Hamiltonian formulation of coupled-mode theory in waveguiding structures

Area of Science:

• Quantum Optics
• Photonics
• Theoretical Physics

Background:

• Coupled mode theory (CMT) traditionally uses orthonormal modes, which is complicated by the relativistic nature of photons.
• Existing CMT formulations face challenges when dealing with modes from different parent structures, like nearby waveguides.

Purpose of the Study:

• To develop a robust Hamiltonian formulation of CMT applicable to coupled modes from distinct parent structures.
• To address the complications arising from the relativistic nature of photons in mode basis selection.
• To generalize the formulation for both linear and nonlinear quantum optics problems.

Main Methods:

• Introduction of "dressed parent modes" as the effective coupled modes of the system.
• Circumvention of basis selection issues by directly constructing dressed modes from parent structure modes.
• Derivation of linear coupled mode equations and generalization to nonlinear regimes.

Main Results:

• Demonstrated a method to construct coupled modes directly from parent structure modes, overcoming basis limitations.
• Successfully derived linear coupled mode equations applicable to various waveguide structures, including photonic crystals.
• Established a pathway for immediate generalization to nonlinear quantum optics.

Conclusions:

• The proposed Hamiltonian formulation with dressed parent modes provides an accurate and versatile approach to CMT.
• This method simplifies the analysis of coupled systems involving different parent structures, applicable to both linear and nonlinear phenomena.
• The formulation is broadly applicable, from simple waveguides to complex photonic crystal structures.