Rare-Earth Dopant Absorption
Absorption coefficient of Er³⁺, Yb³⁺, Tm³⁺ or Ho³⁺ in silica from the cross-section and wt-ppm concentration: α in m⁻¹ and dB/m, absorption length and transmission over a path.
Ytterbium (Yb³⁺) in aluminosilicate glass, M = 173.05 g/mol, silica density 2201 kg/m³. Choosing a line fills its cross-section measured by Melkumov et al. 2004; it varies with the host glass by tens of percent, so enter your sample's own value where you have it.
Concentration is the ion's wt-ppm. Vendors often quote wt % of the oxide (Er₂O₃ → Er: × 0.874, i.e. 2·167.26/382.5) or mol-ppm.
Ion Density N
7.660e+24 m⁻³
1000 wt-ppm
Absorption Coefficient α
20.60 m⁻¹
0.2060 cm⁻¹
Absorption
89.48 dB/m
10·log₁₀(e)·α
Absorption Length 1/α
0.04853 m
path for a 1/e (4.34 dB) drop
Loss over the Path
8.948 dB
0.1 m
Transmitted
12.74 %
e^(−αL), unpumped; no surface reflections
Ytterbium (Yb³⁺): transmission against path length
- Yb 976 nm (pump), entered σ_a
- Yb 1064 nm (signal)
Preset Cross-Sections
| Line | σ_a (×10⁻²⁵ m²) | σ_e (×10⁻²⁵ m²) | Host |
|---|---|---|---|
| Er 980 nm (pump) | 1.7 | 0 | alumino-germanosilicate |
| Er 1480 nm (pump) | 1.7 | 0.5 | alumino-germanosilicate |
| Er 1550 nm (signal) | 2.2 | 3 | alumino-germanosilicate |
| Yb 976 nm (pump) | 26.9 | 29.7 | aluminosilicate |
| Yb 1064 nm (signal) | 0.057 | 3.1 | aluminosilicate |
| Tm 1600 nm (pump) | 3.5 | 0.35 | Al/La-codoped silica |
| Tm 1740 nm (signal) | 2.83 | 3.4 | Al/La-codoped silica |
| Tm 1900 nm (signal) | 0.17 | 2.84 | Al/La-codoped silica |
| Ho 1908 nm (pump) | 1.9 | 1.6 | silica |
| Ho 1946 nm (pump) | 4.3 | 5.9 | silica |
| Ho 1995 nm (signal) | 1.93 | 4.38 | silica |
| Ho 2120 nm (signal) | 0.18 | 1.3 | silica |
Model
N = ppm·10⁻⁶·ρ·N_A/M
α = Γσ_a N; α(dB/m) = 4.343·α
T = e^(−αL)
Small-signal absorption of an unpumped sample: every ion in the ground state. A strong pump bleaches the absorption, and Er at 1550 nm turns into gain once inverted; the Rare-Earth-Doped Fiber Amplifier page models that. Tm is offered in its ³F₄ band only (1600–1900 nm), not at its 790 nm ³H₄ pump band, and Ho in its ⁵I₇ band only (1908–2120 nm). Nd is not offered: its cross-sections need their own sourced data.
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
Small-signal absorption α = Γσ_aN of an unpumped sample (no bleaching or excited-state absorption) and T = e^(−αL) without surface reflections. N from wt-ppm in silica (2201 kg/m³). Preset cross-sections, shared with the fibre amplifier pages, are measured in one fibre each: Er³⁺ in GeO₂–Al₂O₃–SiO₂ (Barnes 1991; 980 nm from Table V, 1480 and 1550 nm read from Fig. 6 to ±0.15e-25 m²), Yb³⁺ in aluminosilicate (Melkumov 2004) and Tm³⁺ in Al/La silica (Agger & Povlsen 2006; ³F₄ band only, 1740 and 1900 nm read from Figs. 3–4) and Ho³⁺ in silica (Le Gouët et al. 2020; ⁵I₇ band only, σ_a(1946) 4.3 ± 0.2e-25 m², 1995 and 2120 nm read from Fig. 1). They vary with the host by tens of percent. Nd is not offered.
- Giles C. R., Desurvire E. (1991). Modeling erbium-doped fiber amplifiers. J. Lightwave Technol. 9, 271–283. Absorption α = Γσ_aN, eqs. (6)–(9). doi:10.1109/50.65886
- 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. GeO₂–Al₂O₃–SiO₂: Table V, Fig. 6(c), τ 10.2 ms; absolute scale ±20 % (Table VI). 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, Appendix 2 (aluminosilicate), τ 0.83 ms. arxiv.org
- Agger S. D., Povlsen J. H. (2006). Emission and absorption cross section of thulium doped silica fibers. Opt. Express 14, 50–57. Al/La-silica fibre Tm1: σ_a(1600) 3.5e-25 m², σ_e/σ_a 0.10 (1600 nm) and 1.20 (1740 nm), τ 650 µs; 1740 and 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. Silica fibre: σ_a(1946) 4.3 ± 0.2e-25 m² (fibre #2, 425 dB/m), σ_e/σ_a 1.40 ± 0.06 at the peaks, 1.9/1.6e-25 m² at 1908 nm, τ 0.7 ms; 1995 and 2120 nm read from the raster Fig. 1. doi:10.1364/OE.394011