zahedpour_2015 ============== S\. Zahedpour, J\. K\. Wahlstrand and H\. M\. Milchberg, "Measurement of the nonlinear refractive index of air constituents at mid\-infrared wavelengths", Opt\. Lett\. 40, 5794 \(2015\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1364/OL\.40\.005794 `__ **Source kind:** measurement **Recorded technique:** Single\-shot supercontinuum spectral interferometry with a tunable pump\: an OPA \(1100\-2600 nm; 1250, 1650, 2200, 2400 nm used\) or the 800\-nm Ti\:sapphire \(and 400 nm\) pump, 500 Hz; supercontinuum probe and reference \(500\-700 nm, from 2 atm Xe; chirped to ~1\.5 ps\), crossing the pump at 2 degrees in a backfilled chamber \(interaction length L ~ 2 mm\); probe parallel or perpendicular to the pump\. Absolute n2 WITHOUT density or intensity measurements\: the electronic phase is referenced to the rotational phase of N2 and O2, whose size is fixed by Delta alpha \(wahlstrand\_2012, 800 nm\); Ar is referenced to N2 \(pp\. 5794\-5796\)\. **Materials in structured entries:** Ar, N2, O2 **Dataset quantities:** nonlinear\_index **Contents:** 10 datasets; 2 formulas. Curation notes -------------- Journal pages 5794\-5797 = PDF pages 1\-4\. Identity checked against the first page\. Air is not curated \(the title's 'air constituents' are N2, O2, Ar\)\. DEFINING EQUATIONS \(Gaussian\)\. Separation, Eq\. \(1\), p\. 5795\: Delta phi\_elec = 3\(Delta phi\_par \+ 2 Delta phi\_perp\)/5, Delta phi\_rot = 2\(Delta phi\_par \- 3 Delta phi\_perp\)/5\. The probe index \(p\. 5795\)\: 'Delta n\_p\(x, t\) = 2 n2 I\(x, t\) \+ int R\(t \- t'\) I\(x, t'\) dt'', 'where I is the pump intensity; n2 is the electronic Kerr coefficient, which depends on the probe and pump wavelengths'\: n2 is the SELF coefficient\. Eq\. \(2\), p\. 5795\: Delta phi\_elec = 2 n2 I0 f\(x, t\) k\_p L; Delta phi\_rot = \(2 pi N k\_p L / n0\) Delta alpha\(lambda\_p\) \(\\_t \- 1/3\) = \(2 pi N k\_p L / n0\) Delta alpha\(lambda\_p\) \[Delta alpha\(lambda\_e\) I0 / \(c hbar\)\] int g\(t \- t'\) f\(x, t'\) dt', g the rescaled rigid\-rotor impulse response; the fit gives mu1 = Delta alpha\(lambda\_p\) Delta alpha\(lambda\_e\) N L I0 and then mu2 = n2 \(Delta alpha\(lambda\_p\) Delta alpha\(lambda\_e\)\)^\-1 \(p\. 5796\)\. Kerr coefficient wanted \(p\. 5796\)\: 'n2\(omega\_e\) = \(12 pi^2 / n0^2 c\) N gamma\(3\)\(omega\_e; omega\_e, \-omega\_e, omega\_e\)'; measured\: 'n2\(omega\_e, omega\_p\) proportional to gamma\(3\)\(omega\_p; omega\_e, \-omega\_e, omega\_p\)'\. Dispersion, Eq\. \(3\), p\. 5796\: gamma\(3\)\(omega\_s; omega1, omega2, omega3\) = gamma0 \(1 \+ a omega\_L^2 \+ b omega\_L^4 \+ \.\.\.\), omega\_L^2 = omega\_s^2 \+ omega1^2 \+ omega2^2 \+ omega3^2; for the pump\-probe case omega\_L^2 = 2 omega\_e^2 \+ 2 omega\_p^2; the difference n2\(omega\_e, omega\_p\) against n2\(omega\_e\) is \< ~6 % for pump and probe longer than ~500 nm and at 400/600 nm \(p\. 5796\)\. DELTA ALPHA DISPERSION \(p\. 5796\)\: Delta alpha\(omega\) ~ Delta alpha\(0\) \+ C omega^2 \(Spackman 1991\); Alms et al\.'s light scattering for N2 is 'well fit by Delta alpha\(0\) = 6\.6 x 10^\-25 cm3 and C = 3\.8 x 10^\-57 cm3 s2', rescaled so that Delta alpha\(800 nm\) equals wahlstrand\_2012's value; Delta alpha\(800\)/Delta alpha\(0\) ~ 1\.03, Delta alpha\(400\)/Delta alpha\(0\) ~ 1\.13\. Table 1 note b\: N2 adjusted for this dispersion; note c\: not adjusted \(Delta alpha\(800 nm\) of \[22\] used at all wavelengths\) \- N2 both ways, O2 unadjusted; note d\: Ar referenced to the N2 measurements \(which N2 column is not stated\)\. Neglecting the dispersion changes N2 by at most ~10 % near 400 nm and \< 2 % elsewhere\. REFERENCE STATE\: 'Scaled to Atmospheric Pressure' \(Table 1 caption\), no temperature\: 101325 Pa, 293\.15 K assumed\. Note the absolute scale is that of wahlstrand\_2012's Delta alpha \(itself on the ~291\-K N2 flow calibration, TC\-281\)\. UNCERTAINTY \(p\. 5796\)\: the residual of the mu1 fits \(standard deviation over x\), laser power drift, and the propagated uncertainty of Delta alpha from \[22\]; larger at 2400 nm \(phase shifts \<= ~30 mrad\)\. Level not stated\. PULSES\. 40 fs at 800 nm \(p\. 5795\); the OPA pulse durations are not stated \(the 2200\-nm envelope is 'longer and more complex', Fig\. 3\)\. Probe central wavelength 600 nm \(p\. 5795\)\. Short pulses \(\< ~50 fs\) see mainly the electronic response in N2 and O2, long ones \(\> ~150 fs\) mainly the rotational one \(p\. 5796\)\. THE \[32\] COLUMN \(Shelton and Rice; computed with Eq\. \(3\) and the ESHG gamma0, a, b of \[33\] = shelton\_1990\)\. Check \(2026\-09\-26\)\: Shelton 1990 Table IV's gamma0, A, B \(library table, N2 57\.20, 1\.003, 1\.852; O2 60\.03, 1\.112, 4\.603; Ar 72\.75, 1\.066, 2\.033\) with a = A/\(2 pi c\)^2, b = B/\(2 pi c\)^4, omega\_L^2 = 2 omega\_e^2 \+ 2 omega\_p^2 \(probe 600 nm\) reproduce all 18 printed values to \+\-0\.5 % \(the printed precision\) with one common factor, n2 = 0\.133 x gamma\(10^\-63 C^4 m^4 J^\-3\) x 10^\-20 cm2/W; this equals N gamma / \(8 eps0^2 c\) with N the ideal\-gas density at 101325 Pa and 293\.15 K \(0\.1331\)\. The self\-Kerr omega\_L^2 = 4 omega\_e^2 does not reproduce the 400\-nm values \(\-12 to \-16 %\)\. So the column is n2\(omega\_e, omega\_p\) of the T\-convention gamma at 1 atm and ~293 K\. use = 'check'\. PRINTED COEFFICIENTS \(formulas below\)\: 'a = 1\.8 x 10^\-33, 3 x 10^\-33 s^2, b = 1\.5 x 10^\-65, 1\.6 x 10^\-65 s^4, and gamma0 = 7\.7 x 10^\-30, 9\.8 x 10^\-30 cm^6 W^\-1 s^\-1 for N2 and Ar'\. From Shelton's A, B\: a\(N2\) = 2\.83e\-33 s^2 \(printed 1\.8e\-33\: misprint, 2\.8 intended \- the \[32\] column is only reproduced with 2\.83e\-33\), a\(Ar\) = 3\.00e\-33, b\(N2\) = 1\.47e\-65, b\(Ar\) = 1\.61e\-65 s^4\. The printed gamma0 are in the ratio of Shelton's \(0\.786\), but with the paper's n2 = \(12 pi^2 / n0^2 c\) N gamma they give n2 ten times the \[32\] column \(7\.6e\-19 against 7\.6e\-20 cm2/W for N2 at 1 atm, 293 K\)\: the gamma0 values or their unit are off by a factor 10 \(TC\-281\); not corrected here\. **Recorded bibliographic identity status:** VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- zahedpour\_2015/n2\_adjusted\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_adjusted_N2", "material": "N2", "file": "N2/zahedpour_2015__n2_adjusted_N2.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "part": "electronic_vibrational", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), N2 note b, 'This Work'\: adjusted for the dispersion of Delta alpha \(Alms et al\. fit, rescaled to 800 nm\)\. Rows are pump wavelengths \(vacuum assumed\)\. Electronic self coefficient; the adiabatic vibrational part is not separated \(40\-fs pump at 800 nm; OPA durations not stated\)\. zahedpour\_2015/n2\_N2 ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_N2", "material": "N2", "file": "N2/zahedpour_2015__n2_N2.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "check", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "part": "electronic_vibrational", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), N2 note c, 'This Work'\: the same measurement without the Delta alpha dispersion adjustment \(Delta alpha\(800 nm\) at all wavelengths\); n2\_adjusted\_N2 is the paper's preferred analysis, this one 'check'\. zahedpour\_2015/n2\_O2 ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_O2", "material": "O2", "file": "O2/zahedpour_2015__n2_O2.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "part": "electronic_vibrational", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), O2 note c, 'This Work'\: not adjusted for Delta alpha dispersion \(none available for O2\)\. Rows are pump wavelengths\. zahedpour\_2015/n2\_Ar ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_Ar", "material": "Ar", "file": "Ar/zahedpour_2015__n2_Ar.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "part": "electronic", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), Ar note d, 'This Work'\: referenced to the N2 measurements \(which N2 column not stated\)\. zahedpour\_2015/shelton\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "shelton_N2", "material": "N2", "file": "N2/zahedpour_2015__shelton_N2.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "check", "reproduced_from": [ "shelton_1990" ], "parameters": { "part": "unknown", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), '\[32\]' column\: n2 computed by the authors with Eq\. \(3\) and Shelton 1990's ESHG gamma0, A, B; reproduced here to \+\-0\.5 % as n2\(omega\_e, omega\_p\) \(probe 600 nm\) = N gamma\_T / \(8 eps0^2 c\) at 1 atm, 293\.15 K \(source notes\)\. A calculation from shelton\_1990's coefficients\: 'check'\. For N2 and O2 Shelton's fit is to gamma^e \(electronic\) of Table III\(a\); for Ar the total\. zahedpour\_2015/shelton\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "shelton_O2", "material": "O2", "file": "O2/zahedpour_2015__shelton_O2.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "check", "reproduced_from": [ "shelton_1990" ], "parameters": { "part": "unknown", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), '\[32\]' column\: n2 computed by the authors with Eq\. \(3\) and Shelton 1990's ESHG gamma0, A, B; reproduced here to \+\-0\.5 % as n2\(omega\_e, omega\_p\) \(probe 600 nm\) = N gamma\_T / \(8 eps0^2 c\) at 1 atm, 293\.15 K \(source notes\)\. A calculation from shelton\_1990's coefficients\: 'check'\. For N2 and O2 Shelton's fit is to gamma^e \(electronic\) of Table III\(a\); for Ar the total\. zahedpour\_2015/shelton\_Ar ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "shelton_Ar", "material": "Ar", "file": "Ar/zahedpour_2015__shelton_Ar.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "check", "reproduced_from": [ "shelton_1990" ], "parameters": { "part": "electronic", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), '\[32\]' column\: n2 computed by the authors with Eq\. \(3\) and Shelton 1990's ESHG gamma0, A, B; reproduced here to \+\-0\.5 % as n2\(omega\_e, omega\_p\) \(probe 600 nm\) = N gamma\_T / \(8 eps0^2 c\) at 1 atm, 293\.15 K \(source notes\)\. A calculation from shelton\_1990's coefficients\: 'check'\. For N2 and O2 Shelton's fit is to gamma^e \(electronic\) of Table III\(a\); for Ar the total\. zahedpour\_2015/n2\_rot\_adjusted\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_rot_adjusted_N2", "material": "N2", "file": "N2/zahedpour_2015__n2_rot_adjusted_N2.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "check", "parameters": { "part": "rotational", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), n2,rot \(N2 note b group\)\: the calculated long\-pulse \(adiabatic\) rotational coefficient from the measured Delta alpha of \[22\] \(wahlstrand\_2012\); n2,eff = n2 \+ n2,rot for very long pulses \(note a\)\. Derived\: 'check'\. zahedpour\_2015/n2\_rot\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_rot_N2", "material": "N2", "file": "N2/zahedpour_2015__n2_rot_N2.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "check", "parameters": { "part": "rotational", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), n2,rot \(N2 note c group\)\: the calculated long\-pulse \(adiabatic\) rotational coefficient from the measured Delta alpha of \[22\] \(wahlstrand\_2012\); n2,eff = n2 \+ n2,rot for very long pulses \(note a\)\. Derived\: 'check'\. zahedpour\_2015/n2\_rot\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_rot_O2", "material": "O2", "file": "O2/zahedpour_2015__n2_rot_O2.csv", "table": "tbl_818914a52f6f", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-20, "abscissa_unit": "nm", "state": { "pressure_pa": 101325.0, "temperature_k": 293.15 }, "use": "check", "parameters": { "part": "rotational", "coefficient": "self", "probe_nm": 600 } } Table 1 \(p\. 5797\), n2,rot \(O2\)\: the calculated long\-pulse \(adiabatic\) rotational coefficient from the measured Delta alpha of \[22\] \(wahlstrand\_2012\); n2,eff = n2 \+ n2,rot for very long pulses \(note a\)\. Derived\: 'check'\. Formulas -------- zahedpour\_2015/bishop\_dispersion\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "bishop_dispersion_N2", "material": "N2", "equation": "Eq. (3), p. 5796, with the coefficients in the text below it", "printed": "gamma(3)(omega_s; omega_1, omega_2, omega_3) = gamma_0 (1 + a omega_L^2 + b omega_L^4 + ...), omega_L^2 = omega_s^2 + omega_1^2 + omega_2^2 + omega_3^2; 'a = 1.8 x 10^-33 ... s^2, b = 1.5 x 10^-65 ... s^4, and gamma_0 = 7.7 x 10^-30 ... cm^6 W^-1 s^-1 for N2'", "variable": "omega: angular frequencies in s^-1; pump-probe case omega_L^2 = 2 omega_e^2 + 2 omega_p^2; self Kerr 4 omega^2", "state": "gamma per molecule; n2(omega_e) = (12 pi^2 / n0^2 c) N gamma (p. 5796)", "density": "none", "range": "below electronic resonance; used here 400-2400 nm pump, 600 nm probe", "coefficients": { "gamma0": "7.7e-30", "a": "1.8e-33", "b": "1.5e-65" }, "check": "The coefficients are shelton_1990's ESHG gamma0, A, B rescaled to angular frequency (a = A/(2 pi c)^2, b = B/(2 pi c)^4): b reproduces (1.47e-65 from B = 1.852e-20 cm^4), a does not (2.83e-33 from A = 1.003e-10 cm^2; printed 1.8e-33). The paper's own [32] column is reproduced to +-0.5 % only with a = 2.83e-33 (checked 2026-09-26): the printed 1.8 is a misprint for 2.8. gamma0 is 10 times too large for the printed n2 formula (source notes)." } Reproduced and rescaled from shelton\_1990 \(Table IV, ESHG fit to gamma^e, He calibration of that paper\); Bishop's \(1989\) form\. Not an independent result\. Corrected form\: a = 2\.83e\-33 s^2 \(from Shelton's A\); gamma0 unresolved \(factor 10\)\. zahedpour\_2015/bishop\_dispersion\_Ar ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "bishop_dispersion_Ar", "material": "Ar", "equation": "Eq. (3), p. 5796, with the coefficients in the text below it", "printed": "as bishop_dispersion_N2; 'a = ... 3 x 10^-33 s^2, b = ... 1.6 x 10^-65 s^4, and gamma_0 = ... 9.8 x 10^-30 cm^6 W^-1 s^-1 for ... Ar'", "variable": "omega: angular frequencies in s^-1; omega_L^2 as bishop_dispersion_N2", "state": "gamma per atom; n2(omega_e) = (12 pi^2 / n0^2 c) N gamma (p. 5796)", "density": "none", "range": "below electronic resonance; used here 400-2400 nm pump, 600 nm probe", "coefficients": { "gamma0": "9.8e-30", "a": "3e-33", "b": "1.6e-65" }, "check": "From shelton_1990 Table IV (A = 1.066e-10 cm^2, B = 2.033e-20 cm^4): a = 3.00e-33 s^2, b = 1.61e-65 s^4, both reproduced. With these the [32] column is reproduced to +-0.5 % (checked 2026-09-26). gamma0 ratio Ar/N2 = 1.27 as Shelton's; absolute gamma0 a factor 10 too large for the printed n2 formula (source notes)." } Reproduced and rescaled from shelton\_1990\. 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 :doc:`review register ` for the checks and decisions.