hollenbeck_2002#
D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function”, J. Opt. Soc. Am. B 19, 2886 (2002)
Source record · Catalogue · Review register
Source kind: fit
Recorded technique: A model, fitted by hand, of the Raman response function and gain spectrum of silica-core fibre: 13 vibrational modes, each a Lorentzian convolved with a Gaussian (intermediate broadening), fitted simultaneously to Stolen et al.’s measured gain spectrum (J. Opt. Soc. Am. B 1, 652 (1984)) and response function (stolen_1989, Fig. 2). The mode frequencies and Gaussian widths start from Walrafen & Krishnan’s 13-Gaussian table (Appl. Opt. 21, 359 (1982)); mainly the amplitudes were adjusted (p. 2890).
Materials in structured entries: SiO2
Dataset quantities: raman_fraction, raman_mode, raman_response_time
Contents: 3 datasets; 1 formula.
Curation notes#
Journal pages 2886-2892 = PDF pages 1-7. Identity checked against the first page. The material is silica-core single-mode fibre (‘most single-mode silica fibers will have frequency and linewidth parameters that are close to those given in Table 1, although the amplitudes of some modes may vary’, p. 2890); no fibre type, dopant or pump wavelength is given.
THE RESPONSE FUNCTION (Eq. (9), p. 2889), verbatim: h_R(t) = sum_{i=1}^{13} (A_i’/omega_{v,i}) exp(-gamma_i t) exp(-Gamma_i^2 t^2/4) sin(omega_{v,i} t) theta(t), with theta the unit step (Eq. (10): 1 for t >= 0, 0 for t < 0); A_i’ the amplitude of mode i, omega_{v,i} its centre vibrational frequency, gamma_i its Lorentzian and Gamma_i its Gaussian linewidth. TABLE 1 FOOTNOTE (p. 2889), verbatim: ‘A_i = A_i’/omega_{v,i}, Gamma_i = pi c x Gaussian FWHM, gamma_i = pi c x Lorentzian FWHM, and omega_{v,i} = 2 pi c x component position.’ So h_R(t) = sum_i A_i exp(-gamma_i t) exp(-Gamma_i^2 t^2/4) sin(omega_{v,i} t) theta(t) with the tabulated A_i; c in cm/s makes the rates s^-1 from the cm^-1 columns. The normalisation of h_R is not stated (the tabulated amplitudes are relative; mode 1 has A = 1.00): a user normalises the integral of h_R to 1 as in stolen_1989 Eq. (6).
WHAT THE WIDTH COLUMNS MEAN. The Lorentzian column is a true FWHM: exp(-gamma t) gives a Lorentzian of angular half width gamma, so FWHM = 2 gamma/(2 pi c) = gamma/(pi c) = the column. The Gaussian column is NOT the FWHM of the Gaussian that Eq. (9) produces: exp(-Gamma^2 t^2/4) transforms to exp(-omega^2/Gamma^2), whose FWHM is 2 sqrt(ln 2) Gamma in angular frequency, i.e. sqrt(ln 2) = 0.8326 times the column in cm^-1; the column is the full width at 1/e. The dataset keeps the paper’s label (parameters.width = ‘fwhm’), and a model must use Gamma_i = pi c x column, as the footnote says (TC-382).
GAUSSIAN-TO-LORENTZIAN RATIO. ‘We chose the ratio of the Gaussian linewidth to the Lorentzian linewidth Gamma_i/gamma_i to be 3’ (p. 2890). Table 1 gives 3.000 +- 0.001 for 12 modes; mode 2 gives 110.42/38.81 = 2.845 (3 x 36.81 = 110.43). 38.81 may be a misprint of 36.81, but the paper does not say every mode has exactly 3, so no erratum is made (TC-382).
CHECK (scratch computation, 2026-09-26): Eq. (9) with Table 1 and the footnote conversions gives a response with its first maximum at 16 fs, near-total decay by ~350-450 fs and a recurrence at ~490-560 fs (the features the paper names for Fig. 7, p. 2889: ‘the initial sharp peak, the phase reversal around 160 fs, and the recurrence around 500 fs’), and a gain spectrum (the sine transform, Eq. (11)) peaking at 448 cm^-1 (the measured peak is at ~440 cm^-1, Fig. 1). Figures 7-9 cannot be compared numerically (not digitised).
OTHER EQUATIONS. Eq. (4), p. 2887: instantaneous polarization P_{S,i} proportional to (1 - f_R) E_S [|E_S|^2 + 2|E_L|^2], with ‘the Raman fraction f_R = 0.18, which can be derived from the maximum Raman gain’ (no citation; stolen_1989’s alpha). Eq. (5): P_{S,R}(t) proportional to i f_R E_L int h_R(t - t’’’) exp[-i(omega_L - omega_S)(t - t’’’)] E_L* E_S dt’’’. Eq. (7), p. 2888 (from Agrawal, Eq. (2.3.42)): g_R(omega) = (omega_0/(c n_0)) f_R chi(3) Im[h_R~(omega)]. Eq. (6)/(8): the single damped oscillator of Blow & Wood, h_R proportional to exp(-t/tau_2) sin(t/tau_1), tau_1 = 1/omega_v = 12.2 fs, tau_2 = 1/gamma = 32 fs (p. 2887). Appendix A (Eq. (A2)): the purely inhomogeneous limit (gamma_i = 0); Appendix B (Eq. (B1)): the purely homogeneous limit (Gamma_i = 0); both with Table 1’s parameters, and both fit worse (pp. 2890-2892).
WALRAFEN & KRISHNAN. ‘Walrafen’s and Krishnan’s Table 1 is mislabeled. They tabulated full widths not half-widths’ (p. 2890). Their spectrum is not corrected for the thermal population factor, unlike Stolen’s (p. 2888). Table 1 here is not a reproduction of theirs (the amplitudes were refitted and the Lorentzian widths are new), so it is this paper’s own model.
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
hollenbeck_2002/modes#
{
"id": "modes",
"material": "SiO2",
"file": "SiO2/hollenbeck_2002__modes.csv",
"table": "tbl_3554e41bb9a2",
"quantity": "raman_mode",
"unit": "arbitrary",
"abscissa": "label",
"columns": {
"shift": "cm-1",
"gaussian_width": "cm-1",
"lorentzian_width": "cm-1"
},
"use": "primary",
"parameters": {
"width": "fwhm",
"form": "fibre",
"model": "intermediate broadening, Eq. (9)"
}
}
Table 1 (p. 2889): the row label is the mode number i; value = ‘Peak Intensity A_i’ (A_i = A_i’/omega_{v,i}, relative units); shift = ‘Component Position’; gaussian_width and lorentzian_width = the columns headed ‘Gaussian FWHM’ and ‘Lorentzian FWHM’. Gamma_i = pi c x gaussian_width and gamma_i = pi c x lorentzian_width enter Eq. (9); the Gaussian column is the full width at 1/e of the Gaussian that Eq. (9) makes, not its FWHM (source notes, TC-382). Mode 2’s width ratio is 2.845, the others 3.000 (TC-382). Fitted by hand; no uncertainty stated.
hollenbeck_2002/f_R#
{
"id": "f_R",
"material": "SiO2",
"table": "text",
"quantity": "raman_fraction",
"abscissa": "label",
"use": "check",
"parameters": {
"definition": "fraction of the nonlinear polarization contributed by the Raman effect: P_{S,i} proportional to (1 - f_R), P_{S,R} proportional to f_R (Eqs. (4), (5))",
"form": "fibre",
"assumed": true
}
}
1 inline points in the source record.
p. 2887, below Eq. (4): ‘the Raman fraction f_R = 0.18, which can be derived from the maximum Raman gain, represents the fraction of the polarization contributed by the Raman effect.’ Stated without a citation; it is the alpha of stolen_1989. A model input here (assumed).
hollenbeck_2002/blow_wood_times#
{
"id": "blow_wood_times",
"material": "SiO2",
"table": "text",
"quantity": "raman_response_time",
"unit": "fs",
"abscissa": "label",
"use": "reproduced",
"reproduced_from": [
"blow_1989"
],
"parameters": {
"constant": "tau1, tau2 (per row)",
"model": "single damped oscillator (Blow and Wood): h_R(t) proportional to exp(-t/tau2) sin(t/tau1)",
"form": "fibre"
}
}
2 inline points in the source record.
p. 2887, relation (6): h_R(t) proportional to exp(-t/tau_2) sin(t/tau_1), tau_1 = 1/omega_v, tau_2 = 1/gamma, ‘taken to be tau_1 = 12.2 fs and tau_2 = 32 fs’ (Blow & Wood, ref. 6). blow_1989 (p. 2666) prints tau_1 = 0.0122 ps and tau_2 = 0.032 ps: agrees.
Formulas#
hollenbeck_2002/response_eq9#
{
"id": "response_eq9",
"material": "SiO2",
"equation": "Eq. (9), p. 2889, with Table 1 and its footnote",
"printed": "h_R(t) = \\sum_{i=1}^{13} \\frac{A_i'}{\\omega_{v,i}} \\exp(-\\gamma_i t) \\exp(-\\Gamma_i^2 t^2/4) \\sin(\\omega_{v,i} t) \\theta(t)",
"variable": "t: time after the impulse (s); A_i = A_i'/omega_{v,i} (Table 1 'Peak Intensity'), omega_{v,i} = 2 pi c x position, Gamma_i = pi c x 'Gaussian FWHM', gamma_i = pi c x 'Lorentzian FWHM' (c in cm/s, columns in cm^-1)",
"coefficients": {
"modes": "hollenbeck_2002/modes (13 rows)",
"width_ratio": "3"
},
"check": "Scratch computation (2026-09-26): first maximum at 16 fs, near-total decay by ~350-450 fs, recurrence at ~490-560 fs; the sine transform (Eq. (11)) peaks at 448 cm^-1 (measured ~440 cm^-1). Qualitative agreement with the features the paper states for Figs. 7 and 8; no printed number to compare."
}
Normalisation not stated. The gain spectrum is Eq. (11), s(omega) = sum_l (A_l’/(2 omega_{v,l})) int_0^inf {cos[(omega_{v,l} - omega) t] - cos[(omega_{v,l} + omega) t]} exp(-gamma_l t) exp(-Gamma_l^2 t^2/4) dt; the real part Eq. (12) with sines and a + sign (p. 2889). Eq. (11) is the sine transform int_0^inf h_R(t) sin(omega t) dt of Eq. (9), which the check computed directly. width_ratio: Gamma_i/gamma_i chosen as 3 (p. 2890).
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.