Hamiltonian formulation of the nonlinear coupled mode equations

Area of Science:

• Nonlinear optics
• Condensed matter physics
• Photonics

Background:

• Coupled mode equations (CME) describe pulse propagation in periodic media.
• Photonic band gaps (PBGs) arise from periodic structures and affect light propagation.
• Kerr media exhibit intensity-dependent refractive indices, leading to nonlinear effects.

Purpose of the Study:

• To derive a canonical Hamiltonian formulation for nonlinear CME in a 1D periodic Kerr medium.
• To investigate the validity of CME for pulse propagation near a photonic band gap.
• To explore the applicability of CME to higher-dimensional photonic band-gap materials.

Main Methods:

• Derivation of a canonical Hamiltonian formulation.
• Analysis of pulse propagation dynamics in a 1D periodic Kerr medium.
• Comparison of CME predictions with the linear dispersion relation of the medium.

Main Results:

• The derived Hamiltonian represents the electromagnetic field energy.
• CME provide an excellent approximation to the dispersion relation, even for large PBGs (25% of Bragg frequency).
• The Hamiltonian formulation offers a new perspective on nonlinear pulse dynamics in periodic media.

Conclusions:

• The canonical Hamiltonian formulation is a valid approach for nonlinear CME in periodic Kerr media.
• CME are effective for describing pulse propagation near PBGs in 1D systems.
• Generalized CME may be suitable for describing 2D and 3D photonic band-gap materials with large index contrasts.