Abstract

Multilayer dielectric (MLD) gratings used in ultrahigh-intensity laser systems often exhibit a laser-induced damage performance below that of their constituent materials. Reduced performance may arise from fabrication- and/or design-related issues. Finite element models were developed to simulate stress waves in MLD grating structures generated by laser-induced damage events. These models specifically investigate the influence of geometric and material parameters on how stress waves can lead to degradation of material structural integrity that can have adverse effects on its optical performance under subsequent laser irradiation: closer impedance matching of the layer materials reduces maximum interface stresses by ~20% to 30%; increasing sole thickness from 50 nm to 500 nm reduces maximum interface stresses by ~50%.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (1)

2017 (1)

2015 (5)

2014 (1)

2013 (1)

2012 (1)

F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

2011 (1)

S. Hocquet, J. Neauport, and N. Bonod, “The role of electric field polarization of the incident laser beam in the short pulse damage mechanism of pulse compression gratings,” Appl. Phys. Lett. 99(6), 061101 (2011).
[Crossref]

2010 (2)

H. T. Nguyen, C. C. Larson, and J. A. Britten, “Improvement of laser damage resistance and diffraction efficiency of multilayer dielectric diffraction gratings by HF etchback linewidth tailoring,” Proc. SPIE 7842, 78421H (2010).
[Crossref]

J. Qiao, A. W. Schmid, L. J. Waxer, T. Nguyen, J. Bunkenburg, C. Kingsley, A. Kozlov, and D. Weiner, “In situ detection and analysis of laser-induced damage on a 1.5-m multilayer-dielectric grating compressor for high-energy, petawatt-class laser systems,” Opt. Express 18(10), 10423–10431 (2010).
[Crossref] [PubMed]

2007 (2)

2006 (2)

B. Ashe, K. L. Marshall, C. Giacofei, A. L. Rigatti, T. J. Kessler, A. W. Schmid, J. B. Oliver, J. Keck, and A. Kozlov, “Evaluation of cleaning methods for multilayer diffraction gratings,” Proc. SPIE 6403, 64030O (2006).
[Crossref]

T. Z. Kosc, A. A. Kozlov, and A. W. Schmid, “Formation of periodic microstructures on multilayer dielectric gratings prior to total ablation,” Opt. Express 14(22), 10921–10929 (2006).
[Crossref] [PubMed]

2004 (1)

C. R. Phipps, J. R. Luke, D. J. Funk, D. S. Moore, J. Glownia, and T. Lippert, “Measurements of laser impulse coupling at 130 fs,” Proc. SPIE 5448, 1201 (2004).
[Crossref]

1999 (1)

J. Kováčik, “Correlation between young’s modulus and porosity in porous materials,” J. Mater. Sci. Lett. 18(13), 1007–1010 (1999).
[Crossref]

1997 (1)

L. Berthe, R. Fabbro, P. Peyre, T. Tollier, and E. Bartnicki, “Shock waves from a water-confined laser-generated plasma,” J. Appl. Phys. 82(6), 2826–2832 (1997).
[Crossref]

1995 (1)

1988 (1)

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[Crossref]

1987 (1)

K. K. Phani and S. K. Niyogi, “Young’s modulus of porous brittle solids,” J. Mater. Sci. 22(1), 257–263 (1987).
[Crossref]

Aiello, A. F.

Alessi, D. A.

Amer, E.

J. E. Field, E. Amer, P. Gren, M. A. Zafar, and S. M. Walley, “High-speed photographic study of laser damage and ablation,” Imaging Sci. J. 63(3), 119–136 (2015).
[Crossref]

Anderson, G. K.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[Crossref]

Ashe, B.

B. Ashe, C. Giacofei, G. Myhre, and A. W. Schmid, “Optimizing a cleaning process for multilayer-dielectric- (mld) diffraction grating,” Proc. SPIE 6720, 67200N (2007).
[Crossref]

B. Ashe, K. L. Marshall, C. Giacofei, A. L. Rigatti, T. J. Kessler, A. W. Schmid, J. B. Oliver, J. Keck, and A. Kozlov, “Evaluation of cleaning methods for multilayer diffraction gratings,” Proc. SPIE 6403, 64030O (2006).
[Crossref]

Balas, M.

Bartnicki, E.

L. Berthe, R. Fabbro, P. Peyre, T. Tollier, and E. Bartnicki, “Shock waves from a water-confined laser-generated plasma,” J. Appl. Phys. 82(6), 2826–2832 (1997).
[Crossref]

Berthe, L.

L. Berthe, R. Fabbro, P. Peyre, T. Tollier, and E. Bartnicki, “Shock waves from a water-confined laser-generated plasma,” J. Appl. Phys. 82(6), 2826–2832 (1997).
[Crossref]

Berthelot, J.

Bonod, N.

S. Hocquet, J. Neauport, and N. Bonod, “The role of electric field polarization of the incident laser beam in the short pulse damage mechanism of pulse compression gratings,” Appl. Phys. Lett. 99(6), 061101 (2011).
[Crossref]

J. Neauport, E. Lavastre, G. Razé, G. Dupuy, N. Bonod, M. Balas, G. de Villele, J. Flamand, S. Kaladgew, and F. Desserouer, “Effect of electric field on laser induced damage threshold of multilayer dielectric gratings,” Opt. Express 15(19), 12508–12522 (2007).
[Crossref] [PubMed]

Bouillet, S.

Boyd, R. D.

Britten, J. A.

Bunkenburg, J.

Carr, C. W.

Chen, W.

F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

Corlis, X. F.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[Crossref]

Cross, D. A.

de Villele, G.

Decker, D.

Demos, S. G.

Desserouer, F.

Dressler, J. G.

Du, Y.

Dupuy, G.

Edwards, N. R.

Fabbro, R.

L. Berthe, R. Fabbro, P. Peyre, T. Tollier, and E. Bartnicki, “Shock waves from a water-confined laser-generated plasma,” J. Appl. Phys. 82(6), 2826–2832 (1997).
[Crossref]

Fair, J. E.

Feit, M. D.

Field, J. E.

J. E. Field, E. Amer, P. Gren, M. A. Zafar, and S. M. Walley, “High-speed photographic study of laser damage and ablation,” Imaging Sci. J. 63(3), 119–136 (2015).
[Crossref]

Flamand, J.

Funk, D. J.

C. R. Phipps, J. R. Luke, D. J. Funk, D. S. Moore, J. Glownia, and T. Lippert, “Measurements of laser impulse coupling at 130 fs,” Proc. SPIE 5448, 1201 (2004).
[Crossref]

Gallais, L.

Giacofei, C.

B. Ashe, C. Giacofei, G. Myhre, and A. W. Schmid, “Optimizing a cleaning process for multilayer-dielectric- (mld) diffraction grating,” Proc. SPIE 6720, 67200N (2007).
[Crossref]

B. Ashe, K. L. Marshall, C. Giacofei, A. L. Rigatti, T. J. Kessler, A. W. Schmid, J. B. Oliver, J. Keck, and A. Kozlov, “Evaluation of cleaning methods for multilayer diffraction gratings,” Proc. SPIE 6403, 64030O (2006).
[Crossref]

Glownia, J.

C. R. Phipps, J. R. Luke, D. J. Funk, D. S. Moore, J. Glownia, and T. Lippert, “Measurements of laser impulse coupling at 130 fs,” Proc. SPIE 5448, 1201 (2004).
[Crossref]

Gourdin, W. H.

Gren, P.

J. E. Field, E. Amer, P. Gren, M. A. Zafar, and S. M. Walley, “High-speed photographic study of laser damage and ablation,” Imaging Sci. J. 63(3), 119–136 (2015).
[Crossref]

Guan, H.

Guo, Y.

Guss, G.

Hackel, R. P.

Haefner, C.

Hao, Y.

Harrison, R. F.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[Crossref]

Haynes, L. C.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[Crossref]

He, H.

F. Kong, Y. Jin, H. Huang, H. Zhang, S. Liu, and H. He, “Laser-induced damage of multilayer dielectric gratings with picosecond laser pulses under vacuum and air,” Opt. Laser Technol. 73, 39–43 (2015).
[Crossref]

F. Kong, Y. Jin, H. Guan, S. Liu, J. Wu, Y. Du, and H. He, “Influence of horizontal damage size of grating ridge on the optical properties of multilayer dielectric gratings,” Appl. Opt. 53(22), 4859–4864 (2014).
[Crossref] [PubMed]

F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

Hocquet, S.

S. Hocquet, J. Neauport, and N. Bonod, “The role of electric field polarization of the incident laser beam in the short pulse damage mechanism of pulse compression gratings,” Appl. Phys. Lett. 99(6), 061101 (2011).
[Crossref]

Howard, H. P.

Huang, H.

F. Kong, Y. Jin, H. Huang, H. Zhang, S. Liu, and H. He, “Laser-induced damage of multilayer dielectric gratings with picosecond laser pulses under vacuum and air,” Opt. Laser Technol. 73, 39–43 (2015).
[Crossref]

Jacobs, S. D.

Jin, Y.

F. Kong, Y. Jin, H. Huang, H. Zhang, S. Liu, and H. He, “Laser-induced damage of multilayer dielectric gratings with picosecond laser pulses under vacuum and air,” Opt. Laser Technol. 73, 39–43 (2015).
[Crossref]

F. Kong, Y. Jin, H. Guan, S. Liu, J. Wu, Y. Du, and H. He, “Influence of horizontal damage size of grating ridge on the optical properties of multilayer dielectric gratings,” Appl. Opt. 53(22), 4859–4864 (2014).
[Crossref] [PubMed]

F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

Kaladgew, S.

Keck, J.

B. Ashe, K. L. Marshall, C. Giacofei, A. L. Rigatti, T. J. Kessler, A. W. Schmid, J. B. Oliver, J. Keck, and A. Kozlov, “Evaluation of cleaning methods for multilayer diffraction gratings,” Proc. SPIE 6403, 64030O (2006).
[Crossref]

Kessler, T. J.

King, T. R.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[Crossref]

Kingsley, C.

Kong, F.

F. Kong, Y. Jin, H. Huang, H. Zhang, S. Liu, and H. He, “Laser-induced damage of multilayer dielectric gratings with picosecond laser pulses under vacuum and air,” Opt. Laser Technol. 73, 39–43 (2015).
[Crossref]

F. Kong, Y. Jin, H. Guan, S. Liu, J. Wu, Y. Du, and H. He, “Influence of horizontal damage size of grating ridge on the optical properties of multilayer dielectric gratings,” Appl. Opt. 53(22), 4859–4864 (2014).
[Crossref] [PubMed]

F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

Kosc, T. Z.

Kovácik, J.

J. Kováčik, “Correlation between young’s modulus and porosity in porous materials,” J. Mater. Sci. Lett. 18(13), 1007–1010 (1999).
[Crossref]

Kozlov, A.

J. Qiao, A. W. Schmid, L. J. Waxer, T. Nguyen, J. Bunkenburg, C. Kingsley, A. Kozlov, and D. Weiner, “In situ detection and analysis of laser-induced damage on a 1.5-m multilayer-dielectric grating compressor for high-energy, petawatt-class laser systems,” Opt. Express 18(10), 10423–10431 (2010).
[Crossref] [PubMed]

B. Ashe, K. L. Marshall, C. Giacofei, A. L. Rigatti, T. J. Kessler, A. W. Schmid, J. B. Oliver, J. Keck, and A. Kozlov, “Evaluation of cleaning methods for multilayer diffraction gratings,” Proc. SPIE 6403, 64030O (2006).
[Crossref]

Kozlov, A. A.

Lamaignère, L.

Lambropoulos, J. C.

Larson, C. C.

H. T. Nguyen, C. C. Larson, and J. A. Britten, “Improvement of laser damage resistance and diffraction efficiency of multilayer dielectric diffraction gratings by HF etchback linewidth tailoring,” Proc. SPIE 7842, 78421H (2010).
[Crossref]

Lavastre, E.

Li, C.

F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

Li, D.

F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

Lippert, T.

C. R. Phipps, J. R. Luke, D. J. Funk, D. S. Moore, J. Glownia, and T. Lippert, “Measurements of laser impulse coupling at 130 fs,” Proc. SPIE 5448, 1201 (2004).
[Crossref]

Liu, S.

F. Kong, Y. Jin, H. Huang, H. Zhang, S. Liu, and H. He, “Laser-induced damage of multilayer dielectric gratings with picosecond laser pulses under vacuum and air,” Opt. Laser Technol. 73, 39–43 (2015).
[Crossref]

F. Kong, Y. Jin, H. Guan, S. Liu, J. Wu, Y. Du, and H. He, “Influence of horizontal damage size of grating ridge on the optical properties of multilayer dielectric gratings,” Appl. Opt. 53(22), 4859–4864 (2014).
[Crossref] [PubMed]

Luke, J. R.

C. R. Phipps, J. R. Luke, D. J. Funk, D. S. Moore, J. Glownia, and T. Lippert, “Measurements of laser impulse coupling at 130 fs,” Proc. SPIE 5448, 1201 (2004).
[Crossref]

Luthi, R.

Manes, K. R.

Manwaring, I. R. T.

Marshall, K. L.

Mehrotra, K.

Moore, D. S.

C. R. Phipps, J. R. Luke, D. J. Funk, D. S. Moore, J. Glownia, and T. Lippert, “Measurements of laser impulse coupling at 130 fs,” Proc. SPIE 5448, 1201 (2004).
[Crossref]

Myhre, G.

B. Ashe, C. Giacofei, G. Myhre, and A. W. Schmid, “Optimizing a cleaning process for multilayer-dielectric- (mld) diffraction grating,” Proc. SPIE 6720, 67200N (2007).
[Crossref]

Neauport, J.

Negres, R. A.

Nguyen, H. T.

H. T. Nguyen, C. C. Larson, and J. A. Britten, “Improvement of laser damage resistance and diffraction efficiency of multilayer dielectric diffraction gratings by HF etchback linewidth tailoring,” Proc. SPIE 7842, 78421H (2010).
[Crossref]

Nguyen, T.

Nissen, J.

Niyogi, S. K.

K. K. Phani and S. K. Niyogi, “Young’s modulus of porous brittle solids,” J. Mater. Sci. 22(1), 257–263 (1987).
[Crossref]

Oliver, J. B.

Osborne, W. Z.

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C. R. Phipps, J. R. Luke, D. J. Funk, D. S. Moore, J. Glownia, and T. Lippert, “Measurements of laser impulse coupling at 130 fs,” Proc. SPIE 5448, 1201 (2004).
[Crossref]

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
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F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

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C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
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Steele, H. S.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
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L. Berthe, R. Fabbro, P. Peyre, T. Tollier, and E. Bartnicki, “Shock waves from a water-confined laser-generated plasma,” J. Appl. Phys. 82(6), 2826–2832 (1997).
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C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
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J. E. Field, E. Amer, P. Gren, M. A. Zafar, and S. M. Walley, “High-speed photographic study of laser damage and ablation,” Imaging Sci. J. 63(3), 119–136 (2015).
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F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

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Weiner, D.

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F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
[Crossref]

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C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, and T. R. King, “Impulse coupling to targets tn vacuum by KrF, Hf, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[Crossref]

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J. E. Field, E. Amer, P. Gren, M. A. Zafar, and S. M. Walley, “High-speed photographic study of laser damage and ablation,” Imaging Sci. J. 63(3), 119–136 (2015).
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F. Kong, Y. Jin, H. Huang, H. Zhang, S. Liu, and H. He, “Laser-induced damage of multilayer dielectric gratings with picosecond laser pulses under vacuum and air,” Opt. Laser Technol. 73, 39–43 (2015).
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F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
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Appl. Opt. (3)

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[Crossref]

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J. Mater. Sci. (1)

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[Crossref]

Opt. Lett. (1)

Optica (1)

Proc. SPIE (5)

F. Kong, Y. Jin, D. Li, W. Chen, M. Zhu, T. Wang, C. Li, H. He, G. Xu, and J. Shao, “Effect of pulse duration on laser induced damage threshold of multilayer dielectric gratings,” Proc. SPIE 8530, 85300L (2012).
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Supplementary Material (2)

NameDescription
» Visualization 1       Visualization 1. Animation of normal stress in the y (verical) direction for a pressure pulse applied normal to the top of a pillar in a dielectric grating structure.
» Visualization 2       Visualization 2. Animation of normal stress in the x (horizontal) direction for a pressure pulse applied normal to the midpoint of the side of a pillar in a dielectric grating structure.

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Figures (7)

Fig. 1
Fig. 1 Finite element model with default geometry, where dsole is the sole thickness; dh and ds are the thicknesses of the hafnia and silica layers, respectively; H is the pillar height; W is the pillar width at mid-height; D is the pillar-to-pillar distance; and θ is the angle of the pillar. The radius of curvature r = 20 nm at all corners. The red + indicates the center point on the first interface and the coordinate axes indicate the x and y directions.
Fig. 2
Fig. 2 Locations of applied pressure pulses, representing locations of laser-induced ablation sites.
Fig. 3
Fig. 3 Contours of normal stress in the y (vertical) direction, σyy, for a compressive pulse applied normal to the top of the pillar, (a) at the time when the peak of the incident compressive wave crosses the first interface and (b) at the time when the peak of the reflected tensile wave crosses the first interface. The tensile stress along the interface could potentially cause delamination. Compressive stresses are blue, tensile stresses are red, and stress-free regions are green. See Visualization 1 for an animation of normal stress in the y direction, σyy.
Fig. 4
Fig. 4 Normal stress at the midpoint of the first interface (point indicated with red + in Fig. 1) versus time for the original geometry with the original bulk silica and hafnia properties (solid black line), changing the first embedded silica layer to alumina (dashed–dotted red line), or using hafnia with 25% porosity (dashed blue line). The shaded oval highlights the tensile (positive) peak that occurs when the wavefront reflected off the second interface crosses the first interface and constructively interferes with a tensile wave propagating directly from the pillar. The material changes considered cause a reduction in impedance mismatch between layers, ultimately reducing this tensile peak by 20% to 30%.
Fig. 5
Fig. 5 Normal stress at the midpoint of the first interface (point indicated with red + in Fig. 1) versus time for the original bulk silica and hafnia properties with a sole thickness of 50 nm (solid black line), 200 nm (dashed–dotted red line), or 500 nm (dashed blue line). Increasing the sole thickness in general reduces both tensile and compressive stresses. The highlighted peaks occur when the wavefront crosses the first interface, before (blue oval highlight) and after (red oval highlight) reflecting off the second interface.
Fig. 6
Fig. 6 Magnitude of peaks of normal stress at the midpoint of the first interface (point indicated in Fig. 1) versus sole thickness. For dsole > 200 nm, the compressive wavefront peaks (blue) decay as d sole 0.5 as shown by the curve fits. The tensile peaks (red) that occur when the wavefront crosses the first interface after reflecting back from the second interface decay approximately as d sole 0.5 as shown by the curve fits. Reflected tensile peak stresses σTapp predicted by Eq. (3) (black) are lower than the finite element model but show a similar dependence on material properties. Solid lines and filled squares indicate original materials, while dashed lines and open squares indicate results for porous hafnia.
Fig. 7
Fig. 7 Contours of maximum principal stress for a compressive pulse applied normal to the midpoint of the left side of the pillar with a temporal duration of (a) 10 ps and (b) 100 ps. See Visualization 2 for an animation of normal stress in the x direction, σ xx .

Tables (1)

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Table 1 Material properties used in the simulations

Equations (3)

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R 12 = Z 2 Z 1 Z 2 + Z 1 and T 12 = 2 Z 2 Z 2 + Z 1 ,
σ c σ l (1+ R sh ) 1 d sole = σ l T sh 1 d sole ,
σ T σ Tapp = σ l T sh R hs T hs 1 d sole +2 d h .

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