zahedpour_2019#
S. Zahedpour, S. W. Hancock and H. M. Milchberg, “Ultrashort infrared 2.5-11 um pulses: spatiotemporal profiles and absolute nonlinear response of air constituents”, Opt. Lett. 44, 843 (2019)
Source record · Catalogue · Review register
DOI: 10.1364/OL.44.000843
Source kind: measurement
Recorded technique: Single-shot supercontinuum spectral interferometry (SSSI), as zahedpour_2015, with a mid- to long-wave infrared pump: non-collinear difference-frequency generation (AgGaS2) of the signal (1.1-1.6 um) and idler (1.6-2.6 um) of an OPA pumped by 36 fs, 8 mJ, ~806 nm pulses (1 kHz Ti:sapphire), tuned over 3.0-10.5 um (Table 1), focused by a BaF2 lens. Supercontinuum probe and reference (400-750 nm, filamentation of ~200 uJ, 806 nm in 2.5 atm Xe; Michelson pair ~2.5 ps apart), copropagating with the pump into a gas cell (BaF2 entrance, fused-silica exit window); window response measured with the cell evacuated; 2 x 10^4 shots per phase image; ~5 fs, ~3 um resolution. Cell at 1 atm for pumps of 3.0-6.5 um and 42 atm for 7.0-11.0 um (the DFG output falls below 10 uJ at 11 um). Probe parallel and perpendicular to the pump separate the electronic and rotational phases. Absolute n2 without density, length or intensity measurements: the electronic phase of N2 and O2 is referenced to their own rotational phase, whose size is fixed by Delta alpha; Ar is referenced to N2 with the same pump pulses (pp. 843-845).
Materials in structured entries: Ar, N2, O2
Dataset quantities: nonlinear_index
Contents: 7 datasets; 0 formulas.
Curation notes#
Journal pages 843-846 = PDF pages 1-4. Identity checked against the first page. No supplement in the library directory (the paper cites none). Air is not curated; the title’s ‘air constituents’ are N2, O2 and Ar, all recorded. The abstract and conclusion say 2.5-11 um; Table 1 covers pump wavelengths 3.0-10.5 um.
DEFINING EQUATIONS (Gaussian units; quoted, checked on the page images). Separation (p. 844, citing Sutherland): Delta phi_par = Delta phi_elec + Delta phi_rot, Delta phi_perp = Delta phi_elec/3 - Delta phi_rot/2, so Delta phi_elec = 3(Delta phi_par + 2 Delta phi_perp)/5 and Delta phi_rot = 2(Delta phi_par - 3 Delta phi_perp)/5. For Ar (p. 844): ‘Delta phi_par(x, t) = 2 n2 I(x, t) k_p L’, ‘n2 is the nonlinear refractive index (electronic)’. Eq. (1), p. 845: Delta phi_elec(x, t) = 2 n2 I0 k_p L (N/N0) f(x, t); Delta phi_rot(x, t) = 2 pi N n0^-1 I0 k_p L Delta alpha(lambda_p) Delta alpha(lambda_pu) g(t) * f(x, t), ‘where n2 and n0 are the nonlinear (electronic) and linear refractive indices at 1 atm’, N the molecular density, N0 ‘the molecular density at 1 atm’, I0 the peak intensity, f the normalised envelope; g(tau) = (-16 pi/(15 hbar c)) sum_j [j(j - 1)/(2j - 1)] (rho_j/(2j + 1) - rho_(j-2)/(2j - 3)) exp(-gamma_(j,j-2) tau) sin(omega_(j,j-2) tau), omega_(j,j-2) = 4 pi c B (2j - 1) (note [19]: the g of zahedpour_2015 was misprinted, its computations used this one). Probe index (p. 845): ‘Delta n_p(x, t) = 2 n2 I(x, t) + int R(t - t’) I(x, t’) dt’’, R(tau) = 2 pi N n0^-1 Delta alpha(lambda_p) Delta alpha(lambda_pu) g(tau): n2 is the SELF coefficient (the probe sees 2 n2, as wahlstrand_2013). Calibration (p. 845): the 2D fit g(t) * Delta phi_elec = mu1 Delta phi_rot gives ‘n2 = mu1 pi n0^-1 N0 Delta alpha(lambda_p) Delta alpha(lambda_pu) = mu1 pi n0^-1 N0 (Delta alpha)^2’ (the library’s Markdown prints n2^-1: a transcription error, the page has n2); Ar: Delta phi_elec^Ar = mu2 Delta phi_elec^N2 ‘gives n2^Ar = n2^N2 mu2 N_Ar/N_N2 from Eq. (1)’. From Eq. (1) the density ratio should be N_N2/N_Ar; the two are equal at equal pressure for an ideal gas (see REFERENCE STATE). No relation between n2 and chi(3) or gamma is given (zahedpour_2015 gives n2 = (12 pi^2/n0^2 c) N gamma).
CALIBRATION ROUTE. N2: ‘Delta alpha_N2(lambda_pu) ~ Delta alpha_N2(0) = 6.6 x 10^-25 cm3 [11]’ (zahedpour_2015’s Alms et al. fit; p. 845) and ‘Delta alpha is taken as spectrally flat’, so (Delta alpha)^2 = (6.6e-25)^2 for the pump and the visible probe alike. O2: ‘we assume Delta alpha(lambda_p) = Delta alpha(lambda_pu) for all analyses, and we use the values of Delta alpha_O2 = 10.2 x 10^-25 cm3 measured in Ref. [15]’ (wahlstrand_2012, 800 nm). Ar: the ratio of its electronic phase to N2’s with the same pump pulses. So the absolute scale is Delta alpha squared: Alms’s light-scattering Delta alpha(0) for N2 (and Ar), wahlstrand_2012’s Delta alpha(800 nm) for O2. This is not quite zahedpour_2015’s N2 scale (Delta alpha rescaled to wahlstrand_2012’s 6.7 at 800 nm, dispersed to the 600-nm probe): with that paper’s C = 3.8e-57 cm3 s2, its product Delta alpha(600) Delta alpha(2400) is 6.87 x 6.52 = 44.8 (1e-50 cm6) against 43.6 here, so this paper’s N2 and Ar values are ~2.7 % below the 2015 scale for the same phases (computed 2026-09-26; within the errors; not corrected).
PART. The electronic part is separated by the polarisation analysis, not by the pulse duration (pulses 77-92 fs at 3.0-6.5 um, up to ~300 fs at the longest wavelengths, Figs. 2 and 4). The paper (p. 843): ‘For the 50-300 fs MIR-LWIR pulses of this experiment, the electronic and rotational responses dominate; the non-resonant Raman vibrational response of O2 and N2 is negligible, owing to insufficient optical bandwidth’. The adiabatic bond-stretch part follows these pulses and is not separated, so N2 and O2 are ‘electronic_vibrational’ (as zahedpour_2015), Ar ‘electronic’. N2 was scanned continuously over 7.5-8.5 um: ‘no signature of two-photon resonant absorption’ (p. 846). Nondegenerate against degenerate (p. 843): ‘the extracted nondegenerate nonlinearity coefficients are within ~5 % of their degenerate MIR counterparts, with even better agreement in the LWIR’ (the pump-probe n2 is n2(omega_pu, omega_p) with a visible probe).
REFERENCE STATE. Table 1: ‘at Atmospheric Pressure’; Eq. (1): n2 ‘at 1 atm’. No temperature: 101325 Pa and 293.15 K assumed. For N2 and O2 the measured density cancels (both phases scale with N), so the 42-atm points need no density: the reference density N0 enters only through n2 = mu1 pi n0^-1 N0 (Delta alpha)^2, whose value (temperature, ideal or real) is not stated. For Ar at 42 atm the density ratio matters: optmat’s equations of state give Z = 0.9941 (N2) and 0.9738 (Ar) at 42 atm, 293.15 K (1 atm: 0.9998, 0.9993), so the real-gas N_N2/N_Ar = 0.980. If the authors used equal (ideal-gas) densities, the Ar values at 7.0-10.5 um are 2.0 % high; if they used the printed N_Ar/N_N2 with real densities, 2 % low the other way round. Not stated, not corrected (TC-312). At 42 atm the O2 rotational dephasing was measured (Fig. 3, below); the paper does not say what dephasing it used for N2.
WAVELENGTHS. Rows are the pump (DFG) wavelengths, ‘determined by measuring the wavelengths of the OPA signal and idler output using a MIR spectrometer ... spanning 1.0-2.5 um’ (Table 1 note a); vacuum assumed (air against vacuum is 3e-4 relative, negligible). The probe is the 400-750 nm supercontinuum (no central wavelength stated; zahedpour_2015’s was 600 nm): probe_nm is not recorded. The 10.5 um column carries note c; Fig. 4 labels its longest traces 10.6 um (whether they are the 10.5 um column is not said).
UNCERTAINTY. The Table 1 errors (+-0.4 to +-0.6, 5-8 %) are not explained in this paper and their level is not stated. They are smaller than the propagated uncertainty of Delta alpha^2 alone (wahlstrand_2012: 6.7 +- 0.3, 10.2 +- 0.4, i.e. 9 % and 8 % on Delta alpha^2), so they are probably the statistical part only; the absolute scale error is larger (TC-312). ‘The values of n2 show no dispersion to within our measurement precision’ (p. 846). The 16 values per gas have mean (standard deviation) N2 7.54 (0.34), O2 8.18 (0.69), Ar 8.54 (0.73) x 1e-20 cm2/W (computed 2026-09-26); the O2 and Ar scatter exceeds the printed errors.
TABLE 1 (tbl_6c9829dbaca8, p. 846; image checked cell by cell 2026-09-26, the CSV matches; the unmapped glyph of validation.json is the caption’s closing parenthesis). Title: ‘Measured Values of n2 (x10^-20 cm2/W) (Electronic) for Major Air Constituents at Atmospheric Pressure’. Last columns: n2,rot (first block; N2 23.0, O2 53.3, Ar 0) and n2,eff(tau_peak) (second block; N2 32.5 (400 fs), O2 64.5 (480 fs), Ar ‘8.5 (average)’). Note b: ‘n2,eff = (k I0 L)^-1 int Delta phi(0, t) f(0, t) dt (int f^2(0, t) dt)^-1, where Delta phi = Delta phi_par(0, t) = Delta phi_elec(0, t) + Delta phi_rot(0, t) using Eq. (1), with I(t) = I0 f(0, t) and f(0, t) = exp(-4 ln 2 (t/t_FWHM)^2). For long pulses (t_FWHM >~ 420 fs for N2 and ~500 fs for O2), n2,eff = n2 + n2,rot. The values shown for n2,eff are maximum at the pulsewidths indicated.’ Ar’s 8.5 is the mean of its 16 measured values (8.54, computed): not written (a derived average; Ar has no rotation). Ar’s n2,rot = 0 is not written.
CHECK OF n2,rot AND n2,eff (2026-09-26, scratch computation, not stored). With the printed g(tau) and R(tau), Delta alpha 6.6 and 10.2 x 10^-25 cm3, B = 2.0 and 1.44 cm^-1 (the values wahlstrand_2012 and wahlstrand_2013 print), nuclear-spin weights (N2 6:3 even:odd; 16O2 odd j only, spin-rotation splitting ignored) and the ideal-gas N at 1 atm: the integral of R (wahlstrand_2012’s long-pulse expression) is 46.4 (N2) and 110.5 (O2) at 293.15 K, 44.3 and 105.5 at 300 K. The printed n2,rot, 23.0 and 53.3, are HALF of it (23.2 and 55.2 at 293.15 K; 22.1 and 52.8 at 300 K). Note b’s n2,eff maximised over Gaussian widths reproduces the printed widths exactly (400 fs N2, 480 fs O2) and the printed values with the same half, n2 + (1/2)<R> (N2 32.7 at 293 K, 31.6 at 300 K, against 32.5; O2 68.3 and 65.6 against 64.5; n2 = 7.6 and 8.2). Taken literally, note b’s Delta phi_par would give 2 n2 + <R> instead. So the printed rotational numbers are self coefficients if the printed g (or R) is twice the self rotational response; wahlstrand_2013 finds the probe’s rotational index equal to the pump’s own (the factor 2 only for the bound electrons), so either g as printed carries an extra factor 2 or the tabulated n2,rot is half the rotational self index. zahedpour_2015’s n2,rot (23, 24, 54) follow the same scale. Unresolved (TC-312); n2,rot and n2,eff are ‘check’.
COMPARISONS (not used in the datasets). zahedpour_2015 at 2400 nm: N2 7.6 +- 1.3, O2 10.0 +- 1.2, Ar 9.9 +- 1.7; its [32] (Shelton ESHG) values at 2400 nm: N2 8.1, O2 8.6, Ar 10.3. Here the means are N2 7.54, O2 8.18, Ar 8.54: N2 agrees with 2015 and is 7 % below Shelton; O2 is 18 % below 2015’s 2.4 um point (1.5 sigma) and 5 % below Shelton; Ar is 14 % below 2015 (0.8 sigma) and 17 % below Shelton’s dispersion (the Ar-to-N2 ratio is 1.13 here against 1.27 for Shelton’s gamma0). Pigeon et al. at 10.6 um (200 ps, total; note c): N2 45 and O2 84 x 1e-20 cm2/W, against n2 + n2,rot = 30.5 (N2) and 61.5 (O2) here (mean n2): 1.5 and 1.4 times higher. Ar at 10.5 um, 7.7 +- 0.5, is consistent with pigeon_2018’s ‘<1’ x 1e-19.
TEXT (not extracted; no quantity). O2 rotational dephasing from the decay of the revivals at 1-41 atm (Fig. 3, p. 845): ‘gamma = 7.4 x 10^9 (s atm)^-1 P(atm)’; at 42 atm gamma^-1 = 3.2 ps; O2 revival period T = (2cB)^-1 = 11.6 ps. Pulse front tilt 7.5 deg at 3.0 um and 2.3 deg at 6.5 um (Fig. 2; computed 7.5 and 2.2 deg, p. 846). No higher-order Kerr terms are measured.
Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
zahedpour_2019/n2_N2#
{
"id": "n2_N2",
"material": "N2",
"file": "N2/zahedpour_2019__n2_N2.csv",
"table": "tbl_6c9829dbaca8",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-20,
"abscissa_unit": "um",
"state": {
"pressure_pa": 101325.0,
"temperature_k": 293.15
},
"use": "primary",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"part": "electronic_vibrational",
"coefficient": "self"
}
}
Table 1 (p. 846), both blocks: rows are pump wavelengths 3.0-6.5 um (measured at 1 atm) and 7.0-10.5 um (measured at 42 atm; the density cancels in the rotational referencing). Electronic self coefficient (with the unseparated adiabatic bond-stretch part), referenced to N2’s rotational phase with Delta alpha = 6.6e-25 cm3 (flat). ‘At atmospheric pressure’; 293.15 K assumed. Pulses 77-300 fs.
zahedpour_2019/n2_O2#
{
"id": "n2_O2",
"material": "O2",
"file": "O2/zahedpour_2019__n2_O2.csv",
"table": "tbl_6c9829dbaca8",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-20,
"abscissa_unit": "um",
"state": {
"pressure_pa": 101325.0,
"temperature_k": 293.15
},
"use": "primary",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"part": "electronic_vibrational",
"coefficient": "self"
}
}
Table 1 (p. 846), both blocks (3.0-6.5 um at 1 atm, 7.0-10.5 um at 42 atm). Electronic self coefficient (adiabatic bond stretch unseparated), referenced to O2’s rotational phase with wahlstrand_2012’s Delta alpha = 10.2e-25 cm3 at pump and probe. ‘At atmospheric pressure’; 293.15 K assumed.
zahedpour_2019/n2_Ar#
{
"id": "n2_Ar",
"material": "Ar",
"file": "Ar/zahedpour_2019__n2_Ar.csv",
"table": "tbl_6c9829dbaca8",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-20,
"abscissa_unit": "um",
"state": {
"pressure_pa": 101325.0,
"temperature_k": 293.15
},
"use": "primary",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"part": "electronic",
"coefficient": "self"
}
}
Table 1 (p. 846), both blocks (3.0-6.5 um at 1 atm, 7.0-10.5 um at 42 atm). Electronic self coefficient, referenced to N2 with the same pump pulses (so on n2_N2’s Delta alpha scale). At 42 atm the Ar/N2 density ratio enters and its treatment is not stated: up to +-2 % on the 7.0-10.5 um rows (source notes). 293.15 K assumed.
zahedpour_2019/n2_rot_N2#
{
"id": "n2_rot_N2",
"material": "N2",
"file": "N2/zahedpour_2019__n2_rot_N2.csv",
"table": "tbl_6c9829dbaca8",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-20,
"abscissa_unit": "um",
"state": {
"pressure_pa": 101325.0,
"temperature_k": 293.15
},
"use": "check",
"parameters": {
"part": "rotational",
"coefficient": "self"
}
}
Table 1 (p. 846), first block, last column n2,rot: the calculated long-pulse (adiabatic) rotational coefficient, n2,eff = n2 + n2,rot for long pulses (note b), from Delta alpha = 6.6e-25 cm3 taken as flat, so one value for every pump wavelength. It is half the integral of the printed R(tau) (source notes: factor 2 unresolved). Derived: ‘check’.
zahedpour_2019/n2_rot_O2#
{
"id": "n2_rot_O2",
"material": "O2",
"file": "O2/zahedpour_2019__n2_rot_O2.csv",
"table": "tbl_6c9829dbaca8",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-20,
"abscissa_unit": "um",
"state": {
"pressure_pa": 101325.0,
"temperature_k": 293.15
},
"use": "check",
"parameters": {
"part": "rotational",
"coefficient": "self"
}
}
Table 1 (p. 846), first block, last column n2,rot, from Delta alpha = 10.2e-25 cm3 (wahlstrand_2012); as n2_rot_N2 (abscissa ‘not stated’). Derived: ‘check’.
zahedpour_2019/n2_eff_N2#
{
"id": "n2_eff_N2",
"material": "N2",
"file": "N2/zahedpour_2019__n2_eff_N2.csv",
"table": "tbl_6c9829dbaca8",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-20,
"abscissa_unit": "um",
"state": {
"pressure_pa": 101325.0,
"temperature_k": 293.15
},
"use": "check",
"parameters": {
"part": "instantaneous",
"coefficient": "self",
"pulse_fs": 400
}
}
Table 1 (p. 846), second block, last column n2,eff(tau_peak) ‘32.5 (400 fs)’: the effective index of note b (electronic plus transient rotational) for a Gaussian pulse, maximal at t_FWHM = 400 fs; calculated from the measured n2 and Delta alpha, one value for all pump wavelengths (abscissa ‘not stated’). Reproduced in a scratch check (source notes). Derived: ‘check’.
zahedpour_2019/n2_eff_O2#
{
"id": "n2_eff_O2",
"material": "O2",
"file": "O2/zahedpour_2019__n2_eff_O2.csv",
"table": "tbl_6c9829dbaca8",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-20,
"abscissa_unit": "um",
"state": {
"pressure_pa": 101325.0,
"temperature_k": 293.15
},
"use": "check",
"parameters": {
"part": "instantaneous",
"coefficient": "self",
"pulse_fs": 480
}
}
Table 1 (p. 846), second block, last column n2,eff(tau_peak) ‘64.5(480 fs)’: as n2_eff_N2, maximal at t_FWHM = 480 fs (abscissa ‘not stated’). Derived: ‘check’.
Formulas#
No formulas are recorded.
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.