wallace_2016

W. C. Wallace, O. Ghafur, C. Khurmi, Satya Sainadh U, J. E. Calvert, D. E. Laban, M. G. Pullen, K. Bartschat, A. N. Grum-Grzhimailo, D. Wells, H. M. Quiney, X. M. Tong, I. V. Litvinyuk, R. T. Sang and D. Kielpinski, “Precise and Accurate Measurements of Strong-Field Photoionization and a Transferable Laser Intensity Calibration Standard”, Phys. Rev. Lett. 117, 053001 (2016)

Source record · Catalogue · Review register

DOI: 10.1103/PhysRevLett.117.053001

Source kind: measurement

Recorded technique: Yields of singly charged ions (Ar+, Kr+, Xe+, and H+ from atomic hydrogen) against the peak intensity of few-cycle pulses: ‘pulses with rapidly varying carrier-envelope phase of 5.5 fs duration (measured at full-width at half maximum of intensity) with a central wavelength of 800 nm’ (pp. 1-2; the carrier-envelope phase is not stabilised), laser bandwidth > 200 nm (p. 4). A well-collimated atomic beam crosses the laser focus; the ions are counted by an ion time-of-flight mass spectrometer with a microchannel plate whose voltage is ‘proportional to the ion yield’ (p. 2). The beam’s diameter ‘is much smaller than the laser Rayleigh range and much larger than the transverse size of the laser beam’, so the focal-volume average is over the two transverse dimensions only (p. 2; the yield curve is P^(2D)). The intensity is calibrated in the same apparatus with atomic hydrogen: the H+ yield is fitted with the focal-volume- and phase-averaged 3D TDSE yield of H, Eq. (1), P = A S(eta_1 I_est), giving eta_1 = 0.641 +- 0.007 and the calibrated intensity I_0 = eta_1 I_est, with I_est = (2P / pi w_0^2) / (f_rep tau_p), Eq. (2) (p. 2). The noble gases flow through the same source with the discharge off.

Materials in structured entries: Ar, Kr, Xe

Dataset quantities: ionization_model_parameter, ionization_yield

Contents: 7 datasets; 1 formula.

Curation notes

Journal pp. 053001-1 to -5 = pdf pages 1-5. Identity checked against pdf p. 1. The Letter’s pages are cited below as p. 1 to p. 5.

WHAT THE YIELDS ARE. Relative yields in arbitrary units: the scale A of Eqs. (1) and (3) ‘rescales the yield to account for both the unknown atomic density and detector efficiency’ (p. 2) and is free for every gas and every model compared (‘A is allowed to vary in order to account for the unknown gas density’, p. 3). Only the shape of the yield against intensity is measured, never a probability. The yields are averaged over the focal volume in two dimensions and over the carrier-envelope phase (‘Effects of carrier-envelope phase on the theoretical yields were negligible’, p. 2). ‘Depletion of the noble-gas atoms to multiply charged states was observed but the effect found to be negligible in this intensity regime’ (p. 3).

ACCURACY (quoted). H: ‘certifying that our data are free of systematic error to within our 2% measurement precision. A value of eta_1 = 0.641 +- 0.007 is found, indicating that we can calibrate the laser peak intensity to a theory-certified accuracy of better than 1.1% without systematic error’; ‘Neglecting background subtraction shifts eta_1 by 3.1%’ (p. 2). The independent estimate I_est is ‘often as large as 50%’ in error (p. 2). Noble gases: ‘While the data are accurate at the 2% level, the theoretical predictions agree with the data only at the tens of percent level, with both ADK rates performing poorly’ (p. 3). I_c: ‘Our values for I_c include the uncertainty in the H+ calibration as well as the fit error, and demonstrate our ability to calibrate the intensity at the 1.3%, 1.5%, and 2.5% levels using Ar, Kr, and Xe as gas targets, respectively [27]’; ref. 27: ‘we have here stated the 1 sigma standard deviation for I_c, i.e., the 68% confidence interval’ (pp. 4, 5). Sensitivity (p. 4): ‘variations of the pulse duration by 10% may cause a rescaling of the overall yield, but cause < 1% shifts in the retrieved intensity’; ‘changes of the wavelength by 50 nm affect the retrieved intensity by < 1%’; good agreement for H ‘with beam M^2 factors as high as 1.5’; pulse-energy fluctuations ‘of at least 0.7% (root mean square)’ are tolerated. Range: ‘The intensity range covered by our calibration, (1-5) x 10^14 W/cm^2’ (p. 4).

THE THEORIES COMPARED (p. 3, figures only). Noble gases: 3D TDSE ‘under the single-active electron approximation with the second-order-split operator method in the energy representation [23,24]’, model potentials of ref. 22 (Tong and Lin 2005, tong_2005) from density functional theory with self-interaction correction; ADK and empirically corrected ADK (ref. 22) ‘using a weighted sum of m_l = 0, 1 for the p-orbital (l = 0) and ground-state ionization potentials’ (as printed: l = 0 for a p orbital is a slip); ‘As a nonrelativistic model was used, the fine-structure splitting was ignored’. With eta fixed to 1 and A free, all three disagree with the data at the tens-of-percent level (Fig. 2(d)-(f), read on the figure: the TDSE lies 20-30 % above the Ar data at 1.3-1.7e14 W/cm^2 and 10-17 % below it above 4e14; ADK lies 60-75 % below the Ar data at the low end). ‘Because of the constrained fit parameter, ADK and empirical ADK (Emp-ADK) fits are weighted more heavily towards the higher intensities ... This results in an artifact where ADK appears to fit better than Emp-ADK. Removing the constraint on the eta parameter vastly improves the fit; however, the retrieved eta is significantly different’ (p. 3).

EXTRACTED. Table I (p. 4): alpha, gamma and I_c of Eq. (4) for Ar, Kr and Xe, by tools/extract/wallace_2016.py. The library’s text layer loses the plus-minus sign of the I_c cells (‘3.86 - 0.05’ for 3.86 +- 0.05) and the plus of the headers (‘Ar’ followed by a thorn for Ar+); both were read on the table image (2026-09-28) and are handled in the script. No misprint is corrected.

NOT EXTRACTED. The measured yields themselves (Fig. 1, H+; Fig. 2(a)-(c), Ar+, Kr+, Xe+, about 40 points per gas between 1.3e14 and 4.7e14 W/cm^2) are in figures only in the Letter. THE SUPPLEMENTAL MATERIAL (ref. 19, ‘Supplementary information’, dated April 26, 2016, 7 pages) is in the library as supplement_01 of the paper’s directory (since 2026-09-28). Its Table I (p. 7) lists 37 measured Kr+ yields, ‘Actual experimental Y_Kr+ data as a function of I_est taken with our apparatus’: the columns I_est (x10^14 W/cm^2, 2.04 to 7.36), Y_Kr+ (arb. units, 0.2349 to 9.6458), sigma_Y (absolute error, 0.0064 to 0.0912) and I_0 (x10^14 W/cm^2, 1.31 to 4.72). I_0/I_est is 0.639-0.643, i.e. I_0 = eta_1 I_est with eta_1 = 0.641 to the two decimals printed. It is the dataset supplement_tableI_yield_Kr (TC-672). No such table is given for Ar+ or Xe+. The supplement also gives (notes only): the error budget of the yields (baseline subtraction, short-term drift by Allan deviation, long-term drift from the power spectral density, dissociation fraction; ‘The long-term drift was the dominant error source’, p. 2; Allan deviation and long-term drift of the H+ yield each 1.4 % at 1.7e14 and 0.6 % at 3.9e14 W/cm^2, pp. 3-4); the geometry (‘the cylindrically symmetric geometry of the interaction region negates the need for focal-volume averaging (FVA) along the propagation direction of the laser’, p. 6); Eq. (S11) for a uniform gas, P^(3D) = int_0^1 P^(2D)(v I_est) dv / (v^(5/2) sqrt(1 - v)) as printed, v = w_0^2 / w(z)^2; the fit of its Table I to Eq. (3) with A and eta_2 free, ‘the weight of each data point should be set equal to the inverse of the error for that point’, giving eta_2 = 0.638 +- 0.008 (p. 6); a 0.5 mm viewport and an off-axis parabolic mirror of 750 mm focal length (p. 5).

Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets

wallace_2016/tableI_shape_Ar

{
  "id": "tableI_shape_Ar",
  "material": "Ar",
  "file": "Ar/wallace_2016__tableI_shape_Ar.csv",
  "table": "tbl_03b3d0db2cd3",
  "quantity": "ionization_model_parameter",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wallace_2016_yield_fit",
    "parameter": "alpha, gamma of Eq. (4), the row label",
    "wavelength_nm": 800,
    "pulse_fs": 5.5,
    "fitted_to": "the authors' 3D TDSE (single active electron), 2D focal-volume average"
  }
}

Table I (p. 4), column ‘Ar+’ (the ion produced; the target is Ar): alpha and gamma of Eq. (4), printed ‘arb. units’ (both are pure numbers in Eq. (4)). ‘The coefficients alpha and gamma are set by fitting Eq. (4) to the 3D TDSE for each gas target’ (p. 4): they are fitted to the authors’ single-active-electron TDSE yield, focal-volume averaged in two dimensions, not to the measurement. No uncertainty is given. Valid only with I_c (tableI_Ic_Ar) over (1-5)e14 W/cm^2, 800 nm, 5.5 fs (the formula eq4_yield and the authors’ warning there).

wallace_2016/tableI_Ic_Ar

{
  "id": "tableI_Ic_Ar",
  "material": "Ar",
  "file": "Ar/wallace_2016__tableI_Ic_Ar.csv",
  "table": "tbl_03b3d0db2cd3",
  "quantity": "ionization_model_parameter",
  "unit": "W/cm2",
  "scale": 100000000000000.0,
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wallace_2016_yield_fit",
    "parameter": "I_c of Eq. (4)",
    "wavelength_nm": 800,
    "pulse_fs": 5.5,
    "fitted_to": "the measured Ar+ yields of Fig. 2(a), alpha and gamma fixed"
  },
  "uncertainty": {
    "column": true,
    "level": "1 sigma"
  }
}

Table I (p. 4), column ‘Ar+’ (the target is Ar): I_c of Eq. (4), printed 3.86 +- 0.05 in 10^14 W/cm^2 (the library’s text layer has ‘3.86 - 0.05’; the plus-minus sign was read on the table image). ‘The value of I_c was determined from the data of Fig. 2 by fixing eta_2 to a value of 1, and substituting I_est with our I_0 values obtained from the H+ fit’; ‘Our values for I_c include the uncertainty in the H+ calibration as well as the fit error’ (p. 4); ‘we have here stated the 1 sigma standard deviation for I_c, i.e., the 68% confidence interval’ (ref. 27, p. 5). 0.05/3.86 = 1.3 %, the ‘calibrate the intensity at the 1.3% ... level’ of p. 4. With alpha and gamma of tableI_shape_Ar.

wallace_2016/tableI_shape_Kr

{
  "id": "tableI_shape_Kr",
  "material": "Kr",
  "file": "Kr/wallace_2016__tableI_shape_Kr.csv",
  "table": "tbl_03b3d0db2cd3",
  "quantity": "ionization_model_parameter",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wallace_2016_yield_fit",
    "parameter": "alpha, gamma of Eq. (4), the row label",
    "wavelength_nm": 800,
    "pulse_fs": 5.5,
    "fitted_to": "the authors' 3D TDSE (single active electron), 2D focal-volume average"
  }
}

Table I (p. 4), column ‘Kr+’ (the ion produced; the target is Kr): alpha and gamma of Eq. (4), printed ‘arb. units’ (both are pure numbers in Eq. (4)). ‘The coefficients alpha and gamma are set by fitting Eq. (4) to the 3D TDSE for each gas target’ (p. 4): they are fitted to the authors’ single-active-electron TDSE yield, focal-volume averaged in two dimensions, not to the measurement. No uncertainty is given. Valid only with I_c (tableI_Ic_Kr) over (1-5)e14 W/cm^2, 800 nm, 5.5 fs (the formula eq4_yield and the authors’ warning there). The supplement restates them (alpha = 4.24, gamma = -2.49, its p. 6).

wallace_2016/tableI_Ic_Kr

{
  "id": "tableI_Ic_Kr",
  "material": "Kr",
  "file": "Kr/wallace_2016__tableI_Ic_Kr.csv",
  "table": "tbl_03b3d0db2cd3",
  "quantity": "ionization_model_parameter",
  "unit": "W/cm2",
  "scale": 100000000000000.0,
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wallace_2016_yield_fit",
    "parameter": "I_c of Eq. (4)",
    "wavelength_nm": 800,
    "pulse_fs": 5.5,
    "fitted_to": "the measured Kr+ yields of Fig. 2(b), alpha and gamma fixed"
  },
  "uncertainty": {
    "column": true,
    "level": "1 sigma"
  }
}

Table I (p. 4), column ‘Kr+’ (the target is Kr): I_c of Eq. (4), printed 2.06 +- 0.03 in 10^14 W/cm^2 (the library’s text layer has ‘2.06 - 0.03’; the plus-minus sign was read on the table image, and the supplement’s p. 6 restates ‘I_c = 2.06 +- 0.03 x 10^14 W/cm^2’). ‘The value of I_c was determined from the data of Fig. 2 by fixing eta_2 to a value of 1, and substituting I_est with our I_0 values obtained from the H+ fit’; ‘Our values for I_c include the uncertainty in the H+ calibration as well as the fit error’ (p. 4); ‘we have here stated the 1 sigma standard deviation for I_c, i.e., the 68% confidence interval’ (ref. 27, p. 5). 0.03/2.06 = 1.5 %. With alpha and gamma of tableI_shape_Kr.

wallace_2016/tableI_shape_Xe

{
  "id": "tableI_shape_Xe",
  "material": "Xe",
  "file": "Xe/wallace_2016__tableI_shape_Xe.csv",
  "table": "tbl_03b3d0db2cd3",
  "quantity": "ionization_model_parameter",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wallace_2016_yield_fit",
    "parameter": "alpha, gamma of Eq. (4), the row label",
    "wavelength_nm": 800,
    "pulse_fs": 5.5,
    "fitted_to": "the authors' 3D TDSE (single active electron), 2D focal-volume average"
  }
}

Table I (p. 4), column ‘Xe+’ (the ion produced; the target is Xe): alpha and gamma of Eq. (4), printed ‘arb. units’ (both are pure numbers in Eq. (4)). ‘The coefficients alpha and gamma are set by fitting Eq. (4) to the 3D TDSE for each gas target’ (p. 4): they are fitted to the authors’ single-active-electron TDSE yield, focal-volume averaged in two dimensions, not to the measurement. No uncertainty is given. Valid only with I_c (tableI_Ic_Xe) over (1-5)e14 W/cm^2, 800 nm, 5.5 fs (the formula eq4_yield and the authors’ warning there). The whole range lies above Xe’s barrier-suppression threshold (‘Xe threshold off scale at 9 x 10^13 W/cm^2’, Fig. 2’s caption).

wallace_2016/tableI_Ic_Xe

{
  "id": "tableI_Ic_Xe",
  "material": "Xe",
  "file": "Xe/wallace_2016__tableI_Ic_Xe.csv",
  "table": "tbl_03b3d0db2cd3",
  "quantity": "ionization_model_parameter",
  "unit": "W/cm2",
  "scale": 100000000000000.0,
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wallace_2016_yield_fit",
    "parameter": "I_c of Eq. (4)",
    "wavelength_nm": 800,
    "pulse_fs": 5.5,
    "fitted_to": "the measured Xe+ yields of Fig. 2(c), alpha and gamma fixed"
  },
  "uncertainty": {
    "column": true,
    "level": "1 sigma"
  }
}

Table I (p. 4), column ‘Xe+’ (the target is Xe): I_c of Eq. (4), printed 1.18 +- 0.03 in 10^14 W/cm^2 (the library’s text layer has ‘1.18 - 0.03’; the plus-minus sign was read on the table image). ‘The value of I_c was determined from the data of Fig. 2 by fixing eta_2 to a value of 1, and substituting I_est with our I_0 values obtained from the H+ fit’; ‘Our values for I_c include the uncertainty in the H+ calibration as well as the fit error’ (p. 4); ‘we have here stated the 1 sigma standard deviation for I_c, i.e., the 68% confidence interval’ (ref. 27, p. 5). 0.03/1.18 = 2.5 %. With alpha and gamma of tableI_shape_Xe.

wallace_2016/supplement_tableI_yield_Kr

{
  "id": "supplement_tableI_yield_Kr",
  "material": "Kr",
  "file": "Kr/wallace_2016__supplement_tableI_yield_Kr.csv",
  "table": "tbl_2cab2b41f725",
  "quantity": "ionization_yield",
  "unit": "arb. units",
  "abscissa": "intensity",
  "abscissa_unit": "W/cm2",
  "columns": {
    "intensity_estimated": "W/cm2"
  },
  "use": "check",
  "parameters": {
    "wavelength_nm": 800,
    "pulse_fs": 5.5,
    "pulse_fs_definition": "full width at half maximum of the intensity",
    "polarization": "linear",
    "cep": "not stabilised (random)",
    "focal_average": "two transverse dimensions",
    "ion": "Kr+"
  },
  "uncertainty": {
    "column": true,
    "level": "not stated"
  }
}

Supplemental Material, Table I (SM p. 7; library supplement_01): ‘Actual experimental Y_Kr+ data as a function of I_est taken with our apparatus. This data can be used to check the implementation of our intensity calibration procedure.’ 37 points, the rows numbered in the order printed (rising intensity). The abscissa is the column I_0, the peak intensity calibrated on the yield of atomic hydrogen, printed in 10^14 W/cm^2 with two decimals (1.31 to 4.72; trailing zeros are not printed: ‘2.8’, ‘3.3’, ‘3.9’); the column intensity_estimated is I_est, the intensity estimated from the laser’s parameters (Eq. (2)), 2.04 to 7.36. I_0/I_est is 0.639 to 0.643: I_0 = eta I_est with eta = 0.641 to the decimals printed; the supplement’s own fit of this table gives eta_2 = 0.638 +- 0.008 (SM p. 6). The yield Y is in arbitrary units (the scale is free) and sigma is its absolute error as printed, ‘abs. error’, 2.7 % of Y at the lowest intensity and 0.9 % at the highest; its level of confidence is not stated in the table (the Letter’s ref. 27 states 1 sigma for I_c). The error budget (SM pp. 2-4): baseline subtraction, short-term drift (Allan deviation), long-term drift, ‘the dominant error source’. The yield is that of Kr+ for a 5.5 fs pulse (FWHM of intensity) at 800 nm, carrier-envelope phase not stabilised, averaged over the two transverse dimensions of the focus (‘the cylindrically symmetric geometry of the interaction region negates the need for focal-volume averaging (FVA) along the propagation direction of the laser’, SM p. 6): not a single-atom probability. The point at I_0 = 3.49e14 W/cm^2 (row 27, Y = 6.9126) is 17 % above the smooth trend of its neighbours (5.3139 at 3.41e14, 6.0803 at 3.56e14): kept as printed, and not excluded. The intensity’s uncertainty is that of the hydrogen calibration, 1.5 % for krypton through I_c (tableI_Ic_Kr); it is common to all points and is not in sigma.

Formulas

wallace_2016/eq4_yield

{
  "id": "eq4_yield",
  "material": "Kr",
  "equation": "Eqs. (3)-(4), p. 4",
  "printed": "P^(2D)_NG+(I_est; A, eta_2) = A S_phenom(eta_2 I_est); S_phenom(eta_2 I_est) = exp[-alpha (eta_2 I_est / I_c)^(-1/2)] / [1 + (eta_2 I_est / I_c)^gamma]; Table I: Ar+ alpha = 2.84, gamma = -3.03, I_c = 3.86 +- 0.05; Kr+ 4.24, -2.49, 2.06 +- 0.03; Xe+ 3.71, -2.69, 1.18 +- 0.03 (alpha, gamma 'arb. units', I_c in 10^14 W/cm^2)",
  "variable": "I_0 = eta_2 I_est: the peak intensity of the pulse at the focus (W/cm^2; 800 nm, 5.5 fs FWHM of intensity, carrier-envelope phase not stabilised); I_est: the intensity estimated from the laser parameters, Eq. (2); eta_2: the intensity rescaling (1 when the intensity is already calibrated); A: the free scale of the yield (arbitrary units); P^(2D): the ion yield averaged over the two transverse dimensions of the focus and over the carrier-envelope phase. The parameters are the datasets tableI_shape_<gas> and tableI_Ic_<gas>.",
  "range": "(1-5) x 10^14 W/cm^2 ('The intensity range covered by our calibration', p. 4); the data of Fig. 2 lie between about 1.3e14 and 4.7e14 W/cm^2",
  "coefficients": {
    "alpha_Ar": "2.84",
    "gamma_Ar": "-3.03",
    "I_c_Ar": "3.86",
    "alpha_Kr": "4.24",
    "gamma_Kr": "-2.49",
    "I_c_Kr": "2.06",
    "alpha_Xe": "3.71",
    "gamma_Xe": "-2.69",
    "I_c_Xe": "1.18"
  },
  "check": "Eq. (4) and Table I checked against the equation and table images (2026-09-28). Against the 37 Kr+ yields of the supplement's Table I (the dataset supplement_tableI_yield_Kr) with Kr's alpha, gamma, I_c, the intensity I_0 as tabulated (eta_2 = 1) and only A fitted (weights 1/sigma^2, A = 190.0; weights 1/sigma, A = 190.3): A S_phenom / Y - 1 lies between -0.166 and +0.097, rms 0.044; seven points beyond 5 % (I_0 = 1.42, 1.52, 1.65, 2.62, 2.88, 3.49 and 4.72e14 W/cm^2; the one at 3.49e14, Y = 6.9126, is an outlier of the data, 17 % above the curve); chi^2 = 369 for 37 points with the printed sigma_Y (computed 2026-09-28, a scratch script). This is the scatter of the 'Phenomenological' residuals of Fig. 2(e). The Ar and Xe parameters cannot be checked: their data are in figures only. The material is nominal: the formula takes each gas's own alpha, gamma and I_c."
}

WHAT WAS FITTED TO WHAT: ‘The coefficients alpha and gamma are set by fitting Eq. (4) to the 3D TDSE for each gas target. The value of I_c was determined from the data of Fig. 2 by fixing eta_2 to a value of 1, and substituting I_est with our I_0 values obtained from the H+ fit’ (p. 4). So alpha and gamma describe the authors’ single-active-electron TDSE yield curve (focal-volume averaged in 2D), and only I_c and the scale A come from the measurement. ‘The value of I_c is insensitive against the removal of any individual data point from the fit data set’. THE AUTHORS’ WARNING: ‘it is important to note that the phenomenological model is introduced purely as a convenience, so that the reader can easily carry out intensity calibration with a closed-form analytic fit function. We emphasize that Eq. (4) is not associated with any model of the ionization physics. Hence, we do not expect it to be valid outside the range of intensities studied here’ (p. 4). It is a curve for this pulse (800 nm, 5.5 fs) and this geometry (2D focal average) only: not a rate, and not a single-atom probability. The scale A is not given for any gas. Fig. 2’s caption calls the curve ‘the phenomenological model given by Eq. (3)’.

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.