Analytic Mode Normalization for the Kerr Nonlinearity Parameter: Prediction of Nonlinear Gain for Leaky Modes.

Based on the resonant-state expansion with analytic mode normalization, we derive a general master equation for the nonlinear pulse propagation in waveguide geometries that is valid for bound and leaky modes. In the single-mode approximation, this equation transforms into the well-known nonlinear Schrödinger equation with a closed expression for the Kerr nonlinearity parameter. The expression for the Kerr nonlinearity parameter can be calculated on the minimal spatial domain that spans only across the regions of spatial inhomogeneities. It agrees with previous vectorial formulations for bound modes, while for 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.

Analytical mode normalization and resonant state expansion for bound and leaky modes in optical fibers - an efficient tool to model transverse disorder.

We adapt the resonant state expansion to optical fibers such as capillary and photonic crystal fibers. As a key requirement of the resonant state expansion and any related perturbative approach, we derive the correct analytical normalization for all modes of these fiber structures, including leaky modes that radiate energy perpendicular to the direction of propagation and have fields that grow with distance from the fiber core. Based on the normalized fiber modes, an eigenvalue equation is derived that allows for calculating the influence of small and large perturbations such as structural disorder on the guiding properties. This is demonstrated for two test systems: a capillary fiber and a photonic crystal fiber.