ettoumi_2010#

W. Ettoumi, Y. Petit, J. Kasparian and J.-P. Wolf, Opt. Express 18, 6613 (2010)

Source record · Catalogue · Review register

Source kind: theory

Recorded technique: Classical anharmonic-oscillator (single resonance, Lorentz) derivation of the frequency dependence of the susceptibility of any order, generalising Miller’s rule; applied to the n2 ... n8 of Loriot et al. (as corrected by Bejot et al.) for N2, O2, air and Ar.

Materials in structured entries: any, any (applied to N2, O2, Ar, air in Fig. 1)

Dataset quantities: None recorded.

Contents: 0 datasets; 2 formulas.

Curation notes#

The library’s metadata calls this copy the submitted manuscript, but its page footers read ‘Received 29 Jan 2010; ... published 15 Mar 2010 (C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 6614’ (the published article). No datasets: the only values are in Fig. 1 (n2 ... n8 of O2, N2, air and Ar against wavelength at atmospheric pressure), a Miller extrapolation of Loriot’s 800 nm values.

Defining equations. Linear: chi(1)(w0) = N e^2 / (m eps0 (w_e^2 - w0^2 + i w0 gamma)) (Eq. 13). General order (Eq. 22-23): chi(q)(w0; w1, ..., wq) = (m eps0^q / (N^q e^(q+1))) Q^(q) prod_(l=0..q) chi(1)(w_l), so that chi(q)(w0; w1...wq) / chi(q)(w0’; w1’...wq’) = prod_l chi(1)(w_l) / prod_l chi(1)(w_l’) (Eq. 23, the generalized Miller formula; for q = 2 it reduces to Miller’s rule, Eq. 1). The polarisation convention is P0^(q)(w0) = eps0 C_q^(u1..uq’) chi(q)(w0; w1, ..., wq) prod E0^(u_l)(w_l) (Eq. 20) with the multinomial degeneracy factor C = q!/(u1! ... uq’!) and E(t) = int E0(w) e^(iwt) dw (cascaded processes excluded). Nonlinear index (section 2.5, p. 6): for degenerate odd order, chi(2p+1)(w0) = [2^(p+1) p! (p + 1)! / (2p + 1)!] [(n0^2(w0) eps0 c^2)^p / n0(w0)] n_2p(w0), ‘which defines the p-th order non-linear refractive index n_2p’ (the factor c^2 is as printed, page image checked; for p = 1 it reads chi(3) = (4/3) n0 eps0 c^2 n2, dimensionally wrong by one power of c and unlike the usual (4/3) n0^2 eps0 c n2: a misprint or a slip, not used here; the ratio formulas do not depend on it). Section 3 (Eqs. 30-32) splits Zhang et al.’s (2008) air Sellmeier into the N2 and O2 formulas of zhang_2008 (reproduced, not recorded again) and applies Eq. (23) per species, combining them by Lorentz-Lorenz for air.

The formulas below are recorded without a form. Check (2026-09-26): Eq. (29) with optmat’s Ar dispersion carries bejot_2010’s 800 nm Ar coefficients to the 1.83 um values of bejot_2010_hollow Table I to within 0.3 % (see that record).

Higher-order Kerr: the paper takes the Loriot/Bejot n4 ... n10 as measured and provides their dispersion; those coefficients are refuted (see loriot_2009) and the Fig. 1 curves are not to be used. The Miller rule itself (Eq. 29 for p = 1, n2 scaling as [(n0^2 - 1)(w)/(n0^2 - 1)(w’)]^4) is the scaling law Pigeon et al. 2018 and Bejot et al. 2010 (hollow core) use; Pigeon et al. (2018, p. 1) note that measurements at 0.4-0.7 um show it underestimates the dispersion of chi(3) (Mizrahi & Shelton).

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

Datasets#

No datasets are recorded.

Formulas#

ettoumi_2010/generalized_miller_n2p#

{
  "id": "generalized_miller_n2p",
  "material": "any (applied to N2, O2, Ar, air in Fig. 1)",
  "equation": "Eq. (28) and (29), p. 6",
  "printed": "chi(2p+1)(w0) / chi(2p+1)(w0') = (Omega(w0')/Omega(w0))^(2p+2) = (chi(1)(w0)/chi(1)(w0'))^(2p+2); n_2p(w0) / n_2p(w0') = ((n0^2(w0) - 1)/(n0^2(w0') - 1))^(2p+2)",
  "variable": "w0, w0': angular frequencies of the degenerate (self-action) process; n0: linear refractive index of the gas (one resonance per species; for a mixture apply per species)",
  "process": "kerr (degenerate self-refraction, all orders 2p + 1)",
  "coefficients": {
    "exponent": "2p + 2"
  },
  "check": "Reproduces bejot_2010_hollow's 1.83 um Ar coefficients from bejot_2010's 800 nm ones to 0.3 % (2026-09-26)."
}

For n2 (p = 1) the exponent is 4. The paper’s Eq. (29) holds ‘if absorption can be neglected’. Derived for a single Lorentz oscillator; Pigeon et al. (2018) report that visible measurements show a stronger chi(3) dispersion than this rule.

ettoumi_2010/generalized_miller_chi_q#

{
  "id": "generalized_miller_chi_q",
  "material": "any",
  "equation": "Eq. (23), p. 5 (three-dimensional form Eqs. 26-27, p. 6)",
  "printed": "chi(q)(w0; w1, ..., wq) / chi(q)(w0'; w1', ..., wq') = prod_(l=0..q) Omega(w_l') / prod_(l=0..q) Omega(w_l) = prod_(l=0..q) chi(1)(w_l) / prod_(l=0..q) chi(1)(w_l')",
  "variable": "Omega(w) = w_e^2 - w^2 + i w gamma (Eq. 13); w0 = -(w1 + ... + wq) the generated frequency (the paper writes the emitted frequency as negative)",
  "process": "any order q, any frequency set (single oscillator)",
  "check": "Not checkable against printed numbers (the paper prints none)."
}

The general Miller rule; for q = 3 it gives the chi(3) of any third-order process (THG, Kerr, ESHG with a zero frequency) from chi(1) at the four frequencies.

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.