Abstract

By coupling statistics and heat transfer, we investigate numerically laser-induced crystal damage by multi-gigawatt nanosecond pulses. Our model is based on the heating of nanometric absorbing defects that may cooperate when sufficiently aggregated. In that configuration, they induce locally a strong increase of temperature that may lead to a subsequent damage. This approach allows to predict cluster size distribution and damage probabilities as a function of the laser fluence. By studying the influence of the pulse duration onto the laser-induced damage threshold, we have established scaling laws that link the critical laser fluence to its pulse duration τ. In particular, this approach provides an explanation to the deviation from the standard τ 1/2 scaling law that has been recently observed in laser-induced damage experiments with KH2PO4 (KDP) crystals [J.J. Adams et al, Proc. of SPIE 5991, 5991R-1 (2005)]. In the present paper, despite the 3D problem is tackled, we focus our attention on a 1D modeling of thermal diffusion that is shown to provide more reliable predictions than the 3D one. These results indicate that absorbers involved in KDP damage may be associated with a collection of planar defects. First general comparisons with some experimental facts have been performed.

© 2007 Optical Society of America

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  1. J. J. De Yoreo, A. K. Burnham, and P. K. Whitman, “Developing KH2PO4 and KD2PO4 crystals for the world’s most powerful laser,” Int. Mater. Rev. 47, 113–152 (2002)
    [CrossRef]
  2. C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004)
    [CrossRef] [PubMed]
  3. J. J. Adamset al, “Wavelength and pulselength dependence of laser conditioning and bulk damage in doubler-cut KH2PO4,” in Laser-induced damage in optical materials, Proc. SPIE 5991, 59911R-1 (2005)
  4. C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in KH2-xD2xPO4 crystals,” Appl. Phys. Lett. 89, 131901 (2006)
    [CrossRef]
  5. C. S. Liu, C. J. Hou, N. Kiousis, S. G. Demos, and H. B. Radousky, “Electronic structure calculations of an oxygen vacancy in KH2PO4,” Phys. Rev. B 72, 134110 (2005)
    [CrossRef]
  6. C. S. Liu, N. Kioussis, S. G. Demos, and H. B. Radousky, “Electron-or hole-assisted reactions of H defects in hydrogen-bonded KDP,” Phys. Rev. Lett. 91, 015505 (2003)
    [CrossRef] [PubMed]
  7. S. G. Demos, M. Staggs, J. J. De Yoreo, and H. B. Radousky, “Imaging of laser-induced reactions of individual defect nanoclusters,” Opt. Lett. 26, 1975–1977 (2001)
    [CrossRef]
  8. S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys 85, 3988–3992 (1999)
    [CrossRef]
  9. J. Dijon, T. Poiroux, and C. Desrumaux, “Nano absorbing centers: a key point in the laser damage of thin film,” in Laser-induced damage in optical materials, Proc. SPIE 2966315 (1997)
  10. C. W. Carr, H. B. Radousky, and S. G. Demos, “Wavelength dependence of laser-induced damage: determining the damage initiation mechanisms,” Phys. Rev. Lett. 91, 127402 (2003)
    [CrossRef] [PubMed]
  11. M. D. Feit, A. M. Rubenchik, M. R. Kozlowski, F. Y. Génin, S. Schwartz, and L. M. Sheehan“Extrapolation of damage test data to predict performance of large area NIF optics at 355nm,” in Laser-induced damage in optical materials, Proc. SPIE 3578, 226–234 (1998)
  12. R. W. Hopper and D. R. Uhlmann, “Mechanism of inclusion damage in laser glass,” J. Appl. Phys. 41, 4023–4037 (1970)
    [CrossRef]
  13. M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulse-length scaling and laser conditioning,” in Laser-induced damage in optical materials, Proc. SPIE 5273, 74–82, (2004)
  14. A. Dyan, F. Enguehard, S. Lallich, H. Piombini, H. Mathis, and G. Duchateau, “Laser-induced damage in KH2PO4 and D2xKH2(1-x)PO4: influence of the laser probed volume and optimization of the conditioning through a revisited thermal approach,” submitted to Optics Communications
  15. J. O. Porteus and S. C. Seitel, “Absolute onset of optical surface damage using distributed defect ensembles,” Appl. Opt. 23, 3796 (1984)
    [CrossRef] [PubMed]
  16. R. M. O’Connell, “Onset threshold analysis of defect-driven surface and bulk laser damage,” Appl. Opt. 314143–4153 (1992)
    [CrossRef] [PubMed]
  17. R. H. Picard, D. Milam, and R. A. Bradbury, “Statistical analysis of defect-caused laser damage in thin films” Appl. Opt. 16, 1563–1571 (1977)
    [CrossRef] [PubMed]
  18. L. Gallais, J. Y. Natoli, and C. Amra, “Statistical study of single and multiple pulse laser-induced damage in glasses,” Opt. Express 10, 1465–1474 (2002)
    [PubMed]
  19. J. Y. Natoli, L. Gallais, H. Akhouayri, and C. Amra, “Laser-induced damage of materials in bulk, thin film, and liquid forms,” Appl. Opt. 41, 3156–3166 (2002)
    [CrossRef] [PubMed]
  20. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford Science Publications, Second Edition (1959)
  21. K. E. Montgomery and F. P. Milanovich, “High-laser-damage-threshold potassium dihydrogen phosphate crystals,” J. Appl. Phys. 683979-82 (1990)
    [CrossRef]
  22. P. K. Bandyopadhyay and L. D. Merkle, “Laser-induced damage in quartz: a study of the influence of impurities and defects,” J. Appl. Phys. 631392–1398 (1988)
    [CrossRef]
  23. F. Bonneauel al, “Study of UV laser interaction with gold nanoparticles embedded in silica,” Appl. Phy. B 75, 803–815 (2002)
    [CrossRef]
  24. N. W. Ashcroft and N. D. Mermin, Solid state physics, Brooks Cole, first edition, 1976
  25. C. Kittel, Introduction to solid state physics, Wiley, seventh edition, 1995
  26. F. Y. Génin, A. Salleo, T. V. Pistor, and L. L. Chase, “Role of light intensification by cracks in optical breakdown on surfaces,” J. Opt. Soc. Am. A 182607-16 (2001)
    [CrossRef]
  27. I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
    [CrossRef]
  28. Z. L. Xia, Z. X. Fan, and J. D. Shao, “A new theory for evaluating the number density of inclusions in films,” Appl. Surf. Sci. 252 (23), 8235–8238 (2006)
    [CrossRef]
  29. S. Papernov and A. W. Schmid, “Correlations between embedded single gold nanoparticles in SiO2 thin film and nanoscale crater formation induced by pulsed-laser radiation,” J. Appl. Phys. 92, 5720–5728 (2002)
    [CrossRef]
  30. M. Runkel, J. DeYoreo, W. Sell, and D. Milam, “Laser conditioning study of KDP on the optical sciences laser using large area beams,” in Laser-induced damage in optical materials, Proc. SPIE 3244, 51 (1998)
  31. E. S. Bliss, “Pulse duration dependence of laser damage mechanism” Opto-electronics 3, 99 (1971)
    [CrossRef]
  32. R. M. Wood, Laser-induced damage of optical materials, Institute Of Physics publishing series in optics ans optoelectronics, Bristol and Philadelphia (2003)
  33. 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 53, 1749–1761 (1996)
    [CrossRef]
  34. A. Dyan and G. Duchateau, CEA, Centre d’Etudes du Ripault, BP 16, 37260 Monts, France, are preparing a manuscript to be called “Laser-induced damage by a nanosecond pulse: a method coupling heat transfer, Mie’s theory and microscopic processes”
  35. H. Goldenberg and C.J. Tranter, “Heat flow in an infinite medium heated by a sphere,” Br. J. Appl. Phys. 3, 296–298 (1952)
    [CrossRef]
  36. M. Sparks, “Theory of laser heating of solids: metals,” J. Appl. Phys. 47, 837–849 (1976)
    [CrossRef]
  37. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Alan Jeffrey Editor, Fifth Edition (1994)

2006 (2)

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in KH2-xD2xPO4 crystals,” Appl. Phys. Lett. 89, 131901 (2006)
[CrossRef]

Z. L. Xia, Z. X. Fan, and J. D. Shao, “A new theory for evaluating the number density of inclusions in films,” Appl. Surf. Sci. 252 (23), 8235–8238 (2006)
[CrossRef]

2005 (2)

J. J. Adamset al, “Wavelength and pulselength dependence of laser conditioning and bulk damage in doubler-cut KH2PO4,” in Laser-induced damage in optical materials, Proc. SPIE 5991, 59911R-1 (2005)

C. S. Liu, C. J. Hou, N. Kiousis, S. G. Demos, and H. B. Radousky, “Electronic structure calculations of an oxygen vacancy in KH2PO4,” Phys. Rev. B 72, 134110 (2005)
[CrossRef]

2004 (2)

C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004)
[CrossRef] [PubMed]

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulse-length scaling and laser conditioning,” in Laser-induced damage in optical materials, Proc. SPIE 5273, 74–82, (2004)

2003 (2)

C. W. Carr, H. B. Radousky, and S. G. Demos, “Wavelength dependence of laser-induced damage: determining the damage initiation mechanisms,” Phys. Rev. Lett. 91, 127402 (2003)
[CrossRef] [PubMed]

C. S. Liu, N. Kioussis, S. G. Demos, and H. B. Radousky, “Electron-or hole-assisted reactions of H defects in hydrogen-bonded KDP,” Phys. Rev. Lett. 91, 015505 (2003)
[CrossRef] [PubMed]

2002 (5)

J. J. De Yoreo, A. K. Burnham, and P. K. Whitman, “Developing KH2PO4 and KD2PO4 crystals for the world’s most powerful laser,” Int. Mater. Rev. 47, 113–152 (2002)
[CrossRef]

L. Gallais, J. Y. Natoli, and C. Amra, “Statistical study of single and multiple pulse laser-induced damage in glasses,” Opt. Express 10, 1465–1474 (2002)
[PubMed]

J. Y. Natoli, L. Gallais, H. Akhouayri, and C. Amra, “Laser-induced damage of materials in bulk, thin film, and liquid forms,” Appl. Opt. 41, 3156–3166 (2002)
[CrossRef] [PubMed]

S. Papernov and A. W. Schmid, “Correlations between embedded single gold nanoparticles in SiO2 thin film and nanoscale crater formation induced by pulsed-laser radiation,” J. Appl. Phys. 92, 5720–5728 (2002)
[CrossRef]

F. Bonneauel al, “Study of UV laser interaction with gold nanoparticles embedded in silica,” Appl. Phy. B 75, 803–815 (2002)
[CrossRef]

2001 (2)

1999 (1)

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys 85, 3988–3992 (1999)
[CrossRef]

1998 (2)

M. D. Feit, A. M. Rubenchik, M. R. Kozlowski, F. Y. Génin, S. Schwartz, and L. M. Sheehan“Extrapolation of damage test data to predict performance of large area NIF optics at 355nm,” in Laser-induced damage in optical materials, Proc. SPIE 3578, 226–234 (1998)

M. Runkel, J. DeYoreo, W. Sell, and D. Milam, “Laser conditioning study of KDP on the optical sciences laser using large area beams,” in Laser-induced damage in optical materials, Proc. SPIE 3244, 51 (1998)

1997 (1)

J. Dijon, T. Poiroux, and C. Desrumaux, “Nano absorbing centers: a key point in the laser damage of thin film,” in Laser-induced damage in optical materials, Proc. SPIE 2966315 (1997)

1996 (1)

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 53, 1749–1761 (1996)
[CrossRef]

1992 (1)

1990 (1)

K. E. Montgomery and F. P. Milanovich, “High-laser-damage-threshold potassium dihydrogen phosphate crystals,” J. Appl. Phys. 683979-82 (1990)
[CrossRef]

1988 (1)

P. K. Bandyopadhyay and L. D. Merkle, “Laser-induced damage in quartz: a study of the influence of impurities and defects,” J. Appl. Phys. 631392–1398 (1988)
[CrossRef]

1984 (1)

1977 (1)

1976 (1)

M. Sparks, “Theory of laser heating of solids: metals,” J. Appl. Phys. 47, 837–849 (1976)
[CrossRef]

1971 (1)

E. S. Bliss, “Pulse duration dependence of laser damage mechanism” Opto-electronics 3, 99 (1971)
[CrossRef]

1970 (1)

R. W. Hopper and D. R. Uhlmann, “Mechanism of inclusion damage in laser glass,” J. Appl. Phys. 41, 4023–4037 (1970)
[CrossRef]

1952 (1)

H. Goldenberg and C.J. Tranter, “Heat flow in an infinite medium heated by a sphere,” Br. J. Appl. Phys. 3, 296–298 (1952)
[CrossRef]

Adams, J. J.

J. J. Adamset al, “Wavelength and pulselength dependence of laser conditioning and bulk damage in doubler-cut KH2PO4,” in Laser-induced damage in optical materials, Proc. SPIE 5991, 59911R-1 (2005)

Akhouayri, H.

Amra, C.

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid state physics, Brooks Cole, first edition, 1976

Bandyopadhyay, P. K.

P. K. Bandyopadhyay and L. D. Merkle, “Laser-induced damage in quartz: a study of the influence of impurities and defects,” J. Appl. Phys. 631392–1398 (1988)
[CrossRef]

Blinc, R.

I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
[CrossRef]

Bliss, E. S.

E. S. Bliss, “Pulse duration dependence of laser damage mechanism” Opto-electronics 3, 99 (1971)
[CrossRef]

Bonneau, F.

F. Bonneauel al, “Study of UV laser interaction with gold nanoparticles embedded in silica,” Appl. Phy. B 75, 803–815 (2002)
[CrossRef]

Bradbury, R. A.

Burnham, A. K.

J. J. De Yoreo, A. K. Burnham, and P. K. Whitman, “Developing KH2PO4 and KD2PO4 crystals for the world’s most powerful laser,” Int. Mater. Rev. 47, 113–152 (2002)
[CrossRef]

Carr, C. W.

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in KH2-xD2xPO4 crystals,” Appl. Phys. Lett. 89, 131901 (2006)
[CrossRef]

C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004)
[CrossRef] [PubMed]

C. W. Carr, H. B. Radousky, and S. G. Demos, “Wavelength dependence of laser-induced damage: determining the damage initiation mechanisms,” Phys. Rev. Lett. 91, 127402 (2003)
[CrossRef] [PubMed]

Carslaw, H.S.

H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford Science Publications, Second Edition (1959)

Chase, L. L.

Copic, M.

I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
[CrossRef]

Demos, S. G.

C. S. Liu, C. J. Hou, N. Kiousis, S. G. Demos, and H. B. Radousky, “Electronic structure calculations of an oxygen vacancy in KH2PO4,” Phys. Rev. B 72, 134110 (2005)
[CrossRef]

C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004)
[CrossRef] [PubMed]

C. W. Carr, H. B. Radousky, and S. G. Demos, “Wavelength dependence of laser-induced damage: determining the damage initiation mechanisms,” Phys. Rev. Lett. 91, 127402 (2003)
[CrossRef] [PubMed]

C. S. Liu, N. Kioussis, S. G. Demos, and H. B. Radousky, “Electron-or hole-assisted reactions of H defects in hydrogen-bonded KDP,” Phys. Rev. Lett. 91, 015505 (2003)
[CrossRef] [PubMed]

S. G. Demos, M. Staggs, J. J. De Yoreo, and H. B. Radousky, “Imaging of laser-induced reactions of individual defect nanoclusters,” Opt. Lett. 26, 1975–1977 (2001)
[CrossRef]

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys 85, 3988–3992 (1999)
[CrossRef]

Desrumaux, C.

J. Dijon, T. Poiroux, and C. Desrumaux, “Nano absorbing centers: a key point in the laser damage of thin film,” in Laser-induced damage in optical materials, Proc. SPIE 2966315 (1997)

DeYoreo, J.

M. Runkel, J. DeYoreo, W. Sell, and D. Milam, “Laser conditioning study of KDP on the optical sciences laser using large area beams,” in Laser-induced damage in optical materials, Proc. SPIE 3244, 51 (1998)

Dijon, J.

J. Dijon, T. Poiroux, and C. Desrumaux, “Nano absorbing centers: a key point in the laser damage of thin film,” in Laser-induced damage in optical materials, Proc. SPIE 2966315 (1997)

Drevensek, I.

I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
[CrossRef]

Duchateau, G.

A. Dyan, F. Enguehard, S. Lallich, H. Piombini, H. Mathis, and G. Duchateau, “Laser-induced damage in KH2PO4 and D2xKH2(1-x)PO4: influence of the laser probed volume and optimization of the conditioning through a revisited thermal approach,” submitted to Optics Communications

A. Dyan and G. Duchateau, CEA, Centre d’Etudes du Ripault, BP 16, 37260 Monts, France, are preparing a manuscript to be called “Laser-induced damage by a nanosecond pulse: a method coupling heat transfer, Mie’s theory and microscopic processes”

Dyan, A.

A. Dyan and G. Duchateau, CEA, Centre d’Etudes du Ripault, BP 16, 37260 Monts, France, are preparing a manuscript to be called “Laser-induced damage by a nanosecond pulse: a method coupling heat transfer, Mie’s theory and microscopic processes”

A. Dyan, F. Enguehard, S. Lallich, H. Piombini, H. Mathis, and G. Duchateau, “Laser-induced damage in KH2PO4 and D2xKH2(1-x)PO4: influence of the laser probed volume and optimization of the conditioning through a revisited thermal approach,” submitted to Optics Communications

Enguehard, F.

A. Dyan, F. Enguehard, S. Lallich, H. Piombini, H. Mathis, and G. Duchateau, “Laser-induced damage in KH2PO4 and D2xKH2(1-x)PO4: influence of the laser probed volume and optimization of the conditioning through a revisited thermal approach,” submitted to Optics Communications

Fally, M.

I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
[CrossRef]

Fan, Z. X.

Z. L. Xia, Z. X. Fan, and J. D. Shao, “A new theory for evaluating the number density of inclusions in films,” Appl. Surf. Sci. 252 (23), 8235–8238 (2006)
[CrossRef]

Feit, M. D.

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in KH2-xD2xPO4 crystals,” Appl. Phys. Lett. 89, 131901 (2006)
[CrossRef]

C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004)
[CrossRef] [PubMed]

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulse-length scaling and laser conditioning,” in Laser-induced damage in optical materials, Proc. SPIE 5273, 74–82, (2004)

M. D. Feit, A. M. Rubenchik, M. R. Kozlowski, F. Y. Génin, S. Schwartz, and L. M. Sheehan“Extrapolation of damage test data to predict performance of large area NIF optics at 355nm,” in Laser-induced damage in optical materials, Proc. SPIE 3578, 226–234 (1998)

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 53, 1749–1761 (1996)
[CrossRef]

Fuith, A.

I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
[CrossRef]

Gallais, L.

Génin, F. Y.

F. Y. Génin, A. Salleo, T. V. Pistor, and L. L. Chase, “Role of light intensification by cracks in optical breakdown on surfaces,” J. Opt. Soc. Am. A 182607-16 (2001)
[CrossRef]

M. D. Feit, A. M. Rubenchik, M. R. Kozlowski, F. Y. Génin, S. Schwartz, and L. M. Sheehan“Extrapolation of damage test data to predict performance of large area NIF optics at 355nm,” in Laser-induced damage in optical materials, Proc. SPIE 3578, 226–234 (1998)

Goldenberg, H.

H. Goldenberg and C.J. Tranter, “Heat flow in an infinite medium heated by a sphere,” Br. J. Appl. Phys. 3, 296–298 (1952)
[CrossRef]

Gradshteyn, I.S.

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Alan Jeffrey Editor, Fifth Edition (1994)

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 53, 1749–1761 (1996)
[CrossRef]

Hopper, R. W.

R. W. Hopper and D. R. Uhlmann, “Mechanism of inclusion damage in laser glass,” J. Appl. Phys. 41, 4023–4037 (1970)
[CrossRef]

Hou, C. J.

C. S. Liu, C. J. Hou, N. Kiousis, S. G. Demos, and H. B. Radousky, “Electronic structure calculations of an oxygen vacancy in KH2PO4,” Phys. Rev. B 72, 134110 (2005)
[CrossRef]

Jaeger, J.C.

H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford Science Publications, Second Edition (1959)

Johnson, M. A.

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in KH2-xD2xPO4 crystals,” Appl. Phys. Lett. 89, 131901 (2006)
[CrossRef]

Kiousis, N.

C. S. Liu, C. J. Hou, N. Kiousis, S. G. Demos, and H. B. Radousky, “Electronic structure calculations of an oxygen vacancy in KH2PO4,” Phys. Rev. B 72, 134110 (2005)
[CrossRef]

Kioussis, N.

C. S. Liu, N. Kioussis, S. G. Demos, and H. B. Radousky, “Electron-or hole-assisted reactions of H defects in hydrogen-bonded KDP,” Phys. Rev. Lett. 91, 015505 (2003)
[CrossRef] [PubMed]

Kittel, C.

C. Kittel, Introduction to solid state physics, Wiley, seventh edition, 1995

Kozlowski, M. R.

M. D. Feit, A. M. Rubenchik, M. R. Kozlowski, F. Y. Génin, S. Schwartz, and L. M. Sheehan“Extrapolation of damage test data to predict performance of large area NIF optics at 355nm,” in Laser-induced damage in optical materials, Proc. SPIE 3578, 226–234 (1998)

Lallich, S.

A. Dyan, F. Enguehard, S. Lallich, H. Piombini, H. Mathis, and G. Duchateau, “Laser-induced damage in KH2PO4 and D2xKH2(1-x)PO4: influence of the laser probed volume and optimization of the conditioning through a revisited thermal approach,” submitted to Optics Communications

Liu, C. S.

C. S. Liu, C. J. Hou, N. Kiousis, S. G. Demos, and H. B. Radousky, “Electronic structure calculations of an oxygen vacancy in KH2PO4,” Phys. Rev. B 72, 134110 (2005)
[CrossRef]

C. S. Liu, N. Kioussis, S. G. Demos, and H. B. Radousky, “Electron-or hole-assisted reactions of H defects in hydrogen-bonded KDP,” Phys. Rev. Lett. 91, 015505 (2003)
[CrossRef] [PubMed]

Mathis, H.

A. Dyan, F. Enguehard, S. Lallich, H. Piombini, H. Mathis, and G. Duchateau, “Laser-induced damage in KH2PO4 and D2xKH2(1-x)PO4: influence of the laser probed volume and optimization of the conditioning through a revisited thermal approach,” submitted to Optics Communications

Merkle, L. D.

P. K. Bandyopadhyay and L. D. Merkle, “Laser-induced damage in quartz: a study of the influence of impurities and defects,” J. Appl. Phys. 631392–1398 (1988)
[CrossRef]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid state physics, Brooks Cole, first edition, 1976

Milam, D.

M. Runkel, J. DeYoreo, W. Sell, and D. Milam, “Laser conditioning study of KDP on the optical sciences laser using large area beams,” in Laser-induced damage in optical materials, Proc. SPIE 3244, 51 (1998)

R. H. Picard, D. Milam, and R. A. Bradbury, “Statistical analysis of defect-caused laser damage in thin films” Appl. Opt. 16, 1563–1571 (1977)
[CrossRef] [PubMed]

Milanovich, F. P.

K. E. Montgomery and F. P. Milanovich, “High-laser-damage-threshold potassium dihydrogen phosphate crystals,” J. Appl. Phys. 683979-82 (1990)
[CrossRef]

Montgomery, K. E.

K. E. Montgomery and F. P. Milanovich, “High-laser-damage-threshold potassium dihydrogen phosphate crystals,” J. Appl. Phys. 683979-82 (1990)
[CrossRef]

Natoli, J. Y.

O’Connell, R. M.

Papernov, S.

S. Papernov and A. W. Schmid, “Correlations between embedded single gold nanoparticles in SiO2 thin film and nanoscale crater formation induced by pulsed-laser radiation,” J. Appl. Phys. 92, 5720–5728 (2002)
[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 53, 1749–1761 (1996)
[CrossRef]

Picard, R. H.

Piombini, H.

A. Dyan, F. Enguehard, S. Lallich, H. Piombini, H. Mathis, and G. Duchateau, “Laser-induced damage in KH2PO4 and D2xKH2(1-x)PO4: influence of the laser probed volume and optimization of the conditioning through a revisited thermal approach,” submitted to Optics Communications

Pistor, T. V.

Poiroux, T.

J. Dijon, T. Poiroux, and C. Desrumaux, “Nano absorbing centers: a key point in the laser damage of thin film,” in Laser-induced damage in optical materials, Proc. SPIE 2966315 (1997)

Porteus, J. O.

Radousky, H. B.

C. S. Liu, C. J. Hou, N. Kiousis, S. G. Demos, and H. B. Radousky, “Electronic structure calculations of an oxygen vacancy in KH2PO4,” Phys. Rev. B 72, 134110 (2005)
[CrossRef]

C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004)
[CrossRef] [PubMed]

C. W. Carr, H. B. Radousky, and S. G. Demos, “Wavelength dependence of laser-induced damage: determining the damage initiation mechanisms,” Phys. Rev. Lett. 91, 127402 (2003)
[CrossRef] [PubMed]

C. S. Liu, N. Kioussis, S. G. Demos, and H. B. Radousky, “Electron-or hole-assisted reactions of H defects in hydrogen-bonded KDP,” Phys. Rev. Lett. 91, 015505 (2003)
[CrossRef] [PubMed]

S. G. Demos, M. Staggs, J. J. De Yoreo, and H. B. Radousky, “Imaging of laser-induced reactions of individual defect nanoclusters,” Opt. Lett. 26, 1975–1977 (2001)
[CrossRef]

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys 85, 3988–3992 (1999)
[CrossRef]

Rubenchik, A. M.

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in KH2-xD2xPO4 crystals,” Appl. Phys. Lett. 89, 131901 (2006)
[CrossRef]

C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004)
[CrossRef] [PubMed]

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulse-length scaling and laser conditioning,” in Laser-induced damage in optical materials, Proc. SPIE 5273, 74–82, (2004)

M. D. Feit, A. M. Rubenchik, M. R. Kozlowski, F. Y. Génin, S. Schwartz, and L. M. Sheehan“Extrapolation of damage test data to predict performance of large area NIF optics at 355nm,” in Laser-induced damage in optical materials, Proc. SPIE 3578, 226–234 (1998)

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 53, 1749–1761 (1996)
[CrossRef]

Runkel, M.

M. Runkel, J. DeYoreo, W. Sell, and D. Milam, “Laser conditioning study of KDP on the optical sciences laser using large area beams,” in Laser-induced damage in optical materials, Proc. SPIE 3244, 51 (1998)

Ryzhik, I.M.

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Alan Jeffrey Editor, Fifth Edition (1994)

Salleo, A.

Schmid, A. W.

S. Papernov and A. W. Schmid, “Correlations between embedded single gold nanoparticles in SiO2 thin film and nanoscale crater formation induced by pulsed-laser radiation,” J. Appl. Phys. 92, 5720–5728 (2002)
[CrossRef]

Schranz, W.

I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
[CrossRef]

Schwartz, S.

M. D. Feit, A. M. Rubenchik, M. R. Kozlowski, F. Y. Génin, S. Schwartz, and L. M. Sheehan“Extrapolation of damage test data to predict performance of large area NIF optics at 355nm,” in Laser-induced damage in optical materials, Proc. SPIE 3578, 226–234 (1998)

Seitel, S. C.

Sell, W.

M. Runkel, J. DeYoreo, W. Sell, and D. Milam, “Laser conditioning study of KDP on the optical sciences laser using large area beams,” in Laser-induced damage in optical materials, Proc. SPIE 3244, 51 (1998)

Shao, J. D.

Z. L. Xia, Z. X. Fan, and J. D. Shao, “A new theory for evaluating the number density of inclusions in films,” Appl. Surf. Sci. 252 (23), 8235–8238 (2006)
[CrossRef]

Sheehan, L. M.

M. D. Feit, A. M. Rubenchik, M. R. Kozlowski, F. Y. Génin, S. Schwartz, and L. M. Sheehan“Extrapolation of damage test data to predict performance of large area NIF optics at 355nm,” in Laser-induced damage in optical materials, Proc. SPIE 3578, 226–234 (1998)

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 53, 1749–1761 (1996)
[CrossRef]

Sparks, M.

M. Sparks, “Theory of laser heating of solids: metals,” J. Appl. Phys. 47, 837–849 (1976)
[CrossRef]

Staggs, M.

S. G. Demos, M. Staggs, J. J. De Yoreo, and H. B. Radousky, “Imaging of laser-induced reactions of individual defect nanoclusters,” Opt. Lett. 26, 1975–1977 (2001)
[CrossRef]

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys 85, 3988–3992 (1999)
[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 53, 1749–1761 (1996)
[CrossRef]

Tranter, C.J.

H. Goldenberg and C.J. Tranter, “Heat flow in an infinite medium heated by a sphere,” Br. J. Appl. Phys. 3, 296–298 (1952)
[CrossRef]

Uhlmann, D. R.

R. W. Hopper and D. R. Uhlmann, “Mechanism of inclusion damage in laser glass,” J. Appl. Phys. 41, 4023–4037 (1970)
[CrossRef]

Warhanek, H.

I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
[CrossRef]

Whitman, P. K.

J. J. De Yoreo, A. K. Burnham, and P. K. Whitman, “Developing KH2PO4 and KD2PO4 crystals for the world’s most powerful laser,” Int. Mater. Rev. 47, 113–152 (2002)
[CrossRef]

Wood, R. M.

R. M. Wood, Laser-induced damage of optical materials, Institute Of Physics publishing series in optics ans optoelectronics, Bristol and Philadelphia (2003)

Xia, Z. L.

Z. L. Xia, Z. X. Fan, and J. D. Shao, “A new theory for evaluating the number density of inclusions in films,” Appl. Surf. Sci. 252 (23), 8235–8238 (2006)
[CrossRef]

Yan, M.

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys 85, 3988–3992 (1999)
[CrossRef]

Yoreo, J. J. De

J. J. De Yoreo, A. K. Burnham, and P. K. Whitman, “Developing KH2PO4 and KD2PO4 crystals for the world’s most powerful laser,” Int. Mater. Rev. 47, 113–152 (2002)
[CrossRef]

S. G. Demos, M. Staggs, J. J. De Yoreo, and H. B. Radousky, “Imaging of laser-induced reactions of individual defect nanoclusters,” Opt. Lett. 26, 1975–1977 (2001)
[CrossRef]

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys 85, 3988–3992 (1999)
[CrossRef]

Zgonik, M.

I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
[CrossRef]

Appl. Opt. (4)

Appl. Phy. B (1)

F. Bonneauel al, “Study of UV laser interaction with gold nanoparticles embedded in silica,” Appl. Phy. B 75, 803–815 (2002)
[CrossRef]

Appl. Phys. Lett. (1)

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in KH2-xD2xPO4 crystals,” Appl. Phys. Lett. 89, 131901 (2006)
[CrossRef]

Appl. Surf. Sci. (1)

Z. L. Xia, Z. X. Fan, and J. D. Shao, “A new theory for evaluating the number density of inclusions in films,” Appl. Surf. Sci. 252 (23), 8235–8238 (2006)
[CrossRef]

Br. J. Appl. Phys. (1)

H. Goldenberg and C.J. Tranter, “Heat flow in an infinite medium heated by a sphere,” Br. J. Appl. Phys. 3, 296–298 (1952)
[CrossRef]

in Laser-induced damage in optical materials, Proc. SPIE (5)

M. Runkel, J. DeYoreo, W. Sell, and D. Milam, “Laser conditioning study of KDP on the optical sciences laser using large area beams,” in Laser-induced damage in optical materials, Proc. SPIE 3244, 51 (1998)

J. J. Adamset al, “Wavelength and pulselength dependence of laser conditioning and bulk damage in doubler-cut KH2PO4,” in Laser-induced damage in optical materials, Proc. SPIE 5991, 59911R-1 (2005)

J. Dijon, T. Poiroux, and C. Desrumaux, “Nano absorbing centers: a key point in the laser damage of thin film,” in Laser-induced damage in optical materials, Proc. SPIE 2966315 (1997)

M. D. Feit, A. M. Rubenchik, M. R. Kozlowski, F. Y. Génin, S. Schwartz, and L. M. Sheehan“Extrapolation of damage test data to predict performance of large area NIF optics at 355nm,” in Laser-induced damage in optical materials, Proc. SPIE 3578, 226–234 (1998)

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulse-length scaling and laser conditioning,” in Laser-induced damage in optical materials, Proc. SPIE 5273, 74–82, (2004)

Int. Mater. Rev. (1)

J. J. De Yoreo, A. K. Burnham, and P. K. Whitman, “Developing KH2PO4 and KD2PO4 crystals for the world’s most powerful laser,” Int. Mater. Rev. 47, 113–152 (2002)
[CrossRef]

J. Appl. Phys (1)

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys 85, 3988–3992 (1999)
[CrossRef]

J. Appl. Phys. (5)

R. W. Hopper and D. R. Uhlmann, “Mechanism of inclusion damage in laser glass,” J. Appl. Phys. 41, 4023–4037 (1970)
[CrossRef]

S. Papernov and A. W. Schmid, “Correlations between embedded single gold nanoparticles in SiO2 thin film and nanoscale crater formation induced by pulsed-laser radiation,” J. Appl. Phys. 92, 5720–5728 (2002)
[CrossRef]

M. Sparks, “Theory of laser heating of solids: metals,” J. Appl. Phys. 47, 837–849 (1976)
[CrossRef]

K. E. Montgomery and F. P. Milanovich, “High-laser-damage-threshold potassium dihydrogen phosphate crystals,” J. Appl. Phys. 683979-82 (1990)
[CrossRef]

P. K. Bandyopadhyay and L. D. Merkle, “Laser-induced damage in quartz: a study of the influence of impurities and defects,” J. Appl. Phys. 631392–1398 (1988)
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (1)

Opto-electronics (1)

E. S. Bliss, “Pulse duration dependence of laser damage mechanism” Opto-electronics 3, 99 (1971)
[CrossRef]

Phys. Rev. B (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 53, 1749–1761 (1996)
[CrossRef]

C. S. Liu, C. J. Hou, N. Kiousis, S. G. Demos, and H. B. Radousky, “Electronic structure calculations of an oxygen vacancy in KH2PO4,” Phys. Rev. B 72, 134110 (2005)
[CrossRef]

Phys. Rev. Lett. (3)

C. S. Liu, N. Kioussis, S. G. Demos, and H. B. Radousky, “Electron-or hole-assisted reactions of H defects in hydrogen-bonded KDP,” Phys. Rev. Lett. 91, 015505 (2003)
[CrossRef] [PubMed]

C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004)
[CrossRef] [PubMed]

C. W. Carr, H. B. Radousky, and S. G. Demos, “Wavelength dependence of laser-induced damage: determining the damage initiation mechanisms,” Phys. Rev. Lett. 91, 127402 (2003)
[CrossRef] [PubMed]

Other (8)

A. Dyan, F. Enguehard, S. Lallich, H. Piombini, H. Mathis, and G. Duchateau, “Laser-induced damage in KH2PO4 and D2xKH2(1-x)PO4: influence of the laser probed volume and optimization of the conditioning through a revisited thermal approach,” submitted to Optics Communications

A. Dyan and G. Duchateau, CEA, Centre d’Etudes du Ripault, BP 16, 37260 Monts, France, are preparing a manuscript to be called “Laser-induced damage by a nanosecond pulse: a method coupling heat transfer, Mie’s theory and microscopic processes”

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Alan Jeffrey Editor, Fifth Edition (1994)

R. M. Wood, Laser-induced damage of optical materials, Institute Of Physics publishing series in optics ans optoelectronics, Bristol and Philadelphia (2003)

I. Drevensek, M. Zgonik, M. Copic, R. Blinc, A. Fuith, W. Schranz, M. Fally, and H. Warhanek, Phys. Rev. B49, 3082–3088 (1994)
[CrossRef]

N. W. Ashcroft and N. D. Mermin, Solid state physics, Brooks Cole, first edition, 1976

C. Kittel, Introduction to solid state physics, Wiley, seventh edition, 1995

H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford Science Publications, Second Edition (1959)

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Figures (9)

Fig. 1.
Fig. 1.

Temperature rise as a function of the cluster size. Full line and dashed lines correspond to 1D and 3D calculations respectively.

Fig. 2.
Fig. 2.

Temperature evolution as a function of the time for different cluster geometries: 1D sphere (solid line), 3D sphere (dashed line) and flat cylinder or enveloppe of a cone (dotted line).

Fig. 3.
Fig. 3.

Spatial temperature evolution resulting from a particular random throwing. 15 ADNS are present, A/ρC = 1013 K.s -1 and τ = 1 ns are used. The temperature rise is enhanced when several ADNS aggregate. Defects positions are shown by vertical arrows.

Fig. 4.
Fig. 4.

Size distribution of clusters derived from a random throwing. Four defects density are considered : nADNS = 50,100,200 and 400 in a domain of size N = 10000. The criterion of clustering d is set to 10nm (value allowing the ADNS to cooperate for τ = 1ns). 10000 drawings have been performed. Sub-figure shows a typical local cluster density as a function of the cluster size.

Fig. 5.
Fig. 5.

Size distribution of clusters derived from distorted throwing in same conditions of Fig. 4 but with d = 0 and nADNS = 100. Biased parameter b values are : 1,10,100,1000,10000,100000 and 1000000.

Fig. 6.
Fig. 6.

Evolution of the damage probability as a function of fluence within the 1D model. Four pulse durations are considered : τ = 250ps,1ns,4ns and 16ns. Parameters are nADNS = 100 and N = 10000. 200 drawings have been performed for each fluence. Sub-figure displays the scaling law exponent as a function of the pulse duration (see text).

Fig. 7.
Fig. 7.

Evolution of the number of ADNS involved in damage as a function of fluence. Four pulse durations are considered : τ = 250ps, 1ns,4ns and 16ns. Parameters are the same of Fig. 6 but with 10000 drawings.

Fig. 8.
Fig. 8.

Damage probability as a function of fluence. (a) Four defects density are considered : n ADNS = 50,100,200 and 400. (b) Three biased parameters are considered : b = 1,10,100. In both cases, calculations have been performed within the 1D model framework using τ = 1ns.

Fig. 9.
Fig. 9.

Evolution of the damage probability as a function of fluence within the 3D model. Five pulse durations are considered : τ = 125ps(a),250ps(b),500ps(c),1ns(d) and 4ns(e). In all cases, nADNS = 1250 and the domain size is 50×50×50. Sub-figure (f) displays the scaling law exponent as a function of the pulse duration (see text).

Tables (1)

Tables Icon

Table 1. Critical fluence and x-exponent as a function of the pulse duration

Equations (41)

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

Δ T 1 D ( x = 0 , t ) = A ρC { t ( t + l 2 2 D ) erfc ( l 2 Dt ) + l ( t πD ) 1 2 exp ( l 2 4 Dt ) }
Δ T 3 D ( r = 0 , t ) = a 2 A DρC { 1 2 2 π [ πDt 2 l exp ( l 2 4 Dt ) π 2 ( Dt l 2 1 2 ) erf ( 1 Dt 2 ) ] }
F c = T c τ ζ f ( τ )
T t = D 2 T x 2 + A ρC i = 1 n ADNS ( x x i )
{ ( x ) = 1 if x [ x a 2 ; x + a 2 ] ( x ) = 0 elsewhere
T ( x , t ) = T 0 + i = 1 n ADNS Δ T ( i ) ( x , t )
Δ T ( i ) t = D 2 Δ T ( i ) x 2 + A ρC ( x x i )
θ 1 D ( x , t ) = 2 λ { 2 ( Dt π ) 1 2 exp ( x 2 4 Dt ) x erfc ( x Dt 2 ) }
θ 3 D ( r , t ) = Aa 3 3 λr erfc ( r 2 Dt )
{ P d = bn d ( bn d + n dl ) ) P dl = n dl ( bn d + n dl ) )
x = α In τ + β
{ Δ T t = D 2 Δ T x 2 + A ρ C if a x a Δ T t = D 2 Δ T x 2 elsewhere
{ Δ T ¯ ʺ q 2 Δ T ¯ = A sDρC if a x a Δ T ¯ ʺ q 2 Δ T ¯ = 0 elsewhere
{ Δ T ¯ ( x , s ) = αe qx if x a Δ T ¯ ( x , s ) = βe qx + γe qx + A s 2 ρ C Δ T ¯ ( x , s ) = δe qx if x a if a x a
Δ T ˉ ( x , s ) = A 2 ρ Cs 2 ( e q ( x a ) e q ( x + a ) )
1 ( e qx s 2 ) = ( t + x 2 2 D ) erfc ( x Dt 2 ) x ( t πD ) 1 2 exp ( x 2 4 Dt )
Δ T ( x , t ) = A 2 ρC { ( t + ( x a ) 2 2 D ) erfc ( x a 2 Dt ) ( x a ) ( t πD ) 1 2 exp ( ( x a ) 2 4 Dt ) ( t + ( x + a ) 2 2 D ) erfc ( x + a 2 Dt ) + ( x + a ) ( t πD ) 1 2 exp ( ( x + a ) 2 4 Dt ) }
Δ T ( x , t ) = A ρC { t 1 2 ( t + ( a x ) 2 2 D ) erfc ( a x 2 Dt ) a x 2 ( t πD ) 1 2 exp ( ( a x ) 2 4 Dt ) 1 2 ( t + ( x + a ) 2 2 D ) erfc ( x + a 2 Dt ) + x + a 2 ( t πD ) 1 2 exp ( ( x + a ) 2 4 Dt ) }
Δ T = ( r = 0 , t ) = a 2 A λ 2 { λ 1 3 λ 2 + 1 6 2 b π 0 + dy exp ( t γ 1 y 2 ) sin y y cos y y ( ( c sin y y cos y ) 2 + b 2 y 2 sin 2 y ) }
0 dy exp ( αy 2 ) y 3 ( sin y y cos y ) = πα 2 exp ( 1 4 α ) π 2 ( α 1 2 ) erf ( 1 2 α )
Δ T ( r = 0 , t ) = a 2 A DρC { 1 2 2 π [ πDt 2 a exp ( a 2 4 Dt ) π 2 ( Dt a 2 1 2 ) erf ( a 2 Dt ) ] }
Δ T ( r , t ) = a 3 A r λ 1 { 1 3 2 π 0 + exp ( D 1 t a 2 y 2 ) y 4 ( sin y y cos y ) sin ( r a y ) dy }
T ( r , t ) ~ a 3 A 3 { 1 r πDt }
θ 1 D t = D 2 θ 1 D x 2 + ( x ) F ( t ) ρC
θ 1 D ( x , s ) = E 2 λ D s exp { x s D }
{ F ( t ) = 1 τ if 0 t τ F ( t ) = 0 elsewhere
θ 1 D ( x , s ) = E 2 λτ D s exp { x s D } 1 e τs s
θ 1 D ( x , s ) = ζ ( x , s ) e τs ζ ( x , s )
ζ ( x , s ) = E 2 λτ D s 3 exp { x s D }
θ 1 D ( x , t ) = E 2 λτ { 2 ( Dt π ) 1 2 exp ( x 2 4 Dt ) x erfc ( x 2 Dt ) }
θ 3 D ( r , t ) = E 4 πλrτ erfc ( r 2 Dt )
Δ T ( r = 0 , t ) = 1 4 3 π a 3 0 a θ 3 D ( r , t ) d r
Δ T ( r = 0 , t ) = A λ 0 a dr r erfc ( r 2 Dt )
erf ( αx ) dx = x erf ( αx ) + e α 2 x 2 α π
Δ T ( r = 0 , t ) = A 2 λ { a 2 ( 1 erf ( αa ) ) ae α 2 a 2 α π + erf ( αa ) 2 α 2 }
Δ T ( r = 0 , t ) = a 2 A λ { 1 2 2 π ( πDt 2 a e a 2 4 Dt π 2 ( Dt a 2 1 2 ) erf ( a 2 Dt ) ) }
Δ T ( r = 0 , z = 0 , t ) A 4 πλ 0 a dr 2 π r Δ z 1 r erfc ( r 2 Dt ) = A Δ z 2 λ 0 a erfc ( r 2 Dt ) dr
Δ T ( r = 0 , z = 0 , t ) = A Δ z 2 λ { a erfc ( a 2 Dt ) 2 Dt π ( e a 2 4 Dt 1 ) }
Δ T ( r = 0 , z = 0 , t ) = 1 4 3 π a 3 0 h dz Δ R 2 πR ( z ) θ 3 D ( r ( z ) , t )
Δ T ( r = 0 , z = 0 , t ) = A 2 λ a Δ R h 2 + a 2 0 h dz erfc z 1 + a 2 h 2 2 Dt
Δ T ( r = 0 , z = 0 , t ) = A 2 λ a Δ R h 2 + a 2 { h erfc ( αh ) 2 Dt π ( e α 2 h 2 1 ) }

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