Abstract

Femtosecond laser exposure of fused silica can lead to non-linear absorption, eventually causing structural modifications in the material. Above a given pulse repetition frequency, the effects from one pulse to the next one become cumulative leading to a localized bulk heating of the substrate, and in turn, to the dissociation of the glass matrix leading to gas bubbles formation. Here, we investigate the dynamics of bubbles formation as a function of the incoming net fluence. In particular, we observe evidences of laser trapping of gas bubbles and the unexpected formation of self-organized nanostructures, resembling nanogratings normally found at much lower repetition rate, i.e. when cumulative effects are absent.

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

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    [Crossref] [PubMed]

2016 (3)

2015 (6)

N. Groothoff, M.-O. Hongler, P. Kazansky, and Y. Bellouard, “Transition and self-healing process between chaotic and self-organized patterns observed during femtosecond laser writing,” Opt. Express 23(13), 16993–17007 (2015).
[Crossref] [PubMed]

T. Meany, M. Grafe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics: Laser written quantum circuits,” Laser Photonics Rev. 9(4), 363–384 (2015).
[Crossref]

L. A. Fernandes, J. R. Grenier, J. S. Aitchison, and P. R. Herman, “Fiber optic stress-independent helical torsion sensor,” Opt. Lett. 40(4), 657–660 (2015).
[Crossref] [PubMed]

T. Yang and Y. Bellouard, “Monolithic transparent 3D dielectrophoretic micro-actuator fabricated by femtosecond laser,” J. Micromech. Microeng. 25(10), 105009 (2015).
[Crossref]

C.-E. Athanasiou and Y. Bellouard, “A monolithic micro-tensile tester for investigating silicon dioxide polymorph micromechanics, fabricated and operated using a femtosecond laser,” Micromachines (Basel) 6(9), 1365–1386 (2015).
[Crossref]

Y. Liao, J. Ni, L. Qiao, M. Huang, Y. Bellouard, K. Sugioka, and Y. Cheng, “High-fidelity visualization of formation of volume nanogratings in porous glass by femtosecond laser irradiation,” Optica 2(4), 329 (2015).
[Crossref]

2014 (5)

V. Tielen and Y. Bellouard, “Three-dimensional glass monolithic microflexure fabricated by femtosecond laser exposure and chemical etching,” Micromachines (Basel) 5(3), 697–710 (2014).
[Crossref]

M. Beresna, M. Gecevičius, and P. G. Kazansky, “Ultrafast laser direct writing and nanostructuring in transparent materials,” Adv. Opt. Photonics 6(3), 293 (2014).
[Crossref]

K. Sugioka and Y. Cheng, “Ultrafast lasers—reliable tools for advanced materials processing,” Light Sci. Appl. 3(4), e149 (2014).
[Crossref]

J. Zhang, M. Gecevičius, M. Beresna, and P. G. Kazansky, “Seemingly unlimited lifetime data storage in nanostructured glass,” Phys. Rev. Lett. 112(3), 033901 (2014).
[Crossref] [PubMed]

K. Cvecek, I. Miyamoto, and M. Schmidt, “Gas bubble formation in fused silica generated by ultra-short laser pulses,” Opt. Express 22(13), 15877–15893 (2014).
[Crossref] [PubMed]

2013 (3)

2012 (4)

M. Lancry, E. Régnier, and B. Poumellec, “Fictive temperature in silica-based glasses and its application to optical fiber manufacturing,” Prog. Mater. Sci. 57(1), 63–94 (2012).
[Crossref]

M. Shimizu, M. Sakakura, M. Ohnishi, M. Yamaji, Y. Shimotsuma, K. Hirao, and K. Miura, “Three-dimensional temperature distribution and modification mechanism in glass during ultrafast laser irradiation at high repetition rates,” Opt. Express 20(2), 934–940 (2012).
[Crossref] [PubMed]

A. Schaap, T. Rohrlack, and Y. Bellouard, “Optical classification of algae species with a glass lab-on-a-chip,” Lab Chip 12(8), 1527–1532 (2012).
[Crossref] [PubMed]

B. Lenssen and Y. Bellouard, “Optically transparent glass micro-actuator fabricated by femtosecond laser exposure and chemical etching,” Appl. Phys. Lett. 101(10), 103503 (2012).
[Crossref]

2011 (3)

2010 (1)

2009 (1)

R. M. Vazquez, R. Osellame, D. Nolli, C. Dongre, H. van den Vlekkert, R. Ramponi, M. Pollnau, and G. Cerullo, “Integration of femtosecond laser written optical waveguides in a lab-on-chip,” Lab Chip 9(1), 91–96 (2009).
[Crossref] [PubMed]

2008 (3)

2006 (3)

R. Graf, A. Fernandez, M. Dubov, H. J. Brueckner, B. N. Chichkov, and A. Apolonski, “Pearl-chain waveguides written at megahertz repetition rate,” Appl. Phys. B 87(1), 21–27 (2006).
[Crossref]

W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89(2), 021106 (2006).
[Crossref]

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

2005 (3)

2004 (3)

2003 (3)

C. B. Schaffer, J. F. Garciá, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76(3), 351–354 (2003).
[Crossref]

R. S. Taylor, C. Hnatovsky, E. Simova, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Femtosecond laser fabrication of nanostructures in silica glass,” Opt. Lett. 28(12), 1043–1045 (2003).
[Crossref] [PubMed]

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91(24), 247405 (2003).
[Crossref] [PubMed]

2002 (2)

R. Osellame, S. Taccheo, G. Cerullo, M. Marangoni, D. Polli, R. Ramponi, P. Laporta, and S. De Silvestri, “Optical gain in Er-Yb doped waveguides fabricated by femtosecond laser pulses,” Electron. Lett. 38(17), 964–965 (2002).
[Crossref]

W. Watanabe and K. Itoh, “Motion of bubble in solid by femtosecond laser pulses,” Opt. Express 10(14), 603–608 (2002).
[Crossref] [PubMed]

2001 (2)

2000 (1)

Y. Sikorski, A. A. Said, P. Bado, R. Maynard, C. Florea, and K. A. Winick, “Optical waveguide amplifier in Nd-doped glass written with near-IR femtosecond laser pulses,” Electron. Lett. 36(3), 226–227 (2000).
[Crossref]

1997 (2)

E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. 71(7), 882–884 (1997).
[Crossref]

A. Agarwal and M. Tomozawa, “Correlation of silica glass properties with the infrared spectra,” J. Non-Cryst. Solids 209(1-2), 166–174 (1997).
[Crossref]

1996 (2)

1995 (1)

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

1994 (1)

D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser‐induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. 64(23), 3071–3073 (1994).
[Crossref]

1960 (1)

S. Spinner and G. W. Cleek, “Temperature Dependence of Young’s Modulus of Vitreous Germania and Silica,” J. Appl. Phys. 31(8), 1407–1410 (1960).
[Crossref]

Agarwal, A.

A. Agarwal and M. Tomozawa, “Correlation of silica glass properties with the infrared spectra,” J. Non-Cryst. Solids 209(1-2), 166–174 (1997).
[Crossref]

Aitchison, J. S.

Apolonski, A.

R. Graf, A. Fernandez, M. Dubov, H. J. Brueckner, B. N. Chichkov, and A. Apolonski, “Pearl-chain waveguides written at megahertz repetition rate,” Appl. Phys. B 87(1), 21–27 (2006).
[Crossref]

Arai, A.

Athanasiou, C.-E.

C.-E. Athanasiou and Y. Bellouard, “A monolithic micro-tensile tester for investigating silicon dioxide polymorph micromechanics, fabricated and operated using a femtosecond laser,” Micromachines (Basel) 6(9), 1365–1386 (2015).
[Crossref]

Bado, P.

Barthel, E.

Bellec, M.

Bellouard, Y.

Y. Bellouard, A. Champion, B. McMillen, S. Mukherjee, R. R. Thomson, C. Pépin, P. Gillet, and Y. Cheng, “Stress-state manipulation in fused silica via femtosecond laser irradiation,” Optica 3(12), 1285 (2016).
[Crossref]

Y. Liao, J. Ni, L. Qiao, M. Huang, Y. Bellouard, K. Sugioka, and Y. Cheng, “High-fidelity visualization of formation of volume nanogratings in porous glass by femtosecond laser irradiation,” Optica 2(4), 329 (2015).
[Crossref]

N. Groothoff, M.-O. Hongler, P. Kazansky, and Y. Bellouard, “Transition and self-healing process between chaotic and self-organized patterns observed during femtosecond laser writing,” Opt. Express 23(13), 16993–17007 (2015).
[Crossref] [PubMed]

C.-E. Athanasiou and Y. Bellouard, “A monolithic micro-tensile tester for investigating silicon dioxide polymorph micromechanics, fabricated and operated using a femtosecond laser,” Micromachines (Basel) 6(9), 1365–1386 (2015).
[Crossref]

T. Yang and Y. Bellouard, “Monolithic transparent 3D dielectrophoretic micro-actuator fabricated by femtosecond laser,” J. Micromech. Microeng. 25(10), 105009 (2015).
[Crossref]

V. Tielen and Y. Bellouard, “Three-dimensional glass monolithic microflexure fabricated by femtosecond laser exposure and chemical etching,” Micromachines (Basel) 5(3), 697–710 (2014).
[Crossref]

B. Lenssen and Y. Bellouard, “Optically transparent glass micro-actuator fabricated by femtosecond laser exposure and chemical etching,” Appl. Phys. Lett. 101(10), 103503 (2012).
[Crossref]

A. Schaap, T. Rohrlack, and Y. Bellouard, “Optical classification of algae species with a glass lab-on-a-chip,” Lab Chip 12(8), 1527–1532 (2012).
[Crossref] [PubMed]

Y. Bellouard and M.-O. Hongler, “Femtosecond-laser generation of self-organized bubble patterns in fused silica,” Opt. Express 19(7), 6807–6821 (2011).
[Crossref] [PubMed]

S. Rajesh and Y. Bellouard, “Towards fast femtosecond laser micromachining of fused silica: The effect of deposited energy,” Opt. Express 18(20), 21490–21497 (2010).
[Crossref] [PubMed]

Y. Bellouard, E. Barthel, A. A. Said, M. Dugan, and P. Bado, “Scanning thermal microscopy and Raman analysis of bulk fused silica exposed to low-energy femtosecond laser pulses,” Opt. Express 16(24), 19520–19534 (2008).
[Crossref] [PubMed]

Y. Bellouard, A. Said, and P. Bado, “Integrating optics and micro-mechanics in a single substrate: a step toward monolithic integration in fused silica,” Opt. Express 13(17), 6635–6644 (2005).
[Crossref] [PubMed]

Y. Bellouard, A. Said, M. Dugan, and P. Bado, “Fabrication of high-aspect ratio, micro-fluidic channels and tunnels using femtosecond laser pulses and chemical etching,” Opt. Express 12(10), 2120–2129 (2004).
[Crossref] [PubMed]

Y. Bellouard, A. Said, M. Dugan, and P. Bado, “Monolithic three dimensional integration of micro-fluidic channels and optical waveguides in fused silica,” in Proceedings of Materials Research Society (MRS) Symposium (2003), pp. 63–68.
[Crossref]

Beresna, M.

J. Zhang, M. Gecevičius, M. Beresna, and P. G. Kazansky, “Seemingly unlimited lifetime data storage in nanostructured glass,” Phys. Rev. Lett. 112(3), 033901 (2014).
[Crossref] [PubMed]

M. Beresna, M. Gecevičius, and P. G. Kazansky, “Ultrafast laser direct writing and nanostructuring in transparent materials,” Adv. Opt. Photonics 6(3), 293 (2014).
[Crossref]

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J. Canning, M. Lancry, K. Cook, A. Weickman, F. Brisset, and B. Poumellec, “Anatomy of a femtosecond laser processed silica waveguide [Invited],” Opt. Mater. Express 1(5), 998–1008 (2011).
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J. Canning, M. Lancry, K. Cook, A. Weickman, F. Brisset, and B. Poumellec, “Anatomy of a femtosecond laser processed silica waveguide [Invited],” Opt. Mater. Express 1(5), 998–1008 (2011).
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J. Canning, M. Lancry, K. Cook, A. Weickman, F. Brisset, and B. Poumellec, “Anatomy of a femtosecond laser processed silica waveguide [Invited],” Opt. Mater. Express 1(5), 998–1008 (2011).
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C. Hnatovsky, R. S. Taylor, P. P. Rajeev, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Pulse duration dependence of femtosecond-laser-fabricated nanogratings in fused silica,” Appl. Phys. Lett. 87(1), 014104 (2005).
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R. M. Vazquez, R. Osellame, D. Nolli, C. Dongre, H. van den Vlekkert, R. Ramponi, M. Pollnau, and G. Cerullo, “Integration of femtosecond laser written optical waveguides in a lab-on-chip,” Lab Chip 9(1), 91–96 (2009).
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R. Graf, A. Fernandez, M. Dubov, H. J. Brueckner, B. N. Chichkov, and A. Apolonski, “Pearl-chain waveguides written at megahertz repetition rate,” Appl. Phys. B 87(1), 21–27 (2006).
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W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89(2), 021106 (2006).
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Kazansky, P. G.

J. Zhang, M. Gecevičius, M. Beresna, and P. G. Kazansky, “Seemingly unlimited lifetime data storage in nanostructured glass,” Phys. Rev. Lett. 112(3), 033901 (2014).
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M. Beresna, M. Gecevičius, and P. G. Kazansky, “Ultrafast laser direct writing and nanostructuring in transparent materials,” Adv. Opt. Photonics 6(3), 293 (2014).
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Korn, G.

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M. Lancry, E. Régnier, and B. Poumellec, “Fictive temperature in silica-based glasses and its application to optical fiber manufacturing,” Prog. Mater. Sci. 57(1), 63–94 (2012).
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J. Canning, M. Lancry, K. Cook, A. Weickman, F. Brisset, and B. Poumellec, “Anatomy of a femtosecond laser processed silica waveguide [Invited],” Opt. Mater. Express 1(5), 998–1008 (2011).
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R. Osellame, S. Taccheo, G. Cerullo, M. Marangoni, D. Polli, R. Ramponi, P. Laporta, and S. De Silvestri, “Optical gain in Er-Yb doped waveguides fabricated by femtosecond laser pulses,” Electron. Lett. 38(17), 964–965 (2002).
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M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
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R. Osellame, S. Taccheo, G. Cerullo, M. Marangoni, D. Polli, R. Ramponi, P. Laporta, and S. De Silvestri, “Optical gain in Er-Yb doped waveguides fabricated by femtosecond laser pulses,” Electron. Lett. 38(17), 964–965 (2002).
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Y. Sikorski, A. A. Said, P. Bado, R. Maynard, C. Florea, and K. A. Winick, “Optical waveguide amplifier in Nd-doped glass written with near-IR femtosecond laser pulses,” Electron. Lett. 36(3), 226–227 (2000).
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C. B. Schaffer, J. F. Garciá, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76(3), 351–354 (2003).
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M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
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D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser‐induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. 64(23), 3071–3073 (1994).
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W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89(2), 021106 (2006).
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R. M. Vazquez, R. Osellame, D. Nolli, C. Dongre, H. van den Vlekkert, R. Ramponi, M. Pollnau, and G. Cerullo, “Integration of femtosecond laser written optical waveguides in a lab-on-chip,” Lab Chip 9(1), 91–96 (2009).
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S. Richter, S. During, F. Burmeister, F. Zimmermann, A. Tünnermann, and S. Nolte, “Formation of periodic disruptions induced by heat accumulation of femtosecond laser pulses,” Opt. Express 21(13), 15452–15463 (2013).
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S. Richter, S. During, A. Tannermann, and S. Nolte, “Bonding of glass with femtosecond laser pulses at high repetition rates,” Appl. Phys., A Mater. Sci. Process. 103(2), 257–261 (2011).
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K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
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Okamoto, Y.

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W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89(2), 021106 (2006).
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R. M. Vazquez, R. Osellame, D. Nolli, C. Dongre, H. van den Vlekkert, R. Ramponi, M. Pollnau, and G. Cerullo, “Integration of femtosecond laser written optical waveguides in a lab-on-chip,” Lab Chip 9(1), 91–96 (2009).
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R. Osellame, S. Taccheo, G. Cerullo, M. Marangoni, D. Polli, R. Ramponi, P. Laporta, and S. De Silvestri, “Optical gain in Er-Yb doped waveguides fabricated by femtosecond laser pulses,” Electron. Lett. 38(17), 964–965 (2002).
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Perez-Leija, A.

T. Meany, M. Grafe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics: Laser written quantum circuits,” Laser Photonics Rev. 9(4), 363–384 (2015).
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B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett. 74(12), 2248–2251 (1995).
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R. Osellame, S. Taccheo, G. Cerullo, M. Marangoni, D. Polli, R. Ramponi, P. Laporta, and S. De Silvestri, “Optical gain in Er-Yb doped waveguides fabricated by femtosecond laser pulses,” Electron. Lett. 38(17), 964–965 (2002).
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R. M. Vazquez, R. Osellame, D. Nolli, C. Dongre, H. van den Vlekkert, R. Ramponi, M. Pollnau, and G. Cerullo, “Integration of femtosecond laser written optical waveguides in a lab-on-chip,” Lab Chip 9(1), 91–96 (2009).
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M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation: In the heart of nanogratings,” Laser Photonics Rev. 1, 10 (2013).

Poumellec, B.

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation: In the heart of nanogratings,” Laser Photonics Rev. 1, 10 (2013).

M. Lancry, E. Régnier, and B. Poumellec, “Fictive temperature in silica-based glasses and its application to optical fiber manufacturing,” Prog. Mater. Sci. 57(1), 63–94 (2012).
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J. Canning, M. Lancry, K. Cook, A. Weickman, F. Brisset, and B. Poumellec, “Anatomy of a femtosecond laser processed silica waveguide [Invited],” Opt. Mater. Express 1(5), 998–1008 (2011).
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Qiu, J.

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91(24), 247405 (2003).
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C. Hnatovsky, R. S. Taylor, P. P. Rajeev, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Pulse duration dependence of femtosecond-laser-fabricated nanogratings in fused silica,” Appl. Phys. Lett. 87(1), 014104 (2005).
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Ramponi, R.

R. M. Vazquez, R. Osellame, D. Nolli, C. Dongre, H. van den Vlekkert, R. Ramponi, M. Pollnau, and G. Cerullo, “Integration of femtosecond laser written optical waveguides in a lab-on-chip,” Lab Chip 9(1), 91–96 (2009).
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R. Osellame, S. Taccheo, G. Cerullo, M. Marangoni, D. Polli, R. Ramponi, P. Laporta, and S. De Silvestri, “Optical gain in Er-Yb doped waveguides fabricated by femtosecond laser pulses,” Electron. Lett. 38(17), 964–965 (2002).
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C. Hnatovsky, R. S. Taylor, P. P. Rajeev, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Pulse duration dependence of femtosecond-laser-fabricated nanogratings in fused silica,” Appl. Phys. Lett. 87(1), 014104 (2005).
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R. S. Taylor, C. Hnatovsky, E. Simova, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Femtosecond laser fabrication of nanostructures in silica glass,” Opt. Lett. 28(12), 1043–1045 (2003).
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M. Lancry, E. Régnier, and B. Poumellec, “Fictive temperature in silica-based glasses and its application to optical fiber manufacturing,” Prog. Mater. Sci. 57(1), 63–94 (2012).
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S. Richter, S. During, F. Burmeister, F. Zimmermann, A. Tünnermann, and S. Nolte, “Formation of periodic disruptions induced by heat accumulation of femtosecond laser pulses,” Opt. Express 21(13), 15452–15463 (2013).
[Crossref] [PubMed]

S. Richter, S. During, A. Tannermann, and S. Nolte, “Bonding of glass with femtosecond laser pulses at high repetition rates,” Appl. Phys., A Mater. Sci. Process. 103(2), 257–261 (2011).
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B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett. 74(12), 2248–2251 (1995).
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Sakakura, M.

Schaap, A.

A. Schaap, T. Rohrlack, and Y. Bellouard, “Optical classification of algae species with a glass lab-on-a-chip,” Lab Chip 12(8), 1527–1532 (2012).
[Crossref] [PubMed]

Schaffer, C. B.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

C. B. Schaffer, J. F. Garciá, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76(3), 351–354 (2003).
[Crossref]

Schmidt, M.

Schulz, W.

Shah, L.

Shimizu, M.

Shimotsuma, Y.

Shore, B. W.

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

Sikorski, Y.

Y. Sikorski, A. A. Said, P. Bado, R. Maynard, C. Florea, and K. A. Winick, “Optical waveguide amplifier in Nd-doped glass written with near-IR femtosecond laser pulses,” Electron. Lett. 36(3), 226–227 (2000).
[Crossref]

Simova, E.

C. Hnatovsky, R. S. Taylor, P. P. Rajeev, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Pulse duration dependence of femtosecond-laser-fabricated nanogratings in fused silica,” Appl. Phys. Lett. 87(1), 014104 (2005).
[Crossref]

R. S. Taylor, C. Hnatovsky, E. Simova, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Femtosecond laser fabrication of nanostructures in silica glass,” Opt. Lett. 28(12), 1043–1045 (2003).
[Crossref] [PubMed]

Spinner, S.

S. Spinner and G. W. Cleek, “Temperature Dependence of Young’s Modulus of Vitreous Germania and Silica,” J. Appl. Phys. 31(8), 1407–1410 (1960).
[Crossref]

Squier, J.

D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser‐induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. 64(23), 3071–3073 (1994).
[Crossref]

Steel, M. J.

T. Meany, M. Grafe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics: Laser written quantum circuits,” Laser Photonics Rev. 9(4), 363–384 (2015).
[Crossref]

Streltsov, A. M.

Stuart, B. C.

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

Sugimoto, N.

Sugioka, K.

Sun, M.

Szameit, A.

T. Meany, M. Grafe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics: Laser written quantum circuits,” Laser Photonics Rev. 9(4), 363–384 (2015).
[Crossref]

Taccheo, S.

R. Osellame, S. Taccheo, G. Cerullo, M. Marangoni, D. Polli, R. Ramponi, P. Laporta, and S. De Silvestri, “Optical gain in Er-Yb doped waveguides fabricated by femtosecond laser pulses,” Electron. Lett. 38(17), 964–965 (2002).
[Crossref]

Tamaki, T.

W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89(2), 021106 (2006).
[Crossref]

Tanabe, R.

Tannermann, A.

S. Richter, S. During, A. Tannermann, and S. Nolte, “Bonding of glass with femtosecond laser pulses at high repetition rates,” Appl. Phys., A Mater. Sci. Process. 103(2), 257–261 (2011).
[Crossref]

Taylor, R. S.

C. Hnatovsky, R. S. Taylor, P. P. Rajeev, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Pulse duration dependence of femtosecond-laser-fabricated nanogratings in fused silica,” Appl. Phys. Lett. 87(1), 014104 (2005).
[Crossref]

R. S. Taylor, C. Hnatovsky, E. Simova, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Femtosecond laser fabrication of nanostructures in silica glass,” Opt. Lett. 28(12), 1043–1045 (2003).
[Crossref] [PubMed]

Thomson, R. R.

Tielen, V.

V. Tielen and Y. Bellouard, “Three-dimensional glass monolithic microflexure fabricated by femtosecond laser exposure and chemical etching,” Micromachines (Basel) 5(3), 697–710 (2014).
[Crossref]

Tomozawa, M.

A. Agarwal and M. Tomozawa, “Correlation of silica glass properties with the infrared spectra,” J. Non-Cryst. Solids 209(1-2), 166–174 (1997).
[Crossref]

Tünnermann, A.

van den Vlekkert, H.

R. M. Vazquez, R. Osellame, D. Nolli, C. Dongre, H. van den Vlekkert, R. Ramponi, M. Pollnau, and G. Cerullo, “Integration of femtosecond laser written optical waveguides in a lab-on-chip,” Lab Chip 9(1), 91–96 (2009).
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Vazquez, R. M.

R. M. Vazquez, R. Osellame, D. Nolli, C. Dongre, H. van den Vlekkert, R. Ramponi, M. Pollnau, and G. Cerullo, “Integration of femtosecond laser written optical waveguides in a lab-on-chip,” Lab Chip 9(1), 91–96 (2009).
[Crossref] [PubMed]

Watanabe, W.

W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89(2), 021106 (2006).
[Crossref]

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

W. Watanabe and K. Itoh, “Motion of bubble in solid by femtosecond laser pulses,” Opt. Express 10(14), 603–608 (2002).
[Crossref] [PubMed]

Weickman, A.

Winick, K. A.

Y. Sikorski, A. A. Said, P. Bado, R. Maynard, C. Florea, and K. A. Winick, “Optical waveguide amplifier in Nd-doped glass written with near-IR femtosecond laser pulses,” Electron. Lett. 36(3), 226–227 (2000).
[Crossref]

Withford, M. J.

T. Meany, M. Grafe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics: Laser written quantum circuits,” Laser Photonics Rev. 9(4), 363–384 (2015).
[Crossref]

Yamaji, M.

Yang, T.

T. Yang and Y. Bellouard, “Monolithic transparent 3D dielectrophoretic micro-actuator fabricated by femtosecond laser,” J. Micromech. Microeng. 25(10), 105009 (2015).
[Crossref]

Yoshino, F.

Zhang, H.

Zhang, J.

J. Zhang, M. Gecevičius, M. Beresna, and P. G. Kazansky, “Seemingly unlimited lifetime data storage in nanostructured glass,” Phys. Rev. Lett. 112(3), 033901 (2014).
[Crossref] [PubMed]

Zhu, J.

Zimmermann, F.

Žukauskas, A.

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
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Adv. Opt. Photonics (1)

M. Beresna, M. Gecevičius, and P. G. Kazansky, “Ultrafast laser direct writing and nanostructuring in transparent materials,” Adv. Opt. Photonics 6(3), 293 (2014).
[Crossref]

Appl. Phys. B (1)

R. Graf, A. Fernandez, M. Dubov, H. J. Brueckner, B. N. Chichkov, and A. Apolonski, “Pearl-chain waveguides written at megahertz repetition rate,” Appl. Phys. B 87(1), 21–27 (2006).
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Appl. Phys. Lett. (5)

D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser‐induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. 64(23), 3071–3073 (1994).
[Crossref]

E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. 71(7), 882–884 (1997).
[Crossref]

C. Hnatovsky, R. S. Taylor, P. P. Rajeev, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Pulse duration dependence of femtosecond-laser-fabricated nanogratings in fused silica,” Appl. Phys. Lett. 87(1), 014104 (2005).
[Crossref]

B. Lenssen and Y. Bellouard, “Optically transparent glass micro-actuator fabricated by femtosecond laser exposure and chemical etching,” Appl. Phys. Lett. 101(10), 103503 (2012).
[Crossref]

W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89(2), 021106 (2006).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (2)

S. Richter, S. During, A. Tannermann, and S. Nolte, “Bonding of glass with femtosecond laser pulses at high repetition rates,” Appl. Phys., A Mater. Sci. Process. 103(2), 257–261 (2011).
[Crossref]

C. B. Schaffer, J. F. Garciá, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76(3), 351–354 (2003).
[Crossref]

Electron. Lett. (2)

R. Osellame, S. Taccheo, G. Cerullo, M. Marangoni, D. Polli, R. Ramponi, P. Laporta, and S. De Silvestri, “Optical gain in Er-Yb doped waveguides fabricated by femtosecond laser pulses,” Electron. Lett. 38(17), 964–965 (2002).
[Crossref]

Y. Sikorski, A. A. Said, P. Bado, R. Maynard, C. Florea, and K. A. Winick, “Optical waveguide amplifier in Nd-doped glass written with near-IR femtosecond laser pulses,” Electron. Lett. 36(3), 226–227 (2000).
[Crossref]

J. Appl. Phys. (1)

S. Spinner and G. W. Cleek, “Temperature Dependence of Young’s Modulus of Vitreous Germania and Silica,” J. Appl. Phys. 31(8), 1407–1410 (1960).
[Crossref]

J. Micromech. Microeng. (1)

T. Yang and Y. Bellouard, “Monolithic transparent 3D dielectrophoretic micro-actuator fabricated by femtosecond laser,” J. Micromech. Microeng. 25(10), 105009 (2015).
[Crossref]

J. Non-Cryst. Solids (2)

D. M. Krol, “Femtosecond laser modification of glass,” J. Non-Cryst. Solids 354(2-9), 416–424 (2008).
[Crossref]

A. Agarwal and M. Tomozawa, “Correlation of silica glass properties with the infrared spectra,” J. Non-Cryst. Solids 209(1-2), 166–174 (1997).
[Crossref]

Lab Chip (2)

R. M. Vazquez, R. Osellame, D. Nolli, C. Dongre, H. van den Vlekkert, R. Ramponi, M. Pollnau, and G. Cerullo, “Integration of femtosecond laser written optical waveguides in a lab-on-chip,” Lab Chip 9(1), 91–96 (2009).
[Crossref] [PubMed]

A. Schaap, T. Rohrlack, and Y. Bellouard, “Optical classification of algae species with a glass lab-on-a-chip,” Lab Chip 12(8), 1527–1532 (2012).
[Crossref] [PubMed]

Laser Photonics Rev. (2)

T. Meany, M. Grafe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics: Laser written quantum circuits,” Laser Photonics Rev. 9(4), 363–384 (2015).
[Crossref]

M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation: In the heart of nanogratings,” Laser Photonics Rev. 1, 10 (2013).

Light Sci. Appl. (2)

K. Sugioka and Y. Cheng, “Ultrafast lasers—reliable tools for advanced materials processing,” Light Sci. Appl. 3(4), e149 (2014).
[Crossref]

M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light Sci. Appl. 5(8), e16133 (2016).
[Crossref]

Micromachines (Basel) (2)

V. Tielen and Y. Bellouard, “Three-dimensional glass monolithic microflexure fabricated by femtosecond laser exposure and chemical etching,” Micromachines (Basel) 5(3), 697–710 (2014).
[Crossref]

C.-E. Athanasiou and Y. Bellouard, “A monolithic micro-tensile tester for investigating silicon dioxide polymorph micromechanics, fabricated and operated using a femtosecond laser,” Micromachines (Basel) 6(9), 1365–1386 (2015).
[Crossref]

MRS Bull. (1)

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Opt. Express (12)

Y. Bellouard and M.-O. Hongler, “Femtosecond-laser generation of self-organized bubble patterns in fused silica,” Opt. Express 19(7), 6807–6821 (2011).
[Crossref] [PubMed]

S. Richter, S. During, F. Burmeister, F. Zimmermann, A. Tünnermann, and S. Nolte, “Formation of periodic disruptions induced by heat accumulation of femtosecond laser pulses,” Opt. Express 21(13), 15452–15463 (2013).
[Crossref] [PubMed]

K. Cvecek, I. Miyamoto, and M. Schmidt, “Gas bubble formation in fused silica generated by ultra-short laser pulses,” Opt. Express 22(13), 15877–15893 (2014).
[Crossref] [PubMed]

M. Shimizu, M. Sakakura, M. Ohnishi, M. Yamaji, Y. Shimotsuma, K. Hirao, and K. Miura, “Three-dimensional temperature distribution and modification mechanism in glass during ultrafast laser irradiation at high repetition rates,” Opt. Express 20(2), 934–940 (2012).
[Crossref] [PubMed]

Y. Bellouard, A. Said, and P. Bado, “Integrating optics and micro-mechanics in a single substrate: a step toward monolithic integration in fused silica,” Opt. Express 13(17), 6635–6644 (2005).
[Crossref] [PubMed]

Y. Bellouard, A. Said, M. Dugan, and P. Bado, “Fabrication of high-aspect ratio, micro-fluidic channels and tunnels using femtosecond laser pulses and chemical etching,” Opt. Express 12(10), 2120–2129 (2004).
[Crossref] [PubMed]

Y. Bellouard, E. Barthel, A. A. Said, M. Dugan, and P. Bado, “Scanning thermal microscopy and Raman analysis of bulk fused silica exposed to low-energy femtosecond laser pulses,” Opt. Express 16(24), 19520–19534 (2008).
[Crossref] [PubMed]

N. Groothoff, M.-O. Hongler, P. Kazansky, and Y. Bellouard, “Transition and self-healing process between chaotic and self-organized patterns observed during femtosecond laser writing,” Opt. Express 23(13), 16993–17007 (2015).
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S. Eaton, H. Zhang, P. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate,” Opt. Express 13(12), 4708–4716 (2005).
[Crossref] [PubMed]

W. Watanabe and K. Itoh, “Motion of bubble in solid by femtosecond laser pulses,” Opt. Express 10(14), 603–608 (2002).
[Crossref] [PubMed]

I. Miyamoto, Y. Okamoto, R. Tanabe, Y. Ito, K. Cvecek, and M. Schmidt, “Mechanism of dynamic plasma motion in internal modification of glass by fs-laser pulses at high pulse repetition rate,” Opt. Express 24(22), 25718–25731 (2016).
[Crossref] [PubMed]

S. Rajesh and Y. Bellouard, “Towards fast femtosecond laser micromachining of fused silica: The effect of deposited energy,” Opt. Express 18(20), 21490–21497 (2010).
[Crossref] [PubMed]

Opt. Lett. (9)

K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21(21), 1729–1731 (1996).
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A. M. Streltsov and N. F. Borrelli, “Fabrication and analysis of a directional coupler written in glass by nanojoule femtosecond laser pulses,” Opt. Lett. 26(1), 42–43 (2001).
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K. Minoshima, A. M. Kowalevicz, I. Hartl, E. P. Ippen, and J. G. Fujimoto, “Photonic device fabrication in glass by use of nonlinear materials processing with a femtosecond laser oscillator,” Opt. Lett. 26(19), 1516–1518 (2001).
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E. Bricchi, B. G. Klappauf, and P. G. Kazansky, “Form birefringence and negative index change created by femtosecond direct writing in transparent materials,” Opt. Lett. 29(1), 119–121 (2004).
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R. S. Taylor, C. Hnatovsky, E. Simova, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Femtosecond laser fabrication of nanostructures in silica glass,” Opt. Lett. 28(12), 1043–1045 (2003).
[Crossref] [PubMed]

Y. Cheng, K. Sugioka, and K. Midorikawa, “Microfluidic laser embedded in glass by three-dimensional femtosecond laser microprocessing,” Opt. Lett. 29(17), 2007–2009 (2004).
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E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, “Three-dimensional optical storage inside transparent materials,” Opt. Lett. 21(24), 2023–2025 (1996).
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Opt. Mater. Express (2)

Optica (2)

Phys. Rev. Lett. (3)

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91(24), 247405 (2003).
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B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett. 74(12), 2248–2251 (1995).
[Crossref] [PubMed]

J. Zhang, M. Gecevičius, M. Beresna, and P. G. Kazansky, “Seemingly unlimited lifetime data storage in nanostructured glass,” Phys. Rev. Lett. 112(3), 033901 (2014).
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Prog. Mater. Sci. (1)

M. Lancry, E. Régnier, and B. Poumellec, “Fictive temperature in silica-based glasses and its application to optical fiber manufacturing,” Prog. Mater. Sci. 57(1), 63–94 (2012).
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Y. Bellouard, A. Said, M. Dugan, and P. Bado, “Monolithic three dimensional integration of micro-fluidic channels and optical waveguides in fused silica,” in Proceedings of Materials Research Society (MRS) Symposium (2003), pp. 63–68.
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Supplementary Material (1)

NameDescription
» Visualization 1       The video illustrates the formation of bubbles during laser exposure of fused silica at high repetition rate. While scanning a line across the specimen, bubbles 'burst' from the laser focus, where non-linear absorption effects and bulk heating take p

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Figures (10)

Fig. 1
Fig. 1 Schematic of the experimental setup: the specimen is translated under a focused femtosecond laser beam emitting pulses at a rate of ~3 MHz. The laser source is an Ytterbium, fiber laser emitting 270 fs-pulses. In these experiments, the laser is linearly polarized. Dynamic observations are done using transverse imaging. The fluctuation of transmitted intensity is monitored to observe the dynamic of formation of bubbles as it will be shown in Fig. 4.
Fig. 2
Fig. 2 a) Optical microscope images of a train of bubbles for moderate deposited energy, for pulse energy of 230 nJ delivered at ~3 MHz. b) SEM image of an interface sliced through the longitudinal plane of the laser track. S represents the scanning direction, k the laser propagation and E the polarization state, respectively. The energies deposited indicated here correspond to a scanning velocity of ~32 mm/s and 10 mm/s respectively.
Fig. 3
Fig. 3 a) Microscope image showing the transition between bubble explosion to a continuous modification while increasing the deposited energy up to 10000 J/mm2 (with yet the same pulse energy 230 nJ and repetition rate ~3 MHz). b) SEM image showing formation of nanostructures between bubble modifications. Inset is a magnification of structures induced between bubble trains. S represents the scanning direction, k the laser propagation and E the polarization state, respectively.
Fig. 4
Fig. 4 Regime of very high energy deposited. a) Microscopic image showing the complete disappearance of bubble patterns at high dose still. b) SEM images of transverse cuts of the specimen. Formation of self-organized periodic nanostructures found in the upper part of the modified zones. (Pulse energy is 230 nJ.) S represents the scanning direction, k the laser propagation and E the polarization state, respectively.
Fig. 5
Fig. 5 Dynamic observation of the nucleation and propelling of gas bubbles in the bulk of the substrate. Top image is a microscope image of the bubbles sequence. The oscilloscope traces below is the corresponding intensity of the beam passing through the specimen during the formation of the bubble (see Fig. 1). As evidenced here, the bubbles shadow the transmitted intensity, making the event seen on the optical image clearly identifiable on the oscilloscope traces. The time resolution is 0.1 ms. (see also Visualization 1)
Fig. 6
Fig. 6 Optical images showing evidences that the bubbles propagates toward the laser beam and not under the action of buoyancy forces. The same substrate was exposed from both surfaces. The direction of propagation of the laser was flipped between the two set of experiments (top and bottom) while the writing direction and orientation of the substrate with respect to gravity was kept the same.
Fig. 7
Fig. 7 Tomography performed by cutting laser exposed lines consisting in intermittent patterns like the one displayed in Figs. 3 and 5. The cut is performed at an angle (as illustrated in the lower left corner) in order to reveal different regions at different depth across the laser affected zones. As the speed decreases and the amount of deposited energy increases, gas bubbles eventually disappears, leaving self-organized structures resembling nanogratings on top on continuously modified zones. On these images, the laser propagated from left to right and the polarization is aligned with the writing direction.
Fig. 8
Fig. 8 Morphologies as a function of pulse and deposited energies. The polarization state of the laser (longitudinal to the writing direction), the repetition rate (~3MHz) and the writing direction are the same for all experiments in this graph. Each symbol represents one experimental data point. The SEM images illustrate typical morphologies found in the three regimes observed. (The scale bar in the SEM micrograph is 2 microns.)
Fig. 9
Fig. 9 Oblique sections of laser exposed zones under three different polarizations (45 degree, 90 degree, and 0 degree, with respect to the writing direction) and three difference writing speed. All the other exposure parameters are kept similar for all the tracks. The pulse energy is 230 nJ. Note the absence of large bubble forming under 45 degree and perpendicular polarization from the writing direction. The sample preparation procedure is the same as for Fig. 7. The surface of the specimen is in this case oriented toward the bottom of the images. A and B are close-up view of particularly noticeable features found for two polarization states.
Fig. 10
Fig. 10 Phenomenological interpretation for the formation of bubbles. We propose an interpretation based on three energy thresholds. The first one is the threshold for non-linear absorption followed by a threshold for bubbles nucleation and growth and then finally, a third threshold corresponding to the collapsing of the interlayers between nanogratings. In this interpretation, step 5 and 6 are eventually suppressed at high exposure dose due to the increase of the Young modulus of the material upon heating [53].

Equations (1)

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Φ d = 4 E p πw_nla ( f v )

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