Brillouin Scattering
Brillouin scattering: frequency shift ν_B = 2nv_A/λ, Stokes wavelength, acoustic wavelength, phonon lifetime and peak gain g_B = 2πn⁷p₁₂²/(cλ²ρv_AΔν_B), for fused silica, As₂Se₃, As₂S₃, diamond or your own medium.
n = 1.45, v_A = 5960 m/s, ρ = 2210 kg/m³, p₁₂ = 0.286; Agrawal, Nonlinear Fiber Optics, 6th ed.. Δν_B is the gain bandwidth at the pump wavelength: it scales as λ_p⁻², so measure or scale it when you change λ.
Brillouin shift ν_B = 2nv_A/λ
11.15 GHz
Stokes wavelength 1550.089 nm
Peak gain g_B
3.650e-11 m/W
= 3.65 cm/GW, Lorentzian of FWHM 20 MHz
Acoustic wavelength Λ = λ/(2n)
0.5345 µm
phonon lifetime T_B = 1/(2πΔν_B) = 7.958 ns
The SBS threshold of a fibre (effective area, length, loss, polarization and source linewidth) is on the SBS threshold calculator.
Brillouin shift vs pump wavelength
Model
ν_B = 2nv_A/λ_p, g_B = 2πn⁷p₁₂² / (cλ_p²ρv_AΔν_B)
Backward Brillouin scattering from one longitudinal acoustic wave of wavelength Λ = λ_p/(2n). The gain line is Lorentzian with the phonon damping as its width; g_B is its peak for a pump much narrower than Δν_B. In a fibre the mode index replaces n and acoustic guiding lowers the overlap, so measured fibre gains are below the bulk value.
Δν_B grows as λ_p⁻², which cancels the λ_p⁻² in g_B: the peak gain of silica stays near 3–5e-11 m/W at all pump wavelengths, while a fixed Δν_B would make it look strongly wavelength dependent.
Each medium other than silica comes with a measurement. As₂Se₃ fibre (Abedin 2005, 1.56 µm): 6.0e-9 m/W measured, 6.09e-9 from the formula, and ν_B = 2nv_A/λ gives 8.10 GHz against the measured 7.95 GHz (+1.9 %); its ρ, v_A and p₁₂ are the paper's Table 1, after Uchida & Niizeki 1973. As₂S₃ waveguide (Pant et al. 2011): 0.715e-9 m/W measured, 0.78e-9 from the formula (0.74e-9 with their mode overlap 0.95). Diamond (Bai et al. 2020, 532 nm; longitudinal phonons along [110], pump polarized along ⟨111⟩): 60 ± 9 cm/GW measured from the laser threshold (79 cm/GW in the abstract), 17.4 cm/GW from the formula at the measured Δν_B of 11.9 ± 4.3 MHz. Its effective p = (p₁₁ + 2p₁₂ − 2p₄₄)/3 = 0.0600 comes from Grimsditch & Ramdas's photoelastic constants; no backscatter polarization gives more than p² = 0.0047. Bai et al.'s p² = 0.0153 is about (2p)², a strain tensor without its factor ½, so the measured gain is 3–4× the elasto-optic estimate. For As₂S₃ and diamond, v_A follows from the measured shift, v_A = ν_Bλ/(2n).
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
Backward scattering from one longitudinal acoustic mode, with a Lorentzian gain line and a pump narrower than Δν_B. Δν_B is an input at the pump wavelength (it scales as λ⁻²). Bulk values: acoustic guiding and the overlap integral of a real fibre are not modelled. The As₂Se₃, As₂S₃ and diamond presets come from papers that measured ν_B, Δν_B and g_B, shown beside the result (formula within 9 % for the glasses; diamond 17.4 against 60 ± 9 cm/GW: the elasto-optic p from Grimsditch & Ramdas underestimates the measured gain 3–4×). The fibre SBS threshold has its own page.
- Agrawal G. P. (2019). Nonlinear Fiber Optics, 6th ed. Academic Press. Brillouin shift ν_B = 2nv_A/λ_p, peak gain g_B and fused-silica n, v_A, ρ₀ and p₁₂. doi:10.1016/C2018-0-01168-8
- Kobyakov A., Sauer M., Chowdhury D. (2010). Stimulated Brillouin scattering in optical fibers. Adv. Opt. Photon. 2, 1–59. doi:10.1364/AOP.2.000001
- Abedin K. S. (2005). Observation of strong stimulated Brillouin scattering in single-mode As₂Se₃ chalcogenide fiber. Opt. Express 13, 10266–10271. Table 1. doi:10.1364/OPEX.13.010266
- Pant R., Poulton C. G., Choi D.-Y., et al. (2011). On-chip stimulated Brillouin scattering. Opt. Express 19, 8285–8290. As₂S₃ n, ρ, p₁₂ and measured ν_B, Δν_B, g_B. doi:10.1364/OE.19.008285
- Bai Z., Williams R. J., Kitzler O., et al. (2020). Diamond Brillouin laser in the visible. APL Photonics 5, 031301. Longitudinal phonons along [110], pump polarized along ⟨111⟩: ν_B 167 GHz, Δν_B 11.9 ± 4.3 MHz, g 60 ± 9 cm/GW from the threshold at 532 nm (79 cm/GW in the abstract). Their p² 0.0153 is about (2p)²; the page uses p = (p₁₁ + 2p₁₂ − 2p₄₄)/3 = 0.0600. doi:10.1063/1.5134907
- Grimsditch M. H., Ramdas A. K. (1975). Brillouin scattering in diamond. Phys. Rev. B 11, 3139–3148. Photoelastic constants p₄₄ = −0.172, p₁₁ − p₁₂ = −0.292, p₁₁ + 2p₁₂ = −0.1640 (abstract). doi:10.1103/PhysRevB.11.3139