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We consider the propagation of Gaussian beams in a waveguide with gain andloss in the paraxial approximation governed by the Schr\"odinger equation. Wederive equations of motion for the beam in the semiclassical limit that arevalid when the waveguide profile is locally well approximated by quadraticfunctions. For Hermitian systems, without any loss or gain, these dynamics aregiven by Hamilton's equations for the center of the beam and its conjugatemomentum. Adding gain and/or loss to the waveguide introduces a non-Hermitiancomponent, causing the width of the Gaussian beam to play an important role inits propagation. Here we show how the width affects the motion of the beam andhow this may be used to filter Gaussian beams located at the same initialposition based on their width.

Propagation of Gaussian beams in the presence of gain and loss