Third-order accuracy#

Read Third-order nonlinearity and Third-order support and resonance limits first. This chapter compares statistical uncertainty, dataset sensitivity and independent measurements. Third-order extrapolation treats evaluation beyond fitted data, and the uncertainty gallery plots the distinction between statistical and working estimates.

What the numbers mean#

The Kerr gamma sets the nonlinear index, \(n_2 = N\gamma_\text{Kerr}/(8\varepsilon_0^2 c)\), and so everything that scales with it. A relative error \(\varepsilon\) in the Kerr gamma is an error \(\varepsilon\) in:

  • the nonlinear phase, the B-integral and the spectral broadening by self-phase modulation;

  • the critical power of self-focusing (which scales as \(1/n_2\));

  • the square of the soliton order (\(N^2 \propto n_2\)), so \(\varepsilon/2\) in the order;

  • the nonlinear phase mismatch of every phase-matched process driven by the pulse (dispersive waves, modulational instability, four-wave mixing).

For the third harmonic the generated field scales with the THG gamma, so its power scales as \(\gamma_\text{THG}^2\): an error \(\varepsilon\) in gamma is \(2\varepsilon\) in the harmonic’s power.

As with the linear index, a level error in gamma is equivalent to a density error, since every third-order effect scales as \(N\gamma\). The linear index is known to 1e-4 or better, far below the uncertainty of a gas fill. Gamma is not: the best absolute scale (the ESHG scale, below) is good to about 1 %, and older values in use are off by 3-30 %. Here the level matters as much as the shape.

The shape is the dependence on the wavelength and on the process. Between a fundamental of 1800 and 400 nm the Kerr gamma rises by 12 % (He) to 86 % (Xe). At 800 nm the THG gamma exceeds the Kerr one by 6 % (He) to 39 % (Xe). A single constant coefficient cannot give both.

Three measures and the working estimate#

Each value below comes with three measures of its uncertainty:

  • Statistical: the one-sigma uncertainty propagated from the fit’s covariance, scaled by the reduced chi-squared where it exceeds one (gamma_sigma_au()). It assumes the dispersion form and the stated uncertainties.

  • Leave-one-out: the largest relative change when one fitted dataset is left out and the gas refitted (tools/fit/nonlinear_robustness.py, records in data/gas/models/nonlinear_robustness/). A calculation’s processes leave together (they share one scale). A refit with fewer values than parameters is not counted: there the priors alone decide and the change measures nothing. An exactly determined refit is counted.

  • Floor: a systematic uncertainty that neither of the other two contains, stated per gas:

    • He, 0.1 %: the allowance added to each of Bishop & Pipin’s calculated values. Shelton & Rice judge the calculation better than 0.1 %, and it is nonrelativistic.

    • Ne, Ar, Kr, Xe, H2, D2, N2, O2, 1 %: the ESHG scale. Every one of Shelton’s values rests on Bishop & Pipin’s helium through measured ratios. Shelton states the ratios to about 0.4 % (0.1-0.7 %) and the curves of Ar, H2 and N2 to about 0.2 %. He then adds that helium pair interactions “may increase chi^(3) … by 1%”. All his values “should be adjusted upward by about 1%” (Ne half as much), “the effective accuracy of our results is only about 1%, even for the atoms and diatomic molecules” (Shelton 1990, p. 13). The adjustment is not applied.

    • CO2, CH4, SF6, 2 %: the same scale, and the fit is to the total ESHG gamma, whose vibrational part Shelton could not separate for these molecules.

    • CO, 5 %: the level rests on one ESHG value, Ward & Miller’s 1730 +- 50 au (2.9 %, electronic plus vibrational).

    • O3, 12 %: the allowance for the method dependence of Naves et al.’s CCSD(T) values.

    • N2O, none beyond the statistical: the n2’s stated uncertainty and the priors on its borrowed dispersion are already in the statistical figure.

The working estimate is the largest of the three, rounded up to the next of 0.1, 0.2, 0.3, 0.5, 0.7, 1, 1.5, 2, 3, 4, 5, 7, 10, 15, 20 %. It is mechanical, and the test recomputes it; the judgement is in the floors and in the notes below.

At 800 nm#

The Kerr gamma (SPM, XPM) and the THG gamma at a fundamental of 800 nm. “Largest when left out” names the dataset, or the calculation (its scale group), whose removal moves the Kerr gamma most.

Gas

Kerr: statistical

Kerr: leave-one-out

Largest when left out

THG: statistical

THG: leave-one-out

Floor

Kerr estimate

THG estimate

He

0.015 %

0.012 %

bishop_1989/omega_thg_zzzz

0.036 %

0.011 %

0.1 %

0.1 %

0.1 %

Ne

0.23 %

0.19 %

shelton_1992/gamma_present

0.23 %

0.25 %

1 %

1 %

1 %

Ar

0.15 %

0.31 %

shelton_1990/table3_Ar_literature

0.16 %

0.4 %

1 %

1 %

1 %

Kr

0.58 %

3.1 %

shelton_1990/table3_Kr_present

0.86 %

0.19 %

1 %

4 %

1 %

Xe

0.39 %

1.4 %

shelton_1990/table3_Xe_present

0.71 %

3.1 %

1 %

1.5 %

4 %

H2

0.1 %

0.15 %

bishop_1991

0.13 %

0.14 %

1 %

1 %

1 %

D2

0.085 %

0.1 %

bishop_1991

0.12 %

0.054 %

1 %

1 %

1 %

N2

0.17 %

0.47 %

hattig_ccsd

0.14 %

0.68 %

1 %

1 %

1 %

O2

0.24 %

1.6 %

shelton_1990/table3_O2_present

0.39 %

0.21 %

1 %

2 %

1 %

CO2

0.51 %

2.6 %

shelton_1990/table3_CO2_present

0.36 %

3.1 %

2 %

3 %

4 %

CH4

0.3 %

0.74 %

shelton_1990/table3_CH4_present

0.28 %

0.092 %

2 %

2 %

2 %

SF6

0.31 %

3.2 %

shelton_1990/table3_SF6_present

0.27 %

0.34 %

2 %

4 %

2 %

CO

2.8 %

10 %

ward_1979/chi3_CO

2.8 %

10 %

5 %

10 %

10 %

N2O

13 %

none

none

14 %

none

0 %

15 %

15 %

O3

10 %

0.082 %

naves_2011/gamma_static_ccsd_t_expt

n/a

n/a

12 %

15 %

n/a (resonant)

At 400 and 1800 nm#

The Kerr gamma at the two ends of the range most used (n/a where outside the model’s declared range):

Gas

400 nm: statistical

leave-one-out

estimate

1800 nm: statistical

leave-one-out

estimate

He

0.12 %

0.046 %

0.2 %

0.017 %

0.0031 %

0.1 %

Ne

0.24 %

0.23 %

1 %

0.27 %

0.63 %

1 %

Ar

0.44 %

0.7 %

1 %

0.23 %

0.37 %

1 %

Kr

3.3 %

0.003 %

4 %

1 %

6.3 %

7 %

Xe

2.9 %

6.8 %

7 %

0.42 %

2.1 %

3 %

H2

0.54 %

0.4 %

1 %

0.11 %

0.11 %

1 %

D2

0.53 %

0.4 %

1 %

0.093 %

0.17 %

1 %

N2

0.41 %

0.93 %

1 %

0.33 %

0.41 %

1 %

O2

1.5 %

0.23 %

1.5 %

0.5 %

3.7 %

4 %

CO2

1.2 %

7.1 %

10 %

1.1 %

5.9 %

7 %

CH4

1.2 %

0.026 %

2 %

0.64 %

1.7 %

2 %

SF6

0.78 %

0.25 %

2 %

0.63 %

6.7 %

7 %

CO

3 %

10 %

10 %

3 %

10 %

10 %

N2O

16 %

none

20 %

14 %

none

15 %

O3

n/a

n/a

n/a

7.3 %

0.1 %

15 %

At 400 nm the Kerr gamma is at the edge of its data. The largest fitted \(W_2\) corresponds to a Kerr fundamental of 374-401 nm for the gases with ESHG data (263 nm for helium’s calculation). For N2O and O3 it corresponds to 664 and 695 nm, so N2O’s 400 nm value rests on its borrowed dispersion. The single legs of the data reach much further than the Kerr diagonal: to the 229-244 nm harmonics of the visible ESHG and calculated THG (152 nm for helium; Third-order support and resonance limits). At 1800 nm the Kerr gamma is within a few per cent of the static limit, fixed by Shelton’s 1064 and 1319 nm values. Leaving those two out, and extrapolating the visible data, moves it by 3.7 % (O2), 5.9 % (CO2), 6.3 % (Kr) and 6.7 % (SF6). This is why those estimates are large.

See Third-order extrapolation.

Gas by gas#

  • He: on theory. The fit reproduces Bishop & Pipin’s four processes with residuals of about 0.015 % rms. Their tabulated values are in the reduced-mass atomic unit of 4He, and are converted to standard atomic units (+0.096 %) since 2026-09-26, as Shelton & Rice do.

  • Ne: Shelton’s 1990 and 1992 ESHG values agree. Hättig & Jørgensen’s CCSD process dependence enters at a fitted scale of 1.030 (their values 3 % above the fit).

  • Ar: Shelton’s 19 rescaled literature values, his two infrared values and Hättig & Jørgensen (scale 1.011) agree (chi-squared 24 for 36).

  • Kr: only Shelton’s six values (two infrared, four rescaled literature), which disagree: chi-squared 20 for 2 degrees of freedom. The statistical uncertainty is scaled up by 3.2, and leaving out the infrared pair moves the Kerr gamma by 3.1 % at 800 nm and 6.3 % at 1800 nm. There is no calculated process dependence (B’ is held at its prior, 0).

  • Xe: Tarazkar et al.’s CCSD calculation (scale 1.009) gives the process dependence. Leaving out the rescaled literature moves the Kerr gamma at 400 nm by 6.8 % and the THG at 800 nm by 3.1 %.

  • H2, D2: Bishop et al.’s 1991 electronic calculation agrees with Shelton’s electronic ESHG (fitted scales 0.997 and 0.995). The vibrational and rotational part is subtracted with Shelton’s own tabulated values.

  • N2: Hättig & Jørgensen (scale 0.994). The vibrational and rotational part is subtracted with Shelton’s formula (Table V).

  • O2: Shelton only, with B’ held at 0. The infrared pair carries the long-wavelength side, hence 4 % at 1800 nm.

  • CO2: Sekino & Bartlett’s CCSD(T) (scale 0.940: their values 6 % below the fit). Leaving out the rescaled literature moves the Kerr gamma at 400 nm by 7.1 %.

  • CH4: Shelton only; his 1319 nm point is excluded, as Shelton excluded it. Hättig & Jørgensen’s CCSD lies 11-15 % below and is a validator; fitting its shape was tried and rejected (TC-314).

  • SF6: Shelton only, with B’ held at 0.

  • CO: the level is Ward & Miller’s ESHG value. Sekino & Bartlett’s CCSD(T) gives the process dependence at a scale of 1.113, so their values are 11 % above the measurement. Without the measurement the level would be theirs, 10 % higher.

  • N2O: one value, Wahlstrand et al.’s 2012 n2 at 800 nm (electronic plus vibrational), with CO2’s A and B as priors of 50 % and 100 %. Nothing else of any kind is in the library.

  • O3: Naves et al.’s CCSD(T) static and Kerr values, 700 nm and longer; the third harmonic of 800 nm falls in the Hartley band and is not given.

What the independent measurements say#

The fits use the ESHG scale and, for the process dependence, calculations. Every other measurement is a validator, compared with the fit at its own process and wavelengths. Each ratio below is the measured gamma over the fitted model, averaged over the dataset’s values; conversions follow Third-order conventions.

Validator

Source

Data / model (mean over the values)

Wahlstrand et al. 2012 (PRA), SSSI n2, 800 nm

wahlstrand_2012

Ar 0.89; N2 0.87; O2 1.03

Wahlstrand et al. 2012 (PRL), SSSI n2, 800 nm

wahlstrand_2012b

He 0.84; Ne 0.94; Ar 0.92; Kr 0.92; Xe 0.89

Wahlstrand et al. 2015, SSSI electronic n2, 800 nm

wahlstrand_2015

H2 1.01; D2 0.93

Zahedpour et al. 2015, SSSI n2, 400-2400 nm

zahedpour_2015

Ar 0.95; N2 (n2_N2) 0.96; N2 (n2_adjusted_N2) 0.96; O2 0.95

Zahedpour et al. 2019, SSSI n2, 3-10.5 um

zahedpour_2019

Ar 0.88; N2 0.99; O2 1.02

Wang et al. 2013, SPM in a hollow fibre

wang_2013

Ne 1.02; Ar (n2_Ar_800) 1.05; Ar (n2_Ar_1800) 0.68

Nibbering et al. 1997, ‘instantaneous’ n2, 800 nm

nibbering_1997

Ar 1.39; Xe 1.30; N2 2.91; O2 6.08; SF6 1.33

Börzsönyi et al. 2010, n2 (header corrected), 800 nm

borzsonyi_2010

Ne 1.98; Ar 1.88; Xe 1.32; N2 0.83

Loriot et al. 2009 (with erratum), n2, 800 nm

loriot_2009

Ar 0.97; N2 1.36; O2 1.86

Geints et al. 2025, self-focusing, 800 nm

geints_2025

CO2 0.97

Pigeon et al. 2018, four-wave mixing, 10.6 um

pigeon_2018

Kr 1.34; Xe 0.88

Lehmeier et al. 1985, THG, 1055 nm

lehmeier_1985

He 0.96; Ne 0.69; Ar 0.77; Kr 0.90; Xe 0.94; N2 0.88

Ward & New 1969, THG, 694 nm

ward_1969

Ne 0.86; Ar 0.93; Kr 1.13; Xe 0.91; H2 0.97; N2 1.03; CO2 1.11

Ward & Elliott 1984, THG, 694 nm

ward_1984

CH4 0.92; SF6 0.98

Ward & Miller 1979, ESHG, 694 nm

ward_1979

H2 0.95; N2 0.97; O2 0.97; CO2 0.97; SF6 0.99; CO 1.00

Shelton & Buckingham 1982, ESHG, 515 nm

shelton_1982

He 0.90

Lundeen et al. 1983, CARS, 532 nm

lundeen_1983

He 0.83; Ar 1.12; Kr 1.20; D2 1.01; N2 1.15; O2 1.39; CO2 1.29; CH4 1.16; SF6 1.31; CO 1.04

Hahn & Lee 1995, CARS, 532 nm

hahn_1995

CO2: not compared (no Stokes wavelength); CH4: not compared (no Stokes wavelength); CO: not compared (no Stokes wavelength)

Buckingham & Dunmur 1968, dc Kerr, 633 nm

buckingham_1968

He 1.21; Ne 0.92; Ar 0.95; Kr 0.99; Xe 1.03; SF6 1.62

Shelton & Rice 1994, dc Kerr (review), 515-633 nm

shelton_1994

He 1.11

Tammer et al. 1992, absolute dc Kerr (field calibrated), 633 nm

tammer_1992

He 1.00; H2 1.03

Buckingham & Orr 1969, dc Kerr, 633 nm

buckingham_1969

CH4 1.03

Gentle et al. 1989, dc Kerr, 633 nm

gentle_1989

CO2 1.63

Hättig & Jørgensen 1998, CCSD Kerr and THG

hattig_1998

CH4 (ccsd_kerr_CH4) 0.89; CH4 (ccsd_thg_CH4) 0.85

Maroulis 1999, CCSD(T) static

maroulis_1999

O3 0.97

Maroulis 1994, static (correlated)

maroulis_1994

O3 1.20

  • The Maryland n2 (single-shot supercontinuum spectral interferometry, SSSI) mostly lie below the ESHG scale: 0.84-1.03 at 800 nm, 0.95-0.96 at 400-2400 nm (2015) and 0.88-1.02 at 3-10.5 µm (2019). Their stated uncertainties are 7-15 %.

    • Their absolute scale is a nitrogen flow whose index corresponds to 290.9 K at 1 atm, and the 2012 and 2015 values are reduced at that density (0.7 % above 293.15 K; TC-316). Zahedpour et al.’s scale comes through the anisotropy and an unstated density, and stays at 293.15 K.

    • Shelton’s suggested +1 % would widen the gap, not close it.

    • The 5-15 % offset is the largest open question for the Kerr level. The fits keep the ESHG scale, which is tied to an ab initio helium.

  • Other n2:

    • Wang et al. agree at 800 nm (1.05 for Ar, 1.02 for Ne) but not at 1800 nm (0.68). Their 1800 nm point is inconsistent with the Maryland group’s flat infrared argon (TC-313, open).

    • Loriot et al.’s argon agrees (0.97). Their molecules include the rotational response (1.36 for N2, 1.86 for O2), as do Nibbering et al.’s “instantaneous” values (N2 2.9, O2 6.1).

    • Börzsönyi et al.’s values are 0.8-2 times the fits’ after correcting their table’s header, which reads 10^-19 where 10^-20 fits their text and their own literature column (an erratum on circumstantial evidence; TC-316). As printed they were 8-20 times.

  • Third-harmonic generation:

    • Lehmeier et al.’s THG lies 4-31 % below the fitted model (Ne 0.69, Ar 0.77, N2 0.88).

    • Ward & New’s and Ward & Elliott’s THG agree within about 10-15 %.

  • An independent ESHG scale: Ward & Miller’s ESHG at 694 nm agrees within 5 % (0.95-1.00).

  • CARS: Lundeen et al.’s absolute values scatter from 0.83 to 1.39, with their stated 13 % scale. Hahn & Lee’s per-amagat values need a Stokes wavelength that is not recorded.

  • dc Kerr:

    • Tammer et al.’s field-calibrated absolute measurement (1992; the field from the Stark effect of OCS) gives helium 1.00 +- 0.02 (TC-401). Helium’s high values in the older dc Kerr data (Buckingham & Dunmur 1.21; Shelton & Rice’s review average 1.11) are those measurements’, not the model’s.

    • Tammer’s H2 (1.03 +- 0.03, electronic plus vibrational, the orientation term subtracted) is the only absolute dc Kerr level independent of both the ESHG scale and helium theory.

    • Buckingham & Orr’s CH4 (1969) agrees (1.03, +-5 %).

    • Buckingham & Dunmur’s atoms agree within 8 %.

    • Molecules with a vibrational part: SF6’s 1.62 presumably contains the vibrational contribution that a static field brings, which the model does not have. CO2’s 1.63 +- 0.42 (Gentle et al.; the orientation term extrapolated away, their uncertainty being the fit’s scatter) is 1.5 sigma high.

  • Theory:

    • Maroulis’s two static O3 values bracket the model (0.97 and 1.20), which is the method dependence the 12 % allows for.

    • Hättig & Jørgensen’s CH4 lies 11-15 % below.

Compared with the state of the art#

  • The reference data are Shelton’s 1990 ESHG scale and his dispersion fits (Table IV: gamma_0, A, B against \(W_2\) for He to Xe, H2, D2, N2, O2), on Bishop & Pipin’s helium. The fits here are built on them and agree within 0.4 % in gamma_0 (TC-314).

  • What is added:

    • the process dependence at fourth order (B’) from ab initio calculations, each at a fitted scale;

    • the conversion of every other source convention into one currency, so that n2, THG, CARS and dc Kerr can be compared with the same curve;

    • a covariance and a leave-one-out record per gas;

    • CO, N2O and O3, for which no dispersion fit existed.

  • The independent absolute n2 of the Maryland group are the modern counterpart. Their stated 7-15 % covers most of the offset from the ESHG scale; the ESHG scale is the more precise.

  • Ab initio: coupled-cluster values for Ne, Ar, Kr, H2 and N2 lie within a few per cent of the ESHG scale (TC-313). Helium’s (Bishop & Pipin) is better than any measurement.

The fits have not been reviewed outside this project.

What the uncertainties leave out#

  • The truncation of the dispersion form: outside the fitted \(W_2\) (Third-order support and resonance limits’s table gives each gas’s shortest Kerr and THG fundamental; the model warns there), beyond the fitted legs (see Third-order extrapolation), and near resonance (O3; the resonance limits of every gas are in Third-order support and resonance limits).

  • Shelton’s possible +1 % helium pair-interaction bias: it is covered by the floor, not applied.

  • The vibrational and rotational parts: the models are electronic, except the unseparated ESHG vibrational part of CO2, CH4 and SF6 and whatever the N2O n2 contains. In propagation the Raman response is separate.

  • Local-field factors: \(n_2\) is given in the dilute limit. The Lorentz local-field factor raises the nonlinear polarization by about \(8(n-1)/3\), or 7e-4 for argon at 1 bar and 800 nm, growing with density.

  • The unresolved disagreements:

    • the Maryland n2 against the ESHG scale;

    • Wang et al.’s 1800 nm argon;

    • Börzsönyi et al.’s values, even corrected;

    • krypton’s internal disagreement.

What would improve it#

The papers listed on the Roadmap, most useful first:

  • an absolute dc Kerr of helium (Tammer et al. 1992) and the older absolute dc Kerr of the molecules;

  • Shelton’s 1987 vibrational gamma of N2 and O2;

  • the Maryland calibration (Wahlstrand et al. 2016);

  • any gamma at all for N2O;

  • the supplementary material behind the O3 calculations.

See the representative four-wave-mixing quartets.