Third-order models and fits#

Read Third-order conventions first. This chapter explains the polynomial model, its time-domain form and fit choices. Use Third-order support and resonance limits and Third-order accuracy to assess a calculation. For a runnable starting point, return to Third-order nonlinearity; commands and configuration maintenance are in Running fits and robustness checks.

The dispersion model#

Below resonance gamma is analytic in the squared frequencies. Bishop showed, and Shelton’s measurements across processes confirm, that to fourth order it depends on the four frequencies only through \(W_2\) and \(W_4\):

\[ \gamma = \gamma_0 \left[1 + A\,W_2 + B\,W_2^2 + B'\,W_4\right]. \]

  • \(\gamma_0\) is the static gamma, in au.

  • \(A\) (in \(10^{-10}\) cm\(^2\)) is the same for every process. To second order in frequency one curve against \(W_2\) serves all processes.

  • \(B\) and \(B'\) (in \(10^{-20}\) cm\(^4\)) are the fourth-order terms.

    • \(B'\) separates the processes: Bishop & De Kee (1996) give it for helium, and calculations of several processes fix it for the other gases.

    • Shelton’s single process-dependent \(B\) is this \(B + B' W_4/W_2^2\). For the ESHG, \(W_4/W_2^2 = 1/2\); for the Kerr effect, 1/4; for the third harmonic, 7/12.

    • Where no calculation gives the process dependence, \(B'\) is held at its prior, 0.

\(W_2\) is the natural variable: THG at 1055 nm has the same \(W_2\) as the Kerr effect at 609 nm, and so nearly the same gamma. The stored model is valid over its declared fundamental-wavelength range and, for any process, while \(W_2\) stays within the fitted data. gamma_au() warns (ModelRangeWarning) outside the wavelength range, and wherever the process’s \(W_2\) exceeds the largest fitted \(W_2\) by more than 10 % (optmat.gas.nonlinear.W2_MARGIN): the third harmonic of a fundamental shorter than the table’s THG limit below, for instance, is an extrapolation of the form, which fails toward resonance.

The fitted parameters (gamma in au; A in 1e-10 cm²; B, B’ in 1e-20 cm⁴), with the largest \(W_2\) among the fitted values and the shortest Kerr and THG fundamentals it corresponds to:

Gas

gamma_0 (au)

A (1e-10 cm²)

B (1e-20 cm⁴)

B’ (1e-20 cm⁴)

Values fitted

chi² / dof

Largest W2 fitted (1e9 cm⁻²)

Shortest Kerr / THG fundamental (nm)

He

43.15

0.448

0.210

-0.043

52

1.24 / 48

5.78

263 / 456

Ne

107.1

0.550

0.298

-0.003

26

7.11 / 21

2.86

374 / 648

Ar

1168

1.115

1.780

-0.233

41

24.4 / 36

2.86

374 / 648

Kr

2602

1.364

3.552

0.000

6

20.3 / 2

2.52

399 / 690

Xe

6836

1.703

7.807

-0.838

10

12 / 5

2.52

399 / 690

H2

684.9

1.288

1.899

-0.151

56

29.4 / 51

2.86

374 / 648

D2

667.2

1.267

1.848

-0.179

46

13.5 / 41

2.64

389 / 674

N2

920.3

1.050

1.563

-0.297

37

38.7 / 32

2.86

374 / 648

O2

962.8

1.112

4.602

0.000

20

5.7 / 16

2.64

389 / 674

CO2

1151

1.219

2.545

-0.203

11

15.1 / 6

2.52

399 / 690

CH4

2590

1.532

4.333

0.000

13

2.04 / 9

2.64

389 / 674

SF6

1429

0.494

2.154

0.000

8

3.37 / 4

2.86

374 / 648

CO

1432

1.402

1.489

0.000

6

0.638 / 2

2.49

401 / 695

N2O

1804

1.250

2.400

0.000

1

0 / -2

0.907

664 / 1150

O3

2712

0.728

3.544

0.000

4

0 / 1

0.829

695 / 1203

The time-domain form#

Because gamma depends on each frequency only through its square and the square of the sum, the model has an exact time-domain form. The operator

\[ \hat D = -\frac{1}{(2\pi c)^2}\frac{\partial^2}{\partial t^2} \]

multiplies a field component of wavenumber \(\nu\) by \(\nu^2\). Applied to one input field it gives \(\nu_i^2\); applied to the product it gives \(\nu_\sigma^2\). The polarization of the model, for any real field \(E(t)\), is then

\[\begin{split} \begin{aligned} P = \frac{N\gamma_0}{6}\Big\{ & E^3 + A\left[\hat D(E^3) + 3E^2\,\hat DE\right] \\ & + B\left[\hat D^2(E^3) + 3E^2\,\hat D^2E + 6\,\hat D\!\left(E^2\,\hat DE\right) + 6E\,(\hat DE)^2\right] + B'\left[\hat D^2(E^3) + 3E^2\,\hat D^2E\right]\Big\}, \end{aligned} \end{split}\]

with \(A\), \(B\), \(B'\) in consistent units. Each term reproduces, for every mixing channel at once (Kerr, cross-phase, the third harmonic and every four-wave-mixing product), the gamma of that channel’s frequencies. Every term is a product of time derivatives of the field and of field products, so on a grid it costs a few FFTs. The polynomial grows as \(\nu^4\): it must only see the band where the model is valid, so the field should be limited to that band or the terms cut off above it (Where the fits are supported, in Third-order support and resonance limits). A propagation code can use this form subject to the same support and resonance limitations.

The fits#

Configuration fields are documented in Running fits and robustness checks.

The fit#

fit_hyperpolarizability() converts every fitted dataset to electronic \(\gamma_T\) and its frequency sums, then minimises

\[ \chi^2 = \sum_{d}\sum_{i\in d}\left(\frac{s_{g(d)}\,\gamma(W_{2,i}, W_{4,i}) - \gamma_i}{\sigma_i}\right)^2 + \sum_\text{priors}\left(\frac{p - p_0}{\sigma_p}\right)^2 \]

by nonlinear least squares:

  • Scales: each dataset with free_scale or systematic belongs to a scale group \(g\), with scale \(s_g\) (1 for the rest). A systematic prior enters as \((s_g - 1)/ \sigma\).

  • Priors: on \(B'\), and where given on \(A\) and \(B\).

  • Covariance: the inverse of \(J^TJ\) at the solution, over the four physical parameters.

  • Statistics: \(\chi^2\) and the degrees of freedom (values minus physical parameters minus scales; the priors are not counted).

  • Uncertainty: gamma_sigma_au() propagates the covariance to any process and wavelength. It scales it by the reduced \(\chi^2\) where that exceeds one (Birge).

The data that are not fitted are validators: validation() reports, per validator, the mean ratio of data to model and its \(\chi^2\), or why it could not be compared (a CARS value without its Stokes wavelength).

Commands, stored records and regeneration order are in Running fits and robustness checks.

The choices, gas by gas#

  • The backbone: Shelton’s 1990 ESHG, on Bishop & Pipin’s helium scale. It gives his own 1064 and 1319 nm values and his rescaling of the visible literature to that scale (shelton_1990/table3_<gas>_present and _literature, and Shelton 1992 for neon).

  • Calculated process dependence, one free scale per calculation (scale_group), so that it informs \(B'\) and the shape but not the level:

    • Hättig & Jørgensen’s CCSD Kerr, ESHG and THG for Ne, Ar and N2;

    • Bishop et al. 1991 (electronic) for H2 and D2;

    • Tarazkar et al. (the q-aug-cc-pVQZ-pp rows) for Xe;

    • Sekino & Bartlett’s CCSD(T) for CO2 and CO.

    A calculated THG enters only for fundamentals of 650 nm and longer, where the fourth-order form holds. The calculations’ nearest fundamental above that is 694.3 nm for all five gases, so the harmonics that enter are at 231.4 nm and longer (not 217 nm, as the manual said before; TC-864).

  • He: on theory alone: Bishop & Pipin’s four processes to 0.1 hartree (455.6 nm), with 0.1 % allowed per value. The measurements are validators.

  • Kr, O2, CH4, SF6: Shelton only, with \(B'\) held at its prior. CH4’s 1319 nm point is excluded, as Shelton excluded it. Fitting Hättig & Jørgensen’s CCSD shape for CH4 was tried and rejected (scale 0.87, \(\chi^2\) 350 for 18; TC-314).

  • CO: Ward & Miller’s ESHG at 694 nm sets the level; Sekino & Bartlett’s four processes the dispersion.

  • N2O: only Wahlstrand et al.’s n2 at 800 nm, so the dispersion is CO2’s, as priors of 50 % on \(A\) and 100 % on \(B\).

  • O3: theory only: Naves et al.’s CCSD(T) static and Kerr values, with 12 % allowed for the method dependence, from 700 nm. Its third harmonic, for fundamentals up to about 1050 nm, falls in the Hartley and Huggins bands and is not given.

The gallery page The second hyperpolarizability: every process on one curve shows the data of every process against \(W_2\) with the fitted curve.

Values#

At 800 nm and 1800 nm, with the statistical one-sigma uncertainty of the 800 nm values. The working uncertainties, which add the leave-one-out spread and the systematic floors, are on the accuracy page.

Gas

gamma_0 (au)

Kerr, 800 nm (au)

THG, 800 nm (au)

Kerr, 1800 nm (au)

n2, 800 nm, 1 atm, 20 °C (1e-20 cm²/W)

He

43.15

44.39 ± 0.0069

47.06 ± 0.017

43.39

0.368

Ne

107.1

110.9 ± 0.26

119.3 ± 0.27

107.8

0.92

Ar

1168

1258 ± 1.9

1480 ± 2.3

1185

10.4

Kr

2602

2860 ± 17

3592 ± 31

2647

23.8

Xe

6836

7767 ± 30

1.078e+04 ± 77

6988

64.8

H2

684.9

745 ± 0.75

893.9 ± 1.2

696

6.18

D2

667.2

724.7 ± 0.62

866.6 ± 1

677.8

6.01

N2

920.3

986.1 ± 1.7

1147 ± 1.6

932.5

8.19

O2

962.8

1047 ± 2.5

1319 ± 5.1

976.7

8.7

CO2

1151

1249 ± 6.3

1512 ± 5.4

1168

10.4

CH4

2590

2882 ± 8.7

3729 ± 11

2641

24

SF6

1429

1485 ± 4.6

1670 ± 4.5

1438

12.5

CO

1432

1566 ± 44

1884 ± 53

1458

13

N2O

1804

1961 ± 2.5e+02

2378 ± 3.4e+02

1832

16.4

O3

2712

2873 ± 3e+02

n/a (resonant)

2738

23.9