Abstract

The transient changes in the optical properties of bulk DKDP material arising from its exposure to high temperatures and pressures associated with localized laser energy deposition are investigated. Two methods for initiation of laser-induced breakdown are used, intrinsic, involving relatively large energy deposition brought about by focusing of the laser beam to high intensities, and extrinsic, arising from more localized deposition due to the presence of pre-existing absorbing damage initiating defects. Each method leads to a very different volume of material being affected, which provides for different material thermal relaxation times to help better understand the processes involved.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Demos, M. Staggs, K. Minoshima, and J. Fujimoto, “Characterization of laser induced damage sites in optical components,” Opt. Express10, 1444–1450 (2002).
    [PubMed]
  2. C. H. Li, X. Ju, X. D. Jiang, J. Huang, X. D. Zhou, Z. Zheng, W. D. Wu, W. G. Zheng, Z. X. Li, B. Y. Wang, and X. H. Yu, “High resolution characterization of modifications in fused silica after exposure to low fluence 355 nm laser at different repetition frequencies,” Opt. Express19, 6439–6449 (2011).
    [CrossRef] [PubMed]
  3. J. D. Musgraves, K. Richardson, and J. Himanshu, “Laser-induced structural modification, its mechanisms, and applications in glassy optical materials,” Opt. Mater. Express1, 921–935 (2011).
    [CrossRef]
  4. V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
    [CrossRef]
  5. R. A. Negres, M. W. Burke, S. B. Sutton, P. DeMange, M. D. Feit, and S. G. Demos, “Evaluation of UV absorption coefficient in laser-modified fused silica,” Appl. Phys. Lett.90, 061115 (2007).
    [CrossRef]
  6. S. O. Kucheyev and S. G. Demos, “Optical defects produced in fused silica during laser-induced breakdown,” Appl. Phys. Lett.82, 3230–3232 (2003).
    [CrossRef]
  7. R. A. Negres, M. D. Feit, and S. G. Demos, “Dynamics of material modifications following laser-breakdown in bulk fused silica,” Opt. Express18, 10642–10649 (2010).
    [CrossRef] [PubMed]
  8. B. Bertussi, P. Cormont, S. Palmier, P. Legros, and J. L. Rullier, “Initiation of laser-induced damage sites in fused silica optical components,” Opt. Express17, 11469–11479 (2009).
    [CrossRef] [PubMed]
  9. Y. Kobayashi, S. Endo, K. Koto, T. Kikegawa, and O. Shimomura, “Phase transitions and amorphization of KH2PO4 at high pressure,” Phys. Rev. B51, 9302–9305 (1995).
    [CrossRef]
  10. 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. B53, 1749–1761 (1996).
    [CrossRef]
  11. C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in K2H(2−x)DxPO4 crystals,” Appl. Phys. Lett.89, 131901 (2006).
    [CrossRef]
  12. C. B. Schaffer, A. Brodeur, J. F. Garcia, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett.26, 93–95 (2001).
    [CrossRef]
  13. C. W. Carr, J. D. Bude, and P. DeMange, “Laser-supported solid-state absorption fronts in silica,” Phys. Rev. B82, 184304 (2010).
    [CrossRef]
  14. S. G. Demos, P. DeMange, R. A. Negres, and M. D. Feit, “Investigation of the electronic and physical properties of defect structures responsible for laser-induced damage in DKDP crystals,” Opt. Express18, 13788–13804 (2010).
    [CrossRef] [PubMed]
  15. S. C. Jones, P. Braunlich, R. T. Casper, X. A. Shen, and P. Kelly, “Recent progress on laser-induced modifications and intrinsic bulk damage of wide-gap optical materials,” Opt. Eng.28, 1039–1068 (1989).
  16. H. Bercegol, P. Grua, D. Hebert, and J. P. Morreeuw, “Progress in the understanding of fracture related laser damage of fused silica,” Proc. SPIE6720, 672003 (2007).
    [CrossRef]
  17. 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]
  18. H. Jiang, J. McNary, H. W. K. Tom, M. Yan, H. B. Radousky, and S. G. Demos, “Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids,” Appl. Phys. Lett.81, 3149–3151 (2002).
    [CrossRef]
  19. A. Salleo, R. Chinsio, and F. Y. Genin, “Crack propagation in fused silica during UV and IR ns-laser illumination,” Proc. SPIE3578, 456–471 (1999).
    [CrossRef]
  20. M. J. Matthews, C. W. Carr, H. A. Bechtel, and R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett.99, 151109 (2011).
    [CrossRef]
  21. M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability, pulselength scaling and laser conditioning,” Proc. SPIE5273, 74–81 (2004).
    [CrossRef]
  22. P. DeMange, C. W. Carr, H. B. Radousky, and S. G. Demos, “System for evaluation of laser-induced damage performance of optical materials for large aperture lasers,” Rev. Sci. Instrum.75, 3298–3301 (2004).
    [CrossRef]
  23. P. DeMange, R. A. Negres, H. B. Radousky, and S. G. Demos, “Differentiation of defect populations responsible for bulk laser-induced damage in potassium dihydrogen phosphate crystals,” Opt. Eng.45, 104205 (2006).
    [CrossRef]
  24. M. Runkel, A. K. Burnham, D. Milam, W. Sell, M. Feit, and A. Rubenchik, “The results of pulse-scaling experiments on rapid-growth DKDP triplers using the optical sciences laser at 351 nm,” Proc. SPIE4347, 359–372 (2001).
    [CrossRef]
  25. C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE6403, 64030K (2006).
    [CrossRef]
  26. R. N. Raman, R. A. Negres, and S. G. Demos, “Time-resolved microscope system to image material response following localized laser energy deposition: exit surface damage in fused silica as a case example,” Opt. Eng.50, 013602 (2011).
    [CrossRef]
  27. P. DeMange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett.30, 221–223 (2005).
    [CrossRef] [PubMed]
  28. R. A. Negres, P. DeMange, and S. G. Demos, “Investigation of laser annealing parameters for optimal laser-damage performance in deuterated potassium dihydrogen phosphate,” Opt. Lett.30, 2766–2768 (2005).
    [CrossRef] [PubMed]
  29. N. Zaitseva and L. Carman, “Rapid growth of KDP-type crystals,” Prog. Cryst. Growth Charact.43, 1–118 (2001).
    [CrossRef]
  30. P. E. Miller, J. D. Bude, T. I. Suratwala, N. Shen, T. A. Laurence, W. A. Steele, J. Menapace, M. D. Feit, and L. L. Wong, “Fracture-induced subbandgap absorption as a precursor to optical damage on fused silica surfaces,” Opt. Lett.35, 2702–2704 (2010).
    [CrossRef] [PubMed]
  31. T. Fang and J. C. Lambropoulos, “Microhardness and indentation fracture of potassium dihydrogen phosphate (KDP),” J. Am. Ceram. Soc.85, 174–178 (2002).
    [CrossRef]

2011

C. H. Li, X. Ju, X. D. Jiang, J. Huang, X. D. Zhou, Z. Zheng, W. D. Wu, W. G. Zheng, Z. X. Li, B. Y. Wang, and X. H. Yu, “High resolution characterization of modifications in fused silica after exposure to low fluence 355 nm laser at different repetition frequencies,” Opt. Express19, 6439–6449 (2011).
[CrossRef] [PubMed]

J. D. Musgraves, K. Richardson, and J. Himanshu, “Laser-induced structural modification, its mechanisms, and applications in glassy optical materials,” Opt. Mater. Express1, 921–935 (2011).
[CrossRef]

V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
[CrossRef]

M. J. Matthews, C. W. Carr, H. A. Bechtel, and R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett.99, 151109 (2011).
[CrossRef]

R. N. Raman, R. A. Negres, and S. G. Demos, “Time-resolved microscope system to image material response following localized laser energy deposition: exit surface damage in fused silica as a case example,” Opt. Eng.50, 013602 (2011).
[CrossRef]

2010

2009

2007

R. A. Negres, M. W. Burke, S. B. Sutton, P. DeMange, M. D. Feit, and S. G. Demos, “Evaluation of UV absorption coefficient in laser-modified fused silica,” Appl. Phys. Lett.90, 061115 (2007).
[CrossRef]

H. Bercegol, P. Grua, D. Hebert, and J. P. Morreeuw, “Progress in the understanding of fracture related laser damage of fused silica,” Proc. SPIE6720, 672003 (2007).
[CrossRef]

2006

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in K2H(2−x)DxPO4 crystals,” Appl. Phys. Lett.89, 131901 (2006).
[CrossRef]

C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE6403, 64030K (2006).
[CrossRef]

P. DeMange, R. A. Negres, H. B. Radousky, and S. G. Demos, “Differentiation of defect populations responsible for bulk laser-induced damage in potassium dihydrogen phosphate crystals,” Opt. Eng.45, 104205 (2006).
[CrossRef]

2005

2004

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability, pulselength scaling and laser conditioning,” Proc. SPIE5273, 74–81 (2004).
[CrossRef]

P. DeMange, C. W. Carr, H. B. Radousky, and S. G. Demos, “System for evaluation of laser-induced damage performance of optical materials for large aperture lasers,” Rev. Sci. Instrum.75, 3298–3301 (2004).
[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]

2003

S. O. Kucheyev and S. G. Demos, “Optical defects produced in fused silica during laser-induced breakdown,” Appl. Phys. Lett.82, 3230–3232 (2003).
[CrossRef]

2002

S. Demos, M. Staggs, K. Minoshima, and J. Fujimoto, “Characterization of laser induced damage sites in optical components,” Opt. Express10, 1444–1450 (2002).
[PubMed]

H. Jiang, J. McNary, H. W. K. Tom, M. Yan, H. B. Radousky, and S. G. Demos, “Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids,” Appl. Phys. Lett.81, 3149–3151 (2002).
[CrossRef]

T. Fang and J. C. Lambropoulos, “Microhardness and indentation fracture of potassium dihydrogen phosphate (KDP),” J. Am. Ceram. Soc.85, 174–178 (2002).
[CrossRef]

2001

M. Runkel, A. K. Burnham, D. Milam, W. Sell, M. Feit, and A. Rubenchik, “The results of pulse-scaling experiments on rapid-growth DKDP triplers using the optical sciences laser at 351 nm,” Proc. SPIE4347, 359–372 (2001).
[CrossRef]

N. Zaitseva and L. Carman, “Rapid growth of KDP-type crystals,” Prog. Cryst. Growth Charact.43, 1–118 (2001).
[CrossRef]

C. B. Schaffer, A. Brodeur, J. F. Garcia, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett.26, 93–95 (2001).
[CrossRef]

1999

A. Salleo, R. Chinsio, and F. Y. Genin, “Crack propagation in fused silica during UV and IR ns-laser illumination,” Proc. SPIE3578, 456–471 (1999).
[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. B53, 1749–1761 (1996).
[CrossRef]

1995

Y. Kobayashi, S. Endo, K. Koto, T. Kikegawa, and O. Shimomura, “Phase transitions and amorphization of KH2PO4 at high pressure,” Phys. Rev. B51, 9302–9305 (1995).
[CrossRef]

1989

S. C. Jones, P. Braunlich, R. T. Casper, X. A. Shen, and P. Kelly, “Recent progress on laser-induced modifications and intrinsic bulk damage of wide-gap optical materials,” Opt. Eng.28, 1039–1068 (1989).

Bechtel, H. A.

M. J. Matthews, C. W. Carr, H. A. Bechtel, and R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett.99, 151109 (2011).
[CrossRef]

Bercegol, H.

H. Bercegol, P. Grua, D. Hebert, and J. P. Morreeuw, “Progress in the understanding of fracture related laser damage of fused silica,” Proc. SPIE6720, 672003 (2007).
[CrossRef]

Bertussi, B.

Braunlich, P.

S. C. Jones, P. Braunlich, R. T. Casper, X. A. Shen, and P. Kelly, “Recent progress on laser-induced modifications and intrinsic bulk damage of wide-gap optical materials,” Opt. Eng.28, 1039–1068 (1989).

Brodeur, A.

Bude, J. D.

C. W. Carr, J. D. Bude, and P. DeMange, “Laser-supported solid-state absorption fronts in silica,” Phys. Rev. B82, 184304 (2010).
[CrossRef]

P. E. Miller, J. D. Bude, T. I. Suratwala, N. Shen, T. A. Laurence, W. A. Steele, J. Menapace, M. D. Feit, and L. L. Wong, “Fracture-induced subbandgap absorption as a precursor to optical damage on fused silica surfaces,” Opt. Lett.35, 2702–2704 (2010).
[CrossRef] [PubMed]

C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE6403, 64030K (2006).
[CrossRef]

Burke, M. W.

R. A. Negres, M. W. Burke, S. B. Sutton, P. DeMange, M. D. Feit, and S. G. Demos, “Evaluation of UV absorption coefficient in laser-modified fused silica,” Appl. Phys. Lett.90, 061115 (2007).
[CrossRef]

Burnham, A. K.

M. Runkel, A. K. Burnham, D. Milam, W. Sell, M. Feit, and A. Rubenchik, “The results of pulse-scaling experiments on rapid-growth DKDP triplers using the optical sciences laser at 351 nm,” Proc. SPIE4347, 359–372 (2001).
[CrossRef]

Carman, L.

N. Zaitseva and L. Carman, “Rapid growth of KDP-type crystals,” Prog. Cryst. Growth Charact.43, 1–118 (2001).
[CrossRef]

Carr, C. W.

M. J. Matthews, C. W. Carr, H. A. Bechtel, and R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett.99, 151109 (2011).
[CrossRef]

C. W. Carr, J. D. Bude, and P. DeMange, “Laser-supported solid-state absorption fronts in silica,” Phys. Rev. B82, 184304 (2010).
[CrossRef]

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in K2H(2−x)DxPO4 crystals,” Appl. Phys. Lett.89, 131901 (2006).
[CrossRef]

C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE6403, 64030K (2006).
[CrossRef]

P. DeMange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett.30, 221–223 (2005).
[CrossRef] [PubMed]

P. DeMange, C. W. Carr, H. B. Radousky, and S. G. Demos, “System for evaluation of laser-induced damage performance of optical materials for large aperture lasers,” Rev. Sci. Instrum.75, 3298–3301 (2004).
[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]

Casper, R. T.

S. C. Jones, P. Braunlich, R. T. Casper, X. A. Shen, and P. Kelly, “Recent progress on laser-induced modifications and intrinsic bulk damage of wide-gap optical materials,” Opt. Eng.28, 1039–1068 (1989).

Chinsio, R.

A. Salleo, R. Chinsio, and F. Y. Genin, “Crack propagation in fused silica during UV and IR ns-laser illumination,” Proc. SPIE3578, 456–471 (1999).
[CrossRef]

Cormont, P.

DeMange, P.

S. G. Demos, P. DeMange, R. A. Negres, and M. D. Feit, “Investigation of the electronic and physical properties of defect structures responsible for laser-induced damage in DKDP crystals,” Opt. Express18, 13788–13804 (2010).
[CrossRef] [PubMed]

C. W. Carr, J. D. Bude, and P. DeMange, “Laser-supported solid-state absorption fronts in silica,” Phys. Rev. B82, 184304 (2010).
[CrossRef]

R. A. Negres, M. W. Burke, S. B. Sutton, P. DeMange, M. D. Feit, and S. G. Demos, “Evaluation of UV absorption coefficient in laser-modified fused silica,” Appl. Phys. Lett.90, 061115 (2007).
[CrossRef]

P. DeMange, R. A. Negres, H. B. Radousky, and S. G. Demos, “Differentiation of defect populations responsible for bulk laser-induced damage in potassium dihydrogen phosphate crystals,” Opt. Eng.45, 104205 (2006).
[CrossRef]

R. A. Negres, P. DeMange, and S. G. Demos, “Investigation of laser annealing parameters for optimal laser-damage performance in deuterated potassium dihydrogen phosphate,” Opt. Lett.30, 2766–2768 (2005).
[CrossRef] [PubMed]

P. DeMange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett.30, 221–223 (2005).
[CrossRef] [PubMed]

P. DeMange, C. W. Carr, H. B. Radousky, and S. G. Demos, “System for evaluation of laser-induced damage performance of optical materials for large aperture lasers,” Rev. Sci. Instrum.75, 3298–3301 (2004).
[CrossRef]

Demos, S.

Demos, S. G.

R. N. Raman, R. A. Negres, and S. G. Demos, “Time-resolved microscope system to image material response following localized laser energy deposition: exit surface damage in fused silica as a case example,” Opt. Eng.50, 013602 (2011).
[CrossRef]

S. G. Demos, P. DeMange, R. A. Negres, and M. D. Feit, “Investigation of the electronic and physical properties of defect structures responsible for laser-induced damage in DKDP crystals,” Opt. Express18, 13788–13804 (2010).
[CrossRef] [PubMed]

R. A. Negres, M. D. Feit, and S. G. Demos, “Dynamics of material modifications following laser-breakdown in bulk fused silica,” Opt. Express18, 10642–10649 (2010).
[CrossRef] [PubMed]

R. A. Negres, M. W. Burke, S. B. Sutton, P. DeMange, M. D. Feit, and S. G. Demos, “Evaluation of UV absorption coefficient in laser-modified fused silica,” Appl. Phys. Lett.90, 061115 (2007).
[CrossRef]

P. DeMange, R. A. Negres, H. B. Radousky, and S. G. Demos, “Differentiation of defect populations responsible for bulk laser-induced damage in potassium dihydrogen phosphate crystals,” Opt. Eng.45, 104205 (2006).
[CrossRef]

R. A. Negres, P. DeMange, and S. G. Demos, “Investigation of laser annealing parameters for optimal laser-damage performance in deuterated potassium dihydrogen phosphate,” Opt. Lett.30, 2766–2768 (2005).
[CrossRef] [PubMed]

P. DeMange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett.30, 221–223 (2005).
[CrossRef] [PubMed]

P. DeMange, C. W. Carr, H. B. Radousky, and S. G. Demos, “System for evaluation of laser-induced damage performance of optical materials for large aperture lasers,” Rev. Sci. Instrum.75, 3298–3301 (2004).
[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]

S. O. Kucheyev and S. G. Demos, “Optical defects produced in fused silica during laser-induced breakdown,” Appl. Phys. Lett.82, 3230–3232 (2003).
[CrossRef]

H. Jiang, J. McNary, H. W. K. Tom, M. Yan, H. B. Radousky, and S. G. Demos, “Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids,” Appl. Phys. Lett.81, 3149–3151 (2002).
[CrossRef]

Endo, S.

Y. Kobayashi, S. Endo, K. Koto, T. Kikegawa, and O. Shimomura, “Phase transitions and amorphization of KH2PO4 at high pressure,” Phys. Rev. B51, 9302–9305 (1995).
[CrossRef]

Fang, T.

T. Fang and J. C. Lambropoulos, “Microhardness and indentation fracture of potassium dihydrogen phosphate (KDP),” J. Am. Ceram. Soc.85, 174–178 (2002).
[CrossRef]

Feit, M.

M. Runkel, A. K. Burnham, D. Milam, W. Sell, M. Feit, and A. Rubenchik, “The results of pulse-scaling experiments on rapid-growth DKDP triplers using the optical sciences laser at 351 nm,” Proc. SPIE4347, 359–372 (2001).
[CrossRef]

Feit, M. D.

P. E. Miller, J. D. Bude, T. I. Suratwala, N. Shen, T. A. Laurence, W. A. Steele, J. Menapace, M. D. Feit, and L. L. Wong, “Fracture-induced subbandgap absorption as a precursor to optical damage on fused silica surfaces,” Opt. Lett.35, 2702–2704 (2010).
[CrossRef] [PubMed]

R. A. Negres, M. D. Feit, and S. G. Demos, “Dynamics of material modifications following laser-breakdown in bulk fused silica,” Opt. Express18, 10642–10649 (2010).
[CrossRef] [PubMed]

S. G. Demos, P. DeMange, R. A. Negres, and M. D. Feit, “Investigation of the electronic and physical properties of defect structures responsible for laser-induced damage in DKDP crystals,” Opt. Express18, 13788–13804 (2010).
[CrossRef] [PubMed]

R. A. Negres, M. W. Burke, S. B. Sutton, P. DeMange, M. D. Feit, and S. G. Demos, “Evaluation of UV absorption coefficient in laser-modified fused silica,” Appl. Phys. Lett.90, 061115 (2007).
[CrossRef]

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in K2H(2−x)DxPO4 crystals,” Appl. Phys. Lett.89, 131901 (2006).
[CrossRef]

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability, pulselength scaling and laser conditioning,” Proc. SPIE5273, 74–81 (2004).
[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]

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. B53, 1749–1761 (1996).
[CrossRef]

Fujimoto, J.

Garcia, J. F.

Genin, F. Y.

A. Salleo, R. Chinsio, and F. Y. Genin, “Crack propagation in fused silica during UV and IR ns-laser illumination,” Proc. SPIE3578, 456–471 (1999).
[CrossRef]

Grua, P.

H. Bercegol, P. Grua, D. Hebert, and J. P. Morreeuw, “Progress in the understanding of fracture related laser damage of fused silica,” Proc. SPIE6720, 672003 (2007).
[CrossRef]

Hebert, D.

H. Bercegol, P. Grua, D. Hebert, and J. P. Morreeuw, “Progress in the understanding of fracture related laser damage of fused silica,” Proc. SPIE6720, 672003 (2007).
[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. B53, 1749–1761 (1996).
[CrossRef]

Himanshu, J.

Hirao, N.

V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
[CrossRef]

Huang, J.

Jiang, H.

H. Jiang, J. McNary, H. W. K. Tom, M. Yan, H. B. Radousky, and S. G. Demos, “Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids,” Appl. Phys. Lett.81, 3149–3151 (2002).
[CrossRef]

Jiang, X. D.

Johnson, M. A.

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in K2H(2−x)DxPO4 crystals,” Appl. Phys. Lett.89, 131901 (2006).
[CrossRef]

Jones, S. C.

S. C. Jones, P. Braunlich, R. T. Casper, X. A. Shen, and P. Kelly, “Recent progress on laser-induced modifications and intrinsic bulk damage of wide-gap optical materials,” Opt. Eng.28, 1039–1068 (1989).

Ju, X.

Juodkazis, S.

V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
[CrossRef]

Kelly, P.

S. C. Jones, P. Braunlich, R. T. Casper, X. A. Shen, and P. Kelly, “Recent progress on laser-induced modifications and intrinsic bulk damage of wide-gap optical materials,” Opt. Eng.28, 1039–1068 (1989).

Kikegawa, T.

Y. Kobayashi, S. Endo, K. Koto, T. Kikegawa, and O. Shimomura, “Phase transitions and amorphization of KH2PO4 at high pressure,” Phys. Rev. B51, 9302–9305 (1995).
[CrossRef]

Kobayashi, Y.

Y. Kobayashi, S. Endo, K. Koto, T. Kikegawa, and O. Shimomura, “Phase transitions and amorphization of KH2PO4 at high pressure,” Phys. Rev. B51, 9302–9305 (1995).
[CrossRef]

Kohara, S.

V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
[CrossRef]

Koto, K.

Y. Kobayashi, S. Endo, K. Koto, T. Kikegawa, and O. Shimomura, “Phase transitions and amorphization of KH2PO4 at high pressure,” Phys. Rev. B51, 9302–9305 (1995).
[CrossRef]

Kucheyev, S. O.

S. O. Kucheyev and S. G. Demos, “Optical defects produced in fused silica during laser-induced breakdown,” Appl. Phys. Lett.82, 3230–3232 (2003).
[CrossRef]

Lambropoulos, J. C.

T. Fang and J. C. Lambropoulos, “Microhardness and indentation fracture of potassium dihydrogen phosphate (KDP),” J. Am. Ceram. Soc.85, 174–178 (2002).
[CrossRef]

Laurence, T. A.

Legros, P.

Li, C. H.

Li, Z. X.

Matthews, M. J.

M. J. Matthews, C. W. Carr, H. A. Bechtel, and R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett.99, 151109 (2011).
[CrossRef]

C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE6403, 64030K (2006).
[CrossRef]

Mazur, E.

McNary, J.

H. Jiang, J. McNary, H. W. K. Tom, M. Yan, H. B. Radousky, and S. G. Demos, “Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids,” Appl. Phys. Lett.81, 3149–3151 (2002).
[CrossRef]

Menapace, J.

Milam, D.

M. Runkel, A. K. Burnham, D. Milam, W. Sell, M. Feit, and A. Rubenchik, “The results of pulse-scaling experiments on rapid-growth DKDP triplers using the optical sciences laser at 351 nm,” Proc. SPIE4347, 359–372 (2001).
[CrossRef]

Miller, P. E.

Minoshima, K.

Mizeikis, V.

V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
[CrossRef]

Morreeuw, J. P.

H. Bercegol, P. Grua, D. Hebert, and J. P. Morreeuw, “Progress in the understanding of fracture related laser damage of fused silica,” Proc. SPIE6720, 672003 (2007).
[CrossRef]

Musgraves, J. D.

Negres, R. A.

R. N. Raman, R. A. Negres, and S. G. Demos, “Time-resolved microscope system to image material response following localized laser energy deposition: exit surface damage in fused silica as a case example,” Opt. Eng.50, 013602 (2011).
[CrossRef]

R. A. Negres, M. D. Feit, and S. G. Demos, “Dynamics of material modifications following laser-breakdown in bulk fused silica,” Opt. Express18, 10642–10649 (2010).
[CrossRef] [PubMed]

S. G. Demos, P. DeMange, R. A. Negres, and M. D. Feit, “Investigation of the electronic and physical properties of defect structures responsible for laser-induced damage in DKDP crystals,” Opt. Express18, 13788–13804 (2010).
[CrossRef] [PubMed]

R. A. Negres, M. W. Burke, S. B. Sutton, P. DeMange, M. D. Feit, and S. G. Demos, “Evaluation of UV absorption coefficient in laser-modified fused silica,” Appl. Phys. Lett.90, 061115 (2007).
[CrossRef]

P. DeMange, R. A. Negres, H. B. Radousky, and S. G. Demos, “Differentiation of defect populations responsible for bulk laser-induced damage in potassium dihydrogen phosphate crystals,” Opt. Eng.45, 104205 (2006).
[CrossRef]

R. A. Negres, P. DeMange, and S. G. Demos, “Investigation of laser annealing parameters for optimal laser-damage performance in deuterated potassium dihydrogen phosphate,” Opt. Lett.30, 2766–2768 (2005).
[CrossRef] [PubMed]

P. DeMange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett.30, 221–223 (2005).
[CrossRef] [PubMed]

Onishi, Y.

V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
[CrossRef]

Palmier, S.

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. B53, 1749–1761 (1996).
[CrossRef]

Radousky, H. B.

P. DeMange, R. A. Negres, H. B. Radousky, and S. G. Demos, “Differentiation of defect populations responsible for bulk laser-induced damage in potassium dihydrogen phosphate crystals,” Opt. Eng.45, 104205 (2006).
[CrossRef]

P. DeMange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett.30, 221–223 (2005).
[CrossRef] [PubMed]

P. DeMange, C. W. Carr, H. B. Radousky, and S. G. Demos, “System for evaluation of laser-induced damage performance of optical materials for large aperture lasers,” Rev. Sci. Instrum.75, 3298–3301 (2004).
[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]

H. Jiang, J. McNary, H. W. K. Tom, M. Yan, H. B. Radousky, and S. G. Demos, “Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids,” Appl. Phys. Lett.81, 3149–3151 (2002).
[CrossRef]

Raman, R. N.

M. J. Matthews, C. W. Carr, H. A. Bechtel, and R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett.99, 151109 (2011).
[CrossRef]

R. N. Raman, R. A. Negres, and S. G. Demos, “Time-resolved microscope system to image material response following localized laser energy deposition: exit surface damage in fused silica as a case example,” Opt. Eng.50, 013602 (2011).
[CrossRef]

Richardson, K.

Rubenchik, A.

M. Runkel, A. K. Burnham, D. Milam, W. Sell, M. Feit, and A. Rubenchik, “The results of pulse-scaling experiments on rapid-growth DKDP triplers using the optical sciences laser at 351 nm,” Proc. SPIE4347, 359–372 (2001).
[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 K2H(2−x)DxPO4 crystals,” Appl. Phys. Lett.89, 131901 (2006).
[CrossRef]

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability, pulselength scaling and laser conditioning,” Proc. SPIE5273, 74–81 (2004).
[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]

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. B53, 1749–1761 (1996).
[CrossRef]

Rullier, J. L.

Runkel, M.

M. Runkel, A. K. Burnham, D. Milam, W. Sell, M. Feit, and A. Rubenchik, “The results of pulse-scaling experiments on rapid-growth DKDP triplers using the optical sciences laser at 351 nm,” Proc. SPIE4347, 359–372 (2001).
[CrossRef]

Saito, A.

V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
[CrossRef]

Salleo, A.

A. Salleo, R. Chinsio, and F. Y. Genin, “Crack propagation in fused silica during UV and IR ns-laser illumination,” Proc. SPIE3578, 456–471 (1999).
[CrossRef]

Schaffer, C. B.

Sell, W.

M. Runkel, A. K. Burnham, D. Milam, W. Sell, M. Feit, and A. Rubenchik, “The results of pulse-scaling experiments on rapid-growth DKDP triplers using the optical sciences laser at 351 nm,” Proc. SPIE4347, 359–372 (2001).
[CrossRef]

Shen, N.

Shen, X. A.

S. C. Jones, P. Braunlich, R. T. Casper, X. A. Shen, and P. Kelly, “Recent progress on laser-induced modifications and intrinsic bulk damage of wide-gap optical materials,” Opt. Eng.28, 1039–1068 (1989).

Shimomura, O.

Y. Kobayashi, S. Endo, K. Koto, T. Kikegawa, and O. Shimomura, “Phase transitions and amorphization of KH2PO4 at high pressure,” Phys. Rev. B51, 9302–9305 (1995).
[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. B53, 1749–1761 (1996).
[CrossRef]

Spaeth, M. L.

C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE6403, 64030K (2006).
[CrossRef]

Staggs, M.

Steele, W. A.

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. B53, 1749–1761 (1996).
[CrossRef]

Suratwala, T. I.

Sutton, S. B.

R. A. Negres, M. W. Burke, S. B. Sutton, P. DeMange, M. D. Feit, and S. G. Demos, “Evaluation of UV absorption coefficient in laser-modified fused silica,” Appl. Phys. Lett.90, 061115 (2007).
[CrossRef]

Tom, H. W. K.

H. Jiang, J. McNary, H. W. K. Tom, M. Yan, H. B. Radousky, and S. G. Demos, “Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids,” Appl. Phys. Lett.81, 3149–3151 (2002).
[CrossRef]

Vailionis, A.

V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
[CrossRef]

Wang, B. Y.

Wong, L. L.

Wu, W. D.

Yan, M.

H. Jiang, J. McNary, H. W. K. Tom, M. Yan, H. B. Radousky, and S. G. Demos, “Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids,” Appl. Phys. Lett.81, 3149–3151 (2002).
[CrossRef]

Yu, X. H.

Zaitseva, N.

N. Zaitseva and L. Carman, “Rapid growth of KDP-type crystals,” Prog. Cryst. Growth Charact.43, 1–118 (2001).
[CrossRef]

Zheng, W. G.

Zheng, Z.

Zhou, X. D.

Appl. Phys. A

V. Mizeikis, S. Kohara, Y. Onishi, N. Hirao, A. Saito, A. Vailionis, and S. Juodkazis, “Synthesis of high-pressure phases of silica by laser-induced optical breakdown,” Appl. Phys. A104, 903–906 (2011).
[CrossRef]

Appl. Phys. Lett.

R. A. Negres, M. W. Burke, S. B. Sutton, P. DeMange, M. D. Feit, and S. G. Demos, “Evaluation of UV absorption coefficient in laser-modified fused silica,” Appl. Phys. Lett.90, 061115 (2007).
[CrossRef]

S. O. Kucheyev and S. G. Demos, “Optical defects produced in fused silica during laser-induced breakdown,” Appl. Phys. Lett.82, 3230–3232 (2003).
[CrossRef]

C. W. Carr, M. D. Feit, M. A. Johnson, and A. M. Rubenchik, “Complex morphology of laser-induced bulk damage in K2H(2−x)DxPO4 crystals,” Appl. Phys. Lett.89, 131901 (2006).
[CrossRef]

H. Jiang, J. McNary, H. W. K. Tom, M. Yan, H. B. Radousky, and S. G. Demos, “Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids,” Appl. Phys. Lett.81, 3149–3151 (2002).
[CrossRef]

M. J. Matthews, C. W. Carr, H. A. Bechtel, and R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett.99, 151109 (2011).
[CrossRef]

J. Am. Ceram. Soc.

T. Fang and J. C. Lambropoulos, “Microhardness and indentation fracture of potassium dihydrogen phosphate (KDP),” J. Am. Ceram. Soc.85, 174–178 (2002).
[CrossRef]

Opt. Eng.

P. DeMange, R. A. Negres, H. B. Radousky, and S. G. Demos, “Differentiation of defect populations responsible for bulk laser-induced damage in potassium dihydrogen phosphate crystals,” Opt. Eng.45, 104205 (2006).
[CrossRef]

R. N. Raman, R. A. Negres, and S. G. Demos, “Time-resolved microscope system to image material response following localized laser energy deposition: exit surface damage in fused silica as a case example,” Opt. Eng.50, 013602 (2011).
[CrossRef]

S. C. Jones, P. Braunlich, R. T. Casper, X. A. Shen, and P. Kelly, “Recent progress on laser-induced modifications and intrinsic bulk damage of wide-gap optical materials,” Opt. Eng.28, 1039–1068 (1989).

Opt. Express

Opt. Lett.

Opt. Mater. Express

Phys. Rev. B

C. W. Carr, J. D. Bude, and P. DeMange, “Laser-supported solid-state absorption fronts in silica,” Phys. Rev. B82, 184304 (2010).
[CrossRef]

Y. Kobayashi, S. Endo, K. Koto, T. Kikegawa, and O. Shimomura, “Phase transitions and amorphization of KH2PO4 at high pressure,” Phys. Rev. B51, 9302–9305 (1995).
[CrossRef]

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. B53, 1749–1761 (1996).
[CrossRef]

Phys. Rev. Lett.

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]

Proc. SPIE

H. Bercegol, P. Grua, D. Hebert, and J. P. Morreeuw, “Progress in the understanding of fracture related laser damage of fused silica,” Proc. SPIE6720, 672003 (2007).
[CrossRef]

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability, pulselength scaling and laser conditioning,” Proc. SPIE5273, 74–81 (2004).
[CrossRef]

A. Salleo, R. Chinsio, and F. Y. Genin, “Crack propagation in fused silica during UV and IR ns-laser illumination,” Proc. SPIE3578, 456–471 (1999).
[CrossRef]

M. Runkel, A. K. Burnham, D. Milam, W. Sell, M. Feit, and A. Rubenchik, “The results of pulse-scaling experiments on rapid-growth DKDP triplers using the optical sciences laser at 351 nm,” Proc. SPIE4347, 359–372 (2001).
[CrossRef]

C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE6403, 64030K (2006).
[CrossRef]

Prog. Cryst. Growth Charact.

N. Zaitseva and L. Carman, “Rapid growth of KDP-type crystals,” Prog. Cryst. Growth Charact.43, 1–118 (2001).
[CrossRef]

Rev. Sci. Instrum.

P. DeMange, C. W. Carr, H. B. Radousky, and S. G. Demos, “System for evaluation of laser-induced damage performance of optical materials for large aperture lasers,” Rev. Sci. Instrum.75, 3298–3301 (2004).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Pump-probe damage testing apparatus in (a) collinear and (b) non-collinear geometries, respectively. Pump and probe pulses are 3-ns and 7-ns in duration, respectively.

Fig. 2
Fig. 2

Typical light-scattering image of spontaneous bulk damage sites in DKDP near the focal region (located on the right hand side) of the spatially overlapping pump and probe pulses at 25 ns delay. The arrows indicate typical damage sites from pump only.

Fig. 3
Fig. 3

Time-resolved microscopic imaging apparatus in trans-illumination geometry. Pump and probe pulses are 3-ns and 180-ps in duration, respectively.

Fig. 4
Fig. 4

Final size (damage cross-section area) of DKDP bulk damage sites observed in the region of spatial overlap between pump and probe pulses versus probe time delay (left panel, semi-log scale) and peak fluence at 25 ns and 80 ns fixed delays (right panel) for (a) spontaneous and (b) intrinsic damage, respectively.

Fig. 5
Fig. 5

Typical ratio images (final divided by transient) of intrinsic bulk damage sites in DKDP for pump and probe time delays between 8 ns and 100 μs. For comparison, all images have the same spatial scale and contrast.

Fig. 6
Fig. 6

Model of transient absorption from Eqs. (1)(3) describing the cross sectional area, A, corresponding to an affected volume (pump-induced) with temperature above a growth threshold Tth: (a) normalized area vs. delay corresponding to different ratios Tth/T0, (b) area vs. delay corresponding to Tth = T0/50 and different initial relative radii, a, of energy deposition volume.

Equations (3)

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

T ( r , t d ) = T 0 ( 1 + 4 D t d a 2 ) 3 / 2 exp ( r 2 a 2 + 4 D t d ) ,
A ( t d , T t h ) = π a 2 τ ln ( T 0 τ 3 / 2 T t h )
τ = 1 + 4 D t d a 2 .

Metrics