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 <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/hollenbeck_2002.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1364/josab\.19\.002886 <https://doi.org/10.1364/josab.19.002886>`__

**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
~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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 :doc:`review register </register>` for the checks and decisions.