We investigate the evolution of optical beam in the nonlocal nonlinear media with loss and gain using the variational approach and the numerical simulation. When the loss gradually changes to the gain, the optical beams can restore to their initial states, the phenomenon we called the "adiabatic propagation". We have proved that, as long as the changing rate of the loss and gain is small enough, the gain can exactly compensate the loss and the adiabatic propagation can occur for any beams with any profiles. However, the optical beams will shed a part of its energy as dispersive waves if they are lumped amplification like the cases of optical pulses in fibers. The numerical simulations agree well with the variational results.
By using the variational approach, we discussed the evolution of Hermite-Gaussian beams of different orders in nonlocal nonlinear media whose characteristic length is set as different functions of propagation distance. We 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.
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