Analytic mode normalization for the Kerr nonlinearity parameter: Prediction of nonlinear gain for leaky modes

Based on resonant-state expansion and analytic mode normalization, we derive a master equation for the nonlinear pulse propagation in waveguide geometries that is capable of treating bounded as well as leaky modes. In the single-mode approximation, this equation transforms into the well-known nonlinear Schroedinger equation with a closed expression for the Kerr nonlinearity parameter. For bound modes, the Kerr nonlinearity parameter agrees with previous vectorial formulations, while the simulations can be restricted to the minimal spatial domain that spans only across the regions of spatial inhomogeneities. In the case of leaky modes, the Kerr nonlinearity parameter turns out to be a complex number with the imaginary part providing either nonlinear loss or even gain for the overall attenuating pulses. This nonlinear gain results in more intense pulse compression and stronger spectral broadening, which is demonstrated here on the example of liquid-filled capillary- type fibers.