Propagation characteristics of modified hollow Gaussian beams in nonlocal media under off-waist incident condition

In this paper, we focus on the evolutions of modified hollow Gaussian beams in highly nonlocal media under the off-waist incident condition. Two alternative analytical expressions describing the beam evolution are deduced, and the analytical expressions of the on-axis intensity, the second-order moment beam width, and the mean radius of curvature of the beam are all given. Numerical simulations are carried out to illustrate the propagation characteristics. The results show that, under the off-waist incident condition, the evolution of the beam width is always periodical whatever the beam power is, even when the beam power is equal to the critical power. The transversal pattern evolution is not symmetric about the middle of the evolution period. These results are much different from those under the on-waist incident condition.