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

Area of Science:

• Nonlinear optics
• Waveguide theory
• Quantum optics

Background:

• Nonlinear pulse propagation is crucial in optical systems.
• Existing models often focus on bound modes, limiting applicability to leaky modes.
• Understanding nonlinear effects in complex waveguide geometries is essential.

Purpose of the Study:

• To derive a general master equation for nonlinear pulse propagation in waveguides.
• To investigate the behavior of both bound and leaky modes.
• To analyze the implications of a complex Kerr nonlinearity parameter for leaky modes.

Main Methods:

• Resonant-state expansion with analytic mode normalization.
• Derivation of a general master equation for nonlinear pulse propagation.
• Single-mode approximation leading to the nonlinear Schrödinger equation.

Main Results:

• A general master equation valid for bound and leaky modes was derived.
• A closed-form expression for the Kerr nonlinearity parameter was obtained.
• For leaky modes, the Kerr nonlinearity parameter is complex, enabling nonlinear gain and enhanced pulse compression/spectral broadening.

Conclusions:

• The derived master equation provides a unified framework for nonlinear pulse propagation in waveguides.
• The complex Kerr nonlinearity in leaky modes offers new possibilities for manipulating optical pulses.
• Demonstrated nonlinear gain in liquid-filled capillary-type fibers highlights practical applications.