Trajectory engineering via a space-fractional Schrödinger equation with dynamic linear index potential**Project supported by the Natural Science Research Project of Anhui Provincal Education Department of China (Grant Nos. KJHS2018B01 and KJ2018A0407), the National Natural Science Foundation of China (Grant No. 11804112), the Natural Science Foundation of Anhui Province of China (Grant No. 1808085QA22), and Start-up Fund of Huangshan University, China

Abstract

We theoretically and numerically study the propagation dynamics of a Gaussian beam modeled by the fractional Schrödinger equation with different dynamic linear potentials. For the limited case α = 1 (α is the Lévy index) in the momentum space, the beam suffers a frequency shift which depends on the applied longitudinal modulation and the involved chirp. While in the real space, by precisely controlling the linear chirp, the beam will exhibit two different evolution characteristics: one is the zigzag trajectory propagation induced by multi-reflection occurring at the zeros of spatial spectrum, the other is diffraction-free propagation. Numerical simulations are in full accordance with the theoretical results. Increase of the Lévy index not only results in the drift of those turning points along the transverse direction, but also leads to the delocalization of the Gaussian beam.