Adiabatic evolution of optical beams of arbitrary shapes in nonlocal nonlinear media

We discuss evolution of Hermite-Gaussian beams of different orders in nonlocal nonlinear media whose characteristic length is set as different functions of propagation distance, using the variational approach. It is proved that as long as the characteristic length varies slowly enough, all the Hermite-Gaussian beams can propagate adiabatically. When the characteristic length gradually comes back to its initial value after changes, all the Hermite-Gaussian beams can adiabatically restore to their own original states. The variational results agree well with the numerical simulations. Arbitrary shaped beams synthesized by Hermite-Gaussian modes can realize adiabatic evolution in nonlocal nonlinear media with gradual characteristic length.