Nonlinear Refractive Index (n₂)
Kerr effect n = n₀ + n₂I: index change, nonlinear phase over a length, fibre nonlinear coefficient γ = 2πn₂/(λA_eff) and the self-focusing critical power, with n₂ of silica fibre, silicon, air and its gases, CS₂ and 17 crystals, glasses and water.
n₂ = 2.60e-16 cm²/W (2.60e-20 m²/W), n₀ = 1.444 at 1550 nm; n₂ Agrawal, Nonlinear Fiber Optics; n₀ Malitson 1965.
Index change Δn = n₂I
2.600e-4
I = 1.00e+16 W/m²
Nonlinear phase φ = k₀n₂IL
10.54 rad
= 3.355π over 10 mm; π at 2.981e+11 W/cm²
Critical power P_cr (Gaussian beam)
9.599 MW
3.77 λ²/(8π n₀ n₂): bulk self-focusing
Fibre nonlinear coefficient γ
1.317 W⁻¹km⁻¹
2πn₂/(λA_eff), A_eff = 80 µm²
Fibre SPM phase per watt
0.00001317 rad/W
γL, lossless length L
Reference values
| Medium | n₂ (cm²/W) | n₀ | λ (nm) | P_cr | Source |
|---|---|---|---|---|---|
| Silicon (1800 nm) | 1.20e-13 | 3.461 | 1800 | 11.7 kW | n₂ Bristow et al. 2007 (peak, 850–2200 nm); n₀ Salzberg & Villa 1957 |
| CS₂ (pulses > 10 ps) | 2.86e-14 | 1.606 | 800 | 20.9 kW | Reichert et al. 2014, Table 1 (800 nm); n₀ Kedenburg et al. 2012, electronic + nuclear (Eq. 18) |
| KTP (E ∥ y) | 2.37e-15 | 1.77 | 1064 | 405.1 kW | DeSalvo et al. 1996, Tables I–II (1064 nm) |
| CS₂ (electronic only) | 2.00e-15 | 1.606 | 800 | 298.9 kW | Reichert et al. 2014, Table 1 (800 nm); n₀ Kedenburg et al. 2012, electronic |
| Tellurite glass (75TeO₂–20ZnO–5Na₂O) | 1.70e-15 | 2 | 1064 | 500.2 kW | DeSalvo et al. 1996, Tables I–II (1064 nm) |
| YVO₄ | 1.50e-15 | 1.97 | 800 | 324.9 kW | Dubietis et al. 2017, Table 1 (800 nm) |
| KGW | 1.10e-15 | 2.02 | 800 | 432.1 kW | Dubietis et al. 2017, Table 1 (800 nm) |
| LiNbO₃ (E ∥ c) | 9.15e-16 | 2.2 | 1064 | 844 kW | DeSalvo et al. 1996, Tables I–II (1064 nm) |
| KBr | 7.95e-16 | 1.53 | 1064 | 1.397 MW | DeSalvo et al. 1996, Tables I–II (1064 nm) |
| YAG | 6.20e-16 | 1.82 | 800 | 850.8 kW | Dubietis et al. 2017, Table 1 (800 nm) |
| Water | 5.70e-16 | 1.33 | 800 | 1.266 MW | Dubietis et al. 2017, Table 1 (800 nm) |
| β-BBO | 5.20e-16 | 1.66 | 800 | 1.112 MW | Dubietis et al. 2017, Table 1 (800 nm) |
| BK7 | 3.75e-16 | 1.51 | 800 | 1.695 MW | Dubietis et al. 2017, Table 1 (800 nm) |
| Sapphire (Al₂O₃, E ∥ c) | 3.11e-16 | 1.752 | 800 | 1.761 MW | n₂(λ) Major et al. 2004, Eq. 2 (270–1550 nm); n₀ = n_e, Malitson & Dodge 1972 |
| Silica fibre (1550 nm) | 2.60e-16 | 1.444 | 1550 | 9.599 MW | n₂ Agrawal, Nonlinear Fiber Optics; n₀ Malitson 1965 |
| Fused silica | 2.40e-16 | 1.45 | 800 | 2.759 MW | Dubietis et al. 2017, Table 1 (800 nm) |
| Fused silica (1024 nm) | 2.18e-16 | 1.45 | 1024 | 4.976 MW | n₂ Wahlstrand & Cruz 2026 (Z-scan, 250 fs); n₀ Malitson 1965 |
| KDP | 2.00e-16 | 1.5 | 800 | 3.2 MW | Dubietis et al. 2017, Table 1 (800 nm) |
| BaF₂ | 1.91e-16 | 1.47 | 800 | 3.419 MW | Dubietis et al. 2017, Table 1 (800 nm) |
| CaF₂ | 1.30e-16 | 1.43 | 800 | 5.164 MW | Dubietis et al. 2017, Table 1 (800 nm) |
| LiF | 8.10e-17 | 1.39 | 800 | 8.527 MW | Dubietis et al. 2017, Table 1 (800 nm) |
| MgF₂ | 5.69e-17 | 1.4 | 1064 | 21.32 MW | DeSalvo et al. 1996, Tables I–II (1064 nm) |
| Ar (gas, 1 atm) | 1.01e-19 | 1 | 800 | 9.503 GW | Zahedpour et al. 2015, Table 1 (800 nm, 1 atm, electronic) |
| O₂ (gas, 1 atm) | 8.10e-20 | 1 | 800 | 11.85 GW | Zahedpour et al. 2015, Table 1 (800 nm, 1 atm, electronic) |
| Air (1 atm) | 7.96e-20 | 1 | 800 | 12.06 GW | Zahedpour et al. 2015 N₂, O₂, Ar weighted by mole fraction |
| N₂ (gas, 1 atm) | 7.90e-20 | 1 | 800 | 12.15 GW | Zahedpour et al. 2015, Table 1 (800 nm, 1 atm, electronic) |
Click a row to select the medium; P_cr at each medium's own wavelength.
Model
n = n₀ + n₂I, φ = (2π/λ) n₂ I L, γ = 2πn₂/(λA_eff)
The electronic Kerr effect is instantaneous and the intensity uniform over L, so φ is the on-axis phase (the B-integral of a slab). In a fibre the same n₂ sets γ through the mode's effective area. n₂ falls slowly towards the infrared below the band gap, so a value measured at one wavelength holds only near it (fused silica: 2.4 × 10⁻¹⁶ cm²/W at 800 nm in Dubietis et al.'s table, 2.18 × 10⁻¹⁶ at 1024 nm in Wahlstrand & Cruz's Z-scan, ±6 %; DeSalvo et al.'s picosecond Z-scans give 2.1 × 10⁻¹⁶ at 1064 nm and 2.2 × 10⁻¹⁶ at 532 nm, ±20 %).
Sapphire follows Major et al.'s empirical double-exponential fit to their picosecond Z-scans with E ∥ c from 550 to 1550 nm (3.3 to 2.8 × 10⁻¹⁶ cm²/W, ±25–30 %); below 550 nm its shape is set mainly by DeSalvo et al.'s 266 and 355 nm points. It gives 5.6 × 10⁻¹⁶ cm²/W at 270 nm, its lower end; below that no n₂ is given, and below about 340 nm (2ħω above the 7.3 eV band gap) sapphire also absorbs two photons. Above 1550 nm n₂ is held at its 1550 nm value (the fit's own tail falls only to 2.7 × 10⁻¹⁶ at 2 µm). n₀ is the extraordinary index.
Silicon's n₂ changes fourfold between 1000 and 2200 nm (4.7 × 10⁻¹⁴ cm²/W at 1220 nm, peak 1.2 × 10⁻¹³ at 1800 nm), and silicon also absorbs two photons up to 2200 nm, which limits the usable nonlinear phase.
Values are for linear polarization; circular polarization gives ⅔ of the electronic n₂ in isotropic media (CS₂'s long-pulse n₂ falls 3.4-fold). Liquids and gases add a slower molecular (orientational) part.
The gases are the electronic n₂ at 1 atm from 40 fs pump–probe measurements (±10 %); n₂ scales with pressure. The electronic part dominates only for pulses shorter than about 50 fs; above about 150 fs the rotation of N₂ and O₂ takes over, and for long pulses it adds 24 and 54 × 10⁻²⁰ cm²/W, three to seven times the electronic value, so P_cr in air drops several-fold. Air is the mole-fraction sum of N₂, O₂ and Ar. CS₂ has two rows: the bound-electronic n₂ (even at 32–165 fs the nuclear part can double the effective n₂), and the plateau above 10 ps, where its collision, libration and diffusive reorientation responses raise n₂ fourteenfold.
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
Instantaneous electronic Kerr effect for linear polarization and uniform intensity over L. n₂ is fixed at the wavelength it was measured at; its dispersion is not modelled, except for sapphire (E ∥ c), which follows Major et al.'s empirical fit over 270–1550 nm (±25–30 %), held at its 1550 nm value above and not given below 270 nm. Gases are electronic only at 1 atm (≲ 50 fs; longer pulses add rotation); the CS₂ > 10 ps row includes the nuclear response. DeSalvo's esu values are converted with n₂(m²/W) = 40π n₂(esu)/(c n₀). P_cr is the Gaussian-beam value 3.77 λ²/(8π n₀ n₂).
- Dubietis A., Tamošauskas G., Šuminas R., Jukna V., Couairon A. (2017). Ultrafast supercontinuum generation in bulk condensed media. Lith. J. Phys. 57(3). Table 1: n₂ and n₀ at 800 nm. doi:10.3952/physics.v57i3.3541
- Agrawal G. P. (2019). Nonlinear Fiber Optics, 6th ed. Academic Press. n₂ = 2.6e-20 m²/W for silica fibre and γ = 2πn₂/(λA_eff). doi:10.1016/C2018-0-01168-8
- Fibich G., Gaeta A. L. (2000). Critical power for self-focusing in bulk media and in hollow waveguides. Opt. Lett. 25, 335–337. doi:10.1364/OL.25.000335
- DeSalvo R., Said A. A., Hagan D. J., Van Stryland E. W., Sheik-Bahae M. (1996). Infrared to ultraviolet measurements of two-photon absorption and n₂ in wide bandgap solids. IEEE J. Quantum Electron. 32, 1324–1333. Tables I–II: n₂ and n₀ at 1064 nm. doi:10.1109/3.511545
- Bristow A. D., Rotenberg N., van Driel H. M. (2007). Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm. Appl. Phys. Lett. 90, 191104. Peak n₂ = 1.2e-13 cm²/W at 1800 nm. doi:10.1063/1.2737359
- Major A., Yoshino F., Nikolakakos I., Aitchison J. S., Smith P. W. E. (2004). Dispersion of the nonlinear refractive index in sapphire. Opt. Lett. 29, 602–604. Eq. 2: empirical n₂(λ) of sapphire for E ∥ c, 270–1550 nm. doi:10.1364/OL.29.000602
- Malitson I. H., Dodge M. J. (1972). Refractive index and birefringence of synthetic sapphire. J. Opt. Soc. Am. 62, 1405 (meeting abstract). n_e of sapphire.
- Salzberg C. D., Villa J. J. (1957). Infrared refractive indexes of silicon, germanium, and modified selenium glass. J. Opt. Soc. Am. 47, 244. n₀ of silicon. doi:10.1364/JOSA.47.000244
- 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
- Zahedpour S., Wahlstrand J. K., Milchberg H. M. (2015). Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths. Opt. Lett. 40, 5794. Table 1: electronic and rotational n₂ of N₂, O₂ and Ar at 1 atm. doi:10.1364/OL.40.005794
- U.S. Standard Atmosphere, 1976 (NOAA, NASA, USAF). Dry-air mole fractions N₂ 0.78084, O₂ 0.209476, Ar 0.00934. ntrs.nasa.gov
- Peck E. R., Reeder K. (1972). Dispersion of Air. J. Opt. Soc. Am. 62, 958. n₀ of the gases. doi:10.1364/josa.62.000958
- Reichert M., Hu H., Ferdinandus M. R., Seidel M., Zhao P., Ensley T. R., Peceli D., Reed J. M., Fishman D. A., Webster S., Hagan D. J., Van Stryland E. W. (2014). Temporal, spectral, and polarization dependence of the nonlinear optical response of carbon disulfide. Optica 1, 436. Table 1 and Eq. 18: CS₂ electronic and long-pulse n₂. doi:10.1364/OPTICA.1.000436
- Kedenburg S., Vieweg M., Gissibl T., Giessen H. (2012). Linear refractive index and absorption measurements of nonlinear optical liquids in the visible and near-infrared spectral region. Opt. Mater. Express 2, 1588. n₀ of CS₂. doi:10.1364/OME.2.001588
- Wahlstrand J. K., Cruz C. D. (2026). Precise measurement of the Kerr coefficient using phase-sensitive pump–probe hyperspectral imaging. Rev. Sci. Instrum. 97(8). Fused silica n₂ = 2.18 ± 0.14 × 10⁻¹⁶ cm²/W at 1024 nm (Z-scan). doi:10.1063/5.0331814
- Boyd R. W. (2020). Nonlinear Optics, 4th ed. Academic Press. The nonlinear index, its polarization dependence (⅔ for circular light, electronic response) and self-focusing. doi:10.1016/C2015-0-05510-1