not influence the evolution period but the intensity distribution. In addition, the intriguing properties are analytically clarified. All these features confirm the promising applications of Airy beams in optical manipulation and optical switch.

The controlling optical frequency in the regime of an optical event horizon by the collision between a soliton and a probe pulse is investigated, and the process by which the probe is converted into the generated pulse (idler) with new optical frequency is reversible. For the two cases in which the probe wave interacts with a bright/dark soliton, a unified analytical formula governing the reversible frequency conversion process is derived that can predict the frequency of the idler very well. This reversible frequency-conversion process is validated by cascaded four-wave mixing (FWM) triggered by a set of quasi-CW fields, strongly suggesting that the process arose from cascaded Bragg scattering. It establishes a link between the nonlinear interactions of a soliton with the probe and FWM of monochromatic quasi-CWs.