Dynamics of circular Airy beams with spatial and frequency modulations in a cubic-quintic nonlinear fractional Schrödinger equation: from linear to soliton control.

The transmission dynamics of a circular Airy beam (CAB) with quadratic phase modulation (QPM) and cross-phase modulation (XPM) in the cubic-quintic nonlinear fractional Schrödinger equation (FSE) optical system is investigated. In the linear case, the energy distribution of the beam is affected by XPM and the focusing position of the beam is influenced by QPM. CAB undergoes splitting and its intensity is shifted as the absolute value of the XPM coefficient (||) increases. When XPM coefficients are opposite to each other, CABs are transmitted in opposite states in space. The degree of interference between beams gradually enhances with the increase of the XPM coefficient, leading to the formation of interference resembling water ripples. In the nonlinear regime, different results (evolving into solitons or undergoing diffraction transmission) are observed in CABs based on cubic-quintic nonlinear combination modes. Furthermore, nonlinear combination modes that can generate solitons and changes in solitons under actions of XPM and QPM are studied in detail. The distribution of solitons can be altered by positive or negative XPM, and solitons exist when QPM coefficients are within a certain range. The spacing and number of solitons can be modified by adjusting the magnitude of the QPM coefficient. The research shows that the control for solitons (number, distribution, and propagation) can be achieved through flexible selection of cubic-quintic nonlinear combination modes and parameter optimization (XPM coefficient, QPM coefficient, Lévy index).