Abstract

The temperature dependence of the laser-induced damage threshold on optical coatings was studied in detail for laser pulses from 123 K to 473 K at different temperature. For pulses longer than a few picoseconds, the laser-induced damage threshold of coated substrates increased with decreasing temperature. This temperature dependence was reversed for pulses shorter than a few picoseconds. We describe the physics models to explain the observed scaling. The electron avalanche is essential to explain the differences in the temperature dependence.

© 2013 Optical Society of America

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  1. A. A. Manenkov, “Fundamental mechanisms of laser-induced damage in optical materials: understanding after a 40-years research,” Proc. SPIE7132, 713202 (2008).
    [CrossRef]
  2. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter53(4), 1749–1761 (1996).
    [CrossRef] [PubMed]
  3. B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett.74(12), 2248–2251 (1995).
    [CrossRef] [PubMed]
  4. M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B71(11), 115109 (2005).
    [CrossRef]
  5. 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(11), 1784–1794 (2001).
    [CrossRef]
  6. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP20, 1307–1314 (1965).
  7. A. Dyan, F. Enguehard, S. Lallich, H. Piombini, and G. Duchateau, “Scaling laws in laser-induced potassium dihydrogen phosphate crystal damage by nanosecond pulses at 3ω,” J. Opt. Soc. Am. B25(6), 1087–1095 (2008).
    [CrossRef]
  8. A. A. Manekov, “New results on avalanche ionization as a laser damage mechanism in transparent solids,” Natl. Bur. Stand. Spec. Publ.541, 455–561 (1978).
  9. L. D. Merkle and D. Kitriotis, “Temperature dependence of laser-induced bulk damage in SiO2 and borosilicate glass,” Phys. Rev. B Condens. Matter38(2), 1473–1482 (1988).
    [CrossRef] [PubMed]
  10. K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, J. Kawanaka, and R. Yasuhara, “Temperature dependence of laser-induced damage threshold in silica glasses,” J. Phys. Conf. Ser.244(3), 032023 (2010).
    [CrossRef]
  11. K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and K. A. Tanaka, “Laser-induced damage thresholds of optical coatings at different temperature,” Proc. SPIE8190, 81900A (2011).
    [CrossRef]
  12. K. Mikami, S. Motokoshi, M. Fujita, T. Somekawa, T. Jitsuno, and K. A. Tanaka, “Temperature dependence of laser induced plasma thresholds and periodic structures by nanosecond infrared laser for copper, iron, and chrome,” Appl. Phys. Express5(6), 062701 (2012).
    [CrossRef]
  13. K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and M. Murakami, “Temperature dependence of nonlinear optical phenomena of silica glasses,” Proc. SPIE7842, 7842X1 (2011).
  14. H. Fröhlich and B. V. Paranjape, “Dielectric breakdown in solids,” Proc. Phys. Soc. B69(1), 21–32 (1956).
    [CrossRef]
  15. M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
    [CrossRef]
  16. W. Hu, Y. C. Shin, and G. King, “Effect of air breakdown with a focusing lens on ultra-short laser ablation,” Appl. Phys. Lett.99(23), 234104 (2011).
    [CrossRef]
  17. J. R. Bettis and R. A. House, II, andA. H. Guenther, “Spot size and pulse duration dependence of laser-induced damage,” Nat. Bur. Stand. Spec. Publ.462, 338–345 (1976).
  18. X. Mao, S. S. Mao, and R. E. Russo, “Imaging femtosecond laser-induced electronic excitation in glass,” Appl. Phys. Lett.82(5), 697–699 (2003).
    [CrossRef]
  19. X. Mao, S.-bor Wen, and R. E. Russo, “Time resolved laser-induced plasma dynamics,” Appl. Surf. Sci.253(15), 6316–6321 (2007).
    [CrossRef]
  20. I. T. Godmanis, A. N. Trukhin, and K. Hubner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi116(1), 279–287 (1983).
    [CrossRef]
  21. J. M. Worlock and P. A. Fleury, “Electric field dependence of optical-phonon frequencies,” Phys. Rev. Lett.19(20), 1176–1179 (1967).
    [CrossRef]
  22. X. Shang, R. Zhang, and P. Ma, “Analysis of avalanche mechanisms in short-pulses laser-induced damage,” Opt. Laser Technol.42(1), 243–246 (2010).
    [CrossRef]
  23. S. Roy and D. Chakravorty, “Electrical conduction in composites of nanosized iron particles and oxide glasses,” J. Mater. Res.9(9), 2314–2318 (1994).
    [CrossRef]
  24. F. Seitz, “On the theory of electron multiplication in crystals,” Phys. Rev.76(9), 1376–1393 (1949).
    [CrossRef]
  25. P. Bräunlich, A. Schmid, and P. Kelly, “Contributions of multi photon absorption to laser-induced damage intrinsic damage in NaCl,” Appl. Phys. Lett.26(4), 150–153 (1975).
    [CrossRef]
  26. W. J. Meath and E. A. Power, “On the importance of permanent moments in multiphoton absorption using perturbation theory,” J. Phys. B17(5), 763–781 (1984).
    [CrossRef]
  27. T. E. Tsai, D. L. Griscom, and E. J. Friebele, “Mechanism of intrinsic Si E’-center photogeneration in high-purity silica,” Phys. Rev. Lett.61(4), 444–446 (1988).
    [CrossRef] [PubMed]
  28. R. Weeks, “Paramagnetic spectra of E’2 center in crystalline quartz,” Phys. Rev.130(2), 570–576 (1963).
    [CrossRef]
  29. K. Arai, H. Imai, H. Hosono, Y. Abe, and H. Imagawa, “Two-photon processes in defect formation by excimer lasers in synthetic silica glass,” Appl. Phys. Lett.53(20), 1891–1893 (1988).
    [CrossRef]
  30. H. Hanafusa, Y. Hibino, and F. Yamamoto, “Formation mechanism of drawing-induced E’ center in silica optical fibers,” J. Appl. Phys.58(3), 1356–1361 (1985).
    [CrossRef]
  31. W. D. Compton and G. W. Arnold, “Radiation effects in fused silica and α-Al2O3,” Discuss. Faraday Soc.31, 130–139 (1961).
    [CrossRef]
  32. L. Pereira, P. Barquinha, E. Fortunato, and R. Martins, “Influence of the oxygen/argon ratio on the properties of sputtered hafnium oxide,” Mater. Sci. Eng. B118(1–3), 210–213 (2005).
    [CrossRef]

2012

K. Mikami, S. Motokoshi, M. Fujita, T. Somekawa, T. Jitsuno, and K. A. Tanaka, “Temperature dependence of laser induced plasma thresholds and periodic structures by nanosecond infrared laser for copper, iron, and chrome,” Appl. Phys. Express5(6), 062701 (2012).
[CrossRef]

2011

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and K. A. Tanaka, “Laser-induced damage thresholds of optical coatings at different temperature,” Proc. SPIE8190, 81900A (2011).
[CrossRef]

W. Hu, Y. C. Shin, and G. King, “Effect of air breakdown with a focusing lens on ultra-short laser ablation,” Appl. Phys. Lett.99(23), 234104 (2011).
[CrossRef]

2010

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, J. Kawanaka, and R. Yasuhara, “Temperature dependence of laser-induced damage threshold in silica glasses,” J. Phys. Conf. Ser.244(3), 032023 (2010).
[CrossRef]

X. Shang, R. Zhang, and P. Ma, “Analysis of avalanche mechanisms in short-pulses laser-induced damage,” Opt. Laser Technol.42(1), 243–246 (2010).
[CrossRef]

2008

A. Dyan, F. Enguehard, S. Lallich, H. Piombini, and G. Duchateau, “Scaling laws in laser-induced potassium dihydrogen phosphate crystal damage by nanosecond pulses at 3ω,” J. Opt. Soc. Am. B25(6), 1087–1095 (2008).
[CrossRef]

A. A. Manenkov, “Fundamental mechanisms of laser-induced damage in optical materials: understanding after a 40-years research,” Proc. SPIE7132, 713202 (2008).
[CrossRef]

2007

X. Mao, S.-bor Wen, and R. E. Russo, “Time resolved laser-induced plasma dynamics,” Appl. Surf. Sci.253(15), 6316–6321 (2007).
[CrossRef]

2005

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B71(11), 115109 (2005).
[CrossRef]

L. Pereira, P. Barquinha, E. Fortunato, and R. Martins, “Influence of the oxygen/argon ratio on the properties of sputtered hafnium oxide,” Mater. Sci. Eng. B118(1–3), 210–213 (2005).
[CrossRef]

2003

X. Mao, S. S. Mao, and R. E. Russo, “Imaging femtosecond laser-induced electronic excitation in glass,” Appl. Phys. Lett.82(5), 697–699 (2003).
[CrossRef]

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(11), 1784–1794 (2001).
[CrossRef]

1996

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter53(4), 1749–1761 (1996).
[CrossRef] [PubMed]

1995

B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett.74(12), 2248–2251 (1995).
[CrossRef] [PubMed]

1994

S. Roy and D. Chakravorty, “Electrical conduction in composites of nanosized iron particles and oxide glasses,” J. Mater. Res.9(9), 2314–2318 (1994).
[CrossRef]

1988

T. E. Tsai, D. L. Griscom, and E. J. Friebele, “Mechanism of intrinsic Si E’-center photogeneration in high-purity silica,” Phys. Rev. Lett.61(4), 444–446 (1988).
[CrossRef] [PubMed]

K. Arai, H. Imai, H. Hosono, Y. Abe, and H. Imagawa, “Two-photon processes in defect formation by excimer lasers in synthetic silica glass,” Appl. Phys. Lett.53(20), 1891–1893 (1988).
[CrossRef]

L. D. Merkle and D. Kitriotis, “Temperature dependence of laser-induced bulk damage in SiO2 and borosilicate glass,” Phys. Rev. B Condens. Matter38(2), 1473–1482 (1988).
[CrossRef] [PubMed]

1985

H. Hanafusa, Y. Hibino, and F. Yamamoto, “Formation mechanism of drawing-induced E’ center in silica optical fibers,” J. Appl. Phys.58(3), 1356–1361 (1985).
[CrossRef]

1984

W. J. Meath and E. A. Power, “On the importance of permanent moments in multiphoton absorption using perturbation theory,” J. Phys. B17(5), 763–781 (1984).
[CrossRef]

1983

I. T. Godmanis, A. N. Trukhin, and K. Hubner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi116(1), 279–287 (1983).
[CrossRef]

1981

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

1978

A. A. Manekov, “New results on avalanche ionization as a laser damage mechanism in transparent solids,” Natl. Bur. Stand. Spec. Publ.541, 455–561 (1978).

1976

J. R. Bettis and R. A. House, II, andA. H. Guenther, “Spot size and pulse duration dependence of laser-induced damage,” Nat. Bur. Stand. Spec. Publ.462, 338–345 (1976).

J. R. Bettis and R. A. House, II, andA. H. Guenther, “Spot size and pulse duration dependence of laser-induced damage,” Nat. Bur. Stand. Spec. Publ.462, 338–345 (1976).

1975

P. Bräunlich, A. Schmid, and P. Kelly, “Contributions of multi photon absorption to laser-induced damage intrinsic damage in NaCl,” Appl. Phys. Lett.26(4), 150–153 (1975).
[CrossRef]

1967

J. M. Worlock and P. A. Fleury, “Electric field dependence of optical-phonon frequencies,” Phys. Rev. Lett.19(20), 1176–1179 (1967).
[CrossRef]

1965

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP20, 1307–1314 (1965).

1963

R. Weeks, “Paramagnetic spectra of E’2 center in crystalline quartz,” Phys. Rev.130(2), 570–576 (1963).
[CrossRef]

1961

W. D. Compton and G. W. Arnold, “Radiation effects in fused silica and α-Al2O3,” Discuss. Faraday Soc.31, 130–139 (1961).
[CrossRef]

1956

H. Fröhlich and B. V. Paranjape, “Dielectric breakdown in solids,” Proc. Phys. Soc. B69(1), 21–32 (1956).
[CrossRef]

1949

F. Seitz, “On the theory of electron multiplication in crystals,” Phys. Rev.76(9), 1376–1393 (1949).
[CrossRef]

Abe, Y.

K. Arai, H. Imai, H. Hosono, Y. Abe, and H. Imagawa, “Two-photon processes in defect formation by excimer lasers in synthetic silica glass,” Appl. Phys. Lett.53(20), 1891–1893 (1988).
[CrossRef]

Arai, K.

K. Arai, H. Imai, H. Hosono, Y. Abe, and H. Imagawa, “Two-photon processes in defect formation by excimer lasers in synthetic silica glass,” Appl. Phys. Lett.53(20), 1891–1893 (1988).
[CrossRef]

Arnold, G. W.

W. D. Compton and G. W. Arnold, “Radiation effects in fused silica and α-Al2O3,” Discuss. Faraday Soc.31, 130–139 (1961).
[CrossRef]

Barquinha, P.

L. Pereira, P. Barquinha, E. Fortunato, and R. Martins, “Influence of the oxygen/argon ratio on the properties of sputtered hafnium oxide,” Mater. Sci. Eng. B118(1–3), 210–213 (2005).
[CrossRef]

Bettis, J. R.

J. R. Bettis and R. A. House, II, andA. H. Guenther, “Spot size and pulse duration dependence of laser-induced damage,” Nat. Bur. Stand. Spec. Publ.462, 338–345 (1976).

Bräunlich, P.

P. Bräunlich, A. Schmid, and P. Kelly, “Contributions of multi photon absorption to laser-induced damage intrinsic damage in NaCl,” Appl. Phys. Lett.26(4), 150–153 (1975).
[CrossRef]

Brodeur, A.

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(11), 1784–1794 (2001).
[CrossRef]

Chakravorty, D.

S. Roy and D. Chakravorty, “Electrical conduction in composites of nanosized iron particles and oxide glasses,” J. Mater. Res.9(9), 2314–2318 (1994).
[CrossRef]

Compton, W. D.

W. D. Compton and G. W. Arnold, “Radiation effects in fused silica and α-Al2O3,” Discuss. Faraday Soc.31, 130–139 (1961).
[CrossRef]

Duchateau, G.

Dyan, A.

Enguehard, F.

Feit, M. D.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter53(4), 1749–1761 (1996).
[CrossRef] [PubMed]

B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett.74(12), 2248–2251 (1995).
[CrossRef] [PubMed]

Fleury, P. A.

J. M. Worlock and P. A. Fleury, “Electric field dependence of optical-phonon frequencies,” Phys. Rev. Lett.19(20), 1176–1179 (1967).
[CrossRef]

Fortunato, E.

L. Pereira, P. Barquinha, E. Fortunato, and R. Martins, “Influence of the oxygen/argon ratio on the properties of sputtered hafnium oxide,” Mater. Sci. Eng. B118(1–3), 210–213 (2005).
[CrossRef]

Friebele, E. J.

T. E. Tsai, D. L. Griscom, and E. J. Friebele, “Mechanism of intrinsic Si E’-center photogeneration in high-purity silica,” Phys. Rev. Lett.61(4), 444–446 (1988).
[CrossRef] [PubMed]

Fröhlich, H.

H. Fröhlich and B. V. Paranjape, “Dielectric breakdown in solids,” Proc. Phys. Soc. B69(1), 21–32 (1956).
[CrossRef]

Fujita, M.

K. Mikami, S. Motokoshi, M. Fujita, T. Somekawa, T. Jitsuno, and K. A. Tanaka, “Temperature dependence of laser induced plasma thresholds and periodic structures by nanosecond infrared laser for copper, iron, and chrome,” Appl. Phys. Express5(6), 062701 (2012).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and K. A. Tanaka, “Laser-induced damage thresholds of optical coatings at different temperature,” Proc. SPIE8190, 81900A (2011).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, J. Kawanaka, and R. Yasuhara, “Temperature dependence of laser-induced damage threshold in silica glasses,” J. Phys. Conf. Ser.244(3), 032023 (2010).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and M. Murakami, “Temperature dependence of nonlinear optical phenomena of silica glasses,” Proc. SPIE7842, 7842X1 (2011).

Godmanis, I. T.

I. T. Godmanis, A. N. Trukhin, and K. Hubner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi116(1), 279–287 (1983).
[CrossRef]

Griscom, D. L.

T. E. Tsai, D. L. Griscom, and E. J. Friebele, “Mechanism of intrinsic Si E’-center photogeneration in high-purity silica,” Phys. Rev. Lett.61(4), 444–446 (1988).
[CrossRef] [PubMed]

Guenther, A. H.

J. R. Bettis and R. A. House, II, andA. H. Guenther, “Spot size and pulse duration dependence of laser-induced damage,” Nat. Bur. Stand. Spec. Publ.462, 338–345 (1976).

Hanafusa, H.

H. Hanafusa, Y. Hibino, and F. Yamamoto, “Formation mechanism of drawing-induced E’ center in silica optical fibers,” J. Appl. Phys.58(3), 1356–1361 (1985).
[CrossRef]

Herman, S.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter53(4), 1749–1761 (1996).
[CrossRef] [PubMed]

Hibino, Y.

H. Hanafusa, Y. Hibino, and F. Yamamoto, “Formation mechanism of drawing-induced E’ center in silica optical fibers,” J. Appl. Phys.58(3), 1356–1361 (1985).
[CrossRef]

Holstein, T.

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

Hosono, H.

K. Arai, H. Imai, H. Hosono, Y. Abe, and H. Imagawa, “Two-photon processes in defect formation by excimer lasers in synthetic silica glass,” Appl. Phys. Lett.53(20), 1891–1893 (1988).
[CrossRef]

House, R. A.

J. R. Bettis and R. A. House, II, andA. H. Guenther, “Spot size and pulse duration dependence of laser-induced damage,” Nat. Bur. Stand. Spec. Publ.462, 338–345 (1976).

Hu, W.

W. Hu, Y. C. Shin, and G. King, “Effect of air breakdown with a focusing lens on ultra-short laser ablation,” Appl. Phys. Lett.99(23), 234104 (2011).
[CrossRef]

Hubner, K.

I. T. Godmanis, A. N. Trukhin, and K. Hubner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi116(1), 279–287 (1983).
[CrossRef]

Imagawa, H.

K. Arai, H. Imai, H. Hosono, Y. Abe, and H. Imagawa, “Two-photon processes in defect formation by excimer lasers in synthetic silica glass,” Appl. Phys. Lett.53(20), 1891–1893 (1988).
[CrossRef]

Imai, H.

K. Arai, H. Imai, H. Hosono, Y. Abe, and H. Imagawa, “Two-photon processes in defect formation by excimer lasers in synthetic silica glass,” Appl. Phys. Lett.53(20), 1891–1893 (1988).
[CrossRef]

Jitsuno, T.

K. Mikami, S. Motokoshi, M. Fujita, T. Somekawa, T. Jitsuno, and K. A. Tanaka, “Temperature dependence of laser induced plasma thresholds and periodic structures by nanosecond infrared laser for copper, iron, and chrome,” Appl. Phys. Express5(6), 062701 (2012).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and K. A. Tanaka, “Laser-induced damage thresholds of optical coatings at different temperature,” Proc. SPIE8190, 81900A (2011).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, J. Kawanaka, and R. Yasuhara, “Temperature dependence of laser-induced damage threshold in silica glasses,” J. Phys. Conf. Ser.244(3), 032023 (2010).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and M. Murakami, “Temperature dependence of nonlinear optical phenomena of silica glasses,” Proc. SPIE7842, 7842X1 (2011).

Kawanaka, J.

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, J. Kawanaka, and R. Yasuhara, “Temperature dependence of laser-induced damage threshold in silica glasses,” J. Phys. Conf. Ser.244(3), 032023 (2010).
[CrossRef]

Keldysh, L. V.

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP20, 1307–1314 (1965).

Kelly, P.

P. Bräunlich, A. Schmid, and P. Kelly, “Contributions of multi photon absorption to laser-induced damage intrinsic damage in NaCl,” Appl. Phys. Lett.26(4), 150–153 (1975).
[CrossRef]

King, D. F.

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

King, G.

W. Hu, Y. C. Shin, and G. King, “Effect of air breakdown with a focusing lens on ultra-short laser ablation,” Appl. Phys. Lett.99(23), 234104 (2011).
[CrossRef]

Kitriotis, D.

L. D. Merkle and D. Kitriotis, “Temperature dependence of laser-induced bulk damage in SiO2 and borosilicate glass,” Phys. Rev. B Condens. Matter38(2), 1473–1482 (1988).
[CrossRef] [PubMed]

Lallich, S.

Liu, J.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B71(11), 115109 (2005).
[CrossRef]

Loh, E.

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

Ma, P.

X. Shang, R. Zhang, and P. Ma, “Analysis of avalanche mechanisms in short-pulses laser-induced damage,” Opt. Laser Technol.42(1), 243–246 (2010).
[CrossRef]

Manekov, A. A.

A. A. Manekov, “New results on avalanche ionization as a laser damage mechanism in transparent solids,” Natl. Bur. Stand. Spec. Publ.541, 455–561 (1978).

Manenkov, A. A.

A. A. Manenkov, “Fundamental mechanisms of laser-induced damage in optical materials: understanding after a 40-years research,” Proc. SPIE7132, 713202 (2008).
[CrossRef]

Mao, S. S.

X. Mao, S. S. Mao, and R. E. Russo, “Imaging femtosecond laser-induced electronic excitation in glass,” Appl. Phys. Lett.82(5), 697–699 (2003).
[CrossRef]

Mao, X.

X. Mao, S.-bor Wen, and R. E. Russo, “Time resolved laser-induced plasma dynamics,” Appl. Surf. Sci.253(15), 6316–6321 (2007).
[CrossRef]

X. Mao, S. S. Mao, and R. E. Russo, “Imaging femtosecond laser-induced electronic excitation in glass,” Appl. Phys. Lett.82(5), 697–699 (2003).
[CrossRef]

Maradudin, A. A.

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

Martins, R.

L. Pereira, P. Barquinha, E. Fortunato, and R. Martins, “Influence of the oxygen/argon ratio on the properties of sputtered hafnium oxide,” Mater. Sci. Eng. B118(1–3), 210–213 (2005).
[CrossRef]

Mazur, E.

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(11), 1784–1794 (2001).
[CrossRef]

Meath, W. J.

W. J. Meath and E. A. Power, “On the importance of permanent moments in multiphoton absorption using perturbation theory,” J. Phys. B17(5), 763–781 (1984).
[CrossRef]

Merkle, L. D.

L. D. Merkle and D. Kitriotis, “Temperature dependence of laser-induced bulk damage in SiO2 and borosilicate glass,” Phys. Rev. B Condens. Matter38(2), 1473–1482 (1988).
[CrossRef] [PubMed]

Mero, M.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B71(11), 115109 (2005).
[CrossRef]

Mikami, K.

K. Mikami, S. Motokoshi, M. Fujita, T. Somekawa, T. Jitsuno, and K. A. Tanaka, “Temperature dependence of laser induced plasma thresholds and periodic structures by nanosecond infrared laser for copper, iron, and chrome,” Appl. Phys. Express5(6), 062701 (2012).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and K. A. Tanaka, “Laser-induced damage thresholds of optical coatings at different temperature,” Proc. SPIE8190, 81900A (2011).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, J. Kawanaka, and R. Yasuhara, “Temperature dependence of laser-induced damage threshold in silica glasses,” J. Phys. Conf. Ser.244(3), 032023 (2010).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and M. Murakami, “Temperature dependence of nonlinear optical phenomena of silica glasses,” Proc. SPIE7842, 7842X1 (2011).

Mills, D. L.

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

Motokoshi, S.

K. Mikami, S. Motokoshi, M. Fujita, T. Somekawa, T. Jitsuno, and K. A. Tanaka, “Temperature dependence of laser induced plasma thresholds and periodic structures by nanosecond infrared laser for copper, iron, and chrome,” Appl. Phys. Express5(6), 062701 (2012).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and K. A. Tanaka, “Laser-induced damage thresholds of optical coatings at different temperature,” Proc. SPIE8190, 81900A (2011).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, J. Kawanaka, and R. Yasuhara, “Temperature dependence of laser-induced damage threshold in silica glasses,” J. Phys. Conf. Ser.244(3), 032023 (2010).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and M. Murakami, “Temperature dependence of nonlinear optical phenomena of silica glasses,” Proc. SPIE7842, 7842X1 (2011).

Murakami, M.

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and M. Murakami, “Temperature dependence of nonlinear optical phenomena of silica glasses,” Proc. SPIE7842, 7842X1 (2011).

Paranjape, B. V.

H. Fröhlich and B. V. Paranjape, “Dielectric breakdown in solids,” Proc. Phys. Soc. B69(1), 21–32 (1956).
[CrossRef]

Pereira, L.

L. Pereira, P. Barquinha, E. Fortunato, and R. Martins, “Influence of the oxygen/argon ratio on the properties of sputtered hafnium oxide,” Mater. Sci. Eng. B118(1–3), 210–213 (2005).
[CrossRef]

Perry, M. D.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter53(4), 1749–1761 (1996).
[CrossRef] [PubMed]

B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett.74(12), 2248–2251 (1995).
[CrossRef] [PubMed]

Piombini, H.

Power, E. A.

W. J. Meath and E. A. Power, “On the importance of permanent moments in multiphoton absorption using perturbation theory,” J. Phys. B17(5), 763–781 (1984).
[CrossRef]

Ristau, D.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B71(11), 115109 (2005).
[CrossRef]

Roy, S.

S. Roy and D. Chakravorty, “Electrical conduction in composites of nanosized iron particles and oxide glasses,” J. Mater. Res.9(9), 2314–2318 (1994).
[CrossRef]

Rubenchik, A. M.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter53(4), 1749–1761 (1996).
[CrossRef] [PubMed]

B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett.74(12), 2248–2251 (1995).
[CrossRef] [PubMed]

Rudolph, W.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B71(11), 115109 (2005).
[CrossRef]

Russo, R. E.

X. Mao, S.-bor Wen, and R. E. Russo, “Time resolved laser-induced plasma dynamics,” Appl. Surf. Sci.253(15), 6316–6321 (2007).
[CrossRef]

X. Mao, S. S. Mao, and R. E. Russo, “Imaging femtosecond laser-induced electronic excitation in glass,” Appl. Phys. Lett.82(5), 697–699 (2003).
[CrossRef]

Schaffer, C. B.

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(11), 1784–1794 (2001).
[CrossRef]

Schmid, A.

P. Bräunlich, A. Schmid, and P. Kelly, “Contributions of multi photon absorption to laser-induced damage intrinsic damage in NaCl,” Appl. Phys. Lett.26(4), 150–153 (1975).
[CrossRef]

Seitz, F.

F. Seitz, “On the theory of electron multiplication in crystals,” Phys. Rev.76(9), 1376–1393 (1949).
[CrossRef]

Sham, L. J.

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

Shang, X.

X. Shang, R. Zhang, and P. Ma, “Analysis of avalanche mechanisms in short-pulses laser-induced damage,” Opt. Laser Technol.42(1), 243–246 (2010).
[CrossRef]

Shin, Y. C.

W. Hu, Y. C. Shin, and G. King, “Effect of air breakdown with a focusing lens on ultra-short laser ablation,” Appl. Phys. Lett.99(23), 234104 (2011).
[CrossRef]

Shore, B. W.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter53(4), 1749–1761 (1996).
[CrossRef] [PubMed]

B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett.74(12), 2248–2251 (1995).
[CrossRef] [PubMed]

Somekawa, T.

K. Mikami, S. Motokoshi, M. Fujita, T. Somekawa, T. Jitsuno, and K. A. Tanaka, “Temperature dependence of laser induced plasma thresholds and periodic structures by nanosecond infrared laser for copper, iron, and chrome,” Appl. Phys. Express5(6), 062701 (2012).
[CrossRef]

Sparks, M.

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

Starke, K.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B71(11), 115109 (2005).
[CrossRef]

Stuart, B. C.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter53(4), 1749–1761 (1996).
[CrossRef] [PubMed]

B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett.74(12), 2248–2251 (1995).
[CrossRef] [PubMed]

Tanaka, K. A.

K. Mikami, S. Motokoshi, M. Fujita, T. Somekawa, T. Jitsuno, and K. A. Tanaka, “Temperature dependence of laser induced plasma thresholds and periodic structures by nanosecond infrared laser for copper, iron, and chrome,” Appl. Phys. Express5(6), 062701 (2012).
[CrossRef]

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and K. A. Tanaka, “Laser-induced damage thresholds of optical coatings at different temperature,” Proc. SPIE8190, 81900A (2011).
[CrossRef]

Trukhin, A. N.

I. T. Godmanis, A. N. Trukhin, and K. Hubner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi116(1), 279–287 (1983).
[CrossRef]

Tsai, T. E.

T. E. Tsai, D. L. Griscom, and E. J. Friebele, “Mechanism of intrinsic Si E’-center photogeneration in high-purity silica,” Phys. Rev. Lett.61(4), 444–446 (1988).
[CrossRef] [PubMed]

Warren, R.

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

Weeks, R.

R. Weeks, “Paramagnetic spectra of E’2 center in crystalline quartz,” Phys. Rev.130(2), 570–576 (1963).
[CrossRef]

Wen, S.-bor

X. Mao, S.-bor Wen, and R. E. Russo, “Time resolved laser-induced plasma dynamics,” Appl. Surf. Sci.253(15), 6316–6321 (2007).
[CrossRef]

Worlock, J. M.

J. M. Worlock and P. A. Fleury, “Electric field dependence of optical-phonon frequencies,” Phys. Rev. Lett.19(20), 1176–1179 (1967).
[CrossRef]

Yamamoto, F.

H. Hanafusa, Y. Hibino, and F. Yamamoto, “Formation mechanism of drawing-induced E’ center in silica optical fibers,” J. Appl. Phys.58(3), 1356–1361 (1985).
[CrossRef]

Yasuhara, R.

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, J. Kawanaka, and R. Yasuhara, “Temperature dependence of laser-induced damage threshold in silica glasses,” J. Phys. Conf. Ser.244(3), 032023 (2010).
[CrossRef]

Zhang, R.

X. Shang, R. Zhang, and P. Ma, “Analysis of avalanche mechanisms in short-pulses laser-induced damage,” Opt. Laser Technol.42(1), 243–246 (2010).
[CrossRef]

Appl. Phys. Express

K. Mikami, S. Motokoshi, M. Fujita, T. Somekawa, T. Jitsuno, and K. A. Tanaka, “Temperature dependence of laser induced plasma thresholds and periodic structures by nanosecond infrared laser for copper, iron, and chrome,” Appl. Phys. Express5(6), 062701 (2012).
[CrossRef]

Appl. Phys. Lett.

W. Hu, Y. C. Shin, and G. King, “Effect of air breakdown with a focusing lens on ultra-short laser ablation,” Appl. Phys. Lett.99(23), 234104 (2011).
[CrossRef]

X. Mao, S. S. Mao, and R. E. Russo, “Imaging femtosecond laser-induced electronic excitation in glass,” Appl. Phys. Lett.82(5), 697–699 (2003).
[CrossRef]

P. Bräunlich, A. Schmid, and P. Kelly, “Contributions of multi photon absorption to laser-induced damage intrinsic damage in NaCl,” Appl. Phys. Lett.26(4), 150–153 (1975).
[CrossRef]

K. Arai, H. Imai, H. Hosono, Y. Abe, and H. Imagawa, “Two-photon processes in defect formation by excimer lasers in synthetic silica glass,” Appl. Phys. Lett.53(20), 1891–1893 (1988).
[CrossRef]

Appl. Surf. Sci.

X. Mao, S.-bor Wen, and R. E. Russo, “Time resolved laser-induced plasma dynamics,” Appl. Surf. Sci.253(15), 6316–6321 (2007).
[CrossRef]

Discuss. Faraday Soc.

W. D. Compton and G. W. Arnold, “Radiation effects in fused silica and α-Al2O3,” Discuss. Faraday Soc.31, 130–139 (1961).
[CrossRef]

J. Appl. Phys.

H. Hanafusa, Y. Hibino, and F. Yamamoto, “Formation mechanism of drawing-induced E’ center in silica optical fibers,” J. Appl. Phys.58(3), 1356–1361 (1985).
[CrossRef]

J. Mater. Res.

S. Roy and D. Chakravorty, “Electrical conduction in composites of nanosized iron particles and oxide glasses,” J. Mater. Res.9(9), 2314–2318 (1994).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. B

W. J. Meath and E. A. Power, “On the importance of permanent moments in multiphoton absorption using perturbation theory,” J. Phys. B17(5), 763–781 (1984).
[CrossRef]

J. Phys. Conf. Ser.

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, J. Kawanaka, and R. Yasuhara, “Temperature dependence of laser-induced damage threshold in silica glasses,” J. Phys. Conf. Ser.244(3), 032023 (2010).
[CrossRef]

Mater. Sci. Eng. B

L. Pereira, P. Barquinha, E. Fortunato, and R. Martins, “Influence of the oxygen/argon ratio on the properties of sputtered hafnium oxide,” Mater. Sci. Eng. B118(1–3), 210–213 (2005).
[CrossRef]

Meas. Sci. Technol.

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(11), 1784–1794 (2001).
[CrossRef]

Nat. Bur. Stand. Spec. Publ.

J. R. Bettis and R. A. House, II, andA. H. Guenther, “Spot size and pulse duration dependence of laser-induced damage,” Nat. Bur. Stand. Spec. Publ.462, 338–345 (1976).

Natl. Bur. Stand. Spec. Publ.

A. A. Manekov, “New results on avalanche ionization as a laser damage mechanism in transparent solids,” Natl. Bur. Stand. Spec. Publ.541, 455–561 (1978).

Opt. Laser Technol.

X. Shang, R. Zhang, and P. Ma, “Analysis of avalanche mechanisms in short-pulses laser-induced damage,” Opt. Laser Technol.42(1), 243–246 (2010).
[CrossRef]

Phys. Rev.

F. Seitz, “On the theory of electron multiplication in crystals,” Phys. Rev.76(9), 1376–1393 (1949).
[CrossRef]

R. Weeks, “Paramagnetic spectra of E’2 center in crystalline quartz,” Phys. Rev.130(2), 570–576 (1963).
[CrossRef]

Phys. Rev. B

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B71(11), 115109 (2005).
[CrossRef]

M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L. J. Sham, E. Loh, and D. F. King, “Theory of electron-avalanche breakdown in solids,” Phys. Rev. B24(6), 3519–3536 (1981).
[CrossRef]

Phys. Rev. B Condens. Matter

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter53(4), 1749–1761 (1996).
[CrossRef] [PubMed]

L. D. Merkle and D. Kitriotis, “Temperature dependence of laser-induced bulk damage in SiO2 and borosilicate glass,” Phys. Rev. B Condens. Matter38(2), 1473–1482 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett.

B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett.74(12), 2248–2251 (1995).
[CrossRef] [PubMed]

T. E. Tsai, D. L. Griscom, and E. J. Friebele, “Mechanism of intrinsic Si E’-center photogeneration in high-purity silica,” Phys. Rev. Lett.61(4), 444–446 (1988).
[CrossRef] [PubMed]

J. M. Worlock and P. A. Fleury, “Electric field dependence of optical-phonon frequencies,” Phys. Rev. Lett.19(20), 1176–1179 (1967).
[CrossRef]

Phys. Status Solidi

I. T. Godmanis, A. N. Trukhin, and K. Hubner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi116(1), 279–287 (1983).
[CrossRef]

Proc. Phys. Soc. B

H. Fröhlich and B. V. Paranjape, “Dielectric breakdown in solids,” Proc. Phys. Soc. B69(1), 21–32 (1956).
[CrossRef]

Proc. SPIE

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and K. A. Tanaka, “Laser-induced damage thresholds of optical coatings at different temperature,” Proc. SPIE8190, 81900A (2011).
[CrossRef]

A. A. Manenkov, “Fundamental mechanisms of laser-induced damage in optical materials: understanding after a 40-years research,” Proc. SPIE7132, 713202 (2008).
[CrossRef]

Sov. Phys. JETP

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP20, 1307–1314 (1965).

Other

K. Mikami, S. Motokoshi, M. Fujita, T. Jitsuno, and M. Murakami, “Temperature dependence of nonlinear optical phenomena of silica glasses,” Proc. SPIE7842, 7842X1 (2011).

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Figures (6)

Fig. 1
Fig. 1

Experimental layout.

Fig. 2
Fig. 2

Optical spectrum at different temperature for (a) SiO2 and (b) HfO2 coating.

Fig. 3
Fig. 3

Temperature dependence of LIDT under (a) 100-fs, (b) 2-ps, (c) 200-ps, and (d) 4-ns laser pulses. Photos for SiO2 coatings at room temperature under (e) 100-fs, (f) 2-ps, (g) 200-ps, and (h) 4-ns laser pulses. Doted circle shows focal spot.

Fig. 4
Fig. 4

Calculated temperature dependence of (a) critical density and (b) multiple rate for silica glass.

Fig. 5
Fig. 5

(a) Measured and calculated temperature dependence of LIDT of SiO2 coating. 5(b) Measured and calculated temperature dependence of LIDT under 2-ps pulses for SiO2 and HfO2 coatings.

Fig. 6
Fig. 6

Calculation result for (a) SiO2 and (b) HfO2 material at 2-ps pulse testing with different corrected value γ in Eq. (9).

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

n cr β ( T ) τVn
n cr( T ) = P 4π e *4 ω ( T ) τ s( ε ) ( 2 N W( T ) +1 )
N W( T ) = { exp( ω ( T ) k 0 T ) } 1
β ( T ) =A E 2 E g ρ ( T )
β ( T ) τV= 2 τ/ τ s( ε )
n ( T ) = n p + n i( T ) + n m
n i( T ) = GL E g V = ( e 2 /m ) τ s( ε ) E 2 ω ( T ) E g V
n m = σ 4 N S ( cε E 2 2ν ) 4 ( π ln2 ) 0.5 τ 4
n cr( T ) γ{ β ( T ) τV( n p + n i( T ) ) }+( 1γ ) n m

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