Beam dynamics in quadratically nonlinear waveguides with gain and losses

A propagation of optical beams in active waveguides with quadratic nonlinearity is considered. It is shown that gain in one harmonic can compensate losses in the other harmonic. As a result of this process, stationary beams can be formed in the system. Exact solutions for stationary modes of a single waveguide are obtained, and their stability is analyzed. A possibility of the Hopf bifurcation that results in emergence of stable periodic regimes in a monomer is demonstrated. Stationary solutions are also found for a dimer with identical waveguides and a dimer with parity-time symmetry. The stability analysis demonstrates that stable beams exist in a wide range of the system parameters.