Rare-Earth-Doped Fiber Amplifier
Ion density from wt-ppm, LP01 overlap, pump and signal absorption, 90 % absorption length, saturation power, gain and noise figure of an Er-, Yb-, Tm- or Ho-doped fiber.
Dopant: Erbium (Er³⁺), M = 167.26 g/mol, in silica (ρ = 2201 kg/m³). Presets fill cross-sections measured in alumino-germanosilicate (Barnes et al. 1991); they vary with the glass by tens of percent, so enter your fibre's own data where you have it.
Ion Density N
7.925e+24 m⁻³
1000 wt-ppm
NA and V
0.2232, V = 1.448
at the signal; LP11 cutoff 933 nm
Overlap Γ
0.807 / 0.509
pump / signal; LP01 power in the core
Pump Absorption
4.724 dB/m
signal 3.853 dB/m (unpumped)
90 % Pump Absorbed
2.117 m
10 dB / α_p, small-signal (bleaching lengthens it)
Saturation Power
0.3819 mW
signal; pump 1.164 mW
Signal Gain
15.60 dB
NF 2.97 dB; full-inversion bound 15.8 dB
Model
N = ppm·10⁻⁶·ρ·N_A/M
Γ = 1 − (U²/V²)(1 − K₀²(W)/K₁²(W)) (LP01, Gloge 1971)
α = 4.343·Γσ_a N dB/m; 90 % absorbed after 10 dB: L₉₀ = 10/α
P_sat = hνA/(Γ(σ_a + σ_e)τ)
Gain and NF: two-level rate equations, solved exactly (Saleh et al. 1990)
Γ assumes doping out to the core edge. Many fibres dope only the centre of the core, which lowers Γ, so here it is an upper bound.
The two-level model covers Er (pumped at 980 or 1480 nm), Yb, Tm pumped in band at 1600 nm (³H₆ ↔ ³F₄) and Ho pumped in band at 1908 nm (⁵I₈ ↔ ⁵I₇). Tm pumped at 790 nm (cross-relaxation), Nd and Er/Yb co-doping need three- or four-level models with energy transfer, so they are not offered. No ASE self-saturation, ion clustering (Le Gouët et al. found 15 % of the Ho ions in pairs) or excited-state absorption; co-propagating pump.
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
Two-level Er, Yb, Tm pumped in band at 1600 nm, or Ho pumped in band at 1908 nm; Γ from the exact LP01 field for a core doped to its edge (an upper bound). Preset cross-sections are measured (Er³⁺: Barnes et al. 1991; Yb³⁺: Melkumov et al. 2004; Tm³⁺: Agger & Povlsen 2006; Ho³⁺: Le Gouët et al. 2020). No ASE self-saturation (gains above ≈ 30 dB are overstated) and no Ho ion pairs. Tm pumped at 790 nm (cross-relaxation), Nd and Er/Yb co-doping are not covered.
- Gloge D. (1971). Weakly Guiding Fibers. Appl. Opt. 10, 2252. LP01 power fraction in the core, Γ = 1 − (U²/V²)(1 − K₀²(W)/K₁²(W)). doi:10.1364/AO.10.002252
- Giles C. R., Desurvire E. (1991). Modeling erbium-doped fiber amplifiers. J. Lightwave Technol. 9, 271. Two-level rate equations, eqs. (6)–(9), and the saturation power hνA/(Γ(σa + σe)τ). doi:10.1109/50.65886
- Saleh A. A. M., Jopson R. M., Evankow J. D., Aspell J. (1990). Modeling of gain in erbium-doped fiber amplifiers. IEEE Photon. Technol. Lett. 2, 714. Closed-form solution for co-propagating beams. doi:10.1109/68.60769
- Barnes W. L., Laming R. I., Tarbox E. J., Morkel P. R. (1991). Absorption and emission cross section of Er³⁺ doped silica fibers. IEEE J. Quantum Electron. 27, 1004–1010. Er preset (GeO₂–Al₂O₃–SiO₂). doi:10.1109/3.83335
- Melkumov M. A., Bufetov I. A., Kravtsov K. S., Shubin A. V., Dianov E. M. (2004). Lasing parameters of ytterbium-doped fibres doped with P₂O₅ and Al₂O₃. Quantum Electron. 34, 843–848; full report arXiv:1502.02885. Yb preset (aluminosilicate). arxiv.org
- Agger S. D., Povlsen J. H. (2006). Emission and absorption cross section of thulium doped silica fibers. Opt. Express 14, 50–57. Tm preset (Al/La silica, fibre Tm1): σ_a(1600) 3.5e-25 m², σ_e/σ_a 0.10 at 1600 nm, τ 650 µs; 1900 nm read from the vector data of Figs. 3–4. doi:10.1364/OPEX.14.000050
- Le Gouët J., Gustave F., Bourdon P., Robin T., Laurent A., Cadier B. (2020). Realization and simulation of high-power holmium doped fiber lasers for long-range transmission. Opt. Express 28, 22307; arXiv:2003.14188. Ho preset (silica): σ_a/σ_e(1908) 1.9/1.6e-25 m² from the text, τ 0.7 ms, fibre #1 N 3.7e25 m⁻³ and Γ 0.76; 2120 nm read from Fig. 1. doi:10.1364/OE.394011