Photonic Crystal Fiber
Effective V-number, single-mode cutoff, effective index, mode-field diameter and chromatic dispersion of a solid-core photonic crystal fiber from hole pitch Λ and diameter d (Saitoh–Koshiba relations).
V_eff (a_eff = Λ/√3)
2.933
d/Λ = 0.600, λ/Λ = 0.344; W = 2.401
Guidance
Multimode
second-mode cutoff λ_c = 3024 nm
Effective index n_eff
1.43513
silica core 1.44402, cladding n_FSM 1.41691
Effective cladding NA
0.279
√(n_co² − n_FSM²)
Mode-field diameter
5.11 µm
A_eff = πw² = 20.5 µm²; Nielsen fit at V_PCF 4.16
Dispersion D
52.5 ps/(nm·km)
anomalous; silica 21.9 + waveguide 30.6
Chromatic Dispersion
Total dispersion of the fundamental mode (blue) and of bulk fused silica (grey, dashed); the gap is the waveguide contribution of the air-hole cladding. The blue curve stops where the fit is invalid (W ≤ 0 or d/Λ outside 0.2–0.8). Zero dispersion near 1089 nm.
Effective V-Number vs Wavelength
The fibre carries one mode where V_eff (amber) is below 2.405 (red, dashed). V_eff levels off at short wavelengths, so for d/Λ below 0.427 it never reaches 2.405: the fibre is endlessly single-mode.
Model
V = A₁ + A₂/(1 + A₃ e^(A₄λ/Λ)), W = B₁ + B₂/(1 + B₃ e^(B₄λ/Λ)), A_i, B_i polynomials in d/Λ
n_eff² = n_silica(λ)² − (V² − W²)(λ/2πa_eff)², a_eff = Λ/√3
D = −(λ/c) d²n_eff/dλ²; w = a_eff(0.65 + 1.619V^−1.5 + 2.879V^−6)
Solid-core fibre with a triangular lattice of air holes. Confinement loss depends on the number of hole rings and is not modelled; for a few rings and small d/Λ it can dominate.
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
Saitoh–Koshiba empirical V and W (FEM fits, 0.2 ≤ d/Λ ≤ 0.8, λ/Λ ≤ 2), a_eff = Λ/√3, silica Sellmeier core; V < 2.405 single-mode; Nielsen et al. PCF spot size from V_PCF = √3 W (V_PCF ≥ 0.5). Blank where the fit gives W ≤ 0. No confinement loss.
- Saitoh K., Koshiba M. (2005). Empirical relations for simple design of photonic crystal fibers. Opt. Express 13, 267. Eqs. (1)–(4), Tables 1–2. doi:10.1364/opex.13.000267
- Koshiba M., Saitoh K. (2004). Applicability of classical optical fiber theories to holey fibers. Opt. Lett. 29, 1739. a_eff = Λ/√3. doi:10.1364/ol.29.001739
- Saitoh K., Tsuchida Y., Koshiba M., Mortensen N. A. (2005). Endlessly single-mode holey fibers: the influence of core design. Opt. Express 13, 10833. ESM limit d/Λ = 0.424. doi:10.1364/opex.13.010833
- Birks T. A., Knight J. C., Russell P. St. J. (1997). Endlessly single-mode photonic crystal fiber. Opt. Lett. 22, 961. doi:10.1364/ol.22.000961
- Nielsen M. D., Mortensen N. A., Folkenberg J. R., Bjarklev A. (2003). Mode-field radius of photonic crystal fibers expressed by the V parameter. Opt. Lett. 28, 2309. w/Λ = 0.7078V^−0.2 + 0.2997V^−1.5 + 0.0037V^−6. doi:10.1364/ol.28.002309
- Malitson I. H. (1965). Interspecimen comparison of the refractive index of fused silica. J. Opt. Soc. Am. 55, 1205. doi:10.1364/josa.55.001205