jing_2012
=========

X\. Jing, Y\. Tian, J\. Zhang, S\. Chen, Y\. Jin, J\. Shao and Z\. Fan, "Modeling validity of femtosecond laser breakdown in wide bandgap dielectrics", Appl\. Surf\. Sci\. 258, 4741 \(2012\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/jing_2012.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1016/j\.apsusc\.2012\.01\.070 <https://doi.org/10.1016/j.apsusc.2012.01.070>`__

**Source kind:** theory

**Recorded technique:** Rate\-equation calculations of the femtosecond breakdown threshold of a 9\-eV dielectric \(fused silica\)\: Keldysh photoionization \(Eqs\. 3\-5\) combined in turn with four avalanche models \(Stuart, Sparks\-Duthler, Thornber, Drude; Eqs\. 6\-9\), a density\-dependent collision time \(Starke, Eq\. 10\) and a 100\-fs relaxation \(Eq\. 11\), with the critical density as the damage criterion; compared with published surface thresholds against pulse duration and wavelength \(Figs\. 7\-8\)\.

**Materials in structured entries:** SiO2

**Dataset quantities:** avalanche\_coefficient, band\_gap, damage\_threshold\_fluence, effective\_mass, electron\_collision\_time, electron\_trapping\_time, ionization\_rate, multiphoton\_ionization\_cross\_section, refractive\_index

**Contents:** 21 datasets; 3 formulas.

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

Journal pages 4741\-4749 = PDF pages 1\-9\. Identity validated\.

MODEL INPUTS \(datasets assumed\)\: E\_g = 9 eV; m\_e\* = 0\.635 m\_e, 'a measured value accepted generally to be accurate for fused silica' \[1\] \(Sudrie et al\., Phys\. Rev\. Lett\. 89, 186601 \(2002\)\), the reduced mass m\_e m\_h/\(m\_e \+ m\_h\) \(p\. 4742\-4743\); n0 = 1\.46 \(Fig\. 4 caption\); tau\_r = 100 fs \(p\. 4744\: 'Li et al\. had indicated that the plasma energy in dielectrics experienced ultra\-fast decay which lasted only ~100 fs \.\.\. In our calculation, the relaxation time tau\_r = 100 fs was applied'\); the Starke collision time Eq\. \(10\); the critical density n\_cr = eps0 m\_e\* omega^2/e^2 = 1\.1e21 cm^\-3 \(Fig\. 4; 1\.106e21 with m\_e\* = 0\.635 m\_e at 800 nm, checked\); e = 1\.6e\-19 C, eps0 = 8\.854e\-12 F/m\.

LITERATURE \(datasets reproduced\), p\. 4742\: 'The effective electron mass of m\_e \[24\], 0\.86 m\_e \[20\], 0\.635 m\_e \[1\], 0\.64 m\_e \[2\], and 0\.5 m\_e \[4,13\] \.\.\. were often used in theoretical calculation of photoionization rate and avalanche ionization rate for fused silica' \(refs\. 24 Kaiser et al\. 2000, 20 Wu et al\. 2005, 1 Sudrie et al\. 2002, 2 couairon\_2005, 4 Vaidyanathan et al\. 1980, 13 Yuan et al\. 2007\); the relaxation time 'between 150 and 250 fs for fused silica was found \[41\-43\]' and 150 fs 'often used' \[1,2,20\] \(a range\: not extracted\); tau\_c 'in the range from 10^\-16 s to 10^\-14 s \[1,8,19,20,35\-37\]' \(range, not extracted\); sigma\_6 of Lenzner et al\. \[11\] 6\.0e\-52 and of Li et al\. \[10\] 3\.0e\-56 cm^\-3 s^\-1 \(cm2/W\)^6 \(p\. 4743\); Stuart's alpha ~ 0\.01 cm2 ps^\-1/GW \(stuart\_1996\); the Sparks model's tau\_c = 1\.36e\-15 s 'In Refs\. \[4,13\]'; the measured collision time 1\.7 fs near 5e19 cm^\-3 \[39\] \(Sun et al\., Opt\. Lett\. 30, 320 \(2005\)\); surface thresholds 2\.8 J/cm2 \('a typical surface damage threshold \.\.\. measured in the experiment with parameters similar to our calculation \[11,16,27\]'\), 1\.5 J/cm2 \(Li et al\. \[10\], 25 fs single pulse\) and 2\.3 J/cm2 \(Jia et al\. \[3\], 800 nm, 150 fs, three shots\) \(pp\. 4745, 4747\)\.

CHECKS \(2026\-09\-26\)\: the fluence 2\.74 J/cm2 = \(pi/ln 2\)^\(1/2\) I0 tau\_p/2 with I0 = 25\.74 TW/cm2 \(a lost glyph in the library text, 'TW/cm2' in the pdf text layer\) and tau\_p = 100 fs \(2\.7399\); sigma\_6 I^6 at 3\.5e13 W/cm2 = 4\.04e34 cm^\-3 s^\-1 \(printed 4\.1e34\); the Starke tau\_c at 5e19 cm^\-3 with m\_e\* = 0\.635 m\_e, E\_g = 9 eV\: 3\.52 fs \(printed 'about 3\.5 fs'; 4\.42 fs with m\_e\); gamma = 1 at 3\.5e13 W/cm2 \(0\.999 with gamma = omega sqrt\(m\_e\* E\_g\)/\(eE\), I = c n0 eps0 E^2/2, n0 = 1\.46\)\. The Keldysh\-limit sigma\_6 = 2\.2e\-47 per volume is couairon\_2005's 9\.6e\-70 per atom times 2\.1e22 cm^\-3 \(2\.0e\-47\), and the paper's W\_MPI \(4\.1e34\) and W\_PI \(1\.6e32\) at 3\.5e13 W/cm2, gamma = 1, are couairon\_2005's \(3\.7e34, 1\.6e32\)\: the paper says 'we calculated' Fig\. 1; the agreement of W\_PI suggests the same implementation or a reproduction \(not cited as such\)\. The Lenzner and Li sigma\_6 here disagree by ~1e7 with couairon\_2005's per\-atom values \(see couairon\_2005's notes\)\.

EFFECTIVE GAP\: Eq\. \(4\) prints E'\_g = E\_g\[1 \+ 1/\(2 gamma^2\)\]; hamam\_2018 prints eps\_eff = eps\_g\[1 \+ 1/\(4 gamma^2\)\]\: the factor differs \(and so do the gamma conventions\); not resolvable here\. Eq\. \(4\)'s \<E'\_g/hbar omega \+ 1\> is typeset as \<E'\_g/\(hbar omega \+ 1\)\> in the library text; as printed in the rendering\.

FIGURES ONLY\: Fig\. 1 \(Keldysh, multiphoton and tunnel rates for 9 eV silica at 800 nm, with the two literature sigma\_6\), Fig\. 2 \(Eq\. 10\), Fig\. 3 \(avalanche rates against field\), Figs\. 4\-9 \(electron density; calculated thresholds against duration, wavelength and collision time; comparison with Lenzner, Jia, Varel, Li\)\.

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

Datasets
--------

jing\_2012/model\_band\_gap
~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "model_band_gap",
     "material": "SiO2",
     "table": "text",
     "quantity": "band_gap",
     "unit": "eV",
     "abscissa": "none",
     "use": "check",
     "parameters": {
       "kind": "assumed",
       "assumed": true
     }
   }

1 inline points in the source record.

'the damage threshold of wide band\-gap dielectric \(9 eV\)' \(p\. 4742\) and Fig\. 4's caption 'band gap \(9 eV\)'; no source\.

jing\_2012/model\_reduced\_mass
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "model_reduced_mass",
     "material": "SiO2",
     "table": "text",
     "quantity": "effective_mass",
     "unit": "m_e",
     "abscissa": "none",
     "use": "check",
     "reproduced_from": [
       "L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis and A. Mysyrowicz, Phys. Rev. Lett. 89, 186601 (2002)"
     ],
     "parameters": {
       "carrier": "reduced",
       "assumed": true
     }
   }

1 inline points in the source record.

p\. 4743\: 'the effective electron mass of m\_e\* = 0\.635 m\_e which is a measured value accepted generally to be accurate for fused silica was used to estimate photoionization rate and avalanche ionization rate \[1\]'; defined as the reduced mass of electron and hole \(p\. 4742\)\.

jing\_2012/mass\_kaiser
~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "mass_kaiser",
     "material": "SiO2",
     "table": "text",
     "quantity": "effective_mass",
     "unit": "m_e",
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "A. Kaiser, B. Rethfeld, M. Vicanek and G. Simon, Phys. Rev. B 61, 11437 (2000)"
     ],
     "parameters": {
       "carrier": "not stated (Jing: 'effective electron mass')"
     }
   }

1 inline points in the source record.

p\. 4742\: 'The effective electron mass of m\_e \[24\] \.\.\.', one of the values 'often used' for fused silica; which mass \(electron or reduced\) the original means is not said here\.

jing\_2012/mass\_wu
~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "mass_wu",
     "material": "SiO2",
     "table": "text",
     "quantity": "effective_mass",
     "unit": "m_e",
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "A. Q. Wu, I. H. Chowdhury and X. Xu, Phys. Rev. B 72, 085128 (2005)"
     ],
     "parameters": {
       "carrier": "not stated (Jing: 'effective electron mass')"
     }
   }

1 inline points in the source record.

p\. 4742\: '0\.86 m\_e \[20\]', as for mass\_kaiser\.

jing\_2012/mass\_sudrie
~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "mass_sudrie",
     "material": "SiO2",
     "table": "text",
     "quantity": "effective_mass",
     "unit": "m_e",
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis and A. Mysyrowicz, Phys. Rev. Lett. 89, 186601 (2002)"
     ],
     "parameters": {
       "carrier": "not stated (Jing: 'effective electron mass')"
     }
   }

1 inline points in the source record.

p\. 4742\: '0\.635 m\_e \[1\]', the value the paper adopts \(model\_reduced\_mass\)\.

jing\_2012/mass\_couairon
~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "mass_couairon",
     "material": "SiO2",
     "table": "text",
     "quantity": "effective_mass",
     "unit": "m_e",
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "couairon_2005"
     ],
     "parameters": {
       "carrier": "reduced"
     }
   }

1 inline points in the source record.

p\. 4742\: '0\.64 m\_e \[2\]'; couairon\_2005 defines it as the electron\-hole reduced mass \(from Fischetti et al\. 1985\)\.

jing\_2012/mass\_vaidyanathan\_yuan
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "mass_vaidyanathan_yuan",
     "material": "SiO2",
     "table": "text",
     "quantity": "effective_mass",
     "unit": "m_e",
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "A. Vaidyanathan, T. W. Walker and A. H. Guenther, IEEE J. Quantum Electron. QE-16, 89 (1980)",
       "L. Yuan, Y. Zhao, G. Shang, C. Wang, H. He, J. Shao and Z. Fan, J. Opt. Soc. Am. B 24, 538 (2007)"
     ],
     "parameters": {
       "carrier": "not stated (Jing: 'effective electron mass')"
     }
   }

1 inline points in the source record.

p\. 4742\: '0\.5 m\_e \[4,13\]', as for mass\_kaiser\.

jing\_2012/model\_refractive\_index
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "model_refractive_index",
     "material": "SiO2",
     "table": "text",
     "quantity": "refractive_index",
     "abscissa_unit": "nm",
     "use": "check",
     "parameters": {
       "assumed": true
     }
   }

1 inline points in the source record.

Fig\. 4 caption\: 'The material parameters of the refractive index n0 = 1\.46 and band gap \(9 eV\) were applied', at 800 nm\.

jing\_2012/model\_relaxation\_time
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "model_relaxation_time",
     "material": "SiO2",
     "table": "text",
     "quantity": "electron_trapping_time",
     "unit": "fs",
     "abscissa": "none",
     "use": "check",
     "reproduced_from": [
       "M. Li, S. Menon, J. P. Nibarger and G. N. Gibson, Phys. Rev. Lett. 82, 2394 (1999)"
     ],
     "parameters": {
       "assumed": true
     }
   }

1 inline points in the source record.

p\. 4744\: the decay of Eq\. \(11\), 'the relaxation time tau\_r = 100 fs was applied', following Li et al\.'s '~100 fs' plasma\-energy decay\.

jing\_2012/sigma6\_keldysh\_limit
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "sigma6_keldysh_limit",
     "material": "SiO2",
     "table": "text",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "cm-3 s-1 (cm2/W)6",
     "abscissa_unit": "nm",
     "use": "check",
     "parameters": {
       "photons": 6
     }
   }

1 inline points in the source record.

p\. 4743\: 'at the laser intensity I \< 4\.0 x 10^12 W/cm2, Keldysh's theory coincides with the multiphoton ionization rate W\_MPI = sigma\_6 I^6, where sigma\_6 = 2\.2 x 10^\-47 cm^\-3 s^\-1 \(cm2/W\)^6', per volume \(printed '\(cm2/w\)^6'\); the weak\-field limit of their Keldysh rate for 9 eV, m\_e\* = 0\.635 m\_e\.

jing\_2012/sigma6\_lenzner
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "sigma6_lenzner",
     "material": "SiO2",
     "table": "text",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "cm-3 s-1 (cm2/W)6",
     "abscissa_unit": "nm",
     "use": "reproduced",
     "reproduced_from": [
       "lenzner_1998"
     ],
     "parameters": {
       "photons": 6
     }
   }

1 inline points in the source record.

p\. 4743\: 'sigma\_6 = 6\.0 x 10^\-52 cm^\-3 s^\-1 \(cm2/w\)^6 for fused silica was estimated by Lenzner et al\. \[11\]'; the same as schaffer\_2001's 6\.0e8 cm^\-3 ps^\-1 \(cm2/TW\)^6\. The original, lenzner\_1998 \(p\. 4078\), prints 6 x 10^\(8 \+\- 0\.9\) cm^\-3 ps^\-1 \(cm2/TW\)^6 = 6\.0e\-52 cm^\-3 s^\-1 \(cm2/W\)^6\: agrees\.

jing\_2012/sigma6\_li
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "sigma6_li",
     "material": "SiO2",
     "table": "text",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "cm-3 s-1 (cm2/W)6",
     "abscissa_unit": "nm",
     "use": "reproduced",
     "reproduced_from": [
       "M. Li, S. Menon, J. P. Nibarger and G. N. Gibson, Phys. Rev. Lett. 82, 2394 (1999)"
     ],
     "parameters": {
       "photons": 6
     }
   }

1 inline points in the source record.

p\. 4743\: 'the value of sigma\_6 = 3\.0 x 10^\-56 cm^\-3 s^\-1 \(cm2/w\)^6 reported by Li et al\. \[10\]'\. couairon\_2005 quotes Li's value per atom as 1\.5e\-71 s^\-1 cm12/W6, ~1e7 larger than this per\-volume value divided by 2\.1e22 cm^\-3\.

jing\_2012/keldysh\_rate
~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "keldysh_rate",
     "material": "SiO2",
     "table": "text",
     "quantity": "ionization_rate",
     "unit": "cm-3 s-1",
     "abscissa": "intensity",
     "abscissa_unit": "W/cm2",
     "use": "check",
     "parameters": {
       "wavelength_nm": 800,
       "polarization": "not stated"
     }
   }

1 inline points in the source record.

p\. 4743\: 'at I = 3\.5 x 10^13 w/cm2 \(gamma = 1\), the multiphoton ionization approximation of W\_MPI = 4\.1 x 10^34 cm^\-3 s^\-1 clearly overestimates the full Keldysh photoionization rate W\_PI = 1\.6 x 10^32 cm^\-3 s^\-1 by 2 orders of magnitude'; the same numbers as couairon\_2005 \(W\_MPI there 3\.7e34\)\.

jing\_2012/alpha\_stuart
~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "alpha_stuart",
     "material": "SiO2",
     "table": "text",
     "quantity": "avalanche_coefficient",
     "unit": "cm2/(ps GW)",
     "abscissa_unit": "nm",
     "use": "reproduced",
     "reproduced_from": [
       "stuart_1996"
     ],
     "parameters": {
       "definition": "alpha in W_Av = alpha I(t) (Eq. 6)"
     }
   }

1 inline points in the source record.

p\. 4743\: 'the avalanche coefficient alpha ~ 0\.01 cm2 ps^\-1/GW was found by numerically solving a Fokker\-Planck equation' \(refs\. 5\-7, Stuart et al\.\); stuart\_1996 gives 0\.011 at 1053 nm and 0\.013 at 526 nm\.

jing\_2012/tau\_c\_sparks
~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tau_c_sparks",
     "material": "SiO2",
     "table": "text",
     "quantity": "electron_collision_time",
     "unit": "s",
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "A. Vaidyanathan, T. W. Walker and A. H. Guenther, IEEE J. Quantum Electron. QE-16, 89 (1980)",
       "L. Yuan, Y. Zhao, G. Shang, C. Wang, H. He, J. Shao and Z. Fan, J. Opt. Soc. Am. B 24, 538 (2007)"
     ]
   }

1 inline points in the source record.

p\. 4743\: 'In Refs\. \[4,13\], tau\_c = 1\.36 x 10^\-15 s was used for their theoretical calculation' with the Sparks\-Duthler rate, Eq\. \(7\)\.

jing\_2012/tau\_c\_starke
~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tau_c_starke",
     "material": "SiO2",
     "table": "text",
     "quantity": "electron_collision_time",
     "unit": "fs",
     "abscissa": "none",
     "use": "check"
   }

1 inline points in the source record.

p\. 4744\: from Eq\. \(10\) \(Starke et al\.\), 'the tau\_c of about 3\.5 fs at the electron density of 5 x 10^19 cm^\-3 was obtained numerically'\. Checked\: 3\.52 fs with m\_e\* = 0\.635 m\_e, E\_g = 9 eV \(2026\-09\-26\)\. The density is in the row id; the abscissa has no density option for this quantity\.

jing\_2012/tau\_c\_sun
~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tau_c_sun",
     "material": "SiO2",
     "table": "text",
     "quantity": "electron_collision_time",
     "unit": "fs",
     "abscissa": "none",
     "use": "reproduced",
     "reproduced_from": [
       "Q. Sun, H. Jiang, Y. Liu, Z. Wu, H. Yang and Q. Gong, Opt. Lett. 30, 320 (2005)"
     ]
   }

1 inline points in the source record.

p\. 4744\: 'it agrees well with the measured collision time of 1\.7 fs at the electron density near 5 x 10^19 cm^\-3 \[39\]'\.

jing\_2012/lidt\_model
~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "lidt_model",
     "material": "SiO2",
     "table": "text",
     "quantity": "damage_threshold_fluence",
     "unit": "J/cm2",
     "abscissa": "duration",
     "abscissa_unit": "fs",
     "use": "check",
     "parameters": {
       "site": "surface",
       "criterion": "calculated: conduction-electron density reaching n_cr = 1.1e21 cm^-3",
       "shots": 1,
       "wavelength_nm": 800
     }
   }

1 inline points in the source record.

p\. 4744\: 'the fluence of 2\.74 J/cm2 is the single pulse LIDT predicted by this combining ionization model' \(Keldysh \+ Drude, 800 nm, 100 fs, critical density criterion, Fig\. 4\)\. Calculated\. Checked\: \(pi/ln 2\)^\(1/2\) I0 tau\_p/2 = 2\.740 J/cm2 with I0 = 25\.74 TW/cm2\.

jing\_2012/lidt\_typical
~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "lidt_typical",
     "material": "SiO2",
     "table": "text",
     "quantity": "damage_threshold_fluence",
     "unit": "J/cm2",
     "abscissa_unit": "nm",
     "use": "reproduced",
     "reproduced_from": [
       "lenzner_1998",
       "A.-C. Tien et al., Phys. Rev. Lett. 82, 3883 (1999)",
       "K. Starke et al., Proc. SPIE 5273, 501 (2004)"
     ],
     "parameters": {
       "site": "surface",
       "criterion": "not stated (literature)"
     }
   }

1 inline points in the source record.

p\. 4745\: 'a typical surface damage threshold of phi' ~ 2\.8 J/cm2 was measured in the experiment with parameters similar to our calculation \[11,16,27\]' \(i\.e\. about 100 fs, 800 nm\); which paper gives exactly 2\.8 is not said\. lenzner\_1998 prints no 2\.8 J/cm2 \(its thresholds are in Fig\. 3; its fitted rate equation gives 3\.8 J/cm2 at 100 fs, computed in lenzner\_1998's notes\)\.

jing\_2012/lidt\_li
~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "lidt_li",
     "material": "SiO2",
     "table": "text",
     "quantity": "damage_threshold_fluence",
     "unit": "J/cm2",
     "abscissa": "duration",
     "abscissa_unit": "fs",
     "use": "reproduced",
     "reproduced_from": [
       "M. Li, S. Menon, J. P. Nibarger and G. N. Gibson, Phys. Rev. Lett. 82, 2394 (1999)"
     ],
     "parameters": {
       "site": "surface",
       "criterion": "not stated (literature)",
       "shots": 1
     }
   }

1 inline points in the source record.

p\. 4747\: 'for a 25 fs single pulse, the threshold was measured to be 1\.5 J/cm2 by Li et al\. \[10\]'; wavelength not stated\.

jing\_2012/lidt\_jia
~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "lidt_jia",
     "material": "SiO2",
     "table": "text",
     "quantity": "damage_threshold_fluence",
     "unit": "J/cm2",
     "abscissa": "duration",
     "abscissa_unit": "fs",
     "use": "reproduced",
     "reproduced_from": [
       "T. Q. Jia, H. X. Chen, M. Huang, F. L. Zhao, X. X. Li, S. Z. Xu, H. Y. Sun, D. H. Feng, C. B. Li, X. F. Wang, R. X. Li, Z. Z. Xu, X. K. He and H. Kuroda, Phys. Rev. B 73, 054105 (2006)"
     ],
     "parameters": {
       "site": "surface",
       "criterion": "not stated (literature)",
       "shots": 3,
       "wavelength_nm": 800
     }
   }

1 inline points in the source record.

p\. 4747\: 'at 800 nm 150 fs three shots pulse the threshold of 2\.3 J/cm2 on fused silica sample was obtained by Jia et al\. \[3\]'\.

Formulas
--------

jing\_2012/keldysh\_rate
~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "keldysh_rate",
     "material": "SiO2",
     "equation": "Eqs. (3)-(5), pp. 4742-4743",
     "printed": "W_PI = (2 omega/9 pi) (omega m_e*/(sqrt(gamma_1) h))^(3/2) Q(gamma, x) exp{-pi <x + 1> [K(gamma_1) - E(gamma_1)]/E(gamma_2)} (3), gamma = omega sqrt(m_e* E_g)/(eE), m_e* = m_e m_h/(m_e + m_h), gamma_1 = gamma^2/(1 + gamma^2), gamma_2 = 1/(1 + gamma^2); W_multpt = (2 omega/9 pi)(m_e* omega/hbar)^(3/2) Phi(sqrt(2 <E'_g/hbar omega + 1> - 2 E'_g/hbar omega)) exp{2 <E'_g/hbar omega + 1> (1 - 1/(4 gamma^2))} (1/(16 gamma^2))^<E'_g/hbar omega + 1> (4), E'_g = E_g [1 + 1/(2 gamma^2)]; W_tunnel = (2 E_g/(9 pi^2 hbar)) (m_e* E_g/hbar^2)^(3/2) (hbar omega/(E_g gamma))^(5/2) exp{-(pi/2)(E_g gamma/hbar omega)(1 - gamma^2/8)} (5)",
     "variable": "K, E: complete elliptic integrals of the first and second kinds; <.> the integer part; Phi the Dawson function; E the field amplitude",
     "coefficients": {
       "E_g": "9 eV",
       "m_e_star": "0.635 m_e"
     },
     "reproduced_from": [
       "L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965)"
     ],
     "check": "gamma = 1 at 3.5e13 W/cm2 with I = c n0 eps0 E^2/2, n0 = 1.46 (0.999, 2026-09-26); the rate values (dataset keldysh_rate) are the reference to reproduce. Eq. (3) prints h (not hbar) under the root and the definition of Q is only partly legible in the library text (the Dawson integral is rendered, Q is not)."
   }

Compare couairon\_2005 Eqs\. \(7\)\-\(8\) and hamam\_2018 Eqs\. \(3\)\-\(4\); the effective gap in Eq\. \(4\) differs from hamam\_2018's by a factor 2 in the 1/gamma^2 term \(source notes\)\.

jing\_2012/avalanche\_models
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "avalanche_models",
     "material": "SiO2",
     "equation": "Eqs. (6)-(9), p. 4743",
     "printed": "W_Av_Stuart = alpha I(t), alpha ~ 0.01 cm2 ps^-1/GW (6); W_Av_Sparks = 0.693 e^2 E^2 tau_c/(m_e* E_g (1 + omega^2 tau_c^2)) - 0.693 hbar omega_p/(E_g tau_L) (7); W_Av_Thornber = (v_s e E/E_g) exp{-E_I/(E (1 + E/E_phonon) + E_kT)} (8); W_Av_Drude = (sigma/E_g) I(t), sigma = (e^2/(c eps0 n0 m_e*)) tau_c/(1 + omega^2 tau_c^2) (9)",
     "variable": "E: field; tau_c the mean collision time; I(t) = (1/2) c n0 eps0 |E(t)|^2",
     "coefficients": {
       "tau_c_sparks": "1.36e-15 s",
       "tau_L": "8.77e-16 s",
       "hbar_omega_p": "0.025 eV",
       "v_s": "2e7 cm/s",
       "E_I": "30 MV/cm",
       "E_phonon": "3.2 MV/cm",
       "E_kT": "0.01 MV/cm"
     },
     "reproduced_from": [
       "stuart_1996",
       "M. Sparks and C. J. Duthler, Fifth Tech. Rep., Xonics Inc. (1975)",
       "K. K. Thornber, J. Appl. Phys. 52, 279 (1981)"
     ],
     "check": "Nothing numeric printed to reproduce beyond the constants."
   }

The dc\-to\-optical scaling E\_th^ac\(omega\) = E\_th^dc \(1 \+ omega^2 tau\_c^2\)^\(1/2\) accompanies Thornber's rate \(p\. 4743\)\.

jing\_2012/rate\_equation\_and\_collision\_time
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "rate_equation_and_collision_time",
     "material": "SiO2",
     "equation": "Eqs. (1), (2), (10), (11), pp. 4742-4744",
     "printed": "n_cr = eps0 m_e* omega^2/e^2 (1); dn_e/dt = W_PI(I(t)) + W_Av(I(t)) n_e(t) - W_rel (2); tau_c = 16 pi eps0^2 sqrt(m_e* (0.1 E_g)^3)/(sqrt(2) e^4 n(t)) (10); W_rel = n(t)/tau_r (11)",
     "variable": "n(t): conduction-electron density; tau_r the relaxation time",
     "coefficients": {
       "tau_r": "100 fs",
       "n_cr": "1.1e21 cm^-3"
     },
     "reproduced_from": [
       "K. Starke, D. Ristau, H. Welling, T. V. Amotchkina, M. Trubetskov, A. A. Tikhonravov and A. S. Chirkin, Proc. SPIE 5273, 501 (2004)"
     ],
     "check": "Eq. (10): 3.52 fs at 5e19 cm^-3 (printed ~3.5 fs); Eq. (1): 1.106e21 cm^-3 with 0.635 m_e at 800 nm (printed 1.1e21) (checked 2026-09-26)."
   }

Eq\. \(10\) is Starke et al\.'s inverse\-bremsstrahlung estimate \(ref\. 27\); the Gaussian pulse I = I0 exp\[\-4 ln\(2\)\(t/tau\_p\)^2\], fluence \(pi/ln 2\)^\(1/2\) I0 tau\_p/2\.

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.