wahlstrand_2018#
J. K. Wahlstrand, S. Zahedpour, A. Bahl, M. Kolesik and H. M. Milchberg, “Bound-Electron Nonlinearity Beyond the Ionization Threshold”, Phys. Rev. Lett. 120, 183901 (2018)
Source record · Catalogue · Review register
DOI: 10.1103/PhysRevLett.120.183901
Source kind: measurement
Recorded technique: 2D+1 single-shot supercontinuum spectral interferometry (the probe image scanned across the spectrometer slit, ~1e4 shots per map) of Ar, Kr, Xe, N2 and O2 in a thin gas target (L_eff ~ 450 um), 42-fs pump (FWHM), supercontinuum probe near 600 nm, 100 ms between pump pulses; the transient index Delta n(x, y, t) is split into a Kerr-like bound-electron term and a plasma term, the plasma term fitted to the measured ionization yield Y = c1 I0^m; intensities calibrated through the Ar Kerr phase (pp. 2-4).
Materials in structured entries: Ar, Kr, N2, O2, Xe
Dataset quantities: ionization_model_parameter, nonlinear_index
Contents: 11 datasets; 1 formula.
Curation notes#
Journal pages 183901-1 to -5 = PDF pages 1-5. Identity checked against the first page.
NO NEW n2 IS TABULATED. The Letter measures the bound-electron response through and above the ionization threshold (up to > 100 TW/cm2, ionization fractions of a few percent to ~10 %) and finds it ‘approximately quadratic in the laser field’ (p. 1, p. 4); the Kerr index shifts Delta n_K against peak intensity (Fig. 4(c), (d)) and the ionization yields (Fig. 3) exist only in figures; the fits c1, m and the other gases’ traces are in the Supplemental Material [27] (in the library since 2026-09-28 as supplement_01; see THE IONISATION YIELDS below).
DEFINITIONS (Gaussian). p. 1: P_NL^b = Delta chi_b E = (Delta n_b / 2 pi) E, Delta chi_b = N alpha_NL, Delta n_b = 2 pi N alpha_NL. p. 3: ‘Delta n_K = 2 n2 I0 is the peak index shift experienced by a probe pulse [8]’, i.e. n2 is the self coefficient. Plasma: Delta n = -N_e / 2 N_cr, N_cr = 3.1e21 cm^-3 at the ~600-nm probe (p. 3). Fit model (p. 3): Delta n(t) = Delta n_K exp(-t^2/tau^2) + Delta n_p (1 + erf[m^(1/2) t / tau]) / 2, tau = tau_FWHM / (2 sqrt(ln 2)), tau_FWHM = 42 fs; Delta n_K is multiplied by 1 - Y/2 to account for the neutral depletion (at most 3 %).
CALIBRATION (p. 3, quoted): ‘we use our previous measurement of the nonlinear refractive index of Ar, n2 = (9.7 +- 1.2) x 10^-20 cm2/W applicable for intensities to nearly the ionization threshold [9], to provide full 2D calibration of our intensity profiles through Delta n(x, y) = 2 n2 I(x, y) ... at subionization intensities of < 50 TW/cm2 ... The uncertainty in the intensity is 13 %, mostly arising from the uncertainty in n2’. [9] = wahlstrand_2012b. That value is recorded as reproduced.
FINDINGS (notes only; no quantity). In Ar, Kr and Xe the Kerr index shift above the ionization threshold is ‘somewhat higher than the curve extrapolated from below the threshold’; ‘Delta n_K rises slightly faster than |E|^2 in the noble gases, but confirmation awaits more accurate measurements’. In N2 it ‘closely follows the |E|^2 curve extrapolated from the subthreshold response [8]’, in O2 it is ‘below the extrapolated curve’ (the bound-electronic/rotational separation complicates both; p. 4). HIGHER-ORDER KERR: the bound response stays quadratic in the field (no saturation or sign change) well beyond the ionization threshold, against the negative higher-order Kerr terms (n4, n6, ...) proposed by Loriot et al. and Bejot et al. [16,17] and the Bree et al. saturation [18]; consistent with the Kramers-Kronig calculation of [9]. The ionization yields agree reasonably with PPT and, for Ar and Kr, with MESA simulations (Fig. 3).
THE IONISATION YIELDS (added 2026-09-28). The Supplemental Material [27] (ionization_SOM_accepted_final.pdf, 11 pages, sha256 4cd891d0...66ef940) is in the library as supplement_01 of this paper; J. C. Travers’s copy is the same file (same sha256). Its pages are cited as SM p. n.
How the yield was obtained (pp. 2-3). The probe’s phase shift is Delta Phi = k Delta n L_eff, ‘The effective gas target thickness L_eff = int [N(z)/N_0] dz ~ 450 um satisfies L_eff/z_p,pr << 1, where N(z) is the gas density profile along the optical path, N_0 is the profile mean density’ (p. 2). ‘For our case of a thin gas target in which the probe experiences negligible refraction, the refractive index profile Delta n(x, y, t) = Delta Phi(x, y, t)/k L_eff gives N_e(x, y) = -2 N_cr Delta n(x, y, t_1) for the axially averaged electron density profile, where t_1 > 50 fs for the noble gases and t_1 > 250 fs for N2 and O2. Here we have used the refractive index shift induced by a low density collisionless plasma Delta n = -N_e/2 N_cr, where N_cr = 3.1 x 10^21 cm^-3 is the critical electron density at the probe central wavelength lambda_pr ~ 600 nm ... The ionization yield Y = N_e/N_0 as a function of intensity is shown in Fig. 3. For each data set (a complete 2D+1 scan of ~10^4 consecutive shots at the same nominal peak laser power), Y (for times t > t_1) was sorted into 20 intensity bins. The points in Fig. 3 are average values for Y in each bin’ (p. 3). So Y is an absolute ionisation fraction (free electrons per initial atom or molecule, all charge states and channels together), local in the transverse plane (no focal-volume average across the beam) and averaged along the thin target; it rests on L_eff and N_0 (from interferometry of the target, SM pp. 1-2) and on N_cr of the probe. The target (SM pp. 1-2): a steel tube with 89 um walls pinched to an outer thickness of 200-400 um, with laser-drilled holes of 80-200 um; the density ‘drops to ~0.5 N_0 near the plane of the hole and falls roughly exponentially, reaching 0.2 N_0 about 100 um beyond the hole’; chamber background below 1 mbar; pump confocal parameter 30 mm, probe 200 mm; pump focused to a 65 um FWHM spot; pump and probe linearly polarised, parallel; pump repetition rate reduced to 20 Hz (SM p. 1; the Letter’s p. 3 has 100 ms between pump pulses). The gas pressure ‘is adjusted to maximize the probe phase shift while limiting ionization-induced refraction’ (SM p. 2) and is not given. The rotational response of N2 and O2 was best fitted ‘assuming a gas temperature of 200 K ... This indicates cooling of the gas as it expands into the vacuum chamber’ (SM p. 8). Data sets: ‘A total of 9 data sets were used for Ar, 4 for Kr, 6 for Xe, 9 for N2, and 8 for O2 to generate Figs. 3 and 4’ (SM p. 8).
The fit (SM p. 8, Eq. (S1), and Table 1 on SM p. 11, which the text calls Table S1): Y = c_1 I_0^m, ‘I_0 is the temporal peak intensity that occurred at a particular x, y location’. Extracted by tools/extract/wahlstrand_2018.py (datasets yield_fit_c1_<gas>, yield_fit_m_<gas>; the formula yield_fit). Table 1 also lists m_MPI (Ar 11, Kr 9, Xe 7, N2 11, O2 7; not fitted, notes only) and two columns of n2 (cm2/W): ‘full intensity range from below to above ionization threshold’, Ar (12.2 +- 1.6)e-20, Kr (2.7 +- 0.4)e-19, Xe (7.1 +- 0.9)e-19, N2 (7.4 +- 1.0)e-20, O2 (5.8 +- 0.8)e-20, ‘extracted from fits of the bound response to a linear intensity dependence for datasets spanning the full intensity range’ (SM p. 11), and ‘Below ionization threshold’ from refs. 8 and 9, Ar (9.7 +- 1.2)e-20, Kr (2.2 +- 0.4)e-19, Xe (5.8 +- 1.1)e-19, N2 (7.4 +- 0.9)e-20, O2 (9.5 +- 1.2)e-20. The n2 columns are NOT CURATED yet (they are third-order data: the full-range values are new effective values of this work, the others reproductions); so the heading ‘NO NEW n2 IS TABULATED’ above holds for the Letter only.
Accuracy of the yields (quoted): ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13% as discussed in the main text’ (Fig. 3’s caption, p. 3); ‘The uncertainty in the intensity is 13%, mostly arising from the uncertainty in n2, with the residual uncertainty due to shot-to-shot fluctuations in the measured phase shift of ~3 mrad’ (p. 3). No uncertainty of c_1 or m is given, and none of L_eff, N_0 or N_cr. The scatter of the points of Fig. 3 about the fits, read on the figure, is about a factor 1.5 either way for Ar, Kr and N2 below 1e-3 and tighter above. With m about 5, 13 % in intensity is a factor 1.13^5 = 1.8 in yield at a given nominal intensity.
Fig. 3’s models: ‘PPT [28] rates are shown as solid lines’ ([28] = Perelomov, Popov and Terent’ev, Sov. Phys. JETP 23, 924 (1966)); ‘The single-active-electron Perelomov-Popov-Terent’ev (PPT) model [28] (solid lines) shows ionization yields in reasonable agreement with the curves of Fig. 3’ (p. 3). THE PPT CURVE IS NOT SPECIFIED: neither the Letter nor the supplement says which form of the rate was used (the Coulomb correction, the sum over m, the number of electrons, the structure constants, the ionisation energies, the treatment of the molecules), nor how the yield was integrated over the pulse. It cannot be reproduced from the paper and is not a reference. MESA (refs. 29-33): open circles for Ar and Kr, figure only.
Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
wahlstrand_2018/calibration_n2_Ar#
{
"id": "calibration_n2_Ar",
"material": "Ar",
"table": "text",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-20,
"abscissa_unit": "nm",
"state": {
"pressure_pa": 101325.0,
"temperature_k": 293.15
},
"use": "reproduced",
"reproduced_from": [
"wahlstrand_2012b"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"part": "electronic",
"coefficient": "self",
"pulse_fs": 40
}
}
1 inline points in the source record.
p. 3: ‘our previous measurement of the nonlinear refractive index of Ar, n2 = (9.7 +- 1.2) x 10^-20 cm2/W’ ([9] = wahlstrand_2012b), used here to calibrate the intensity. State and parameters as in the original (1 atm, 293.15 K assumed).
wahlstrand_2018/yield_fit_c1_Ar#
{
"id": "yield_fit_c1_Ar",
"material": "Ar",
"file": "Ar/wahlstrand_2018__yield_fit_c1_Ar.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "(cm2/TW)^m",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "c_1 of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear",
"intensity_unit": "TW/cm2"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row Ar: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. c_1, printed 2.05 x 10^-13 under the header ‘c_1 [(cm^2/TW)^m]’ (read on the table image; the text layer loses the superscripts), with m of yield_fit_m_Ar: the unit depends on the fitted exponent m = 5.14, so that c_1 I_0^m is dimensionless with I_0 in TW/cm^2. It is as printed and is not converted. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 45-133 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
wahlstrand_2018/yield_fit_m_Ar#
{
"id": "yield_fit_m_Ar",
"material": "Ar",
"file": "Ar/wahlstrand_2018__yield_fit_m_Ar.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "1",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "m of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row Ar: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. m, the column ‘m_fit’, printed 5.14; the multiphoton order printed beside it is m_MPI = 11 (‘the best fit values for m are notably smaller than their corresponding multiphoton ionization values’, SM p. 8). With c_1 of yield_fit_c1_Ar. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 45-133 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
wahlstrand_2018/yield_fit_c1_Kr#
{
"id": "yield_fit_c1_Kr",
"material": "Kr",
"file": "Kr/wahlstrand_2018__yield_fit_c1_Kr.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "(cm2/TW)^m",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "c_1 of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear",
"intensity_unit": "TW/cm2"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row Kr: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. c_1, printed 5.17 x 10^-11 under the header ‘c_1 [(cm^2/TW)^m]’ (read on the table image; the text layer loses the superscripts), with m of yield_fit_m_Kr: the unit depends on the fitted exponent m = 4.43, so that c_1 I_0^m is dimensionless with I_0 in TW/cm^2. It is as printed and is not converted. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 35-105 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
wahlstrand_2018/yield_fit_m_Kr#
{
"id": "yield_fit_m_Kr",
"material": "Kr",
"file": "Kr/wahlstrand_2018__yield_fit_m_Kr.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "1",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "m of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row Kr: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. m, the column ‘m_fit’, printed 4.43; the multiphoton order printed beside it is m_MPI = 9 (‘the best fit values for m are notably smaller than their corresponding multiphoton ionization values’, SM p. 8). With c_1 of yield_fit_c1_Kr. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 35-105 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
wahlstrand_2018/yield_fit_c1_Xe#
{
"id": "yield_fit_c1_Xe",
"material": "Xe",
"file": "Xe/wahlstrand_2018__yield_fit_c1_Xe.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "(cm2/TW)^m",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "c_1 of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear",
"intensity_unit": "TW/cm2"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row Xe: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. c_1, printed 5.47 x 10^-10 under the header ‘c_1 [(cm^2/TW)^m]’ (read on the table image; the text layer loses the superscripts), with m of yield_fit_m_Xe: the unit depends on the fitted exponent m = 4.36, so that c_1 I_0^m is dimensionless with I_0 in TW/cm^2. It is as printed and is not converted. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 18-78 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
wahlstrand_2018/yield_fit_m_Xe#
{
"id": "yield_fit_m_Xe",
"material": "Xe",
"file": "Xe/wahlstrand_2018__yield_fit_m_Xe.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "1",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "m of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row Xe: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. m, the column ‘m_fit’, printed 4.36; the multiphoton order printed beside it is m_MPI = 7 (‘the best fit values for m are notably smaller than their corresponding multiphoton ionization values’, SM p. 8). With c_1 of yield_fit_c1_Xe. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 18-78 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
wahlstrand_2018/yield_fit_c1_N2#
{
"id": "yield_fit_c1_N2",
"material": "N2",
"file": "N2/wahlstrand_2018__yield_fit_c1_N2.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "(cm2/TW)^m",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "c_1 of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear",
"intensity_unit": "TW/cm2"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row N2: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. c_1, printed 5.7 x 10^-16 under the header ‘c_1 [(cm^2/TW)^m]’ (read on the table image; the text layer loses the superscripts), with m of yield_fit_m_N2: the unit depends on the fitted exponent m = 6.28, so that c_1 I_0^m is dimensionless with I_0 in TW/cm^2. It is as printed and is not converted. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 75-125 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
wahlstrand_2018/yield_fit_m_N2#
{
"id": "yield_fit_m_N2",
"material": "N2",
"file": "N2/wahlstrand_2018__yield_fit_m_N2.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "1",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "m of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row N2: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. m, the column ‘m_fit’, printed 6.28; the multiphoton order printed beside it is m_MPI = 11 (‘the best fit values for m are notably smaller than their corresponding multiphoton ionization values’, SM p. 8). With c_1 of yield_fit_c1_N2. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 75-125 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
wahlstrand_2018/yield_fit_c1_O2#
{
"id": "yield_fit_c1_O2",
"material": "O2",
"file": "O2/wahlstrand_2018__yield_fit_c1_O2.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "(cm2/TW)^m",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "c_1 of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear",
"intensity_unit": "TW/cm2"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row O2: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. c_1, printed 9.6 x 10^-11 under the header ‘c_1 [(cm^2/TW)^m]’ (read on the table image; the text layer loses the superscripts), with m of yield_fit_m_O2: the unit depends on the fitted exponent m = 4.09, so that c_1 I_0^m is dimensionless with I_0 in TW/cm^2. It is as printed and is not converted. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 40-105 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
wahlstrand_2018/yield_fit_m_O2#
{
"id": "yield_fit_m_O2",
"material": "O2",
"file": "O2/wahlstrand_2018__yield_fit_m_O2.csv",
"table": "tbl_668d3655bf2e",
"quantity": "ionization_model_parameter",
"unit": "1",
"abscissa": "label",
"use": "check",
"parameters": {
"model": "wahlstrand_2018_yield_fit",
"parameter": "m of Eq. (S1)",
"wavelength_nm": 800,
"pulse_fs": 42,
"polarization": "linear"
}
}
Supplemental Material, Table 1 (SM p. 11; library supplement_01), row O2: the fit Y = c_1 I_0^m, Eq. (S1), to the measured ionisation yield of Fig. 3 (the formula yield_fit), I_0 the temporal peak intensity in TW/cm^2 of the 42 fs (FWHM of intensity), 800 nm pump. m, the column ‘m_fit’, printed 4.09; the multiphoton order printed beside it is m_MPI = 7 (‘the best fit values for m are notably smaller than their corresponding multiphoton ionization values’, SM p. 8). With c_1 of yield_fit_c1_O2. No uncertainty is given; ‘The accuracy of the ionization yield measurement is set by the vertical scatter of the points, while the intensity accuracy is ~13%’ (Fig. 3’s caption, p. 3). The fitted range is not stated; the points of Fig. 3 span about 40-105 TW/cm^2 (read on the figure). A validator of yields over that range, not a rate model: nothing is fitted to it.
Formulas#
wahlstrand_2018/yield_fit#
{
"id": "yield_fit",
"material": "Ar",
"equation": "Eq. (S1), Supplemental Material p. 8, with its Table 1 (p. 11); the Letter's p. 4 and Fig. 3",
"printed": "Y = c_1 I_0^m; Table 1, c_1 [(cm^2/TW)^m] and m_fit: Ar 2.05 x 10^-13, 5.14; Kr 5.17 x 10^-11, 4.43; Xe 5.47 x 10^-10, 4.36; N2 5.7 x 10^-16, 6.28; O2 9.6 x 10^-11, 4.09",
"variable": "I_0: the temporal peak of the cycle-averaged intensity of the pump at the transverse position of the probe ray, in TW/cm^2 (1 TW/cm^2 = 1e12 W/cm^2), of a pulse at 800 nm whose intensity envelope is 'well fit by I(t) = I_0 exp(-t^2/tau^2) where tau = tau_FWHM/(2 sqrt(ln 2)) corresponds to our pulse full width at half maximum tau_FWHM = 42 fs' (p. 4), linearly polarised; Y = N_e/N_0: the ionisation fraction after the pulse (t > 50 fs for the noble gases, t >= 250 fs for N2 and O2), dimensionless. The parameters are the datasets yield_fit_c1_<gas> and yield_fit_m_<gas>.",
"range": "Not stated. 'The measured ionization yield Y over the intensity range measured here is very well-fit by the expression' (SM p. 8). The points of Fig. 3, read on the figure (2026-09-28), span about 45-133 TW/cm^2 for Ar (Y 1e-4 to 1.6e-2), 35-105 for Kr (3.5e-4 to 4e-2), 18-78 for Xe (1.4e-4 to 7e-2), 75-125 for N2 (6e-4 to 9e-3) and 40-105 for O2 (5e-4 to 1.6e-2).",
"coefficients": {
"c1_Ar": "2.05e-13",
"m_Ar": "5.14",
"c1_Kr": "5.17e-11",
"m_Kr": "4.43",
"c1_Xe": "5.47e-10",
"m_Xe": "4.36",
"c1_N2": "5.7e-16",
"m_N2": "6.28",
"c1_O2": "9.6e-11",
"m_O2": "4.09"
},
"check": "Table 1 checked against the table image and the rendered page (SM p. 11, 2026-09-28), every cell of the columns Gas, c_1, m_fit and m_MPI. The fits evaluated (computed 2026-09-28): Ar 3.9e-3 at 100 and 1.7e-2 at 133 TW/cm^2; Kr 1.05e-2 at 75 and 4.7e-2 at 105; Xe 1.4e-2 at 50 and 9.7e-2 at 78; N2 2.1e-3 at 100 and 8.4e-3 at 125; O2 4.5e-3 at 75 and 1.8e-2 at 105. These agree with the dashed lines and the points of Fig. 3 as far as the figure can be read (about 30 %), which also confirms the unit of I_0 (TW/cm^2) and the positive exponent of the unit of c_1. The measured points themselves are in the figure only and are not digitised."
}
NOT A MODEL OF IONISATION: ‘The point of the expression used for Y is not to advance a multiphoton-ionization (MPI)-like model for ionization; it is to provide an analytic model fit to the ionization yield data to enable separation of the bound and free electron contributions. In fact, as seen in [27], the best fit values for m are notably smaller than their corresponding MPI values for each species, indicating the significant contribution of tunneling ionization’ (p. 4; SM p. 8 likewise). A power law in the peak intensity for this pulse (42 fs, 800 nm) over the range measured: it has no saturation and must not be extrapolated. In the Letter’s transient model the rate is w(t) = c_2 (I_0 exp(-t^2/tau^2))^m, ‘for which the yield is Y(t -> infinity) = c_1 I_0^m’ (p. 4): c_2 = c_1 sqrt(m/pi)/tau follows and is not printed. UNIT OF c_1: the image of Table 1 has ‘c_1 [(cm^2/TW)^m]’, which makes Y dimensionless with I_0 in TW/cm^2; the library’s text layer has the exponent as ‘-m’ (a lost superscript). The material is nominal: the formula takes each gas’s own c_1 and m.
Corrections#
No dataset errata are recorded in this source record.
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.