Fiber Taper Calculation
Waist and profile of a heat-and-pull fiber taper from the hot-zone length and elongation, where the light leaves the core, and whether the taper is adiabatic.
Waist Diameter
27.9 µm
taper ratio r₀/r_w = 4.482; core Ø there 1.83 µm
Taper Lengths
20.00 mm
waist 5.00 mm + 2 transitions of 7.50 mm
Core–Cladding Transition
49.4 µm
cladding Ø where LP01 n_eff = n₂ (V = 0.818; small-V estimate √(2/(ln S + ¼)) 0.820)
Waist Mode (LP01)
n_eff 1.44350
guided by the cladding–surround boundary; V_core 0.462, V_clad 58.9
Evanescent Decay Length
0.237 µm
1/e field depth into the surround, λ/(2π√(n_eff² − n₃²))
Adiabaticity (worst Ω/Ω_d)
0.50
below Love's delineation; worst at Ø 60.2 µm
The waist guide is multimode (V_clad = 58.9); an adiabatic taper still keeps the light in its fundamental mode. It becomes single-mode below Ø 1.14µm (V_clad < 2.405).
Taper Profile (Birks–Li)
Adiabaticity: Taper Angle vs Love's Delineation Curve
The taper is locally adiabatic where the taper angle (amber) lies below the delineation curve (green). The curve dips where LP01 leaves the core, near Ø 49 µm.
Model
r_w = r₀ (1 + αx/L₀)^(−1/2α) → r₀ e^(−x/2L₀) (α = 0); L_w = L₀ + αx; z₀ = (1 − α)x/2
Ω(r) = r (r/r₀)^(2α) / ((1 − α)L₀); Ω_d = ρ(β₁ − β₂)/2π
transition: J₀(V)/(V J₁(V)) = ln S + K₀(W)/(W K₁(W))
Shape: Birks and Li's volume-conservation model. A hot zone of length L₀ + αx softens the fibre while it is stretched by x; α = 0 is a constant hot zone (exponential transitions), α = −0.5 gives linear transitions.
Modes: the core and cladding shrink together inside the surround (index n₃). The circularly symmetric LP0m modes of this core–cladding–surround step profile are solved exactly in the scalar approximation. LP01 starts in the core; once its n_eff falls below n₂ it is guided by the cladding–surround boundary. Love's length-scale criterion compares the local taper length r/Ω with the LP01–LP02 beat length; the taper is adiabatic where Ω < Ω_d (LP02 replaced by the radiation edge kn₃ when it is not guided). The criterion is approximate, to a factor of order one.
Not included: vector (HE/EH) corrections at the cladding–surround boundary (important for waists near the wavelength), dopant diffusion and the real hot-zone temperature profile, non-circular tapers (which couple to LP1m), and the loss value itself (the criterion only says where coupling becomes significant).
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
Birks–Li volume-conservation shape (exact for its hot-zone model); scalar three-layer LP0m modes along the taper; Love's length-scale adiabaticity criterion, approximate to a factor of order one.
- Birks T. A., Li Y. W. (1992). The shape of fiber tapers. J. Lightwave Technol. 10, 432. Waist radius, transition profile and lengths for a hot zone L₀ + αx. doi:10.1109/50.134196
- Love J. D., Henry W. M., Stewart W. J., Black R. J., Lacroix S., Gonthier F. (1991). Tapered single-mode fibres and devices. Part 1: Adiabaticity criteria. IEE Proc. J 138, 343. Delineation curve Ω = ρ(β₁ − β₂)/2π. doi:10.1049/ip-j.1991.0060
- Snyder A. W., Love J. D. (1983). Optical Waveguide Theory. Chapman & Hall (Springer reprint 1984). Scalar modes of step profiles; TE01/TM01 cutoff at V = 2.405. doi:10.1007/978-1-4613-2813-1