stolen_1989 =========== R\. H\. Stolen, J\. P\. Gordon, W\. J\. Tomlinson and H\. A\. Haus, "Raman response function of silica\-core fibers", J\. Opt\. Soc\. Am\. B 6, 1159 \(1989\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1364/josab\.6\.001159 `__ **Source kind:** measurement **Recorded technique:** The Raman response function of silica\-core fibre computed by the sine transform, Eq\. \(9\), of the imaginary third\-order susceptibility taken from room\-temperature spontaneous Raman\-scattering spectra of silica\-core fibres \(ref\. 15\: Stolen, Lee and Jain, J\. Opt\. Soc\. Am\. B 1, 652 \(1984\); Stolen and Bosch, Phys\. Rev\. Lett\. 48, 805 \(1982\)\), converted to gain by dividing by 1 \+ n\(Omega, T\), n the thermal phonon population factor \(p\. 1161\)\. The Raman fraction is estimated from Raman cross\-section and gain measurements \(refs\. 6, 16\)\. The response is then used in nonlinear\-Schroedinger simulations \(pp\. 1162\-1164\)\. **Materials in structured entries:** SiO2 **Dataset quantities:** nonlinear\_index, raman\_fraction, raman\_response\_time **Contents:** 5 datasets; 1 formula. Curation notes -------------- Journal pages 1159\-1166 = PDF pages 1\-8\. Identity checked against the first page\. The material is 'pure silica\-core fibers'\: 'Some of the details of the gain spectrum will vary from fiber to fiber depending on dopants in the core as well as drawing temperature, but these data are a fairly representative curve' \(p\. 1161\)\. DEFINITIONS \(all waves linearly polarized, parallel polarizations; p\. 1161\)\. Expression \(6\), p\. 1160\: N2 \|u\(t\)\|^2 \-\> N\_2inf \|u\(t\)\|^2 \+ N\_2R int\_{\-inf}^{t} dt' \|u\(t'\)\|^2 f\(t \- t'\), 'where the response function f\(t \- t'\) is real and is normalized such that its integral is unity'\. N\_2inf is the \(nearly instantaneous\) electronic part, N\_2R the delayed part\. Eq\. \(7\)\: N2\(Omega\) = N\_2R int\_0^inf f\(s\) exp\(i Omega s\) ds, Omega = omega\_p \- omega\_s\. Eq\. \(5\)\: the Raman power gain g = \(2 omega\_p/c\) N2''\(Omega\)\. Eq\. \(9\), p\. 1161\: N\_2R f\(s\) = \(2/pi\) int\_0^inf dOmega N2'\(Omega\) cos\(Omega s\) = \(2/pi\) int\_0^inf dOmega N2''\(Omega\) sin\(Omega s\)\. 'In the opposite limit of long pulses all the nonlinearity is essentially instantaneous, with a strength N2\(0\) = N\_20 = N\_2inf \+ N\_2R\. It is N\_20 that is usually measured in self\-focusing or self\-phase modulation experiments \.\.\. N\_2inf = N\_20\(1 \- alpha\) and N\_2R = alpha N\_20' \(p\. 1161\)\. The relation to Hellwarth's response\: 'the present N\_2R f\(t \- t'\) is equal to the expression a\(t \- s\) \+ b\(t \- s\) of Refs\. 5 and 6 times a factor 2 pi/n' \(p\. 1165\)\. THE RESPONSE FUNCTION ITSELF is given only as Fig\. 2 \(and Fig\. 3 for spectral features removed\); not digitised\. The gain spectrum \(Fig\. 1\(a\), normalised to unity at its peak at 440 cm^\-1\) and its Kramers\-Kronig real part \(Fig\. 1\(b\)\) are figures only\. Fig\. 3\: \(a\) full gain and response, \(b\) sharp features removed, \(c\) the 604 cm^\-1 line, \(d\) bands beyond 750 cm^\-1, \(e\) a Lorentzian through the half\-gain points\. The text calls the line both 604 \(caption\) and 603 cm^\-1 \(p\. 1162\)\. EXPONENTIAL MODELS\. Eq\. \(2\) with Eq\. \(4\), p\. 1160\: N2'\(Omega\) = N\_2inf \+ N\_2R/\(1 \+ Omega^2 tau^2\), N2''\(Omega\) = N\_2R Omega tau/\(1 \+ Omega^2 tau^2\); tau = 76 fs matches the peak of the silica gain curve \(Fig\. 1, dashed\)\. In the pulse\-compression simulation \(p\. 1162\) an artificial tau = 391 fs with 40 % of the nonlinearity was used because the 122\-fs grid could not resolve the real response \('This produces a peak Raman gain close to that of fused silica'\)\. Both are recorded as check values, the second as assumed\. NONLINEAR INDEX\. 'For silica the nonlinear index N2 is 3\.2 x 10^\-16 cm2/W\. \[Note that in electrostatic units the nonlinear index is 1\.1 x 10^\-13 esu and is related to N2 by the factor \(4 pi/nc\) x 10^7\.\]' \(pp\. 1159\-1160\), and n2 = 1\.1e\-13 esu 'for silica' in Eq\. \(13\) \(p\. 1162\)\. No reference, wavelength or measurement is given\: a model input\. Check\: 1\.1e\-13 x 4 pi/\(1\.45 x 2\.998e10\) x 1e7 = 3\.18e\-16 cm2/W with n = 1\.45 \(n is not stated\)\. Which part it is \(N\_20 or N\_2inf\) is not said\. SOLITON SELF\-FREQUENCY SHIFT \(p\. 1164, a check of the normalisation\)\: 9\.2 MHz/m at z/z0 = 4\.0 for a 40\-fs sech pulse at 1\.5 um with 15 ps/\(nm km\); Gordon's formula gives 8\.5 MHz/m for a 45\-fs pulse\. Not a material property; not extracted\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- stolen\_1989/alpha ~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha", "material": "SiO2", "table": "text", "quantity": "raman_fraction", "abscissa": "label", "use": "primary", "parameters": { "definition": "alpha = N_2R/N_20: the fraction of the long-pulse nonlinear index N_20 = N_2inf + N_2R that comes from the delayed Raman response, f(t) normalised to unit integral; parallel linear polarizations (Eq. (6), p. 1161)", "form": "fibre" } } 1 inline points in the source record. p\. 1161\-1162\: 'From Raman\-scattering cross\-section and gain measurements, we estimate that the fractional contribution of the Raman response of N\_20 is alpha = 0\.18\.' Repeated p\. 1164\: 'The Raman contribution represents 18% of N\_20\.' An estimate from refs\. 6 \(Hellwarth, Cherlow and Yang 1975\) and 16 \(Stolen and Ippen 1973\); no uncertainty stated\. stolen\_1989/alpha\_simulation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_simulation", "material": "SiO2", "table": "text", "quantity": "raman_fraction", "abscissa": "label", "use": "check", "parameters": { "definition": "fraction of the nonlinearity with the exponential response of Eq. (2) (tau = 391 fs), chosen to give silica's peak Raman gain", "form": "fibre", "assumed": true } } 1 inline points in the source record. p\. 1162\: 'we chose a time constant tau = 391 fsec, and we assumed that 40% of the nonlinearity of the fiber had the exponential response\. This produces a peak Raman gain close to that of fused silica'\. An artificial model input for a coarse \(122 fs\) simulation grid, not a property of silica\. Printed as '40%'\. stolen\_1989/tau\_exponential ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "tau_exponential", "material": "SiO2", "table": "text", "quantity": "raman_response_time", "unit": "fs", "abscissa": "label", "use": "check", "parameters": { "constant": "tau", "model": "single exponential relaxation: N_2R/tau int |u(t')|^2 exp[-(t - t')/tau] dt' (Eq. (2))", "form": "fibre" } } 1 inline points in the source record. p\. 1160 and Fig\. 1 caption\: the relaxation time of the exponential response, Eqs\. \(2\) and \(4\), 'chosen to match the peak of the Raman\-gain curve of fused silica'\. Illustrative\: the exponential model does not reproduce silica's gain shape \(Fig\. 1\)\. stolen\_1989/tau\_simulation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "tau_simulation", "material": "SiO2", "table": "text", "quantity": "raman_response_time", "unit": "fs", "abscissa": "label", "use": "check", "parameters": { "constant": "tau", "model": "single exponential relaxation (Eq. (2)), with 40 % of the nonlinearity", "form": "fibre", "assumed": true } } 1 inline points in the source record. p\. 1162\: the artificially large time constant of the pulse\-compression simulation \(with 40 % of the nonlinearity; see alpha\_simulation\)\. Not a property of silica\. stolen\_1989/n2 ~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2", "material": "SiO2", "table": "text", "quantity": "nonlinear_index", "unit": "cm2/W", "scale": 1e-16, "abscissa_unit": "nm", "use": "check", "parameters": { "part": "unknown", "coefficient": "self", "form": "fibre", "assumed": true } } 1 inline points in the source record. p\. 1159\: 'For silica the nonlinear index N2 is 3\.2 x 10^\-16 cm2/W' \(and 1\.1 x 10^\-13 esu, p\. 1160, related by \(4 pi/nc\) x 10^7\)\. No reference, wavelength or part given; used as a model input for the nonlinear Schroedinger equation \(source notes\)\. Formulas -------- stolen\_1989/low\_frequency\_gain ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "low_frequency_gain", "material": "SiO2", "equation": "Eq. (10), p. 1161", "printed": "g = (a \\Omega + b \\Omega^3) cm/W, a = 7.67 \\times 10^{-4}, b = 1.33 \\times 10^{-6}", "variable": "Omega: frequency shift in cm^-1 (valid below 25 cm^-1); g: the gain spectrum normalised to unity at its maximum at 440 cm^-1", "range": "Omega < 25 cm^-1", "coefficients": { "a": "7.67e-4", "b": "1.33e-6" }, "check": "No printed value to compare. At 25 cm^-1 it gives 0.0192 + 0.0208 = 0.040 of the peak (computed 2026-09-26)." } The fit to the imaginary susceptibility below 25 cm^\-1, joined to the measured spectrum above\. 'This curve is within the experimental uncertainty of the actual gain for a pump wavelength of 1\.0 um\.' The unit printed, cm/W, sits oddly with the statement that the spectrum is normalised to unity at 440 cm^\-1; the coefficients are recorded as printed \(TC\-383\)\. 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.