schaffer_2001#
C. B. Schaffer, A. Brodeur and E. Mazur, “Laser-induced breakdown and damage in bulk transparent materials induced by tightly focused femtosecond laser pulses”, Meas. Sci. Technol. 12, 1784 (2001)
Source record · Catalogue · Review register
DOI: 10.1088/0957-0233/12/11/305
Source kind: measurement
Recorded technique: Bulk (not surface) damage thresholds of SF11, Corning 0211, fused silica and CaF2 with 110-fs pulses at 400 and 800 nm, focused inside the sample by microscope objectives of NA 0.25-1.4 at the depth of best focus (ray-traced for spherical aberration). Energy thresholds by dark-field He:Ne scattering after 3000 pulses per site (Sec. 3.4), checked by DIC microscopy (Sec. 3.1) and transmission (Sec. 3.2). The threshold intensity I_th is fitted to the energy thresholds at several NA with Eq. (7), E_th = I_th tau lambda^2 / (pi NA^2 + I_th lambda^2 / P_cr), which includes weak self-focusing through P_cr (pp. 1789-1792).
Materials in structured entries: BK7, CaF2, Corning-0211, SF11, SiO2
Dataset quantities: avalanche_coefficient, band_gap, critical_power, damage_threshold_energy, damage_threshold_intensity, ionization_yield, multiphoton_ionization_cross_section
Contents: 28 datasets; 3 formulas.
Curation notes#
Journal pages 1784-1794 = PDF pages 2-12 (PDF page 1 is the publisher’s download cover). The PDF prints no usable DOI; identity from the library’s bibliographic search, title and authors checked against page 1784.
TABLE 1 (p. 1793, tbl_dbd271b58058, image checked): band gap, and at 400 and 800 nm, 110 fs, the fitted I_th and ‘the critical powers, P_cr, used in the fit to equation (7)’. SF11 at 400 nm is ‘—’: ‘We do not observe damage in SF11 using 400 nm laser pulses with laser energy up to several times the threshold for other materials’, attributed to two-photon absorption before the focus (p. 1792). Origin of the critical powers (p. 1792): ‘The lines represent fits to equation (7) using critical powers from the literature [34] or measured using the technique described in section 3.3. For Corning 0211 ... the critical power at 800 nm was left as a free parameter. Our fits yield P_cr = 1.5 MW, in good agreement with the 1.6 MW value for BK7 ... We used the known scaling of the critical power with laser wavelength (equation (4)) to obtain a value for Corning 0211 at 400 nm.’ Checked: 1.5 MW x (400/800)^2 = 0.375 MW, printed 0.4. Which of the other P_cr are from ref. 34 (Brodeur and Chin 1998) and which from Sec. 3.3 is not said. Sec. 3.3 (p. 1790): self-focusing thresholds in fused silica measured as white-light-continuum thresholds with a 0.20-m lens, ‘140 nJ at 400 nm and 580 nJ at 800 nm’ (energies; the conversion to power is not given, not extracted). The band gaps have no cited source; they are recorded as assumed model inputs. Eq. (4) is P_cr = 3.77 lambda^2 / (8 pi n0 n2) (Marburger, ref. 31).
TABLE 2 (p. 1793, tbl_ec8b126f418c) is scrambled in the library; transcribed from the image in the script, which checks the transcription against the scrambled characters. Keldysh parameter gamma (Eq. 1) and electron density N from Keldysh’s full formula at the Table 1 threshold intensities, avalanche excluded. gamma, not a curated quantity: 400 nm: Corning 0211 2.7, fused silica 2.5, CaF2 2.3 (SF11 ‘—’); 800 nm: SF11 1.2, Corning 0211 1.0, fused silica 1.2, CaF2 1.2. N is recorded as ionization_yield (electron density per cm3, check) against the threshold intensity. The caption cites ‘equation (37) in [30]’, but ref. 30 is Boyd’s Nonlinear Optics; the text (pp. 1785, 1793) says ‘equation (37) in [8]’, Keldysh: the caption’s reference number is wrong. Critical densities for comparison (caption, p. 1793): 6.9e21 cm^-3 (400 nm), 1.7e21 (800 nm); omega^2 eps0 m_e/e^2 gives 6.97e21 and 1.74e21 (checked 2026-09-26: the 400-nm value is truncated, not rounded). CHECK of gamma (2026-09-26): Eq. (1), gamma = (omega/e)[m c n eps0 E_g/I]^(1/2), with the Table 1 E_g and I_th, requires m n = 0.72-0.76 m_e for Corning 0211, silica and CaF2 at both wavelengths (0.83 m_e for SF11); m (‘the reduced mass’) and n are not given, so the absolute values cannot be checked, but the 400/800-nm ratios agree with 2 sqrt(I_800/I_400) to 1-2 % (2.70 v 2.73, 2.08 v 2.10, 1.92 v 1.89).
TEXT VALUES (datasets): DIC threshold in Corning 0211 at 1.4 NA, 6.6 nJ for single and multiple shots (p. 1790; the wavelength of Sec. 3.1 is not stated; at 0.25 NA the threshold falls with the number of shots, incubation); transmission-drop thresholds in fused silica at 0.65 NA, 25 nJ at 800 nm and 20 nJ at 400 nm (p. 1790, Fig. 5); the additional 800-nm dark-field thresholds in Corning 0211 at 1.0 NA (7.0 nJ) and 1.4 NA (5.0 nJ) used in the fit (p. 1792); the avalanche coefficients needed to reach the critical density with Keldysh photoionization, 1.6 (800 nm, fused silica), 2.0 (800 nm, CaF2), 1.3 (400 nm, fused silica), 1.4 cm2/J (400 nm, CaF2), and 4.5 cm2/J (fused silica, 800 nm) with Lenzner’s sigma_6 = 6.0 x 10^8 cm^-3 ps^-1 (cm2/TW)^6 (p. 1793). ‘Values range from 4.0 cm2 J-1 to 10 cm2 J-1’ for the literature fused-silica avalanche coefficient at 800 nm (p. 1793, no citation: not extracted).
FIGURES ONLY: Fig. 2, the Keldysh, multiphoton and tunnelling rates for fused silica (7.5 eV) at 800 nm against intensity; Fig. 5, transmission of 110-fs pulses at 400 and 800 nm through fused silica (0.65 NA) against energy, with the self-focusing energies marked; Fig. 7, the energy thresholds against NA (0.25-1.4) at 400 and 800 nm for the four materials with the Eq. (7) fits.
STATED UNCERTAINTY: none for Table 1. ‘The three different techniques used to determine the threshold for damage yield results that agree within experimental error’ (p. 1790). Polarization not stated. The surface/bulk comparison: ‘For fused silica we find that the intensity threshold for bulk material damage ... agrees very well with the thresholds for surface ablation found by other groups. The bulk threshold for CaF2, however, is larger than the previously measured surface threshold [21] by about a factor of two’ (pp. 1789-1790).
REPRODUCTIONS: Lenzner et al. 1998’s sigma_6 (reproduced dataset); Stuart et al.’s factor of two between 526 and 1053 nm (stuart_1996, text only). Reproduced by hamam_2018 (ref. 10, the NA limit 0.5) and jing_2012 (ref. 9).
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
schaffer_2001/band_gap_SF11#
{
"id": "band_gap_SF11",
"material": "SF11",
"file": "SF11/schaffer_2001__band_gap_SF11.csv",
"table": "tbl_dbd271b58058",
"quantity": "band_gap",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "assumed",
"assumed": true
}
}
Table 1 ‘Band gap (eV)’, the value used in the Keldysh calculations (Table 2); no source cited.
schaffer_2001/band_gap_Corning0211#
{
"id": "band_gap_Corning0211",
"material": "Corning-0211",
"file": "Corning-0211/schaffer_2001__band_gap_Corning0211.csv",
"table": "tbl_dbd271b58058",
"quantity": "band_gap",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "assumed",
"assumed": true
}
}
Table 1, as for SF11.
schaffer_2001/band_gap_SiO2#
{
"id": "band_gap_SiO2",
"material": "SiO2",
"file": "SiO2/schaffer_2001__band_gap_SiO2.csv",
"table": "tbl_dbd271b58058",
"quantity": "band_gap",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "assumed",
"assumed": true
}
}
Table 1, as for SF11; also ‘fused silica (7.5 eV band-gap)’ in Fig. 2 (p. 1785). Grade of the fused silica not stated.
schaffer_2001/band_gap_CaF2#
{
"id": "band_gap_CaF2",
"material": "CaF2",
"file": "CaF2/schaffer_2001__band_gap_CaF2.csv",
"table": "tbl_dbd271b58058",
"quantity": "band_gap",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "assumed",
"assumed": true
}
}
Table 1, as for SF11.
schaffer_2001/threshold_SF11#
{
"id": "threshold_SF11",
"material": "SF11",
"file": "SF11/schaffer_2001__threshold_SF11.csv",
"table": "tbl_dbd271b58058",
"quantity": "damage_threshold_intensity",
"unit": "W/cm2",
"scale": 10000000000000.0,
"abscissa_unit": "nm",
"use": "primary",
"parameters": {
"site": "bulk",
"criterion": "increase of dark-field He:Ne scattering after 3000 pulses on one site (Sec. 3.4)",
"shots": 3000,
"pulse_fs": 110,
"numerical_aperture": "0.25-1.4 (I_th fitted over several objectives with Eq. 7)"
}
}
Table 1 I_th, 800 nm only (no damage at 400 nm). Fitted with Eq. (7) to the energy thresholds at several NA (Fig. 7). Uncertainty not stated.
schaffer_2001/threshold_Corning0211#
{
"id": "threshold_Corning0211",
"material": "Corning-0211",
"file": "Corning-0211/schaffer_2001__threshold_Corning0211.csv",
"table": "tbl_dbd271b58058",
"quantity": "damage_threshold_intensity",
"unit": "W/cm2",
"scale": 10000000000000.0,
"abscissa_unit": "nm",
"use": "primary",
"parameters": {
"site": "bulk",
"criterion": "increase of dark-field He:Ne scattering after 3000 pulses on one site (Sec. 3.4)",
"shots": 3000,
"pulse_fs": 110,
"numerical_aperture": "0.25-1.4 (I_th fitted over several objectives with Eq. 7)"
}
}
Table 1 I_th at 400 and 800 nm; the 800-nm fit includes the 1.0 and 1.4 NA thresholds not shown in Fig. 7 (p. 1792).
schaffer_2001/threshold_SiO2#
{
"id": "threshold_SiO2",
"material": "SiO2",
"file": "SiO2/schaffer_2001__threshold_SiO2.csv",
"table": "tbl_dbd271b58058",
"quantity": "damage_threshold_intensity",
"unit": "W/cm2",
"scale": 10000000000000.0,
"abscissa_unit": "nm",
"use": "primary",
"parameters": {
"site": "bulk",
"criterion": "increase of dark-field He:Ne scattering after 3000 pulses on one site (Sec. 3.4)",
"shots": 3000,
"pulse_fs": 110,
"numerical_aperture": "0.25-1.4 (I_th fitted over several objectives with Eq. 7)"
}
}
Table 1 I_th at 400 and 800 nm.
schaffer_2001/threshold_CaF2#
{
"id": "threshold_CaF2",
"material": "CaF2",
"file": "CaF2/schaffer_2001__threshold_CaF2.csv",
"table": "tbl_dbd271b58058",
"quantity": "damage_threshold_intensity",
"unit": "W/cm2",
"scale": 10000000000000.0,
"abscissa_unit": "nm",
"use": "primary",
"parameters": {
"site": "bulk",
"criterion": "increase of dark-field He:Ne scattering after 3000 pulses on one site (Sec. 3.4)",
"shots": 3000,
"pulse_fs": 110,
"numerical_aperture": "0.25-1.4 (I_th fitted over several objectives with Eq. 7)"
}
}
Table 1 I_th at 400 and 800 nm (the 400-nm threshold is the larger, unlike the glasses, p. 1792).
schaffer_2001/critical_power_SF11#
{
"id": "critical_power_SF11",
"material": "SF11",
"file": "SF11/schaffer_2001__critical_power_SF11.csv",
"table": "tbl_dbd271b58058",
"quantity": "critical_power",
"unit": "MW",
"abscissa_unit": "nm",
"use": "check",
"reproduced_from": [
"A. Brodeur and S. L. Chin, Phys. Rev. Lett. 80, 4406 (1998) (ref. 34), or the paper's own continuum-threshold measurement (Sec. 3.3): not said which, per value"
],
"parameters": {
"definition": "the P_cr used in the fit of Eq. (7); Eq. (4) is P_cr = 3.77 lambda^2/(8 pi n0 n2)",
"assumed": true
}
}
Table 1 P_cr at 800 nm, an input of the Eq. (7) fit (source notes).
schaffer_2001/critical_power_fit_Corning0211#
{
"id": "critical_power_fit_Corning0211",
"material": "Corning-0211",
"file": "Corning-0211/schaffer_2001__critical_power_fit_Corning0211.csv",
"table": "tbl_dbd271b58058",
"quantity": "critical_power",
"unit": "MW",
"abscissa_unit": "nm",
"use": "primary",
"parameters": {
"definition": "free parameter of the fit of Eq. (7) to the 800-nm bulk damage energies at NA 0.25-1.4"
}
}
Table 1 P_cr at 800 nm: ‘the critical power at 800 nm was left as a free parameter. Our fits yield P_cr = 1.5 MW, in good agreement with the 1.6 MW value for BK7’ (p. 1792). Derived from the NA dependence of the damage energy, not from a self-focusing measurement.
schaffer_2001/critical_power_scaled_Corning0211#
{
"id": "critical_power_scaled_Corning0211",
"material": "Corning-0211",
"file": "Corning-0211/schaffer_2001__critical_power_scaled_Corning0211.csv",
"table": "tbl_dbd271b58058",
"quantity": "critical_power",
"unit": "MW",
"abscissa_unit": "nm",
"use": "check",
"parameters": {
"definition": "the 800-nm fit value scaled as lambda^2 (Eq. 4)",
"assumed": true
}
}
Table 1 P_cr at 400 nm, scaled from the 800-nm fit with Eq. (4): 1.5 MW x (400/800)^2 = 0.375 MW, printed 0.4 (checked 2026-09-26).
schaffer_2001/critical_power_SiO2#
{
"id": "critical_power_SiO2",
"material": "SiO2",
"file": "SiO2/schaffer_2001__critical_power_SiO2.csv",
"table": "tbl_dbd271b58058",
"quantity": "critical_power",
"unit": "MW",
"abscissa_unit": "nm",
"use": "check",
"reproduced_from": [
"A. Brodeur and S. L. Chin, Phys. Rev. Lett. 80, 4406 (1998) (ref. 34), or the paper's own continuum-threshold measurement (Sec. 3.3): not said which, per value"
],
"parameters": {
"definition": "the P_cr used in the fit of Eq. (7); Eq. (4) is P_cr = 3.77 lambda^2/(8 pi n0 n2)",
"assumed": true
}
}
Table 1 P_cr at 400 and 800 nm, inputs of the Eq. (7) fit. The Sec. 3.3 continuum thresholds in fused silica are 140 nJ (400 nm) and 580 nJ (800 nm), energies of 110-fs pulses (p. 1790).
schaffer_2001/critical_power_CaF2#
{
"id": "critical_power_CaF2",
"material": "CaF2",
"file": "CaF2/schaffer_2001__critical_power_CaF2.csv",
"table": "tbl_dbd271b58058",
"quantity": "critical_power",
"unit": "MW",
"abscissa_unit": "nm",
"use": "check",
"reproduced_from": [
"A. Brodeur and S. L. Chin, Phys. Rev. Lett. 80, 4406 (1998) (ref. 34), or the paper's own continuum-threshold measurement (Sec. 3.3): not said which, per value"
],
"parameters": {
"definition": "the P_cr used in the fit of Eq. (7); Eq. (4) is P_cr = 3.77 lambda^2/(8 pi n0 n2)",
"assumed": true
}
}
Table 1 P_cr at 400 and 800 nm, inputs of the Eq. (7) fit.
schaffer_2001/critical_power_BK7#
{
"id": "critical_power_BK7",
"material": "BK7",
"table": "text",
"quantity": "critical_power",
"unit": "MW",
"abscissa_unit": "nm",
"use": "reproduced",
"reproduced_from": [
"uncited in the sentence (p. 1792); the paper's literature critical powers are from ref. 34, A. Brodeur and S. L. Chin, Phys. Rev. Lett. 80, 4406 (1998)"
],
"parameters": {
"definition": "not stated (a literature value)"
}
}
1 inline points in the source record.
‘in good agreement with the 1.6 MW value for BK7, a glass with composition and band-gap similar to Corning 0211’ (p. 1792), in the discussion of the 800-nm fit. Whether the historical BK7 or N-BK7 is meant is not said; BK7 as written.
schaffer_2001/keldysh_density_SF11_800#
{
"id": "keldysh_density_SF11_800",
"material": "SF11",
"file": "SF11/schaffer_2001__keldysh_density_SF11_800.csv",
"table": "tbl_ec8b126f418c",
"quantity": "ionization_yield",
"unit": "cm-3",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"parameters": {
"wavelength_nm": 800,
"pulse_fs": 110,
"polarization": "not stated"
}
}
Table 2 (transcribed from the image; library table scrambled): conduction-electron density from Keldysh’s full photoionization rate (his Eq. 37), avalanche excluded, at the Table 1 threshold intensity (the abscissa, taken from Table 1 by the script); gamma = 1.2. A calculation, not a measurement.
schaffer_2001/keldysh_density_Corning0211_400#
{
"id": "keldysh_density_Corning0211_400",
"material": "Corning-0211",
"file": "Corning-0211/schaffer_2001__keldysh_density_Corning0211_400.csv",
"table": "tbl_ec8b126f418c",
"quantity": "ionization_yield",
"unit": "cm-3",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"parameters": {
"wavelength_nm": 400,
"pulse_fs": 110,
"polarization": "not stated"
}
}
Table 2, as for SF11; gamma = 2.7. The text counts Corning 0211 at 400 nm among the cases where ‘photoionization alone is sufficient to produce a critical density plasma’ (p. 1793), but 5.2e21 cm^-3 is below the 400-nm critical density the caption gives, 6.9e21 cm^-3 (SF11 and Corning 0211 at 800 nm, 3.0e21 and 2.7e21, do exceed 1.7e21).
schaffer_2001/keldysh_density_Corning0211_800#
{
"id": "keldysh_density_Corning0211_800",
"material": "Corning-0211",
"file": "Corning-0211/schaffer_2001__keldysh_density_Corning0211_800.csv",
"table": "tbl_ec8b126f418c",
"quantity": "ionization_yield",
"unit": "cm-3",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"parameters": {
"wavelength_nm": 800,
"pulse_fs": 110,
"polarization": "not stated"
}
}
Table 2, as for SF11; gamma = 1.0.
schaffer_2001/keldysh_density_SiO2_400#
{
"id": "keldysh_density_SiO2_400",
"material": "SiO2",
"file": "SiO2/schaffer_2001__keldysh_density_SiO2_400.csv",
"table": "tbl_ec8b126f418c",
"quantity": "ionization_yield",
"unit": "cm-3",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"parameters": {
"wavelength_nm": 400,
"pulse_fs": 110,
"polarization": "not stated"
}
}
Table 2, as for SF11; gamma = 2.5.
schaffer_2001/keldysh_density_SiO2_800#
{
"id": "keldysh_density_SiO2_800",
"material": "SiO2",
"file": "SiO2/schaffer_2001__keldysh_density_SiO2_800.csv",
"table": "tbl_ec8b126f418c",
"quantity": "ionization_yield",
"unit": "cm-3",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"parameters": {
"wavelength_nm": 800,
"pulse_fs": 110,
"polarization": "not stated"
}
}
Table 2, as for SF11; gamma = 1.2.
schaffer_2001/keldysh_density_CaF2_400#
{
"id": "keldysh_density_CaF2_400",
"material": "CaF2",
"file": "CaF2/schaffer_2001__keldysh_density_CaF2_400.csv",
"table": "tbl_ec8b126f418c",
"quantity": "ionization_yield",
"unit": "cm-3",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"parameters": {
"wavelength_nm": 400,
"pulse_fs": 110,
"polarization": "not stated"
}
}
Table 2, as for SF11; gamma = 2.3.
schaffer_2001/keldysh_density_CaF2_800#
{
"id": "keldysh_density_CaF2_800",
"material": "CaF2",
"file": "CaF2/schaffer_2001__keldysh_density_CaF2_800.csv",
"table": "tbl_ec8b126f418c",
"quantity": "ionization_yield",
"unit": "cm-3",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"parameters": {
"wavelength_nm": 800,
"pulse_fs": 110,
"polarization": "not stated"
}
}
Table 2, as for SF11; gamma = 1.2.
schaffer_2001/energy_dic_Corning0211#
{
"id": "energy_dic_Corning0211",
"material": "Corning-0211",
"table": "text",
"quantity": "damage_threshold_energy",
"unit": "nJ",
"abscissa": "duration",
"abscissa_unit": "fs",
"use": "primary",
"parameters": {
"site": "bulk",
"criterion": "any structure visible by DIC microscopy (1.4 NA oil-immersion objective)",
"shots": "single and multiple (the same threshold at 1.4 NA)",
"numerical_aperture": 1.4
}
}
1 inline points in the source record.
Sec. 3.1, p. 1790: ‘No material change is observed for a laser pulse energy below 6 nJ ... At 6.6 nJ we observe both single-shot and multiple-shot damage and so we conclude that the energy threshold for single-shot and multiple-shot damage in Corning 0211 at 1.4 NA is 6.6 nJ.’ 1-kHz laser; the wavelength is not stated in Sec. 3.1.
schaffer_2001/energy_darkfield_Corning0211#
{
"id": "energy_darkfield_Corning0211",
"material": "Corning-0211",
"table": "text",
"quantity": "damage_threshold_energy",
"unit": "nJ",
"abscissa_unit": "nm",
"columns": {
"numerical_aperture": "1"
},
"use": "primary",
"parameters": {
"site": "bulk",
"criterion": "increase of dark-field He:Ne scattering after 3000 pulses on one site (Sec. 3.4)",
"shots": 3000,
"pulse_fs": 110
}
}
2 inline points in the source record.
p. 1792: ‘For Corning 0211, there are additional 800 nm energy thresholds at 1.0 NA (7.0 nJ) and 1.4 NA (5.0 nJ) that are not shown in figure 7, but were used in the fitting procedure.’ The other energy thresholds (Fig. 7) are figure-only.
schaffer_2001/energy_transmission_SiO2#
{
"id": "energy_transmission_SiO2",
"material": "SiO2",
"table": "text",
"quantity": "damage_threshold_energy",
"unit": "nJ",
"abscissa_unit": "nm",
"use": "primary",
"parameters": {
"site": "bulk",
"criterion": "onset of the drop in transmission of the pulse (Sec. 3.2)",
"pulse_fs": 110,
"numerical_aperture": 0.65
}
}
2 inline points in the source record.
Sec. 3.2, p. 1790 (Fig. 5): ‘The energy at which the transmission begins to drop, corresponding to the energy threshold for optical breakdown and damage, is 25 nJ for 800 nm pulses. Using 400 nm pulses, we find an energy threshold of 20 nJ.’
schaffer_2001/avalanche_SiO2#
{
"id": "avalanche_SiO2",
"material": "SiO2",
"table": "text",
"quantity": "avalanche_coefficient",
"unit": "cm2/J",
"abscissa_unit": "nm",
"use": "check",
"parameters": {
"definition": "alpha in dN/dt = alpha I N + P(I) (Eq. 8) that reaches the critical density at the Table 1 I_th, with Keldysh's P(I)"
}
}
2 inline points in the source record.
p. 1793: derived, not measured: ‘Equation (8) allows us to calculate the avalanche coefficients that are necessary to produce a critical density plasma ... To determine the photoionization rate, P(I), we use the full Keldysh formula. This procedure yields ... 1.6 cm2 J-1 (800 nm, fused silica) ... 1.3 cm2 J-1 (400 nm, fused silica)’.
schaffer_2001/avalanche_CaF2#
{
"id": "avalanche_CaF2",
"material": "CaF2",
"table": "text",
"quantity": "avalanche_coefficient",
"unit": "cm2/J",
"abscissa_unit": "nm",
"use": "check",
"parameters": {
"definition": "alpha in dN/dt = alpha I N + P(I) (Eq. 8) that reaches the critical density at the Table 1 I_th, with Keldysh's P(I)"
}
}
2 inline points in the source record.
p. 1793, as for fused silica: 2.0 cm2/J (800 nm), 1.4 cm2/J (400 nm).
schaffer_2001/avalanche_lenzner_SiO2#
{
"id": "avalanche_lenzner_SiO2",
"material": "SiO2",
"table": "text",
"quantity": "avalanche_coefficient",
"unit": "cm2/J",
"abscissa_unit": "nm",
"use": "check",
"parameters": {
"definition": "alpha in dN/dt = alpha I N + P(I) (Eq. 8) that reaches the critical density at the Table 1 I_th, with P(I) = sigma_6 I^6 (Lenzner et al. 1998)"
}
}
1 inline points in the source record.
p. 1793: ‘Using the multiphoton ionization rate suggested by Lenzner, sigma_6 = 6.0 x 10^8 cm^-3 ps^-1 (cm2 TW^-1)^6, we find from our measured threshold intensity an avalanche ionization coefficient of 4.5 cm2 J-1’.
schaffer_2001/sigma6_lenzner_SiO2#
{
"id": "sigma6_lenzner_SiO2",
"material": "SiO2",
"table": "text",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm-3 ps-1 (cm2/TW)6",
"abscissa_unit": "nm",
"use": "reproduced",
"reproduced_from": [
"lenzner_1998"
],
"parameters": {
"photons": 6
}
}
1 inline points in the source record.
p. 1793, ‘the multiphoton ionization rate suggested by Lenzner’. A rate per volume, P(I) = sigma_6 I^6, not per atom: 6.0e8 cm^-3 ps^-1 (cm2/TW)^6 = 6.0e-52 cm^-3 s^-1 (cm2/W)^6, the value jing_2012 also quotes for Lenzner. The wavelength of Lenzner’s fit is not stated here. The original, lenzner_1998 (p. 4078), prints 6 x 10^(8 +- 0.9) cm^-3 ps^-1 (cm2/TW)^6 for fused silica at 780 nm: agrees (the exponent’s +-0.9 is not quoted here).
Formulas#
schaffer_2001/keldysh_parameter#
{
"id": "keldysh_parameter",
"equation": "Eq. (1), p. 1785",
"printed": "gamma = (omega/e) [m c n eps0 E_g / I]^(1/2)",
"variable": "omega: laser angular frequency; I the intensity at the focus; m, e 'the reduced mass and charge of the electron'; n the refractive index; E_g the band gap",
"coefficients": {
"transition": "gamma ~ 1.5 (multiphoton above, tunnelling below)"
},
"check": "Table 2's gamma with Table 1's E_g and I_th require m n = 0.72-0.76 m_e (0.83 for SF11); the 400/800-nm ratios agree with 2 sqrt(I_800/I_400) to 1-2 % (checked 2026-09-26). m and n are not given."
}
‘When the Keldysh parameter is larger (smaller) than about 1.5, photoionization is a multiphoton (tunnelling) process’ (p. 1785).
schaffer_2001/energy_threshold#
{
"id": "energy_threshold",
"equation": "Eqs. (4), (5), (7), pp. 1788-1789",
"printed": "P_cr = 3.77 lambda^2 / (8 pi n0 n2) (4); I_sf = I / (1 - P/P_cr) (5); E_th = I_th tau lambda^2 / (pi NA^2 + I_th lambda^2 / P_cr) (7)",
"variable": "lambda: wavelength; tau the pulse duration; NA the objective's numerical aperture; I_th the threshold intensity",
"coefficients": {
"factor": "3.77"
},
"check": "Not reproducible: the energy thresholds are in Fig. 7 only."
}
Eq. (5) is Soileau et al.’s weak self-focusing correction (ref. 29), valid below about P_cr/4.
schaffer_2001/rate_equation#
{
"id": "rate_equation",
"equation": "Eqs. (2), (8), pp. 1786, 1792",
"printed": "dN/dt = eta N (2), eta = alpha I (Stuart); dN/dt = alpha I N + P(I) (8), P(I) = sigma_k I^k",
"variable": "N: conduction-electron density; I the intensity; alpha the avalanche coefficient; P(I) the photoionization rate",
"coefficients": {
"alpha": "fitted per material (datasets avalanche_*)"
},
"reproduced_from": [
"stuart_1996"
],
"check": "The alpha values are derived by the paper from the measured I_th; nothing further printed to reproduce."
}
The multiphoton form P(I) = sigma_k I^k with k the smallest integer with k hbar omega >= E_g (p. 1785).
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 review register for the checks and decisions.