Side-Polished Fiber
Evanescent power beyond the polished face, overlay index shift and polished-coupler transfer of a side-polished single-mode fiber.
Power beyond the face
1.12 %
V = 2.077, n_eff = 1.44638
Intensity at the face
18.2 %
of the core-boundary value; field 1/e length a/W (intensity a/2W) = 2.97 µm
Overlay shift Δn_eff
-1.56e-4
first-order, η(n_ov² − n₂²)/(2n_eff)
Polished coupler
46.3 %
coupled to a twin fibre; κ₀ = 514 m⁻¹, L_eff = 1.46 mm
The coupler result assumes the gap between the two polished faces is index-matched to the cladding (n₂); with this overlay it is indicative only.
Evanescent Power and Coupler Transfer vs Remaining Cladding
Model
ψ(r) = J₀(U) K₀(Wr/a)/K₀(W) in the cladding; η = ∫_(x > a+d) ψ² dA / ∫ψ² dA
κ(D) = (NA/(n₁a)) (U²/V³) K₀(WD/a)/K₁²(W), D(z) = 2(a + d) + z²/R, P₂ = sin²(∫κ dz)
The exact LP01 mode of the unpolished step-index fibre (eigenvalues U, W) gives the field beyond the polished face, which lies a distance d from the core boundary. Its tail decays as K₀(Wr/a), faster near the core than a pure exponential. η is the fraction of the mode power on the far side of the face; an overlay of index n_ov shifts n_eff to first order by η(n_ov² − n₂²)/(2n_eff).
Coupler: two identical fibres, each bent to radius R in a block and polished, are placed face to face. Their core spacing grows as z²/R away from the centre, so the coupling κ (Snyder's coefficient) acts over an effective length close to Digonnet and Shaw's √(πaR/W). The fibres are identical, so the transfer is complete when ∫κ dz = π/2.
Not included: the change of the mode itself near the face (first order overestimates the shift for n_ov < n₂ and underestimates it as n_ov approaches n_eff), polarization (scalar LP model; polished-fibre polarizers rely on it), polishing scatter and roughness, and a gap index different from n₂ in the coupler.
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
Exact LP01 K₀ tail: power beyond the face by quadrature, first-order overlay shift of n_eff; twin-fibre coupler, Snyder κ over D(z) = D₀ + z²/R. Scalar; mode change near the face not modelled.
- Gloge D. (1971). Weakly guiding fibers. Appl. Opt. 10, 2252. LP01 field, eigenvalue equation and power normalization. doi:10.1364/ao.10.002252
- Snyder A. W. (1972). Coupled-mode theory for optical fibers. J. Opt. Soc. Am. 62, 1267. Coupling coefficient of two identical step-index cores. doi:10.1364/josa.62.001267
- Digonnet M. J. F., Shaw H. J. (1982). Analysis of a tunable single mode optical fiber coupler. IEEE J. Quantum Electron. 18, 746. Polished-coupler geometry and effective interaction length. doi:10.1109/jqe.1982.1071574
- Snyder A. W., Love J. D. (1983). Optical Waveguide Theory. Chapman & Hall (Springer reprint 1984). Step-profile modes, perturbation theory and weakly coupled fibres. doi:10.1007/978-1-4613-2813-1