D-Shaped Fiber
Evanescent power at the flat, external-index shift of the effective index and refractometric sensitivity of a D-shaped single-mode fiber.
Power beyond the flat
0.251 %
V = 2.280, n_eff = 1.44670
Intensity at the flat
4.82 %
of the core-boundary value; 3.5 µm of cladding left
Index shift Δn_eff
-2.67e-4
external index 1.333 vs cladding, first order; near the limit of first-order validity
Sensitivity ∂n_eff/∂n_ext
2.31e-3
MFD 10.08 µm (Petermann II)
Evanescent Power vs Flat Position
Model
ψ(r) = J₀(U) K₀(Wr/a)/K₀(W) for r > a; η = ∫_(x > h) ψ² dA / ∫ψ² dA
Δn_eff ≈ η (n_ext² − n₂²) / (2 n_eff), ∂n_eff/∂n_ext ≈ η n_ext / n_eff
The flat is a plane at distance h from the core centre. The exact LP01 mode of the round step-index fibre (eigenvalues U and W) gives η, the fraction of the mode power beyond that plane; its tail decays as K₀(Wr/a). Replacing the cladding beyond the flat by a medium of index n_ext perturbs n_eff to first order by η(n_ext² − n₂²)/(2n_eff), which sets the refractometric sensitivity of a D-fibre sensor.
First order uses the unperturbed mode. By the variational principle it overestimates the shift for n_ext < n₂ (water, air) and underestimates the shift and the sensitivity as n_ext approaches n_eff.
Not included: the change of the mode itself (see above), polarization (scalar LP model), the form birefringence of the D shape (it needs a vector mode solver; commercial polarization-maintaining D-fibres use an elliptical core), and roughness of the flat.
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 flat by quadrature. First-order index shift: too large for n_ext < n₂, too small as n_ext → n_eff (warned). Scalar; no D-shape birefringence.
- 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., 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