Analytic Expressions for the Interaction of Modified Hollow Gaussian Beams in Highly Nonlocal Nonlinear Media

We obtain two equivalent analytic expressions describing the propagation of two interacting modified hollow Gaussian beams in highly nonlocal nonlinear media. We carry out some numerical simulations to illustrate their interaction properties, such as the transverse intensity distributions and the on-axis intensity evolutions. We find that the evolutions of two interacting modified hollow Gaussian beams in highly nonlocal nonlinear media are periodical. The two beams always attract each other for the in-phase case and even for the out-of-phase case. The difference is that a prominent intensity peak appears at the center of the interacting region for the in-phase case, while it is zero for the out-of-phase case.