ettoumi_2010 ============ W\. Ettoumi, Y\. Petit, J\. Kasparian and J\.\-P\. Wolf, Opt\. Express 18, 6613 \(2010\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1364/oe\.18\.006613 `__ **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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 :doc:`review register ` for the checks and decisions.