# Nonlinear Index

Definition: a parameter for quantifying the Kerr nonlinearity of a medium

Alternative term: nonlinear refractive index

German: nichtlinearer Brechungsindex

Units: m^{2}/W

Formula symbol: <$n_2$>

Author: Dr. Rüdiger Paschotta

When light with high intensity propagates through a medium, this causes nonlinear effects. The simplest of these is the Kerr effect, which can be described as a change (usually an increase) in the refractive index in proportion to the optical intensity <$I$>

$$\Delta n = {n_2}\;I$$with the *nonlinear refractive index* (in short: *nonlinear index*) <$n_2$>. The units of that quantity are m^{2}/W (or cm^{2}/W) in the SI system, but in older literature one finds <$n_2$> values in esu units. For the conversion of such units, the equation

can be used, where <$n$> is the refractive index.

The nonlinear index results from the <$\chi^{(3)}$> nonlinearity of the material and can be calculated as:

$$n_2 = \frac{3 \; \chi^{(3)}}{4 \; \varepsilon_0 \; c \; n_0\rm \; Re (n_0)}$$That shows that a high nonlinear index usually results from a high <$\chi^{(3)}$> value, but can also be increased if the refractive index gets very small, e.g. due to a phonon resonance effect or in photonic metamaterials. Both involved quantities can be somewhat wavelength-dependent.

Normally, nonlinear indices are measured for linearly polarized light. For circularly polarized light in a not birefringent medium, the Kerr effect is weaker by one third.

## Additional Complications

According to the equation <$\Delta n = {n_2}\;I$>, the temporal pulse would remain unaffected by the nonlinear interaction. In reality, however, there is the self-steepening effect, which delays the pulse peak with respect to weaker parts of the pulse. It is particularly relevant for broadband pulses and can be interpreted essentially as resulting from the frequency dependence of the nonlinearity.

At very high optical intensities, the equation above for the nonlinear index change may need a higher-order correction. For example, one may have a term proportional to the index squared, with a negative coefficient, so that the Kerr effect saturates.

In addition to the Kerr effect (a purely electronic nonlinearity), *electrostriction* can significantly contribute to the value of the nonlinear index [9, 10]. Here, the electric field of light causes density variations (acoustic waves) which themselves influence the refractive index via the photoelastic effect. That mechanism, however, involves a significant time delay and is thus relevant only for relatively slow power modulations, but not for ultrashort pulses. In optical fibers, the contribution of electrostriction at low (megahertz) frequencies is typically of the order of 10–20% of that of the Kerr effect, but can strongly depend on the material.

## Values of the Nonlinear Refractive Index

For transparent crystals and glasses, <$n_2$> is typically of the order of 10^{−16} cm^{2}/W to 10^{−14} cm^{2}/W. Fused silica, as used e.g. in silica fibers, has a particularly low nonlinear index of 2.7 · 10^{−16} cm^{2}/W for wavelengths around 1.5 μm, whereas some chalcogenide glasses exhibit several hundred times higher values.

Semiconductor materials also have very high nonlinear index values.
It has been shown that the nonlinear index scales in proportion to the inverse fourth power of the band gap energy, but also depends on the proximity of the test wavelength to the bandgap [5]. The nonlinearity can also be negative (*self-defocusing nonlinearity*), particularly for photon energies above ≈ 70% of the bandgap energy.

Material | nonlinear index | Reference |
---|---|---|

air | 1.22 · 10^{−22} m^{2}/W at 308 nm, 1 atm | [2] |

fused silica | 2.19 · 10^{−20} m^{2}/W at 1030 nm | [17] |

sapphire | 2.8 · 10^{−20} m^{2}/W at 1550 nm | [12] |

yttrium aluminum garnet (Y_{3}Al_{5}O_{12}) | 6.13 · 10^{−20} m^{2}/W at 1030 nm | [17] |

yttrium vanadate (YVO_{4}), <$n_\rm{o}$> | 15.6 · 10^{−20} m^{2}/W at 1030 nm | [17] |

yttrium vanadate (YVO_{4}), <$n_\rm{e}$> | 14.9 · 10^{−20} m^{2}/W at 1030 nm | [17] |

calcite (CaCO_{3}), <$n_\rm{o}$> | 3.22 · 10^{−20} m^{2}/W at 1030 nm | [17] |

calcite (CaCO_{3}), <$n_\rm{e}$> | 2.12 · 10^{−20} m^{2}/W at 1030 nm | [17] |

calcium fluoride (CaF_{2}) | 1.71 · 10^{−20} m^{2}/W at 1030 nm | [17] |

Materials with a high nonlinear index often have a small bandgap energy, and therefore also often exhibit strong two-photon absorption (TPA). For some applications such as channel conversion in telecom systems, this is detrimental, and a figure of merit such as <$n_2 / \beta$> (where <$\beta$> is the TPA coefficient) can be used to compare different materials.

A very high nonlinear refractive index can also result for materials with near-zero refractive index. For example, indium tin oxide (ITO) exhibits huge <$n_2$> values in the wavelength region around 1250 nm [15]. Similar effects have been observed in aluminum-doped zinc oxide (AZO) in the 1300-nm region [16]. Note that such extremely strong nonlinearities can still exhibit an ultrafast response, i.e., on a sub-picosecond time scale. They may be useful for nonlinear signal processing, for example.

Note that even air exhibits some Kerr nonlinearity, with a nonlinear index around 5 · 10^{−19} cm^{2}/W for visible light and atmospheric pressure.

## Measurement of Nonlinear Index

The measurement of the nonlinear index of bulk samples is often done with the z-scan technique, which is based on self-focusing via the Kerr lens.

Another possibility for cases with very strong refractive index is to use pump–probe measurements: a pulsed pump beam causes time-dependent phase changes which result in spectral changes, and those can be measured [18].

The nonlinearity of optical fibers can be quantified by measuring spectral broadening resulting from self-phase modulation. Note, however, that the polarization state may be scrambled in a non-polarization-maintaining fiber, and this can affect the result. Also, the result is a kind of average over the material properties of the fiber core and the cladding.

### Bibliography

[1] | M. J. Weber, D. Milam. and W. L. Smith, “Nonlinear refractive index of glasses and crystals”, Opt. Eng. 17, 463 (1978), DOI:10.1117/12.7972266 |

[2] | Y. Shimoji et al., “Direct measurement of the nonlinear refractive index of air”, J. Opt. Soc. Am. B 6 (11), 1994 (1989), DOI:10.1364/JOSAB.6.001994 |

[3] | M. Sheik-Bahae et al., “Sensitive measurement of optical nonlinearities using a single beam”, IEEE J. Quantum Electron. 26 (4), 760 (1990), DOI:10.1109/3.53394 |

[4] | D. W. Hall et al., “Nonlinear optical susceptibilities of high-index glasses”, App. Phys. Lett. 54 (14), 1293 (1989), DOI:10.1063/1.100697 |

[5] | M. Sheik-Bahae et al., “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption”, Phys. Rev. Lett. 65 (1), 96 (1990), DOI:10.1103/PhysRevLett.65.96 |

[6] | K. S. Kim et al., “Measurement of the nonlinear index of silica-core and dispersion-shifted fibers”, Opt. Lett. 19 (4), 257 (1994), DOI:10.1364/OL.19.000257 |

[7] | T. Kato et al., “Measurement of the nonlinear refractive index in optical fiber by the cross-phase-modulation method with depolarized pump light”, Opt. Lett. 20 (9), 988 (1995), DOI:10.1364/OL.20.000988 |

[8] | T. Kato, Y. Suetsugu and M. Nishimura, “Estimation of nonlinear refractive index in various silica-based glasses for optical fibers”, Opt. Lett. 20 (22), 2279 (1995), DOI:10.1364/OL.20.002279 |

[9] | E. L. Buckland and R. W. Boyd, “Electrostrictive contribution to the intensity-dependent refractive index of optical fibers”, Opt. Lett. 21 (15), 1117 (1996), DOI:10.1364/OL.21.001117 |

[10] | E. L. Buckland and R. W. Boyd, “Measurement of the frequency response of the electrostrictive nonlinearity in optical fibers”, Opt. Lett. 22 (10), 676 (1997), DOI:10.1364/OL.22.000676 |

[11] | I. P. Nikolakakos et al., “Broadband characterization of the nonlinear optical properties of common reference materials”, IEEE Sel. Top. Quantum Electron. 10 (5), 1164 (2004), DOI:10.1109/JSTQE.2004.836007 |

[12] | A. Major et al., “Dispersion of the nonlinear refractive index in sapphire”, Opt. Lett. 29 (6), 602 (2004), DOI:10.1364/OL.29.000602 |

[13] | Shahraam Afshar V. et al., “Small core optical waveguides are more nonlinear than expected: experimental confirmation”, Opt. Lett. 34 (22), 3577 (2009), DOI:10.1364/OL.34.003577 |

[14] | I. Dancus et al., “Single shot interferometric method for measuring the nonlinear refractive index”, Opt. Express 21 (25), 31303 (2013), DOI:10.1364/OE.21.031303 |

[15] | M. Z. Alam, I. De Leon and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region”, Science 352 (6287), 795 (2016), DOI:10.1126/science.aae0330 |

[16] | L. Caspani et al., “Enhanced nonlinear refractive index in ε-near-zero materials”, Phys. Rev. Lett. 116, 233901 (2016), DOI:10.1103/PhysRevLett.116.233901 |

[17] | P. Kabacinski et al., “Nonlinear refractive index measurement by SPM-induced phase regression”, Opt. Express 27 (8), 11018 (2019), DOI:10.1364/OE.27.011018 |

[18] | Y. Zhou et al., “Broadband frequency translation through time refraction in an epsilon-near-zero material”, Nature Communications 11, 2180 (2020), DOI:10.1038/s41467-020-15682-2 |

[19] | G. N. Patwardhan et al., “Nonlinear refractive index of solids in mid-infrared”, Opt. Lett. 46 (8), 1824 (2021), DOI:10.1364/OL.421469 |

[20] | R. Schiek, “Nonlinear refractive index in silica glass”, Opt. Mater. Express 13 (6), 1727 (2023), DOI:10.1364/OME.489520 |

[21] | B. Maingot et al., “Measurement of nonlinear refractive indices of bulk and liquid crystals by nonlinear chirped interferometry”, Opt. Lett. 48 (12), 3243 (2023), DOI:10.1364/OL.487261 |

See also: Kerr effect, self-phase modulation, Kerr lens, refractive index, B integral, nonlinearities, delayed nonlinear response, z-scan measurements

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2020-07-04

What is the nonlinear refractive index of chalcogenide glass?

The author's answer:

There are plenty of different chalcogenide glasses, which also differ substantially in terms of their nonlinear index. Generally, it is substantially higher than for silica glass.