hellwarth_1990#
R. W. Hellwarth, D. M. Pennington and M. A. Henesian, “Indices governing optical self-focusing and self-induced changes in the state of polarization in N2, O2, H2, and Ar gases”, Phys. Rev. A 41, 2766 (1990)
Source record · Catalogue · Review register
Source kind: compilation
Recorded technique: Analysis of published dc Kerr constants (Carusotto et al. 1985 at 515 nm, absolute and relative to He; older 633-nm values of Buckingham and co-workers) and ESHG ratios to He (Mizrahi and Shelton 1985) on Bishop and Lam’s He theory, through the Born-Oppenheimer form of the third-order response (electronic sigma plus nuclear response functions a(t), b(t)), with two dispersion formulas (A: linear-index Sellmeier poles; B: one fitted nonlinear pole omega_0 per gas, tensor-component-specific), to give the long-wavelength (LWL) parameters and the degenerate four-wave-mixing c coefficients (self-focusing c1111, polarization change c1122) from 261 to 1053 nm (secs. III-VI). No new measurement.
Materials in structured entries: Ar, H2, He, N2, O2
Dataset quantities: hyperpolarizability_ratio, kerr_constant, nonlinear_susceptibility
Contents: 54 datasets; 9 formulas.
Curation notes#
Journal pages 2766-2777 = PDF pages 1-12 (Table VII is on p. 2774). Identity checked against the first page. N2, O2, Ar, H2, He; air (0.78 N2 + 0.21 O2) is not curated: its rows are skipped.
UNITS AND DEFINITIONS (Gaussian esu throughout, p. 2766-2768). Eqs. (1)-(2): real fields E(r, t) = Re sum_i^+ E(r, omega_i) exp(-i omega_i t), P likewise (full complex amplitudes, ‘some authors express the fields in terms of Fourier amplitudes which are one-half of the ordinary E and P amplitudes’); n = (1 + 4 pi chi)^(1/2). Eq. (4): P_alpha^NL (omega) = D c_(alpha beta gamma delta)(omega_i, omega_j, omega_k, omega_l) E_beta(omega_j) E_gamma(omega_k) E_delta(omega_l), omega_i + omega_j + omega_k + omega_l = 0, omega = -omega_i, ‘The degeneracy factor D ... takes on the values 1, 3, or 6 when either all three, or any two, or none of the last three frequency arguments ... are equal’ (the Maker-Terhune c tensor; ‘sometimes denoted chi(3) ... and sometimes written 4 chi(3) when field conventions differ by a factor of 2’). Isotropic form, Eq. (8): c = d d c1122 + d d c1212 + d d c1221. Eq. (11): Delta n = n2 <E^2> = (1/2) n2 E*(omega).E(omega); Eq. (12): n2 = (12 pi / n) c1111(-omega, -omega, omega, omega) (self-focusing index, esu, per mean square field). ESHG: c1111(-2 omega, 0, omega, omega) (D = 3); dc Kerr: c1221(-omega, omega, 0, 0) (D = 3); THG: c1111(-3 omega, omega, omega, omega) (D = 1). Born-Oppenheimer form, Eq. (13): P^NL(t) = (1/2) sigma E(t) E^2(t) + int ds [a(t - s) E(t) E^2(s) + b(t - s) E(s) E(t).E(s)], sigma the electronic (‘nuclei frozen’), a and b the isotropic (vibrational Raman) and anisotropic (rotational) nuclear responses, A_nu, B_nu their Fourier transforms (Eq. (14)). LWL relations: Eq. (15) c1111(-omega, -omega, omega, omega) = sigma/8 + (A0 + B0)/6; Eq. (16) A0 = -B0/3 (pure vibrational Q branch neglected); Eq. (17) c1122 = (sigma + 2 B0)/24, c1212 = (sigma + 2 A0 + B0)/24; Eq. (18) ESHG and THG c1111 = sigma/8; Eq. (19) (n_par - n_perp) / (2 pi E0^2) = 24 c1221(-omega, omega, 0, 0)/n (dc Kerr; Carusotto et al.’s K = (n_par - n_perp) rho E0^2 as printed, p. 2768); Eq. (20) 24 c1221(-omega, omega, 0, 0)_LWL = sigma + B0. Eq. (21), p. 2769: ‘5.036 x 10^-40 gamma_zzzz(omega_4, omega_1, omega_2, omega_3) a.u. = 24 rho^-1 c1111(omega_4, ...) esu’ (Bishop and Lam’s per-atom gamma in atomic units), and p. 2770: ‘When converted to esu, the gamma parameter of Ref. 16 [Boyle et al. 1966 = boyle_1966] equals 3 sigma/rho’. So sigma = rho gamma_T(esu)/3 with gamma_T the Taylor-series gamma, 1 au = 5.036e-40 esu. Check (2026-09-26): He sigma = 1.76e-19 esu at 20 C, 760 Torr (ideal rho = 2.5035e19 cm^-3) gives gamma = 41.9 au, Bishop and Lam’s value; the Bishop-Pipin update raises it 2.9 % (note 15, p. 2776) to ~43.1; Ar sigma = 47.4 gives 1128 au (x 1.029 = 1161).
TABLE I (p. 2770; image checked; tools/extract/hellwarth_1990.py): dc Kerr (n_par - n_perp)/(2 pi E0^2) in 1e-19 esu at 20 C, 760 Torr; ‘Values in parentheses are limits of error of last figures given by source. All other values have estimated error limits of +-10 %’. Footnotes: a, Carusotto et al.’s value divided by their He value; b, that ratio times the He theory (1.83 at 515 nm, 1.80 at 633 nm, g); c, Carusotto et al.’s absolute value (Nuovo Cimento 5D, 328 (1985); not in the library; 2-3 % errors, He +-7 %, their He 9 % above Bishop and Lam, p. 2770); d, Buckingham et al., Trans. Faraday Soc. 66, 1548 (1970) (no He calibration); e, Buckingham and Dunmur (1968) = buckingham_1968, ‘as quoted ... divided by the helium value also given there’; f, Buckingham and Orr, Proc. R. Soc. A 305, 259 (1968) (no He calibration); g, Bishop and Lam’s per-atom value (ideal gas, ~+-3 %). Average: c weighted three times b. ERRATUM: H2 ‘Normalized to He’ printed 3.2; b = a x 1.83 holds for N2 (129 -> 236), O2 (281 -> 514), Ar (25.5 -> 46.7) and gives 59/1.83 = 32.2 for H2. Checks: N2 average (236 + 3 x 257)/4 = 251.75 (252 printed), Ar 49.7, H2 62.8 (63); O2 (515 + 3 x 559)/4 = 548.0 against 549 printed (1 unit; not corrected).
TABLE II (p. 2771, CSV; image checked): sigma (1e-19 esu, 20 C, 760 Torr) = Mizrahi and Shelton’s ESHG ratio to He, extrapolated to the LWL with Eq. (22) and multiplied by He’s 1.76 (Bishop and Lam); errors in parentheses (absolute: 36.5(1.4) = +- 1.4). Eq. (22): c1111(-2 omega, omega, omega, 0) = (sigma/8)(1 - 6 omega^2/omega_0^2)^-1, the MS data (700-450 nm) spanning 0.12-0.35 omega_0.
TABLES V AND VII (pp. 2772, 2774): sigma + B0 (= 24 c1221 LWL) from each Kerr value divided by F^A(-omega, omega, 0, 0) = (1 - omega^2/omega_s^2)^-2 (V) or F_1221^B = (1 - 13 omega^2 / 6 omega_0^2)^-1 (VII); parentheses = relative weights of the average (footnotes b of V, a of VII), not errors. Ar: the average (44.4, 42.9) is replaced in the calculations by 47.4, ‘directly from sigma of Table II’ (+-5 %) (row ‘used’). Estimated uncertainty of these values ‘~+-10 %’ (VII caption, V footnote c). Structural fixes in the script (averages misplaced in the cells of Table V; Table VII’s H2 49.7(3) printed without its wavelength: 633 nm, it is 52.5/F_1221^B(633)).
TABLES VI AND VIII (pp. 2773, 2775): DFWM c coefficients (-omega, -omega, omega, omega), ‘omega(cm^-1) = lambda^-1’, 1e-19 esu, 20 C, 760 Torr, from formula A (VI: F^A = (1 - omega^2/omega_s^2)^-4; c1122 = [2(sigma + B0) - sigma] F^A/24, c1111 = [(sigma + B0)/9 + sigma/72] F^A) and formula B (VIII: c1122 = [2(sigma + B0) - sigma] F_1122^B/24, c1212 = [(sigma + B0)/72 + sigma/36] F_1212^B, c1111 = c1122 + 2 c1212; notes a and b of Table VIII say ‘Table VIII’ for the sigma + B0 source, meaning Table VII). Uncertainty ‘+-10 %’ (Kerr), Ar +-5 % (VIII caption); formula A ‘at least +-10 %’. CHECK (2026-09-26, scratch script): with sigma (Table II), sigma + B0 (Table V averages, Ar 47.4) and omega_s (Table III), formula A reproduces all 20 curated entries of Table VI to <= 0.6 %; with Table VII’s averages and omega_0 (Table IV), formula B reproduces Table VIII’s c1122 and c1212 to <= 0.3 %, except Ar at 261 nm: c1212 printed 2.45 against 3.14 computed (and so c1111 9.74 against 11.11; the printed row is self-consistent, 4.84 + 2 x 2.45 = 9.74). Not corrected (no printed alternative); flagged in the dataset notes. O2 1053 nm c1111 printed 58.2 against c1122 + 2 c1212 = 58.54 (and 58.52 computed): a 0.6 % inconsistency, not corrected. Air c1111 at 333 and 522 nm (104, 61.8) disagree with 0.78 N2 + 0.21 O2 (57.3, 39.3): air skipped anyway.
CONVERSION CHECK (for the reader; not data): Ar c1111(1053 nm, VIII) = 6.09e-19 esu -> n2 = 12 pi c1111 = 2.30e-17 esu per <E^2>; with I = (c n / 8 pi)|E|^2 and Delta n = (1/2) n2 |E|^2 this is 4 pi n2 / c = 9.6e-20 cm2/W (x 1e7 from erg/s to W) at 20 C, 1 atm.
TEXT (notes only): air at 1053 nm and 20 C: c1122(-omega, -omega, omega, omega) = (24 +- 3) x 10^-19 esu (Eq. (33), formula A 23.86, B 23.8) and n2 = (1.2 +- 0.2) x 10^-16 esu (Eq. (34), B value 32.55 of c1111); Ar n2 5.4 times less than air’s, its ICSP coefficient 11 times less; 1-ns pulses may see a c1221 of air ~20 % lower (Raman linewidths ~3 GHz) (p. 2776). Shimoji et al.’s 308-nm air n2 ~2.9e-16 esu per atm at 300 K against 1.6e-16 (A) and 3.9e-16 (B) at 333 nm.
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
hellwarth_1990/kerr_ratio_carusotto_N2#
{
"id": "kerr_ratio_carusotto_N2",
"material": "N2",
"file": "N2/hellwarth_1990__kerr_ratio_carusotto_N2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"scale": 1.0,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"reference_material": "He",
"process": "dc_kerr",
"part": "total"
}
}
Table I, ‘Normalized to He’ (footnote a): Carusotto et al.’s 515-nm Kerr constant divided by their He value, at the same density. A ratio of dc Kerr responses (electronic plus orientational), not of electronic gammas.
hellwarth_1990/kerr_ratio_carusotto_O2#
{
"id": "kerr_ratio_carusotto_O2",
"material": "O2",
"file": "O2/hellwarth_1990__kerr_ratio_carusotto_O2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"scale": 1.0,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"reference_material": "He",
"process": "dc_kerr",
"part": "total"
}
}
Table I, ‘Normalized to He’ (footnote a): Carusotto et al.’s 515-nm Kerr constant divided by their He value, at the same density. A ratio of dc Kerr responses (electronic plus orientational), not of electronic gammas.
hellwarth_1990/kerr_ratio_carusotto_Ar#
{
"id": "kerr_ratio_carusotto_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__kerr_ratio_carusotto_Ar.csv",
"table": "tbl_223eea4ff99e",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"scale": 1.0,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"reference_material": "He",
"process": "dc_kerr",
"part": "total"
}
}
Table I, ‘Normalized to He’ (footnote a): Carusotto et al.’s 515-nm Kerr constant divided by their He value, at the same density. A ratio of dc Kerr responses (electronic plus orientational), not of electronic gammas.
hellwarth_1990/kerr_ratio_carusotto_H2#
{
"id": "kerr_ratio_carusotto_H2",
"material": "H2",
"file": "H2/hellwarth_1990__kerr_ratio_carusotto_H2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"scale": 1.0,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"reference_material": "He",
"process": "dc_kerr",
"part": "total"
}
}
Table I, ‘Normalized to He’ (footnote a): Carusotto et al.’s 515-nm Kerr constant divided by their He value, at the same density. A ratio of dc Kerr responses (electronic plus orientational), not of electronic gammas. ERRATUM: printed 3.2, corrected to 32.2 (source notes).
hellwarth_1990/kerr_ratio_buckingham_dunmur_Ar#
{
"id": "kerr_ratio_buckingham_dunmur_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__kerr_ratio_buckingham_dunmur_Ar.csv",
"table": "tbl_223eea4ff99e",
"quantity": "hyperpolarizability_ratio",
"unit": "1",
"scale": 1.0,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"buckingham_1968"
],
"uncertainty": {
"column": true,
"level": "bound"
},
"parameters": {
"reference_material": "He",
"process": "dc_kerr",
"part": "total"
}
}
Table I, ‘Normalized to He’ at 633 nm (footnote e): Buckingham and Dunmur’s Ar value divided by their He value, 22(3) = 22 +- 3 (limits of error).
hellwarth_1990/kerr_he_scaled_N2#
{
"id": "kerr_he_scaled_N2",
"material": "N2",
"file": "N2/hellwarth_1990__kerr_he_scaled_N2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Absolute’ with footnote b: the He-normalized ratio times the He theory value (1.83 at 515 nm, 1.80 at 633 nm): this paper’s rescaling of Carusotto (515 nm) or Buckingham and Dunmur (Ar, 633 nm). Derived: ‘check’. Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_carusotto_N2#
{
"id": "kerr_carusotto_N2",
"material": "N2",
"file": "N2/hellwarth_1990__kerr_carusotto_N2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Absolute’ with footnote c: Carusotto et al.’s absolute Kerr constant at 515 nm as given there (their errors 2-3 %, He +-7 %). Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_average_N2#
{
"id": "kerr_average_N2",
"material": "N2",
"file": "N2/hellwarth_1990__kerr_average_N2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"carusotto_1985",
"buckingham_1970"
],
"uncertainty": {
"column": true,
"level": "bound",
"relative_rows": {
"515": 0.1
}
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Average’: this paper’s evaluation (absolute value weighted 3 to 1 against the He-scaled one); the 633-nm entry repeats the single value there. Derived from reproduced values: ‘check’. Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_he_scaled_O2#
{
"id": "kerr_he_scaled_O2",
"material": "O2",
"file": "O2/hellwarth_1990__kerr_he_scaled_O2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Absolute’ with footnote b: the He-normalized ratio times the He theory value (1.83 at 515 nm, 1.80 at 633 nm): this paper’s rescaling of Carusotto (515 nm) or Buckingham and Dunmur (Ar, 633 nm). Derived: ‘check’. Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_carusotto_O2#
{
"id": "kerr_carusotto_O2",
"material": "O2",
"file": "O2/hellwarth_1990__kerr_carusotto_O2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Absolute’ with footnote c: Carusotto et al.’s absolute Kerr constant at 515 nm as given there (their errors 2-3 %, He +-7 %). Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_average_O2#
{
"id": "kerr_average_O2",
"material": "O2",
"file": "O2/hellwarth_1990__kerr_average_O2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Average’: this paper’s evaluation (absolute value weighted 3 to 1 against the He-scaled one); the 633-nm entry repeats the single value there. Derived from reproduced values: ‘check’. Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_he_scaled_Ar#
{
"id": "kerr_he_scaled_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__kerr_he_scaled_Ar.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"carusotto_1985",
"buckingham_1968"
],
"uncertainty": {
"column": true,
"level": "bound",
"relative_rows": {
"515": 0.1
}
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Absolute’ with footnote b: the He-normalized ratio times the He theory value (1.83 at 515 nm, 1.80 at 633 nm): this paper’s rescaling of Carusotto (515 nm) or Buckingham and Dunmur (Ar, 633 nm). Derived: ‘check’. Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_carusotto_Ar#
{
"id": "kerr_carusotto_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__kerr_carusotto_Ar.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Absolute’ with footnote c: Carusotto et al.’s absolute Kerr constant at 515 nm as given there (their errors 2-3 %, He +-7 %). Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_average_Ar#
{
"id": "kerr_average_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__kerr_average_Ar.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"carusotto_1985",
"buckingham_1968"
],
"uncertainty": {
"column": true,
"level": "bound",
"relative_rows": {
"515": 0.1
}
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Average’: this paper’s evaluation (absolute value weighted 3 to 1 against the He-scaled one); the 633-nm entry repeats the single value there. Derived from reproduced values: ‘check’. Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_he_scaled_H2#
{
"id": "kerr_he_scaled_H2",
"material": "H2",
"file": "H2/hellwarth_1990__kerr_he_scaled_H2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Absolute’ with footnote b: the He-normalized ratio times the He theory value (1.83 at 515 nm, 1.80 at 633 nm): this paper’s rescaling of Carusotto (515 nm) or Buckingham and Dunmur (Ar, 633 nm). Derived: ‘check’. Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_carusotto_H2#
{
"id": "kerr_carusotto_H2",
"material": "H2",
"file": "H2/hellwarth_1990__kerr_carusotto_H2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Absolute’ with footnote c: Carusotto et al.’s absolute Kerr constant at 515 nm as given there (their errors 2-3 %, He +-7 %). Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_average_H2#
{
"id": "kerr_average_H2",
"material": "H2",
"file": "H2/hellwarth_1990__kerr_average_H2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"carusotto_1985",
"buckingham_1968b"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Average’: this paper’s evaluation (absolute value weighted 3 to 1 against the He-scaled one); the 633-nm entry repeats the single value there. Derived from reproduced values: ‘check’. Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_carusotto_He#
{
"id": "kerr_carusotto_He",
"material": "He",
"file": "He/hellwarth_1990__kerr_carusotto_He.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"carusotto_1985"
],
"uncertainty": {
"column": true,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, ‘Absolute’ with footnote c: Carusotto et al.’s absolute Kerr constant at 515 nm as given there (their errors 2-3 %, He +-7 %). Unit: (n_par - n_perp)/(2 pi E0^2) in esu (= 24 c1221(-omega, omega, 0, 0)/n, Eq. (19)).
hellwarth_1990/kerr_buckingham_1970_N2#
{
"id": "kerr_buckingham_1970_N2",
"material": "N2",
"file": "N2/hellwarth_1990__kerr_buckingham_1970_N2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"buckingham_1970"
],
"uncertainty": {
"column": true,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, 633 nm, footnote d: 251(36) = 251 +- 36 (limits of error), from Buckingham et al. (1970), ‘which has no value for He by which to calibrate the static electric field’.
hellwarth_1990/kerr_buckingham_orr_H2#
{
"id": "kerr_buckingham_orr_H2",
"material": "H2",
"file": "H2/hellwarth_1990__kerr_buckingham_orr_H2.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"buckingham_1968b"
],
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, 633 nm, footnote f: from Buckingham and Orr (1968), ‘which has no value for He by which to calibrate the static electric field’.
hellwarth_1990/kerr_bishop_lam_He#
{
"id": "kerr_bishop_lam_He",
"material": "He",
"file": "He/hellwarth_1990__kerr_bishop_lam_He.csv",
"table": "tbl_223eea4ff99e",
"quantity": "kerr_constant",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "ideal",
"use": "reproduced",
"reproduced_from": [
"D. M. Bishop and B. Lam, Phys. Rev. A 37, 464 (1988)"
],
"uncertainty": {
"relative": 0.03,
"level": "not stated"
},
"parameters": {
"temperature_k": 293.15
}
}
Table I, He ‘Absolute’ footnote g: ‘From the value per atom calculated from Eqs. (14) and (16) of Ref. 15 [Bishop and Lam], assuming ideal gas law; error estimated as ~+-3 %’. Rows 515, 633 nm and inf (static). Theory, the calibration of every ‘b’ value.
hellwarth_1990/sigma_N2#
{
"id": "sigma_N2",
"material": "N2",
"file": "N2/hellwarth_1990__sigma_N2.csv",
"table": "tbl_0c2959460533",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"mizrahi_1985_arn2o2"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "static",
"convention": "hellwarth_sigma",
"part": "electronic"
}
}
Table II: sigma, the electronic parameter of Eq. (13), in the long-wavelength limit (row LWL, wavelength inf): Mizrahi and Shelton’s ESHG ratio to He extrapolated with Eq. (22) and multiplied by He’s 1.76. Derived from mizrahi’s ratios: ‘check’. Error in parentheses (absolute).
hellwarth_1990/sigma_O2#
{
"id": "sigma_O2",
"material": "O2",
"file": "O2/hellwarth_1990__sigma_O2.csv",
"table": "tbl_0c2959460533",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"mizrahi_1985_arn2o2"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "static",
"convention": "hellwarth_sigma",
"part": "electronic"
}
}
Table II: sigma, the electronic parameter of Eq. (13), in the long-wavelength limit (row LWL, wavelength inf): Mizrahi and Shelton’s ESHG ratio to He extrapolated with Eq. (22) and multiplied by He’s 1.76. Derived from mizrahi’s ratios: ‘check’. Error in parentheses (absolute).
hellwarth_1990/sigma_Ar#
{
"id": "sigma_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__sigma_Ar.csv",
"table": "tbl_0c2959460533",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"mizrahi_1985_arn2o2"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "static",
"convention": "hellwarth_sigma",
"part": "electronic"
}
}
Table II: sigma, the electronic parameter of Eq. (13), in the long-wavelength limit (row LWL, wavelength inf): Mizrahi and Shelton’s ESHG ratio to He extrapolated with Eq. (22) and multiplied by He’s 1.76. Derived from mizrahi’s ratios: ‘check’. Error in parentheses (absolute).
hellwarth_1990/sigma_H2#
{
"id": "sigma_H2",
"material": "H2",
"file": "H2/hellwarth_1990__sigma_H2.csv",
"table": "tbl_0c2959460533",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"reproduced_from": [
"mizrahi_1985_h2d2"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "static",
"convention": "hellwarth_sigma",
"part": "electronic"
}
}
Table II: sigma, the electronic parameter of Eq. (13), in the long-wavelength limit (row LWL, wavelength inf): Mizrahi and Shelton’s ESHG ratio to He extrapolated with Eq. (22) and multiplied by He’s 1.76. Derived from mizrahi’s ratios: ‘check’. Error in parentheses (absolute). Note a: ‘See also Ref. 21’ (Mizrahi and Shelton, Phys. Rev. A 32, 3454 (1985) = mizrahi_1985_h2d2).
hellwarth_1990/sigma_He#
{
"id": "sigma_He",
"material": "He",
"file": "He/hellwarth_1990__sigma_He.csv",
"table": "tbl_0c2959460533",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"D. M. Bishop and B. Lam, Phys. Rev. A 37, 464 (1988)"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "static",
"convention": "hellwarth_sigma",
"part": "electronic"
}
}
Table II, He: 1.76(0.05), Bishop and Lam’s theory (Ref. 15), the calibration of the other sigma values. Note 15 (p. 2776): Bishop and Pipin’s corrections raise it by 2.9 %.
hellwarth_1990/kerr_lwl_A_N2#
{
"id": "kerr_lwl_A_N2",
"material": "N2",
"file": "N2/hellwarth_1990__kerr_lwl_A_N2.csv",
"table": "tbl_2ee7689e92b6",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "dc_kerr",
"convention": "hellwarth_sigma",
"part": "total"
}
}
Table V: sigma + B0 = 24 c1221(-omega, omega, 0, 0)_LWL (Eq. (20)) extrapolated with formula A from the Kerr value at the wavelength named in the row id (515, 633 nm); row ‘average’ the paper’s weighted average. All rows are long-wavelength limits (wavelength inf). Weights in parentheses are dropped (kept in ‘printed’). Derived: ‘check’. Estimated uncertainty ~+-10 %.
hellwarth_1990/kerr_lwl_B_N2#
{
"id": "kerr_lwl_B_N2",
"material": "N2",
"file": "N2/hellwarth_1990__kerr_lwl_B_N2.csv",
"table": "tbl_3b8085ea2a30",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "dc_kerr",
"convention": "hellwarth_sigma",
"part": "total"
}
}
Table VII: sigma + B0 = 24 c1221(-omega, omega, 0, 0)_LWL (Eq. (20)) extrapolated with formula B from the Kerr value at the wavelength named in the row id (515, 633 nm); row ‘average’ the paper’s weighted average. All rows are long-wavelength limits (wavelength inf). Weights in parentheses are dropped (kept in ‘printed’). Derived: ‘check’. Estimated uncertainty ~+-10 %.
hellwarth_1990/kerr_lwl_A_O2#
{
"id": "kerr_lwl_A_O2",
"material": "O2",
"file": "O2/hellwarth_1990__kerr_lwl_A_O2.csv",
"table": "tbl_2ee7689e92b6",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "dc_kerr",
"convention": "hellwarth_sigma",
"part": "total"
}
}
Table V: sigma + B0 = 24 c1221(-omega, omega, 0, 0)_LWL (Eq. (20)) extrapolated with formula A from the Kerr value at the wavelength named in the row id (515, 633 nm); row ‘average’ the paper’s weighted average. All rows are long-wavelength limits (wavelength inf). Weights in parentheses are dropped (kept in ‘printed’). Derived: ‘check’. Estimated uncertainty ~+-10 %.
hellwarth_1990/kerr_lwl_B_O2#
{
"id": "kerr_lwl_B_O2",
"material": "O2",
"file": "O2/hellwarth_1990__kerr_lwl_B_O2.csv",
"table": "tbl_3b8085ea2a30",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "dc_kerr",
"convention": "hellwarth_sigma",
"part": "total"
}
}
Table VII: sigma + B0 = 24 c1221(-omega, omega, 0, 0)_LWL (Eq. (20)) extrapolated with formula B from the Kerr value at the wavelength named in the row id (515, 633 nm); row ‘average’ the paper’s weighted average. All rows are long-wavelength limits (wavelength inf). Weights in parentheses are dropped (kept in ‘printed’). Derived: ‘check’. Estimated uncertainty ~+-10 %.
hellwarth_1990/kerr_lwl_A_Ar#
{
"id": "kerr_lwl_A_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__kerr_lwl_A_Ar.csv",
"table": "tbl_2ee7689e92b6",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "dc_kerr",
"convention": "hellwarth_sigma",
"part": "total"
}
}
Table V: sigma + B0 = 24 c1221(-omega, omega, 0, 0)_LWL (Eq. (20)) extrapolated with formula A from the Kerr value at the wavelength named in the row id (515, 633 nm); row ‘average’ the paper’s weighted average. All rows are long-wavelength limits (wavelength inf). Row ‘used’ = 47.4, the Ar value taken from sigma (Table II) for the calculations (footnote f). Weights in parentheses are dropped (kept in ‘printed’). Derived: ‘check’. Estimated uncertainty ~+-10 %.
hellwarth_1990/kerr_lwl_B_Ar#
{
"id": "kerr_lwl_B_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__kerr_lwl_B_Ar.csv",
"table": "tbl_3b8085ea2a30",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "dc_kerr",
"convention": "hellwarth_sigma",
"part": "total"
}
}
Table VII: sigma + B0 = 24 c1221(-omega, omega, 0, 0)_LWL (Eq. (20)) extrapolated with formula B from the Kerr value at the wavelength named in the row id (515, 633 nm); row ‘average’ the paper’s weighted average. All rows are long-wavelength limits (wavelength inf). Row ‘used’ = 47.4, the Ar value taken from sigma (Table II) for the calculations (footnote b). Weights in parentheses are dropped (kept in ‘printed’). Derived: ‘check’. Estimated uncertainty ~+-10 %.
hellwarth_1990/kerr_lwl_A_H2#
{
"id": "kerr_lwl_A_H2",
"material": "H2",
"file": "H2/hellwarth_1990__kerr_lwl_A_H2.csv",
"table": "tbl_2ee7689e92b6",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "dc_kerr",
"convention": "hellwarth_sigma",
"part": "total"
}
}
Table V: sigma + B0 = 24 c1221(-omega, omega, 0, 0)_LWL (Eq. (20)) extrapolated with formula A from the Kerr value at the wavelength named in the row id (515, 633 nm); row ‘average’ the paper’s weighted average. All rows are long-wavelength limits (wavelength inf). Weights in parentheses are dropped (kept in ‘printed’). Derived: ‘check’. Estimated uncertainty ~+-10 %.
hellwarth_1990/kerr_lwl_B_H2#
{
"id": "kerr_lwl_B_H2",
"material": "H2",
"file": "H2/hellwarth_1990__kerr_lwl_B_H2.csv",
"table": "tbl_3b8085ea2a30",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "dc_kerr",
"convention": "hellwarth_sigma",
"part": "total"
}
}
Table VII: sigma + B0 = 24 c1221(-omega, omega, 0, 0)_LWL (Eq. (20)) extrapolated with formula B from the Kerr value at the wavelength named in the row id (515, 633 nm); row ‘average’ the paper’s weighted average. All rows are long-wavelength limits (wavelength inf). Weights in parentheses are dropped (kept in ‘printed’). Derived: ‘check’. Estimated uncertainty ~+-10 %.
hellwarth_1990/dfwm_A_c1122_N2#
{
"id": "dfwm_A_c1122_N2",
"material": "N2",
"file": "N2/hellwarth_1990__dfwm_A_c1122_N2.csv",
"table": "tbl_03c27c55bc84",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1122"
}
}
Table VI: c1122(-omega, -omega, omega, omega) from formula A, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_A_c1111_N2#
{
"id": "dfwm_A_c1111_N2",
"material": "N2",
"file": "N2/hellwarth_1990__dfwm_A_c1111_N2.csv",
"table": "tbl_03c27c55bc84",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1111"
}
}
Table VI: c1111(-omega, -omega, omega, omega) from formula A, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1122_N2#
{
"id": "dfwm_B_c1122_N2",
"material": "N2",
"file": "N2/hellwarth_1990__dfwm_B_c1122_N2.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1122"
}
}
Table VIII: c1122(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1212_N2#
{
"id": "dfwm_B_c1212_N2",
"material": "N2",
"file": "N2/hellwarth_1990__dfwm_B_c1212_N2.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1212"
}
}
Table VIII: c1212(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1111_N2#
{
"id": "dfwm_B_c1111_N2",
"material": "N2",
"file": "N2/hellwarth_1990__dfwm_B_c1111_N2.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1111"
}
}
Table VIII: c1111(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_A_c1122_O2#
{
"id": "dfwm_A_c1122_O2",
"material": "O2",
"file": "O2/hellwarth_1990__dfwm_A_c1122_O2.csv",
"table": "tbl_03c27c55bc84",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1122"
}
}
Table VI: c1122(-omega, -omega, omega, omega) from formula A, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_A_c1111_O2#
{
"id": "dfwm_A_c1111_O2",
"material": "O2",
"file": "O2/hellwarth_1990__dfwm_A_c1111_O2.csv",
"table": "tbl_03c27c55bc84",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1111"
}
}
Table VI: c1111(-omega, -omega, omega, omega) from formula A, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1122_O2#
{
"id": "dfwm_B_c1122_O2",
"material": "O2",
"file": "O2/hellwarth_1990__dfwm_B_c1122_O2.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1122"
}
}
Table VIII: c1122(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1212_O2#
{
"id": "dfwm_B_c1212_O2",
"material": "O2",
"file": "O2/hellwarth_1990__dfwm_B_c1212_O2.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1212"
}
}
Table VIII: c1212(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1111_O2#
{
"id": "dfwm_B_c1111_O2",
"material": "O2",
"file": "O2/hellwarth_1990__dfwm_B_c1111_O2.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1111"
}
}
Table VIII: c1111(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths. The 1053-nm value 58.2 is 0.6 % below c1122 + 2 c1212 = 58.54 (source notes).
hellwarth_1990/dfwm_A_c1122_Ar#
{
"id": "dfwm_A_c1122_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__dfwm_A_c1122_Ar.csv",
"table": "tbl_03c27c55bc84",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1122"
}
}
Table VI: c1122(-omega, -omega, omega, omega) from formula A, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_A_c1111_Ar#
{
"id": "dfwm_A_c1111_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__dfwm_A_c1111_Ar.csv",
"table": "tbl_03c27c55bc84",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1111"
}
}
Table VI: c1111(-omega, -omega, omega, omega) from formula A, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1122_Ar#
{
"id": "dfwm_B_c1122_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__dfwm_B_c1122_Ar.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.05,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1122"
}
}
Table VIII: c1122(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1212_Ar#
{
"id": "dfwm_B_c1212_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__dfwm_B_c1212_Ar.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.05,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1212"
}
}
Table VIII: c1212(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths. The 261-nm row does not follow formula B (c1212 3.14 and c1111 11.11 computed against 2.45 and 9.74 printed; every other entry agrees to <= 0.6 %): doubtful (source notes, TC-283).
hellwarth_1990/dfwm_B_c1111_Ar#
{
"id": "dfwm_B_c1111_Ar",
"material": "Ar",
"file": "Ar/hellwarth_1990__dfwm_B_c1111_Ar.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.05,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1111"
}
}
Table VIII: c1111(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths. The 261-nm row does not follow formula B (c1212 3.14 and c1111 11.11 computed against 2.45 and 9.74 printed; every other entry agrees to <= 0.6 %): doubtful (source notes, TC-283).
hellwarth_1990/dfwm_A_c1122_H2#
{
"id": "dfwm_A_c1122_H2",
"material": "H2",
"file": "H2/hellwarth_1990__dfwm_A_c1122_H2.csv",
"table": "tbl_03c27c55bc84",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1122"
}
}
Table VI: c1122(-omega, -omega, omega, omega) from formula A, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_A_c1111_H2#
{
"id": "dfwm_A_c1111_H2",
"material": "H2",
"file": "H2/hellwarth_1990__dfwm_A_c1111_H2.csv",
"table": "tbl_03c27c55bc84",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1111"
}
}
Table VI: c1111(-omega, -omega, omega, omega) from formula A, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1122_H2#
{
"id": "dfwm_B_c1122_H2",
"material": "H2",
"file": "H2/hellwarth_1990__dfwm_B_c1122_H2.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1122"
}
}
Table VIII: c1122(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1212_H2#
{
"id": "dfwm_B_c1212_H2",
"material": "H2",
"file": "H2/hellwarth_1990__dfwm_B_c1212_H2.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1212"
}
}
Table VIII: c1212(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
hellwarth_1990/dfwm_B_c1111_H2#
{
"id": "dfwm_B_c1111_H2",
"material": "H2",
"file": "H2/hellwarth_1990__dfwm_B_c1111_H2.csv",
"table": "tbl_757adefa584f",
"quantity": "nonlinear_susceptibility",
"unit": "esu",
"scale": 1e-19,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_c": 20.0
},
"density": "unknown",
"use": "check",
"uncertainty": {
"relative": 0.1,
"level": "bound"
},
"parameters": {
"process": "kerr",
"convention": "maker_terhune_c",
"part": "total",
"component": "1111"
}
}
Table VIII: c1111(-omega, -omega, omega, omega) from formula B, rows by wavelength (nm) and LWL (inf). n2 = 12 pi c1111 / n (Eq. (12)). Calculated from Kerr and ESHG data (the long-pulse response: electronic plus adiabatic rotational and vibrational): ‘check’. Uncertainty ‘+-10 %’ (Ar in Table VIII +-5 %); formula A’s larger at short wavelengths.
Formulas#
hellwarth_1990/formula_A_N2#
{
"id": "formula_A_N2",
"material": "N2",
"equation": "Eqs. (23)-(25), p. 2772-2773, with Table III (p. 2771) and the LWL values of Tables II and V",
"printed": "c(ijkl) = c(ijkl)_LWL F^A(ijkl), F^A(omega_4, omega_1, omega_2, omega_3) = prod_(i=1..4) (1 - omega_i^2 / omega_s^2)^-1; c1122(-w,-w,w,w)_LWL = [2(sigma + B0) - sigma]/24, c1111_LWL = (sigma + B0)/9 + sigma/72, c1221(-w,w,0,0)_LWL = (sigma + B0)/24",
"variable": "omega: wavenumber in cm^-1 (omega = 1/lambda, Table VI caption); omega_s in 10^3 cm^-1",
"state": "20 C, 760 Torr (c in 1e-19 esu)",
"density": "unknown",
"range": "omega_L < ~omega_s/2; Table VI down to 261-333 nm ('the shortest wavelengths tabulated are those below which formula B is known to have larger than 10 % error')",
"coefficients": {
"omega_s": "131e3",
"sigma": "36.5",
"sigma_plus_B0": "242"
},
"check": "Reproduces every Table VI entry of N2 (c1122, c1111) to <= 0.6 % (checked 2026-09-26)."
}
omega_s: Table III, ‘taken from Ref. 20’ (Landolt-Boernstein, Table 2863): the single-pole Sellmeier frequency of the linear index, reproduced. sigma + B0: Table V average.
hellwarth_1990/formula_B_N2#
{
"id": "formula_B_N2",
"material": "N2",
"equation": "Eqs. (26)-(31), p. 2773-2774, with Table IV (p. 2771) and the LWL values of Tables II and VII",
"printed": "c1122(ijkl) = c1122(ijkl)_LWL F^B_1122(ijkl), F^B_1122 = (1 - omega_1122^2 / omega_0^2)^-1, omega_1122^2 = [13 omega_L^2 + 6(omega_i omega_j + omega_k omega_l)]/12 (likewise 1212 with omega_i omega_k + omega_j omega_l, 1221 with omega_i omega_l + omega_j omega_k), omega_L^2 = omega_i^2 + omega_j^2 + omega_k^2 + omega_l^2",
"variable": "omega: wavenumber in cm^-1; omega_0 in 10^3 cm^-1",
"state": "20 C, 760 Torr (c in 1e-19 esu)",
"density": "unknown",
"range": "omega_L < 2 omega_0 / 3 (He: < 10 % at 2 omega_0/3, < 3 % at omega_0/2); Table VIII down to 3/omega_0",
"coefficients": {
"omega_0": "115e3",
"omega_0_uncertainty": "10e3",
"sigma": "36.5",
"sigma_plus_B0": "240"
},
"check": "Reproduces Table VIII's N2 c1122, c1212 and c1111 to <= 0.6 % (checked 2026-09-26)."
}
omega_0: Table IV, fitted to Mizrahi and Shelton’s ESHG dispersion with Eq. (22); ‘the uncertainties quoted span the values which give a satisfactory fit’. sigma + B0: Table VII average. Special cases: ESHG c1111(2w,w,w,0) = (sigma/24)[1/(1 - 7 w^2/w0^2) + 2/(1 - 11 w^2/2 w0^2)] (Eq. (32)), approximated by Eq. (22) (sigma/8)(1 - 6 w^2/w0^2)^-1 within 0.2 % over the MS range; dc Kerr F^B_1221(-w,w,0,0) = (1 - 13 w^2/6 w0^2)^-1 (Table VII caption).
hellwarth_1990/formula_A_O2#
{
"id": "formula_A_O2",
"material": "O2",
"equation": "Eqs. (23)-(25), p. 2772-2773, with Table III (p. 2771) and the LWL values of Tables II and V",
"printed": "c(ijkl) = c(ijkl)_LWL F^A(ijkl), F^A(omega_4, omega_1, omega_2, omega_3) = prod_(i=1..4) (1 - omega_i^2 / omega_s^2)^-1; c1122(-w,-w,w,w)_LWL = [2(sigma + B0) - sigma]/24, c1111_LWL = (sigma + B0)/9 + sigma/72, c1221(-w,w,0,0)_LWL = (sigma + B0)/24",
"variable": "omega: wavenumber in cm^-1 (omega = 1/lambda, Table VI caption); omega_s in 10^3 cm^-1",
"state": "20 C, 760 Torr (c in 1e-19 esu)",
"density": "unknown",
"range": "omega_L < ~omega_s/2; Table VI down to 261-333 nm ('the shortest wavelengths tabulated are those below which formula B is known to have larger than 10 % error')",
"coefficients": {
"omega_s": "113e3",
"sigma": "37.8",
"sigma_plus_B0": "517"
},
"check": "Reproduces every Table VI entry of O2 (c1122, c1111) to <= 0.6 % (checked 2026-09-26)."
}
omega_s: Table III, ‘taken from Ref. 20’ (Landolt-Boernstein, Table 2863): the single-pole Sellmeier frequency of the linear index, reproduced. sigma + B0: Table V average.
hellwarth_1990/formula_B_O2#
{
"id": "formula_B_O2",
"material": "O2",
"equation": "Eqs. (26)-(31), p. 2773-2774, with Table IV (p. 2771) and the LWL values of Tables II and VII",
"printed": "c1122(ijkl) = c1122(ijkl)_LWL F^B_1122(ijkl), F^B_1122 = (1 - omega_1122^2 / omega_0^2)^-1, omega_1122^2 = [13 omega_L^2 + 6(omega_i omega_j + omega_k omega_l)]/12 (likewise 1212 with omega_i omega_k + omega_j omega_l, 1221 with omega_i omega_l + omega_j omega_k), omega_L^2 = omega_i^2 + omega_j^2 + omega_k^2 + omega_l^2",
"variable": "omega: wavenumber in cm^-1; omega_0 in 10^3 cm^-1",
"state": "20 C, 760 Torr (c in 1e-19 esu)",
"density": "unknown",
"range": "omega_L < 2 omega_0 / 3 (He: < 10 % at 2 omega_0/3, < 3 % at omega_0/2); Table VIII down to 3/omega_0",
"coefficients": {
"omega_0": "90e3",
"omega_0_uncertainty": "5e3",
"sigma": "37.8",
"sigma_plus_B0": "494"
},
"check": "Reproduces Table VIII's O2 c1122, c1212 and c1111 to <= 0.6 % (checked 2026-09-26)."
}
omega_0: Table IV, fitted to Mizrahi and Shelton’s ESHG dispersion with Eq. (22); ‘the uncertainties quoted span the values which give a satisfactory fit’. sigma + B0: Table VII average. Special cases: ESHG c1111(2w,w,w,0) = (sigma/24)[1/(1 - 7 w^2/w0^2) + 2/(1 - 11 w^2/2 w0^2)] (Eq. (32)), approximated by Eq. (22) (sigma/8)(1 - 6 w^2/w0^2)^-1 within 0.2 % over the MS range; dc Kerr F^B_1221(-w,w,0,0) = (1 - 13 w^2/6 w0^2)^-1 (Table VII caption).
hellwarth_1990/formula_A_Ar#
{
"id": "formula_A_Ar",
"material": "Ar",
"equation": "Eqs. (23)-(25), p. 2772-2773, with Table III (p. 2771) and the LWL values of Tables II and V",
"printed": "c(ijkl) = c(ijkl)_LWL F^A(ijkl), F^A(omega_4, omega_1, omega_2, omega_3) = prod_(i=1..4) (1 - omega_i^2 / omega_s^2)^-1; c1122(-w,-w,w,w)_LWL = [2(sigma + B0) - sigma]/24, c1111_LWL = (sigma + B0)/9 + sigma/72, c1221(-w,w,0,0)_LWL = (sigma + B0)/24",
"variable": "omega: wavenumber in cm^-1 (omega = 1/lambda, Table VI caption); omega_s in 10^3 cm^-1",
"state": "20 C, 760 Torr (c in 1e-19 esu)",
"density": "unknown",
"range": "omega_L < ~omega_s/2; Table VI down to 261-333 nm ('the shortest wavelengths tabulated are those below which formula B is known to have larger than 10 % error')",
"coefficients": {
"omega_s": "130e3",
"sigma": "47.4",
"sigma_plus_B0": "47.4"
},
"check": "Reproduces every Table VI entry of Ar (c1122, c1111) to <= 0.6 % (checked 2026-09-26)."
}
omega_s: Table III, ‘taken from Ref. 20’ (Landolt-Boernstein, Table 2863): the single-pole Sellmeier frequency of the linear index, reproduced. sigma + B0: Table V average (Ar: 47.4 from sigma, footnote f).
hellwarth_1990/formula_B_Ar#
{
"id": "formula_B_Ar",
"material": "Ar",
"equation": "Eqs. (26)-(31), p. 2773-2774, with Table IV (p. 2771) and the LWL values of Tables II and VII",
"printed": "c1122(ijkl) = c1122(ijkl)_LWL F^B_1122(ijkl), F^B_1122 = (1 - omega_1122^2 / omega_0^2)^-1, omega_1122^2 = [13 omega_L^2 + 6(omega_i omega_j + omega_k omega_l)]/12 (likewise 1212 with omega_i omega_k + omega_j omega_l, 1221 with omega_i omega_l + omega_j omega_k), omega_L^2 = omega_i^2 + omega_j^2 + omega_k^2 + omega_l^2",
"variable": "omega: wavenumber in cm^-1; omega_0 in 10^3 cm^-1",
"state": "20 C, 760 Torr (c in 1e-19 esu)",
"density": "unknown",
"range": "omega_L < 2 omega_0 / 3 (He: < 10 % at 2 omega_0/3, < 3 % at omega_0/2); Table VIII down to 3/omega_0",
"coefficients": {
"omega_0": "115e3",
"omega_0_uncertainty": "10e3",
"sigma": "47.4",
"sigma_plus_B0": "47.4"
},
"check": "Reproduces Table VIII's Ar c1122, c1212 and c1111 to <= 0.6 %, except the 261-nm c1212 (3.14 computed, 2.45 printed) and hence c1111 (checked 2026-09-26)."
}
omega_0: Table IV, fitted to Mizrahi and Shelton’s ESHG dispersion with Eq. (22); ‘the uncertainties quoted span the values which give a satisfactory fit’. sigma + B0: Table VII average (Ar: 47.4 from sigma, footnote b). Special cases: ESHG c1111(2w,w,w,0) = (sigma/24)[1/(1 - 7 w^2/w0^2) + 2/(1 - 11 w^2/2 w0^2)] (Eq. (32)), approximated by Eq. (22) (sigma/8)(1 - 6 w^2/w0^2)^-1 within 0.2 % over the MS range; dc Kerr F^B_1221(-w,w,0,0) = (1 - 13 w^2/6 w0^2)^-1 (Table VII caption).
hellwarth_1990/formula_A_H2#
{
"id": "formula_A_H2",
"material": "H2",
"equation": "Eqs. (23)-(25), p. 2772-2773, with Table III (p. 2771) and the LWL values of Tables II and V",
"printed": "c(ijkl) = c(ijkl)_LWL F^A(ijkl), F^A(omega_4, omega_1, omega_2, omega_3) = prod_(i=1..4) (1 - omega_i^2 / omega_s^2)^-1; c1122(-w,-w,w,w)_LWL = [2(sigma + B0) - sigma]/24, c1111_LWL = (sigma + B0)/9 + sigma/72, c1221(-w,w,0,0)_LWL = (sigma + B0)/24",
"variable": "omega: wavenumber in cm^-1 (omega = 1/lambda, Table VI caption); omega_s in 10^3 cm^-1",
"state": "20 C, 760 Torr (c in 1e-19 esu)",
"density": "unknown",
"range": "omega_L < ~omega_s/2; Table VI down to 261-333 nm ('the shortest wavelengths tabulated are those below which formula B is known to have larger than 10 % error')",
"coefficients": {
"omega_s": "111e3",
"sigma": "27.9",
"sigma_plus_B0": "55.8"
},
"check": "Reproduces every Table VI entry of H2 (c1122, c1111) to <= 0.6 % (checked 2026-09-26)."
}
omega_s: Table III, ‘taken from Ref. 20’ (Landolt-Boernstein, Table 2863): the single-pole Sellmeier frequency of the linear index, reproduced. sigma + B0: Table V average.
hellwarth_1990/formula_B_H2#
{
"id": "formula_B_H2",
"material": "H2",
"equation": "Eqs. (26)-(31), p. 2773-2774, with Table IV (p. 2771) and the LWL values of Tables II and VII",
"printed": "c1122(ijkl) = c1122(ijkl)_LWL F^B_1122(ijkl), F^B_1122 = (1 - omega_1122^2 / omega_0^2)^-1, omega_1122^2 = [13 omega_L^2 + 6(omega_i omega_j + omega_k omega_l)]/12 (likewise 1212 with omega_i omega_k + omega_j omega_l, 1221 with omega_i omega_l + omega_j omega_k), omega_L^2 = omega_i^2 + omega_j^2 + omega_k^2 + omega_l^2",
"variable": "omega: wavenumber in cm^-1; omega_0 in 10^3 cm^-1",
"state": "20 C, 760 Torr (c in 1e-19 esu)",
"density": "unknown",
"range": "omega_L < 2 omega_0 / 3 (He: < 10 % at 2 omega_0/3, < 3 % at omega_0/2); Table VIII down to 3/omega_0",
"coefficients": {
"omega_0": "100e3",
"omega_0_uncertainty": "10e3",
"sigma": "27.9",
"sigma_plus_B0": "54.8"
},
"check": "Reproduces Table VIII's H2 c1122, c1212 and c1111 to <= 0.6 % (checked 2026-09-26)."
}
omega_0: Table IV, fitted to Mizrahi and Shelton’s ESHG dispersion with Eq. (22); ‘the uncertainties quoted span the values which give a satisfactory fit’. sigma + B0: Table VII average. Special cases: ESHG c1111(2w,w,w,0) = (sigma/24)[1/(1 - 7 w^2/w0^2) + 2/(1 - 11 w^2/2 w0^2)] (Eq. (32)), approximated by Eq. (22) (sigma/8)(1 - 6 w^2/w0^2)^-1 within 0.2 % over the MS range; dc Kerr F^B_1221(-w,w,0,0) = (1 - 13 w^2/6 w0^2)^-1 (Table VII caption).
hellwarth_1990/formula_B_He#
{
"id": "formula_B_He",
"material": "He",
"equation": "Eqs. (26)-(31) and Table IV (p. 2771, note a: 'Fit to the calculations of Ref. 15')",
"printed": "as formula_B_N2, omega_0 = 158 +- 5 (10^3 cm^-1); text p. 2774: 'the choice omega_0 = 161146 cm^-1 reproduced all the BL results to better than 10 % at omega_L = 2 omega_0/3, and to better than 3 % at omega_L = omega_0/2 ... In the limit of low frequencies, the asymptotic data were fit even better ... by omega_0 = 153009 cm^-1'",
"variable": "omega: wavenumber in cm^-1",
"state": "20 C, 760 Torr (c in 1e-19 esu)",
"density": "ideal",
"range": "omega_L < 2 omega_0 / 3",
"coefficients": {
"omega_0": "158e3",
"omega_0_uncertainty": "5e3",
"sigma": "1.76"
},
"check": "Not checked against Bishop and Lam (not in the library). Table VII's He values 1.83 (515 nm) and 1.80 (633 nm) = 1.76 x F^B_1221 'agree with the values calculated in BL ... to within 0.5 %' (footnote c)."
}
The He dispersion that justifies formula B; He’s LWL c1122 = c1212 = c1221 = sigma/24 (A0 = B0 = 0).
Corrections#
[
{
"dataset": "kerr_ratio_carusotto_H2",
"row": "515",
"column": "value",
"printed": "3.2",
"corrected": "32.2",
"reason": "A lost digit: the He-normalized H2 Kerr constant is printed 3.2.",
"evidence": "Table I's footnote b values are the normalized value times 1.83 (the He theory at 515 nm): N2 129 -> 236, O2 281 -> 514 (515 printed), Ar 25.5 -> 46.7; for H2 the printed b value 59 requires 59 / 1.83 = 32.2, and Carusotto's absolute H2 64.1 over their He 1.99 is 32.2. Table image checked: '3.2^a'."
}
]
Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.