lin_2007
========

Q\. Lin, O\. J\. Painter and G\. P\. Agrawal, "Nonlinear optical phenomena in silicon waveguides\: modeling and applications", Opt\. Express 15, 16604 \(2007\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/lin_2007.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1364/oe\.15\.016604 <https://doi.org/10.1364/oe.15.016604>`__

**Source kind:** compilation

**Recorded technique:** A theoretical review of third\-order nonlinear propagation in silicon\-on\-insulator \(SOI\) waveguides\: the tensor chi\(3\) of crystalline silicon \(electronic Kerr and TPA, the one\-phonon Raman response\), the Drude free\-carrier index and absorption, the carrier continuity equation, and the resulting generalized nonlinear Schroedinger and coupled\-wave equations\. It measures nothing; its material numbers are quoted from the literature or chosen as typical\.

**Materials in structured entries:** Si

**Dataset quantities:** band\_gap, electron\_trapping\_time, raman\_linewidth, raman\_response\_time, refractive\_index, two\_photon\_absorption\_coefficient

**Contents:** 9 datasets; 2 formulas.

Curation notes
--------------

Journal pages 16604\-16644 = PDF pages 1\-41 \(journal page = PDF page \+ 16603; the references occupy PDF pages 1\-7\)\. Identity checked against the first page\. The material is crystalline silicon \(Si\); the waveguide\-specific numbers \(modal indices, waveguide GVD, effective areas\) are not material properties and are not extracted\. There are no tables; every value below is from the text\.

EXTRACTED \(datasets below\)\: the band gap 'near 1\.12 eV' \(p\. 16610; 'E\_g ~ 1\.12 eV for silicon corresponds to a wavelength of 1\.1 um', p\. 16613\); the material index 'n\_0 = 3\.48' at 1\.55 um \(p\. 16618; 'around 3\.48', p\. 16622\); the typical TPA coefficient 'beta\_T ~ 0\.5 cm/GW' used in an estimate near 1550 nm \(p\. 16624\); the one\-phonon Raman gain FWHM 'Gamma\_R/pi ~ 105 GHz at room temperature' \(p\. 16612, ref\. 108\) and the times of Eq\. \(41\), 'tau\_2 ~ 3 ps', 'tau\_1 ~ 10 fs' \(p\. 16620\); the effective carrier lifetimes\: typical 1 ns \(p\. 16632, '~1 ns', p\. 16631\), 25 ns \(ref\. 33\), 100 ns reduced to 1\.9 ns by helium implantation \(ref\. 140\), and the bound \< 60 ns for net Raman gain with 0\.2 dB/cm loss \(p\. 16633\)\.

NOT EXTRACTED \(no quantity, or not a number for silicon\)\: the Raman shift 'Omega\_R/2pi = 15\.6 THz at room temperature' \(p\. 16612, refs\. 108\-110; raman\_shift takes cm^\-1 only, and the paper prints THz\); its temperature dependence \('Omega\_R/2pi decreases by ~40 GHz, while Gamma\_R/pi increases by ~10 GHz, if temperature increases by 50 C', p\. 16612\); the phonon lifetime 'about 3 ps' \(p\. 16612\); the second\-order Raman peaks near 29 THz \(two TO phonons\) and 9 THz \(two TA phonons\), 'more than 50 and 200 times smaller' \(p\. 16612\); Brillouin 'nearly two orders of magnitude smaller' than first\-order Raman \(p\. 16612\); the Raman gain coefficient 'in the range of 4\.3 to 76 cm/GW' in the 1\.55\-um band \(p\. 16620, refs\. 6, 27, 29, 30, 32, 34, 41, 145, 146; no quantity for a Raman gain coefficient, TC\-384\); the Raman index 'n\_2^R ~ c g\_R/\(4 omega\_s\) ~ 1\.2 x 10^\-4 cm2/GW' \(p\. 16641\); the nonlinear anisotropy 'rho is close to 1\.27 in the telecom band, with a negligible imaginary part' \(p\. 16614, ref\. 87; rho = 3 chi\_1122/chi\_1111, Eq\. \(7\)\); the nonlinear figure of merit F\_n = n2/\(lambda beta\_T\) 'only about 0\.3' at 1\.55 um \(p\. 16613\); n2 'more than 100 times larger in the 1\.55\-um region than that of fused silica', peaking near 1\.8\-1\.9 um, always positive below E\_g \(p\. 16613\); mobilities '100\-1000 cm2/\(V s\) for densities of up to 10^18 cm^\-3' \(p\. 16614\); the SiO2 zero\-dispersion wavelength 'around 1\.276 um' \(p\. 16622, no quantity\)\. No numerical n2 is given\.

DEFINITIONS\. Eq\. \(8\), p\. 16613\: \(omega/c\) n2 \+ \(i/2\) beta\_T = 3 omega chi\_1111^e/\(4 eps0 c^2 n0^2\), with P\(3\) of Eq\. \(1\) in Butcher and Cotter's convention \(3 eps0/\(4 \(2 pi\)^2\) \.\.\.\)\. Eq\. \(3\)\: H\_R\(Omega\) = Omega\_R^2/\(Omega\_R^2 \- Omega^2 \- 2 i Gamma\_R Omega\)\. Eq\. \(41\), p\. 16620\: h\_R\(t\) = Omega\_R^2 tau\_1 exp\(\-t/tau\_2\) sin\(t/tau\_1\), tau\_2 = 1/Gamma\_R, tau\_1 = 1/\(Omega\_R^2 \- Gamma\_R^2\)^\(1/2\) ~ 1/Omega\_R\. The TPA power loss dP/dz = \-beta\_T \|A\|^4/a \(p\. 16624\), i\.e\. dI/dz = \-beta\_T I^2\. Free\-carrier index and absorption\: Drude, Eqs\. \(16\)\-\(17\); the empirical 1550\-nm formulas \(18\)\-\(19\) and Eq\. \(20\) are recorded as formulas below\. Carrier dynamics\: dN/dt = G \- N/tau\_0, Eq\. \(36\), with tau\_0 the effective lifetime \(recombination, diffusion and drift\)\.

**Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets
--------

lin\_2007/band\_gap
~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "band_gap",
     "material": "Si",
     "table": "text",
     "quantity": "band_gap",
     "unit": "eV",
     "abscissa": "label",
     "use": "check",
     "parameters": {
       "kind": "optical"
     }
   }

1 inline points in the source record.

p\. 16610\: 'Silicon crystals, with a band gap near 1\.12 eV'; p\. 16613\: 'E\_g ~ 1\.12 eV for silicon corresponds to a wavelength of 1\.1 um'; the indirect gap \(TPA 'has to be assisted by phonons \.\.\. because of the indirect nature of the band gap', p\. 16613\)\. Quoted without a citation\.

lin\_2007/n0
~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "n0",
     "material": "Si",
     "table": "text",
     "quantity": "refractive_index",
     "abscissa_unit": "um",
     "use": "check"
   }

1 inline points in the source record.

p\. 16618\: 'the material index \(n\_0 = 3\.48\)' at lambda = 1\.55 um \(also 'around 3\.48', p\. 16622\)\. Quoted without a citation or temperature\.

lin\_2007/beta\_T
~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "beta_T",
     "material": "Si",
     "table": "text",
     "quantity": "two_photon_absorption_coefficient",
     "unit": "cm/GW",
     "abscissa_unit": "nm",
     "use": "check",
     "parameters": {
       "assumed": true
     }
   }

1 inline points in the source record.

p\. 16624\: 'With r\_a \<\< 1 and beta\_T ~ 0\.5 cm/GW, alpha\_fm \<\< 0\.4 dB/cm at a peak intensity of 0\.2 GW/cm2', 'using typical parameter values at wavelengths near 1550 nm'\. A typical value for an estimate, uncited\. Definition\: dP/dz = \-beta\_T \|A\|^4/a \(p\. 16624\)\.

lin\_2007/raman\_linewidth
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "raman_linewidth",
     "material": "Si",
     "table": "text",
     "quantity": "raman_linewidth",
     "unit": "GHz",
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "P. A. Temple and C. E. Hathaway, Phys. Rev. B 7, 3685-3697 (1973)"
     ],
     "parameters": {
       "transition": "first-order Raman, zone-centre optical phonon (Gamma_25')",
       "width": "fwhm"
     }
   }

1 inline points in the source record.

p\. 16612\: 'Gamma\_R is related inversely to the phonon lifetime \(about 3 ps\), resulting in a full width at half\-maximum \(FWHM\) of the Raman\-gain spectrum of Gamma\_R/pi ~ 105 GHz at room temperature \[108\]\.' The Raman shift is 15\.6 THz \(not extracted\: printed in THz\)\.

lin\_2007/raman\_tau1
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "raman_tau1",
     "material": "Si",
     "table": "text",
     "quantity": "raman_response_time",
     "unit": "fs",
     "abscissa": "label",
     "use": "check",
     "parameters": {
       "constant": "tau1",
       "model": "h_R(t) = Omega_R^2 tau_1 exp(-t/tau_2) sin(t/tau_1), tau_2 = 1/Gamma_R, tau_1 = 1/(Omega_R^2 - Gamma_R^2)^(1/2) (Eq. (41))"
     }
   }

1 inline points in the source record.

p\. 16620\: 'the Raman shift of 15\.6 THz corresponds to tau\_1 ~ 10 fs\.' Derived by the paper from the quoted shift\.

lin\_2007/raman\_tau2
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "raman_tau2",
     "material": "Si",
     "table": "text",
     "quantity": "raman_response_time",
     "unit": "ps",
     "abscissa": "label",
     "use": "check",
     "parameters": {
       "constant": "tau2",
       "model": "h_R(t) = Omega_R^2 tau_1 exp(-t/tau_2) sin(t/tau_1), tau_2 = 1/Gamma_R, tau_1 = 1/(Omega_R^2 - Gamma_R^2)^(1/2) (Eq. (41))"
     }
   }

1 inline points in the source record.

p\. 16620\: 'The Raman\-gain bandwidth of 105 GHz in silicon corresponds to a response time of tau\_2 ~ 3 ps\.' Derived by the paper from the quoted width \(also 'the phonon lifetime \(about 3 ps\)', p\. 16612\)\.

lin\_2007/carrier\_lifetime\_typical
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "carrier_lifetime_typical",
     "material": "Si",
     "table": "text",
     "quantity": "electron_trapping_time",
     "unit": "ns",
     "abscissa": "label",
     "use": "check",
     "parameters": {
       "assumed": true
     }
   }

1 inline points in the source record.

p\. 16632\: the estimate 'for an SOI waveguide with tau\_0 = 1 ns'; p\. 16631\: 'the carrier lifetime \(typically ~1 ns\)'; p\. 16619\: 'For SOI waveguides commonly used for photonic applications, the effective lifetime varies from subnanosecond \[54, 90, 141\] to tens of nanoseconds \[41, 42\]\.' tau\_0 is the effective lifetime of Eq\. \(36\), including recombination, diffusion and drift out of the waveguide core\: a property of the waveguide, not of bulk silicon\.

lin\_2007/carrier\_lifetime\_rong
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "carrier_lifetime_rong",
     "material": "Si",
     "table": "text",
     "quantity": "electron_trapping_time",
     "unit": "ns",
     "abscissa": "label",
     "use": "reproduced",
     "reproduced_from": [
       "H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, Appl. Phys. Lett. 85, 2196-2198 (2004)"
     ]
   }

1 inline points in the source record.

p\. 16633\: 'positive Raman amplification with CW pumping was observed inside a waveguide with a linear loss of 0\.22 dB/cm and a carrier lifetime of 25 ns \[33\]\.' An SOI\-waveguide effective lifetime\.

lin\_2007/carrier\_lifetime\_implanted
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "carrier_lifetime_implanted",
     "material": "Si",
     "table": "text",
     "quantity": "electron_trapping_time",
     "unit": "ns",
     "abscissa": "label",
     "use": "reproduced",
     "reproduced_from": [
       "Y. Liu and H. K. Tsang, Opt. Lett. 31, 1714-1716 (2006)"
     ]
   }

2 inline points in the source record.

p\. 16633\: 'the effective lifetime could be reduced in one experiment from 100 ns to 1\.9 ns through helium\-ion implantation, thereby enabling positive Raman amplification \[140\]\.' SOI\-waveguide effective lifetimes\.

Formulas
--------

lin\_2007/free\_carrier\_index\_1550
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "free_carrier_index_1550",
     "material": "Si",
     "equation": "Eq. (18), p. 16615",
     "printed": "n_f(\\omega_r, N_e, N_h) = -(8.8 \\times 10^{-4} N_e + 8.5 N_h^{0.8}) \\times 10^{-18}",
     "variable": "N_e, N_h: electron and hole densities in cm^-3; at lambda_r = 2 pi c/omega_r = 1550 nm",
     "coefficients": {
       "electron": "8.8e-4",
       "hole": "8.5",
       "hole_exponent": "0.8",
       "scale": "1e-18"
     },
     "check": "The paper's statements follow (computed 2026-09-26): the hole term over the electron term is 3.85 at 1e17 cm^-3 and 6.1 at 1e16 cm^-3 (printed 'about 3.8 and 6.1'); with holes taken as 5 times the electron term, sigma_n = -6 x 8.8e-22 = -5.3e-21 cm3 (printed zeta ~ -5.3e-21 cm3)."
   }

Reproduced from Soref and Bennett \(IEEE J\. Quantum Electron\. 23, 123 \(1987\); Proc\. SPIE 704, 32 \(1987\)\), refs\. 81 and 135, 'it is common to employ the following empirical formulas'\. For TPA\-generated carriers \(N\_e = N\_h = N\), Eq\. \(20\)\: n\_f = sigma\_n N with sigma\_n = zeta \(omega\_r/omega\)^2, zeta ~ \-5\.3e\-21 cm3 \(the paper's assumption that holes contribute 5 times the electrons\)\.

lin\_2007/free\_carrier\_absorption\_1550
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "free_carrier_absorption_1550",
     "material": "Si",
     "equation": "Eq. (19), p. 16615",
     "printed": "\\alpha_f(\\omega_r, N_e, N_h) = (8.5 N_e + 6.0 N_h) \\times 10^{-18}",
     "variable": "N_e, N_h: electron and hole densities in cm^-3; alpha_f in cm^-1; at 1550 nm",
     "coefficients": {
       "electron": "8.5",
       "hole": "6.0",
       "scale": "1e-18"
     },
     "check": "With N_e = N_h = N it gives alpha_f = 1.45e-17 N cm^-1, the printed sigma_a = 1.45 x 10^-17 cm2 of Eq. (20) at omega = omega_r (computed 2026-09-26)."
   }

Reproduced from Soref and Bennett \(refs\. 81, 135\)\. Eq\. \(20\)\: alpha\_f = sigma\_a N, sigma\_a = 1\.45 x 10^\-17 \(omega\_r/omega\)^2 cm2, the Drude lambda^2 scaling applied to the empirical coefficient\.

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.