Manipulating circular Airy beam dynamics with quadratic phase modulation in fractional systems under some diffraction modulations and potentials.

Based on a split-step Fourier algorithm, the transmission of circular Airy beams with quadratic phase modulation (QPM) is investigated in the fractional Schrödinger equation (FSE) under diffraction modulations (periodic modulation, linear modulation and power function modulation) and external potentials (parabolic potential and linear potential). The results show that QPM is able to change the focusing position and intensity, as well as the transmission trajectory of the beam. In a periodic modulation, the circular Airy beam (CAB) exhibits periodic variation characteristics, and the beam splitting is retarded under the action of the QPM. The self-focusing distance of the beam is significantly reduced, and its transmission trajectory and beam width are altered by the QPM under the linear modulation. The CAB progressively evolves into a non-diffraction beam under the power function modulation, and the QPM is able to reduce the light intensity and increase the beam width as the Lévy index decreases. In a parabolic potential, CABs display autofocusing and defocusing behavior, and the QPM affects the intensity distribution and optical width of the beam. The CAB is deflected and evolves periodically in a linear potential. The beam width increases and gradually stabilizes with the addition of the QPM. The propagation of CABs controlled with QPM in parabolic and linear potentials is also analyzed in the frequency domain. The results demonstrate that we can control the transmission of CABs in an FSE optical system by rationally setting parameters such as QPM, modulation coefficients, and external potentials.