Coupled-Mode Power Transfer
Coupled-mode power transfer between two waveguides: coupled and through power, maximum transfer κ²/ψ² and the length that reaches it, power along the coupler and the response to the mismatch Δβ.
Coupled power P₂
0.2297 mW
22.97 % of P₀
Through power P₁
0.7703 mW
77.03 % of P₀
Maximum transfer κ²/ψ²
99.01 %
first reached at π/(2ψ) = 3.126 mm
ψ = √(κ² + (Δβ/2)²)
0.5025 mm⁻¹
exchange period π/ψ = 6.252 mm
Two parallel guides exchange power through their overlapping evanescent fields. Launching P₀ into guide 1, the coupled-mode equations give P₂(z) = P₀(κ²/ψ²) sin²(ψz) and P₁ = P₀ − P₂, with ψ = √(κ² + (Δβ/2)²). Identical guides (Δβ = 0) transfer all the power after the coupling length L_c = π/(2κ). A mismatch speeds up the exchange but caps it at κ²/ψ², which is why a coupler made of unequal guides acts as a wavelength filter. For κ computed from the core spacing of two fibres, see the fiber coupler page.
Power along the coupler
- Coupled guide
- Through guide
- This length
Coupled fraction at length L vs mismatch
- Coupled fraction
- This Δβ
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Lossless co-directional coupled-mode theory for two modes with constant κ and mismatch Δβ: P₂ = (κ/ψ)² sin²(ψL), ψ = √(κ² + (Δβ/2)²). Not modelled: taper profile, κ(λ), polarization, excess loss.
- Yariv A. (1973). Coupled-mode theory for guided-wave optics. IEEE J. Quantum Electron. 9, 919. Two-mode co-directional coupling. doi:10.1109/JQE.1973.1077767