coupledmode

Coupled-mode theory (CMT) is a mathematical framework used to describe the exchange of energy between interacting modes in systems where the modes are weakly but non-negligibly coupled. It is widely applied in photonics, acoustics, microwave engineering, and mechanical and quantum systems to model how energy transfers between resonant modes, or between a mode and external channels.

In its common form, the system is reduced to a small set of slowly varying complex amplitudes a1(t), a2(t), etc., each representing a mode. The time evolution is described by coupled differential equations that include intrinsic losses, detunings from resonance, and a coupling coefficient between modes. A simple two-mode example is:

da1/dt = (-γ1 - iΔ1) a1 - i κ a2 + s1_in

da2/dt = (-γ2 - iΔ2) a2 - i κ a1 + s2_in

Outputs are taken as linear combinations of inputs and mode amplitudes, for example through s_out terms that depend on s_in and on a1, a2.

Solving the equations yields the steady-state response and the frequency-domain transmission or reflection spectra. The coupling strength κ, detunings Δ1, Δ2, and loss rates γ1, γ2 determine phenomena such as normal-mode splitting, resonance shifts, and the efficiency of power transfer between modes. Different regimes appear: weak coupling produces small energy exchange and broad features, while strong coupling can create split resonances and enhanced cross-coupling.

Applications include optical microring and microdisk resonators, directional couplers, filters, modulators, and sensors in integrated photonics, as well as analogous systems in acoustics and microwave circuits. CMT provides a tractable, intuitive model that complements full numerical simulations and helps interpret experimental spectra.