lenzner_1998
============

M\. Lenzner, J\. Krüger, S\. Sartania, Z\. Cheng, Ch\. Spielmann, G\. Mourou, W\. Kautek, and F\. Krausz, "Femtosecond optical breakdown in dielectrics", Phys\. Rev\. Lett\. 80, 4076\-4079 \(1998\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/lenzner_1998.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1103/physrevlett\.80\.4076 <https://doi.org/10.1103/physrevlett.80.4076>`__

**Source kind:** measurement

**Recorded technique:** Front\-surface optical breakdown \(ablation\) of fused silica \(FS, Corning 7940\) and barium aluminum borosilicate glass \(BBS, Corning 7059\), both with drawn surfaces \(roughness \< 10 and \< 13 nm\), by linearly polarized 780\-nm pulses of 5 fs to 5 ps\: a Ti\:sapphire amplifier \(20 fs, 1 mJ, 1 kHz\) and a hollow\-fibre chirped\-mirror compressor \(5\-20 fs bandwidth\-limited by the argon pressure; \>= 20 fs by evacuating the fibre and stretching\), the fibre acting as a spatial filter \(M^2 \< 1\.1\); focused to a 1/e2 diameter of ~30 um, imaged on a CCD\. Each site exposed to 50 pulses at one fluence; the ablated volume per pulse V\_a against on\-axis fluence F is linear, and its regression line extrapolated to V\_a = 0 gives the threshold F\_th \(Fig\. 2\)\. Ablation depths from light and electron microscopy \(Figs\. 4\-5\)\. In air \(sub\-10\-fs breakdown in vacuum gave the same F\_th; note 13\)\.

**Materials in structured entries:** Corning\-7059, SiO2

**Dataset quantities:** avalanche\_coefficient, band\_gap, effective\_mass, multiphoton\_ionization\_cross\_section

**Contents:** 8 datasets; 1 formula.

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

Journal pages 4076\-4079 = pdf pages 1\-4\. Identity checked against the first page\.

WHAT WAS MEASURED \(figures\)\. Fig\. 3 \(p\. 4077\)\: F\_th of FS and BBS against the pulse duration tau \(FWHM\) at 780 nm, 5 fs to 5 ps, 50 pulses per site, with the fits of Eq\. \(1\)\. Error bars 'represent standard deviations'; 'the absolute \(systematic\) error of the measurements is less than \+\-15%' \(p\. 4077; the extraction prints '615%', the '6' being the lost '\+\-'\)\. Not digitised\. In the text\: 'we report F\_th ~ 1\.5 J/cm^2 in fused silica for tau \<= 10 fs' \(p\. 4076\), against Stuart et al\.'s prediction of \< 0\.1 J/cm^2 for tau ~ 10 fs; for tau \> 100 fs the FS thresholds agree with Varel et al\. \(tau \> 190 fs\) 'but exceed by approximately a factor of 2 those reported in' Stuart et al\. \(stuart\_1996\), and 'Starting from the same level at tau \>= 1 ps, F\_th in BBS becomes by a factor as large as ~3 smaller as compared to FS for tau \<= 20 fs' \(p\. 4077\)\. The 1\.5 J/cm^2 is not recorded as a dataset \(it spans 5\-10 fs; no single duration\)\. Damage site\: the front surface; criterion\: extrapolated onset of ablation \(zero ablated volume\); 50\-on\-1\. Fig\. 2\: V\_a against F at two durations in FS\. Figs\. 4\-5\: accumulated ablation depth against the number of pulses at \(5 \+\- 1\) J/cm^2 \(FS\) and \(6\.2 \+\- 0\.7\) J/cm^2 \(BBS\); per pulse d\_a\(220 fs\) = \(170 \+\- 35\) nm and d\_a\(5 fs\) = \(200 \+\- 10\) nm for FS, \(270 \+\- 65\) and \(115 \+\- 8\) nm for BBS, 'the uncertainties representing a 99% confidence interval' \(p\. 4078; no quantity\)\. Fig\. 1\: an SEM image of a 5\-fs crater at 6 F\_th\.

THE MODEL \(Eq\. 1, p\. 4078\), from Stuart et al\.\: dn/dt = alpha I\(t\) n\(t\) \+ sigma\_k I^k, 'alpha is the avalanche coefficient, and sigma\_k is the k\-photon absorption cross section with the smallest k satisfying k hbar omega \>= Delta'; n\_th = 1e21 cm^\-3 \('near the plasma critical density'\); chi^2 fits to the F\_th data for tau \>= 20 fs\: 'alpha\_fs = \(4 \+\- 0\.6\) cm^2/J and sigma\_6 = 6 x 10^\(8 \+\- 0\.9\) cm^\-3 ps^\-1 \(cm^2/TW\)^6 for FS and alpha\_bbs = \(1\.2 \+\- 0\.4\) cm^2/J and sigma\_3 = 7 x 10^\(17 \+\- 0\.5\) cm^\-3 ps^\-1 \(cm^2/TW\)^3 for BBS\. The errors given are related to a factor of 2 increase in the chi^2 merit function with respect to its minimum value' \(p\. 4078\)\. sigma\_k is a rate per unit volume \(the equation has no atom density\)\: 6 x 10^8 cm^\-3 ps^\-1 \(cm^2/TW\)^6 = 6\.0e\-52 cm^\-3 s^\-1 \(cm^2/W\)^6, and 7e17 cm^\-3 ps^\-1 \(cm^2/TW\)^3 = 7e\-7 cm^\-3 s^\-1 \(cm^2/W\)^3\. 'Whereas alpha\_fs is in reasonable agreement with the prediction of the kinetic theory of Stuart et al\., sigma\_6 and sigma\_3 are by some 6 and more than 2 orders of magnitude smaller as compared to the prediction of the Keldysh formula' \(the Keldysh rate evaluated with m\_eff = 0\.5 m for FS, Fischetti and DiMaria, and m\_eff = m for BBS; note 16\)\. Band gaps \(p\. 4077\)\: 'Delta\_fs ~ 9 eV and Delta\_bbs ~ 4 eV', no source\. Keldysh parameter gamma = omega sqrt\(2 m Delta\)/eE ~ 0\.5 for both materials at 5 fs \(p\. 4078\)\. Photoionised fraction at F = F\_th \(p\. 4079\)\: n\_p\(500 fs\)/n\_th ~ 4e\-8 and n\_p\(50 fs\)/n\_th ~ 1\.5e\-4 for FS, 0\.09 and 0\.35 for BBS, so 'optical breakdown in fused silica is dominated by the avalanche process down to the 10\-fs regime, whereas in BBS \.\.\. multiphoton ionization takes over for pulse durations below 100 fs'\. Note 18\: 'our data \.\.\. are inconsistent with a sqrt\(I\) scaling of the avalanche as proposed' by Du et al\. \(du\_1994\)\.

CHECK OF THE FIT \(computed 2026\-09\-27\)\: Eq\. \(1\) with the printed alpha and sigma\_k, n\_th = 1e21 cm^\-3, a Gaussian pulse of the stated FWHM and the peak \(on\-axis\) fluence gives F\_th\(FS\) = 1\.08, 1\.58, 2\.17, 3\.06, 3\.79, 5\.60 J/cm^2 at 5, 10, 20, 50, 100, 500 fs \(the text's ~1\.5 J/cm^2 at \<= 10 fs is reproduced\), F\_th\(BBS\) = 0\.58, 0\.87, 1\.41, 1\.97, 3\.72 J/cm^2 at 10, 20, 50, 100, 500 fs, and photoionised fractions at threshold of 4\.7e\-4 \(50 fs\) and 1\.8e\-7 \(500 fs\) for FS, 0\.40 and 0\.073 for BBS, against the printed 1\.5e\-4, 4e\-8, 0\.35 and 0\.09\: the same order of magnitude, not the printed digits \(the paper does not state its pulse shape; a sech^2 pulse gives 4\.0e\-4, 1\.45e\-7, 0\.39 and 0\.068\); well within sigma\_6's stated factor 10^\(\+\-0\.9\)\.

REPRODUCTIONS OF THIS PAPER\: schaffer\_2001 and jing\_2012 \(sigma\_6 = 6\.0e\-52 cm^\-3 s^\-1 \(cm^2/W\)^6, i\.e\. this paper's value converted\), couairon\_2005 \(sigma\_6 = 3 x 10^\(\-67 \+\- 0\.9\) s^\-1 cm^12/W^6 'per atom', which is not this value divided by an atom density\: 6\.0e\-52/2\.1e22 = 2\.9e\-74; a factor 1e7 apart\), jing\_2012 \(a threshold comparison\), dubietis\_2019\.

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

Datasets
--------

lenzner\_1998/avalanche\_SiO2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "avalanche_SiO2",
     "material": "SiO2",
     "table": "text",
     "quantity": "avalanche_coefficient",
     "unit": "cm2/J",
     "abscissa_unit": "nm",
     "use": "primary",
     "uncertainty": {
       "column": true,
       "level": "not stated"
     },
     "parameters": {
       "definition": "alpha in dn/dt = alpha I(t) n(t) + sigma_k I^k (Eq. 1), with n_th = 1e21 cm^-3 at threshold",
       "grade": "Corning 7940",
       "technique": "fit of Eq. (1) to the surface ablation thresholds against pulse duration (tau >= 20 fs)",
       "form": "bulk"
     }
   }

1 inline points in the source record.

p\. 4078\: 'alpha\_fs = \(4 \+\- 0\.6\) cm^2/J', fitted with sigma\_6 to the fused\-silica thresholds for tau \>= 20 fs \(Fig\. 3, 50 pulses per site\); the error is 'related to a factor of 2 increase in the chi^2 merit function'\. Corning 7940\. 'in reasonable agreement with the prediction of the kinetic theory of Stuart et al\.' \(stuart\_1996\)\.

lenzner\_1998/avalanche\_Corning\-7059
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "avalanche_Corning-7059",
     "material": "Corning-7059",
     "table": "text",
     "quantity": "avalanche_coefficient",
     "unit": "cm2/J",
     "abscissa_unit": "nm",
     "use": "primary",
     "uncertainty": {
       "column": true,
       "level": "not stated"
     },
     "parameters": {
       "definition": "alpha in dn/dt = alpha I(t) n(t) + sigma_k I^k (Eq. 1), with n_th = 1e21 cm^-3 at threshold",
       "grade": "Corning 7059",
       "technique": "fit of Eq. (1) to the surface ablation thresholds against pulse duration (tau >= 20 fs)",
       "form": "bulk"
     }
   }

1 inline points in the source record.

p\. 4078\: 'alpha\_bbs = \(1\.2 \+\- 0\.4\) cm^2/J', fitted with sigma\_3 to the BBS \(Corning 7059\) thresholds for tau \>= 20 fs; the error as for alpha\_fs\.

lenzner\_1998/sigma6\_SiO2
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "sigma6_SiO2",
     "material": "SiO2",
     "table": "text",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "cm-3 ps-1 (cm2/TW)6",
     "abscissa_unit": "nm",
     "use": "primary",
     "parameters": {
       "photons": 6,
       "grade": "Corning 7940",
       "technique": "fit of Eq. (1) to the surface ablation thresholds against pulse duration (tau >= 20 fs)",
       "form": "bulk"
     }
   }

1 inline points in the source record.

p\. 4078\: 'sigma\_6 = 6 x 10^\(8 \+\- 0\.9\) cm^\-3 ps^\-1 \(cm^2/TW\)^6 for FS'\: the six\-photon term of Eq\. \(1\), a generation rate per unit volume \(no atom density\), = 6\.0e\-52 cm^\-3 s^\-1 \(cm^2/W\)^6\. The uncertainty is in the exponent \(a factor 10^0\.9 = 7\.9 either way\), from a factor\-2 increase of chi^2; the format cannot express it as a sigma\. Six orders of magnitude below the Keldysh rate \(m\_eff = 0\.5 m\)\.

lenzner\_1998/sigma3\_Corning\-7059
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "sigma3_Corning-7059",
     "material": "Corning-7059",
     "table": "text",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "cm-3 ps-1 (cm2/TW)3",
     "abscissa_unit": "nm",
     "use": "primary",
     "parameters": {
       "photons": 3,
       "grade": "Corning 7059",
       "technique": "fit of Eq. (1) to the surface ablation thresholds against pulse duration (tau >= 20 fs)",
       "form": "bulk"
     }
   }

1 inline points in the source record.

p\. 4078\: 'sigma\_3 = 7 x 10^\(17 \+\- 0\.5\) cm^\-3 ps^\-1 \(cm^2/TW\)^3 for BBS'\: the three\-photon term of Eq\. \(1\), per unit volume, = 7e\-7 cm^\-3 s^\-1 \(cm^2/W\)^3\. Uncertainty a factor 10^0\.5 = 3\.2 either way \(not expressible as a sigma\)\. 'more than 2 orders of magnitude smaller' than the Keldysh rate \(m\_eff = m\)\.

lenzner\_1998/band\_gap\_SiO2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "band_gap_SiO2",
     "material": "SiO2",
     "table": "text",
     "quantity": "band_gap",
     "unit": "eV",
     "abscissa": "none",
     "use": "check",
     "parameters": {
       "kind": "not stated"
     }
   }

1 inline points in the source record.

p\. 4077\: 'The band\-gap energies of the two materials are Delta\_fs ~ 9 eV and Delta\_bbs ~ 4 eV', approximate, no source or method; it sets k = 6 in Eq\. \(1\) at 780 nm \(1\.59 eV photons\)\.

lenzner\_1998/band\_gap\_Corning\-7059
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "band_gap_Corning-7059",
     "material": "Corning-7059",
     "table": "text",
     "quantity": "band_gap",
     "unit": "eV",
     "abscissa": "none",
     "use": "check",
     "parameters": {
       "kind": "not stated"
     }
   }

1 inline points in the source record.

p\. 4077\: 'Delta\_bbs ~ 4 eV', approximate, no source or method; it sets k = 3 in Eq\. \(1\)\.

lenzner\_1998/mass\_SiO2
~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "mass_SiO2",
     "material": "SiO2",
     "table": "text",
     "quantity": "effective_mass",
     "unit": "m_e",
     "abscissa": "none",
     "use": "check",
     "reproduced_from": [
       "M. V. Fischetti and D. J. DiMaria, Solid State Electron. 31, 629 (1988)"
     ],
     "parameters": {
       "carrier": "electron",
       "assumed": true
     }
   }

1 inline points in the source record.

Note 16 \(p\. 4079\)\: 'For calculating the Keldysh MPI rate we have approximated the reduced electron\-hole mass with the effective electron mass m\_eff\. For FS we used m\_eff = 0\.5 m' \(Fischetti and DiMaria, printed 'Fischetti and D\. D\. DiMaria'\)\. A model input of the Keldysh comparison, not of Eq\. \(1\)'s fit\.

lenzner\_1998/mass\_Corning\-7059
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "mass_Corning-7059",
     "material": "Corning-7059",
     "table": "text",
     "quantity": "effective_mass",
     "unit": "m_e",
     "abscissa": "none",
     "use": "check",
     "parameters": {
       "carrier": "electron",
       "assumed": true
     }
   }

1 inline points in the source record.

Note 16 \(p\. 4079\)\: 'm\_eff = m was taken for BBS', the effective electron mass standing for the reduced mass in the Keldysh rate; an assumption without a source\.

Formulas
--------

lenzner\_1998/rate\_equation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "rate_equation",
     "equation": "Eq. (1), p. 4078",
     "printed": "\\frac{dn}{dt} = \\alpha I(t) n(t) + \\sigma_k I^k",
     "variable": "n: free-electron density (cm^-3); I(t): laser intensity (TW/cm^2 with sigma_k's unit); k: the smallest integer with k hbar omega >= Delta; breakdown when n reaches n_th = 1e21 cm^-3",
     "range": "fitted to tau >= 20 fs; for shorter pulses the perturbative MPI term 'tends to fail' as the Keldysh parameter approaches 1 (dotted curves of Fig. 3)",
     "coefficients": {
       "alpha_FS": "4 cm^2/J",
       "sigma_6_FS": "6 x 10^(8 +- 0.9) cm^-3 ps^-1 (cm^2/TW)^6",
       "alpha_BBS": "1.2 cm^2/J",
       "sigma_3_BBS": "7 x 10^(17 +- 0.5) cm^-3 ps^-1 (cm^2/TW)^3",
       "n_th": "1e21 cm^-3"
     },
     "check": "Solved for a Gaussian pulse of the stated FWHM with the peak fluence (computed 2026-09-27): F_th(FS) = 1.58 J/cm^2 at 10 fs (the text: ~1.5 J/cm^2 for tau <= 10 fs); photoionised fraction at threshold 4.7e-4 and 1.8e-7 at 50 and 500 fs (printed 1.5e-4 and 4e-8) for FS and 0.40 and 0.073 (printed 0.35 and 0.09) for BBS: order-of-magnitude agreement (source notes)."
   }

The rate equation of Stuart et al\. \(stuart\_1996, Ref\. 6\); no recombination term\.

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.