Abstract

Femtosecond laser exposure in the non-ablative regime induces a variety of bulk structural modifications, in which anisotropy may depend on the polarization of the writing beam. In this work, we investigate the correlation between polarization state and stress anisotropy. In particular, we introduce a methodology that allows for rapid analysis and visualization of laser-induced stress anisotropy in glasses and crystals. Using radial and azimuthal polarization, we also demonstrate stress states that are nearly isotropic.

© 2015 Optical Society of America

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  1. Y. Cheng, K. Sugioka, K. Midorikawa, M. Masuda, K. Toyoda, M. Kawachi, and K. Shihoyama, “Control of the cross-sectional shape of a hollow microchannel embedded in photostructurable glass by use of a femtosecond laser,” Opt. Lett. 28(1), 55–57 (2003).
    [Crossref] [PubMed]
  2. K. Sugioka, Y. Hanada, and K. Midorikawa, “Three-dimensional femtosecond laser micromachining of photosensitive glass for biomicrochips,” Laser Photon. Rev. 4(3), 386–400 (2010).
    [Crossref]
  3. J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82(15), 2371–2373 (2003).
    [Crossref]
  4. K. Minoshima, A. Kowalevicz, E. Ippen, and J. Fujimoto, “Fabrication of coupled mode photonic devices in glass by nonlinear femtosecond laser materials processing,” Opt. Express 10(15), 645–652 (2002).
    [Crossref] [PubMed]
  5. G. A. Torchia, C. Mendez, I. Arias, L. Roso, A. Rodenas, and D. Jaque, “Laser gain in femtosecond microstructured Nd:MgO:LiNbO3crystals,” Appl. Phys. B-Lasers O. 83(4), 559–563 (2006).
    [Crossref]
  6. 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]
  7. G. A. Torchia, P. F. Meilán, A. Ródenas, D. Jaque, C. Mendez, and L. Roso, “Femtosecond laser written surface waveguides fabricated in Nd:YAG ceramics,” Opt. Express 15(20), 13266–13271 (2007).
    [Crossref] [PubMed]
  8. P. Nandi, G. Jose, C. Jayakrishnan, S. Debbarma, K. Chalapathi, K. Alti, A. K. Dharmadhikari, J. A. Dharmadhikari, and D. Mathur, “Femtosecond laser written channel waveguides in tellurite glass,” Opt. Express 14(25), 12145–12150 (2006).
    [Crossref] [PubMed]
  9. Y. Tokuda, M. Saito, M. Takahashi, and K. Yamada, “Waveguide formation in niobium tellurite glasses by pico- and femtosecond laser pulses,” J. Non-Cryst. Solids 326–327, 472–475 (2003).
    [Crossref]
  10. S. Juodkazis and H. Misawa, “Laser processing of sapphire by strongly focused femtosecond pulses,” Appl. Phys. Adv. Mater. 93(4), 857–861 (2008).
  11. B. McMillen, K. Chen, H. An, S. Fleming, V. Hartwell, and D. Snoke, “Waveguiding and nonlinear optical properties of three-dimensional waveguides in LiTaO3 written by high-repetition rate ultrafast laser,” Appl. Phys. Lett. 93(11), 111106 (2008).
    [Crossref]
  12. S. Hasegawa and Y. Hayasaki, “Holographic Vector Wave Femtosecond Laser Processing,” Int. J. Optomechatronics 8(2), 73–88 (2014).
    [Crossref]
  13. 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).
    [Crossref] [PubMed]
  14. 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]
  15. 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).
    [Crossref] [PubMed]
  16. V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
    [Crossref] [PubMed]
  17. J. Bai, G. Cheng, X. Long, Y. Wang, W. Zhao, G. Chen, R. Stoian, and R. Hui, “Polarization behavior of femtosecond laser written optical waveguides in Ti:Sapphire,” Opt. Express 20(14), 15035–15044 (2012).
    [Crossref] [PubMed]
  18. F. Liang, R. Vallée, and S. L. Chin, “Mechanism of nanograting formation on the surface of fused silica,” Opt. Express 20(4), 4389–4396 (2012).
    [Crossref] [PubMed]
  19. R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: Recent advances,” Prog. Quantum Electron. 38(3), 119–156 (2014).
    [Crossref]
  20. 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).
    [Crossref] [PubMed]
  21. E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. 71(7), 882–884 (1997).
    [Crossref]
  22. S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
    [Crossref] [PubMed]
  23. 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]
  24. M. Ams, G. D. Marshall, and M. J. Withford, “Study of the influence of femtosecond laser polarisation on direct writing of waveguides,” Opt. Express 14(26), 13158–13163 (2006).
    [Crossref] [PubMed]
  25. Y. Bellouard, M. Dugan, A. A. Said, and P. Bado, “Thermal conductivity contrast measurement of fused silica exposed to low-energy femtosecond laser pulses,” Appl. Phys. Lett. 89(16), 161911 (2006).
    [Crossref]
  26. Y. Bellouard, T. Colomb, C. Depeursinge, M. Dugan, A. A. Said, and P. Bado, “Nanoindentation and birefringence measurements on fused silica specimen exposed to low-energy femtosecond pulses,” Opt. Express 14(18), 8360–8366 (2006).
    [Crossref] [PubMed]
  27. A. Champion and Y. Bellouard, “Direct volume variation measurements in fused silica specimens exposed to femtosecond laser,” Opt. Mater. 2(6), 789–798 (2012).
    [Crossref]
  28. B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11(9), 1070–1079 (2003).
    [Crossref] [PubMed]
  29. A. Champion, M. Beresna, P. Kazansky, and Y. Bellouard, “Stress distribution around femtosecond laser affected zones: effect of nanogratings orientation,” Opt. Express 21(21), 24942–24951 (2013).
    [Crossref] [PubMed]
  30. F. Madani-Grasset and Y. Bellouard, “Femtosecond laser micromachining of fused silica molds,” Opt. Express 18(21), 21826–21840 (2010).
    [Crossref] [PubMed]
  31. M. Tomozawa and T. Takamori, “Relation of Surface Structure of Glass to HF Acid Attack and Stress State,” J. Am. Ceram. Soc. 62(7–8), 370–373 (1979).
    [Crossref]
  32. V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29(12), 1312–1314 (2004).
    [Crossref] [PubMed]
  33. D. Grobnic, S. J. Mihailov, and C. W. Smelser, “Localized High Birefringence Induced in SMF-28 Fiber by Femtosecond IR Laser Exposure of the Cladding,” J. Lightwave Technol. 25(8), 1996–2001 (2007).
    [Crossref]
  34. P. Yang, G. R. Burns, J. Guo, T. S. Luk, and G. A. Vawter, “Femtosecond laser-pulse-induced birefringence in optically isotropic glass,” J. Appl. Phys. 95(10), 5280–5283 (2004).
    [Crossref]
  35. L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3–6), 333–339 (2001).
    [Crossref]
  36. G. Zhou, A. Jesacher, M. Booth, T. Wilson, A. Ródenas, D. Jaque, and M. Gu, “Axial birefringence induced focus splitting in lithium niobate,” Opt. Express 17(20), 17970–17975 (2009).
    [Crossref] [PubMed]
  37. A. Champion, Y. Bellouard, G. Mindaugas, M. Beresna, and P. G. Kazansky, “Role of stress in the chemical etching of fused silica exposed to low-energy femtosecond laser pulses,” Proc. SPIE 7925, 79250Z (2011).
    [Crossref]
  38. A. Rodenas, A. Benayas, J. R. Macdonald, J. Zhang, D. Y. Tang, D. Jaque, and A. K. Kar, “Direct laser writing of near-IR step-index buried channel waveguides in rare earth doped YAG,” Opt. Lett. 36(17), 3395–3397 (2011).
    [Crossref] [PubMed]
  39. P. S. Salter and M. J. Booth, “Dynamic control of directional asymmetry observed in ultrafast laser direct writing,” Appl. Phys. Lett. 101(14), 141109 (2012).
    [Crossref]
  40. P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
    [Crossref]
  41. W. Yang, P. G. Kazansky, and Y. P. Svirko, “Non-reciprocal ultrafast laser writing,” Nat. Photonics 2(2), 99–104 (2008).
    [Crossref]
  42. J. F. Nye, Physical Properties of Crystals (Oxford University, 1985).
  43. T. N. Vasudevan and R. S. Krishnan, “Dispersion of the stress-optic coefficient in glasses,” J. Phys. D Appl. Phys. 5(12), 2283–2287 (1972).
    [Crossref]
  44. M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011).
    [Crossref]
  45. 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]

2014 (2)

S. Hasegawa and Y. Hayasaki, “Holographic Vector Wave Femtosecond Laser Processing,” Int. J. Optomechatronics 8(2), 73–88 (2014).
[Crossref]

R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: Recent advances,” Prog. Quantum Electron. 38(3), 119–156 (2014).
[Crossref]

2013 (1)

2012 (5)

A. Champion and Y. Bellouard, “Direct volume variation measurements in fused silica specimens exposed to femtosecond laser,” Opt. Mater. 2(6), 789–798 (2012).
[Crossref]

J. Bai, G. Cheng, X. Long, Y. Wang, W. Zhao, G. Chen, R. Stoian, and R. Hui, “Polarization behavior of femtosecond laser written optical waveguides in Ti:Sapphire,” Opt. Express 20(14), 15035–15044 (2012).
[Crossref] [PubMed]

F. Liang, R. Vallée, and S. L. Chin, “Mechanism of nanograting formation on the surface of fused silica,” Opt. Express 20(4), 4389–4396 (2012).
[Crossref] [PubMed]

P. S. Salter and M. J. Booth, “Dynamic control of directional asymmetry observed in ultrafast laser direct writing,” Appl. Phys. Lett. 101(14), 141109 (2012).
[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]

2011 (3)

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011).
[Crossref]

A. Champion, Y. Bellouard, G. Mindaugas, M. Beresna, and P. G. Kazansky, “Role of stress in the chemical etching of fused silica exposed to low-energy femtosecond laser pulses,” Proc. SPIE 7925, 79250Z (2011).
[Crossref]

A. Rodenas, A. Benayas, J. R. Macdonald, J. Zhang, D. Y. Tang, D. Jaque, and A. K. Kar, “Direct laser writing of near-IR step-index buried channel waveguides in rare earth doped YAG,” Opt. Lett. 36(17), 3395–3397 (2011).
[Crossref] [PubMed]

2010 (2)

F. Madani-Grasset and Y. Bellouard, “Femtosecond laser micromachining of fused silica molds,” Opt. Express 18(21), 21826–21840 (2010).
[Crossref] [PubMed]

K. Sugioka, Y. Hanada, and K. Midorikawa, “Three-dimensional femtosecond laser micromachining of photosensitive glass for biomicrochips,” Laser Photon. Rev. 4(3), 386–400 (2010).
[Crossref]

2009 (1)

2008 (3)

W. Yang, P. G. Kazansky, and Y. P. Svirko, “Non-reciprocal ultrafast laser writing,” Nat. Photonics 2(2), 99–104 (2008).
[Crossref]

S. Juodkazis and H. Misawa, “Laser processing of sapphire by strongly focused femtosecond pulses,” Appl. Phys. Adv. Mater. 93(4), 857–861 (2008).

B. McMillen, K. Chen, H. An, S. Fleming, V. Hartwell, and D. Snoke, “Waveguiding and nonlinear optical properties of three-dimensional waveguides in LiTaO3 written by high-repetition rate ultrafast laser,” Appl. Phys. Lett. 93(11), 111106 (2008).
[Crossref]

2007 (3)

2006 (7)

M. Ams, G. D. Marshall, and M. J. Withford, “Study of the influence of femtosecond laser polarisation on direct writing of waveguides,” Opt. Express 14(26), 13158–13163 (2006).
[Crossref] [PubMed]

Y. Bellouard, M. Dugan, A. A. Said, and P. Bado, “Thermal conductivity contrast measurement of fused silica exposed to low-energy femtosecond laser pulses,” Appl. Phys. Lett. 89(16), 161911 (2006).
[Crossref]

Y. Bellouard, T. Colomb, C. Depeursinge, M. Dugan, A. A. Said, and P. Bado, “Nanoindentation and birefringence measurements on fused silica specimen exposed to low-energy femtosecond pulses,” Opt. Express 14(18), 8360–8366 (2006).
[Crossref] [PubMed]

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

P. Nandi, G. Jose, C. Jayakrishnan, S. Debbarma, K. Chalapathi, K. Alti, A. K. Dharmadhikari, J. A. Dharmadhikari, and D. Mathur, “Femtosecond laser written channel waveguides in tellurite glass,” Opt. Express 14(25), 12145–12150 (2006).
[Crossref] [PubMed]

G. A. Torchia, C. Mendez, I. Arias, L. Roso, A. Rodenas, and D. Jaque, “Laser gain in femtosecond microstructured Nd:MgO:LiNbO3crystals,” Appl. Phys. B-Lasers O. 83(4), 559–563 (2006).
[Crossref]

V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
[Crossref] [PubMed]

2004 (4)

2003 (5)

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11(9), 1070–1079 (2003).
[Crossref] [PubMed]

Y. Cheng, K. Sugioka, K. Midorikawa, M. Masuda, K. Toyoda, M. Kawachi, and K. Shihoyama, “Control of the cross-sectional shape of a hollow microchannel embedded in photostructurable glass by use of a femtosecond laser,” Opt. Lett. 28(1), 55–57 (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]

J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82(15), 2371–2373 (2003).
[Crossref]

Y. Tokuda, M. Saito, M. Takahashi, and K. Yamada, “Waveguide formation in niobium tellurite glasses by pico- and femtosecond laser pulses,” J. Non-Cryst. Solids 326–327, 472–475 (2003).
[Crossref]

2002 (1)

2001 (1)

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3–6), 333–339 (2001).
[Crossref]

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 (1)

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

1996 (2)

1979 (1)

M. Tomozawa and T. Takamori, “Relation of Surface Structure of Glass to HF Acid Attack and Stress State,” J. Am. Ceram. Soc. 62(7–8), 370–373 (1979).
[Crossref]

1972 (1)

T. N. Vasudevan and R. S. Krishnan, “Dispersion of the stress-optic coefficient in glasses,” J. Phys. D Appl. Phys. 5(12), 2283–2287 (1972).
[Crossref]

Alti, K.

Ams, M.

An, H.

B. McMillen, K. Chen, H. An, S. Fleming, V. Hartwell, and D. Snoke, “Waveguiding and nonlinear optical properties of three-dimensional waveguides in LiTaO3 written by high-repetition rate ultrafast laser,” Appl. Phys. Lett. 93(11), 111106 (2008).
[Crossref]

Arai, A.

P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
[Crossref]

Arias, I.

G. A. Torchia, C. Mendez, I. Arias, L. Roso, A. Rodenas, and D. Jaque, “Laser gain in femtosecond microstructured Nd:MgO:LiNbO3crystals,” Appl. Phys. B-Lasers O. 83(4), 559–563 (2006).
[Crossref]

Bado, P.

Y. Bellouard, M. Dugan, A. A. Said, and P. Bado, “Thermal conductivity contrast measurement of fused silica exposed to low-energy femtosecond laser pulses,” Appl. Phys. Lett. 89(16), 161911 (2006).
[Crossref]

Y. Bellouard, T. Colomb, C. Depeursinge, M. Dugan, A. A. Said, and P. Bado, “Nanoindentation and birefringence measurements on fused silica specimen exposed to low-energy femtosecond pulses,” Opt. Express 14(18), 8360–8366 (2006).
[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. 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]

Bai, J.

Bellouard, Y.

A. Champion, M. Beresna, P. Kazansky, and Y. Bellouard, “Stress distribution around femtosecond laser affected zones: effect of nanogratings orientation,” Opt. Express 21(21), 24942–24951 (2013).
[Crossref] [PubMed]

A. Champion and Y. Bellouard, “Direct volume variation measurements in fused silica specimens exposed to femtosecond laser,” Opt. Mater. 2(6), 789–798 (2012).
[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. Champion, Y. Bellouard, G. Mindaugas, M. Beresna, and P. G. Kazansky, “Role of stress in the chemical etching of fused silica exposed to low-energy femtosecond laser pulses,” Proc. SPIE 7925, 79250Z (2011).
[Crossref]

F. Madani-Grasset and Y. Bellouard, “Femtosecond laser micromachining of fused silica molds,” Opt. Express 18(21), 21826–21840 (2010).
[Crossref] [PubMed]

Y. Bellouard, M. Dugan, A. A. Said, and P. Bado, “Thermal conductivity contrast measurement of fused silica exposed to low-energy femtosecond laser pulses,” Appl. Phys. Lett. 89(16), 161911 (2006).
[Crossref]

Y. Bellouard, T. Colomb, C. Depeursinge, M. Dugan, A. A. Said, and P. Bado, “Nanoindentation and birefringence measurements on fused silica specimen exposed to low-energy femtosecond pulses,” Opt. Express 14(18), 8360–8366 (2006).
[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]

Benayas, A.

Beresna, M.

A. Champion, M. Beresna, P. Kazansky, and Y. Bellouard, “Stress distribution around femtosecond laser affected zones: effect of nanogratings orientation,” Opt. Express 21(21), 24942–24951 (2013).
[Crossref] [PubMed]

A. Champion, Y. Bellouard, G. Mindaugas, M. Beresna, and P. G. Kazansky, “Role of stress in the chemical etching of fused silica exposed to low-energy femtosecond laser pulses,” Proc. SPIE 7925, 79250Z (2011).
[Crossref]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011).
[Crossref]

Bhardwaj, V. R.

V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
[Crossref] [PubMed]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29(12), 1312–1314 (2004).
[Crossref] [PubMed]

Booth, M.

Booth, M. J.

P. S. Salter and M. J. Booth, “Dynamic control of directional asymmetry observed in ultrafast laser direct writing,” Appl. Phys. Lett. 101(14), 141109 (2012).
[Crossref]

Bovatsek, J.

P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
[Crossref]

Bricchi, E.

P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
[Crossref]

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).
[Crossref] [PubMed]

Buividas, R.

R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: Recent advances,” Prog. Quantum Electron. 38(3), 119–156 (2014).
[Crossref]

Burns, G. R.

P. Yang, G. R. Burns, J. Guo, T. S. Luk, and G. A. Vawter, “Femtosecond laser-pulse-induced birefringence in optically isotropic glass,” J. Appl. Phys. 95(10), 5280–5283 (2004).
[Crossref]

Callan, J. P.

Chalapathi, K.

Champion, A.

A. Champion, M. Beresna, P. Kazansky, and Y. Bellouard, “Stress distribution around femtosecond laser affected zones: effect of nanogratings orientation,” Opt. Express 21(21), 24942–24951 (2013).
[Crossref] [PubMed]

A. Champion and Y. Bellouard, “Direct volume variation measurements in fused silica specimens exposed to femtosecond laser,” Opt. Mater. 2(6), 789–798 (2012).
[Crossref]

A. Champion, Y. Bellouard, G. Mindaugas, M. Beresna, and P. G. Kazansky, “Role of stress in the chemical etching of fused silica exposed to low-energy femtosecond laser pulses,” Proc. SPIE 7925, 79250Z (2011).
[Crossref]

Chan, J. W.

J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82(15), 2371–2373 (2003).
[Crossref]

Chen, G.

Chen, K.

B. McMillen, K. Chen, H. An, S. Fleming, V. Hartwell, and D. Snoke, “Waveguiding and nonlinear optical properties of three-dimensional waveguides in LiTaO3 written by high-repetition rate ultrafast laser,” Appl. Phys. Lett. 93(11), 111106 (2008).
[Crossref]

Cheng, G.

Cheng, Y.

Chin, S. L.

Colomb, T.

Corkum, P. B.

V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
[Crossref] [PubMed]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29(12), 1312–1314 (2004).
[Crossref] [PubMed]

Davis, K. M.

Debbarma, S.

Depeursinge, C.

Dharmadhikari, A. K.

Dharmadhikari, J. A.

Dugan, M.

Finlay, R. J.

Fleming, S.

B. McMillen, K. Chen, H. An, S. Fleming, V. Hartwell, and D. Snoke, “Waveguiding and nonlinear optical properties of three-dimensional waveguides in LiTaO3 written by high-repetition rate ultrafast laser,” Appl. Phys. Lett. 93(11), 111106 (2008).
[Crossref]

Florea, C.

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]

Franco, M.

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11(9), 1070–1079 (2003).
[Crossref] [PubMed]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3–6), 333–339 (2001).
[Crossref]

Fujimoto, J.

Gamaly, E. G.

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

Gecevicius, M.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011).
[Crossref]

Gertus, T.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011).
[Crossref]

Glezer, E. N.

Grobnic, D.

Gu, M.

Guo, J.

P. Yang, G. R. Burns, J. Guo, T. S. Luk, and G. A. Vawter, “Femtosecond laser-pulse-induced birefringence in optically isotropic glass,” J. Appl. Phys. 95(10), 5280–5283 (2004).
[Crossref]

Hallo, L.

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

Hanada, Y.

K. Sugioka, Y. Hanada, and K. Midorikawa, “Three-dimensional femtosecond laser micromachining of photosensitive glass for biomicrochips,” Laser Photon. Rev. 4(3), 386–400 (2010).
[Crossref]

Hartwell, V.

B. McMillen, K. Chen, H. An, S. Fleming, V. Hartwell, and D. Snoke, “Waveguiding and nonlinear optical properties of three-dimensional waveguides in LiTaO3 written by high-repetition rate ultrafast laser,” Appl. Phys. Lett. 93(11), 111106 (2008).
[Crossref]

Hasegawa, S.

S. Hasegawa and Y. Hayasaki, “Holographic Vector Wave Femtosecond Laser Processing,” Int. J. Optomechatronics 8(2), 73–88 (2014).
[Crossref]

Hayasaki, Y.

S. Hasegawa and Y. Hayasaki, “Holographic Vector Wave Femtosecond Laser Processing,” Int. J. Optomechatronics 8(2), 73–88 (2014).
[Crossref]

Hayden, J. S.

J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82(15), 2371–2373 (2003).
[Crossref]

Her, T. H.

Hirao, K.

P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
[Crossref]

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]

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).
[Crossref] [PubMed]

Hnatovsky, C.

V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
[Crossref] [PubMed]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29(12), 1312–1314 (2004).
[Crossref] [PubMed]

Huang, L.

Hui, R.

Huser, T. R.

J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82(15), 2371–2373 (2003).
[Crossref]

Ippen, E.

Jaque, D.

Jayakrishnan, C.

Jesacher, A.

Jose, G.

Juodkazis, S.

R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: Recent advances,” Prog. Quantum Electron. 38(3), 119–156 (2014).
[Crossref]

S. Juodkazis and H. Misawa, “Laser processing of sapphire by strongly focused femtosecond pulses,” Appl. Phys. Adv. Mater. 93(4), 857–861 (2008).

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

Kar, A. K.

Kawachi, M.

Kazansky, P.

A. Champion, M. Beresna, P. Kazansky, and Y. Bellouard, “Stress distribution around femtosecond laser affected zones: effect of nanogratings orientation,” Opt. Express 21(21), 24942–24951 (2013).
[Crossref] [PubMed]

P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
[Crossref]

Kazansky, P. G.

A. Champion, Y. Bellouard, G. Mindaugas, M. Beresna, and P. G. Kazansky, “Role of stress in the chemical etching of fused silica exposed to low-energy femtosecond laser pulses,” Proc. SPIE 7925, 79250Z (2011).
[Crossref]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011).
[Crossref]

W. Yang, P. G. Kazansky, and Y. P. Svirko, “Non-reciprocal ultrafast laser writing,” Nat. Photonics 2(2), 99–104 (2008).
[Crossref]

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).
[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]

Klappauf, B. G.

Kowalevicz, A.

Krishnan, R. S.

T. N. Vasudevan and R. S. Krishnan, “Dispersion of the stress-optic coefficient in glasses,” J. Phys. D Appl. Phys. 5(12), 2283–2287 (1972).
[Crossref]

Krol, D. M.

J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82(15), 2371–2373 (2003).
[Crossref]

Lenssen, B.

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]

Liang, F.

Long, X.

Luk, T. S.

P. Yang, G. R. Burns, J. Guo, T. S. Luk, and G. A. Vawter, “Femtosecond laser-pulse-induced birefringence in optically isotropic glass,” J. Appl. Phys. 95(10), 5280–5283 (2004).
[Crossref]

Luther-Davies, B.

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

Macdonald, J. R.

Madani-Grasset, F.

Marshall, G. D.

Masuda, M.

Mathur, D.

Maynard, R.

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]

Mazur, E.

McMillen, B.

B. McMillen, K. Chen, H. An, S. Fleming, V. Hartwell, and D. Snoke, “Waveguiding and nonlinear optical properties of three-dimensional waveguides in LiTaO3 written by high-repetition rate ultrafast laser,” Appl. Phys. Lett. 93(11), 111106 (2008).
[Crossref]

Meilán, P. F.

Mendez, C.

G. A. Torchia, P. F. Meilán, A. Ródenas, D. Jaque, C. Mendez, and L. Roso, “Femtosecond laser written surface waveguides fabricated in Nd:YAG ceramics,” Opt. Express 15(20), 13266–13271 (2007).
[Crossref] [PubMed]

G. A. Torchia, C. Mendez, I. Arias, L. Roso, A. Rodenas, and D. Jaque, “Laser gain in femtosecond microstructured Nd:MgO:LiNbO3crystals,” Appl. Phys. B-Lasers O. 83(4), 559–563 (2006).
[Crossref]

Midorikawa, K.

Mihailov, S. J.

Mikutis, M.

R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: Recent advances,” Prog. Quantum Electron. 38(3), 119–156 (2014).
[Crossref]

Milosavljevic, M.

Mindaugas, G.

A. Champion, Y. Bellouard, G. Mindaugas, M. Beresna, and P. G. Kazansky, “Role of stress in the chemical etching of fused silica exposed to low-energy femtosecond laser pulses,” Proc. SPIE 7925, 79250Z (2011).
[Crossref]

Minoshima, K.

Misawa, H.

S. Juodkazis and H. Misawa, “Laser processing of sapphire by strongly focused femtosecond pulses,” Appl. Phys. Adv. Mater. 93(4), 857–861 (2008).

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

Miura, K.

P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
[Crossref]

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).
[Crossref] [PubMed]

Mysyrowicz, A.

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11(9), 1070–1079 (2003).
[Crossref] [PubMed]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3–6), 333–339 (2001).
[Crossref]

Nandi, P.

Nicolai, P.

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

Nishimura, K.

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

Poumellec, B.

Prade, B.

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11(9), 1070–1079 (2003).
[Crossref] [PubMed]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3–6), 333–339 (2001).
[Crossref]

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).
[Crossref] [PubMed]

Rajeev, P. P.

V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
[Crossref] [PubMed]

Rayner, D. M.

V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
[Crossref] [PubMed]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29(12), 1312–1314 (2004).
[Crossref] [PubMed]

Risbud, S. H.

J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82(15), 2371–2373 (2003).
[Crossref]

Rodenas, A.

A. Rodenas, A. Benayas, J. R. Macdonald, J. Zhang, D. Y. Tang, D. Jaque, and A. K. Kar, “Direct laser writing of near-IR step-index buried channel waveguides in rare earth doped YAG,” Opt. Lett. 36(17), 3395–3397 (2011).
[Crossref] [PubMed]

G. A. Torchia, C. Mendez, I. Arias, L. Roso, A. Rodenas, and D. Jaque, “Laser gain in femtosecond microstructured Nd:MgO:LiNbO3crystals,” Appl. Phys. B-Lasers O. 83(4), 559–563 (2006).
[Crossref]

Ródenas, A.

Roso, L.

G. A. Torchia, P. F. Meilán, A. Ródenas, D. Jaque, C. Mendez, and L. Roso, “Femtosecond laser written surface waveguides fabricated in Nd:YAG ceramics,” Opt. Express 15(20), 13266–13271 (2007).
[Crossref] [PubMed]

G. A. Torchia, C. Mendez, I. Arias, L. Roso, A. Rodenas, and D. Jaque, “Laser gain in femtosecond microstructured Nd:MgO:LiNbO3crystals,” Appl. Phys. B-Lasers O. 83(4), 559–563 (2006).
[Crossref]

Said, A.

Said, A. A.

Y. Bellouard, M. Dugan, A. A. Said, and P. Bado, “Thermal conductivity contrast measurement of fused silica exposed to low-energy femtosecond laser pulses,” Appl. Phys. Lett. 89(16), 161911 (2006).
[Crossref]

Y. Bellouard, T. Colomb, C. Depeursinge, M. Dugan, A. A. Said, and P. Bado, “Nanoindentation and birefringence measurements on fused silica specimen exposed to low-energy femtosecond pulses,” Opt. Express 14(18), 8360–8366 (2006).
[Crossref] [PubMed]

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]

Saito, M.

Y. Tokuda, M. Saito, M. Takahashi, and K. Yamada, “Waveguide formation in niobium tellurite glasses by pico- and femtosecond laser pulses,” J. Non-Cryst. Solids 326–327, 472–475 (2003).
[Crossref]

Salter, P. S.

P. S. Salter and M. J. Booth, “Dynamic control of directional asymmetry observed in ultrafast laser direct writing,” Appl. Phys. Lett. 101(14), 141109 (2012).
[Crossref]

Shihoyama, K.

Shimotsuma, Y.

P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
[Crossref]

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]

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.

V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
[Crossref] [PubMed]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29(12), 1312–1314 (2004).
[Crossref] [PubMed]

Smelser, C. W.

Snoke, D.

B. McMillen, K. Chen, H. An, S. Fleming, V. Hartwell, and D. Snoke, “Waveguiding and nonlinear optical properties of three-dimensional waveguides in LiTaO3 written by high-repetition rate ultrafast laser,” Appl. Phys. Lett. 93(11), 111106 (2008).
[Crossref]

Stoian, R.

Sudrie, L.

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11(9), 1070–1079 (2003).
[Crossref] [PubMed]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3–6), 333–339 (2001).
[Crossref]

Sugimoto, N.

Sugioka, K.

Svirko, Y. P.

W. Yang, P. G. Kazansky, and Y. P. Svirko, “Non-reciprocal ultrafast laser writing,” Nat. Photonics 2(2), 99–104 (2008).
[Crossref]

Takahashi, M.

Y. Tokuda, M. Saito, M. Takahashi, and K. Yamada, “Waveguide formation in niobium tellurite glasses by pico- and femtosecond laser pulses,” J. Non-Cryst. Solids 326–327, 472–475 (2003).
[Crossref]

Takamori, T.

M. Tomozawa and T. Takamori, “Relation of Surface Structure of Glass to HF Acid Attack and Stress State,” J. Am. Ceram. Soc. 62(7–8), 370–373 (1979).
[Crossref]

Tanaka, S.

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

Tang, D. Y.

Taylor, R. S.

V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
[Crossref] [PubMed]

V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, “Stress in femtosecond-laser-written waveguides in fused silica,” Opt. Lett. 29(12), 1312–1314 (2004).
[Crossref] [PubMed]

Tikhonchuk, V. T.

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[Crossref] [PubMed]

Tokuda, Y.

Y. Tokuda, M. Saito, M. Takahashi, and K. Yamada, “Waveguide formation in niobium tellurite glasses by pico- and femtosecond laser pulses,” J. Non-Cryst. Solids 326–327, 472–475 (2003).
[Crossref]

Tomozawa, M.

M. Tomozawa and T. Takamori, “Relation of Surface Structure of Glass to HF Acid Attack and Stress State,” J. Am. Ceram. Soc. 62(7–8), 370–373 (1979).
[Crossref]

Torchia, G. A.

G. A. Torchia, P. F. Meilán, A. Ródenas, D. Jaque, C. Mendez, and L. Roso, “Femtosecond laser written surface waveguides fabricated in Nd:YAG ceramics,” Opt. Express 15(20), 13266–13271 (2007).
[Crossref] [PubMed]

G. A. Torchia, C. Mendez, I. Arias, L. Roso, A. Rodenas, and D. Jaque, “Laser gain in femtosecond microstructured Nd:MgO:LiNbO3crystals,” Appl. Phys. B-Lasers O. 83(4), 559–563 (2006).
[Crossref]

Toyoda, K.

Vallée, R.

Vasudevan, T. N.

T. N. Vasudevan and R. S. Krishnan, “Dispersion of the stress-optic coefficient in glasses,” J. Phys. D Appl. Phys. 5(12), 2283–2287 (1972).
[Crossref]

Vawter, G. A.

P. Yang, G. R. Burns, J. Guo, T. S. Luk, and G. A. Vawter, “Femtosecond laser-pulse-induced birefringence in optically isotropic glass,” J. Appl. Phys. 95(10), 5280–5283 (2004).
[Crossref]

Wang, Y.

Wilson, T.

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.

Yamada, K.

Y. Tokuda, M. Saito, M. Takahashi, and K. Yamada, “Waveguide formation in niobium tellurite glasses by pico- and femtosecond laser pulses,” J. Non-Cryst. Solids 326–327, 472–475 (2003).
[Crossref]

Yang, P.

P. Yang, G. R. Burns, J. Guo, T. S. Luk, and G. A. Vawter, “Femtosecond laser-pulse-induced birefringence in optically isotropic glass,” J. Appl. Phys. 95(10), 5280–5283 (2004).
[Crossref]

Yang, W.

W. Yang, P. G. Kazansky, and Y. P. Svirko, “Non-reciprocal ultrafast laser writing,” Nat. Photonics 2(2), 99–104 (2008).
[Crossref]

P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
[Crossref]

Zhang, J.

Zhao, W.

Zhou, G.

Appl. Phys. Adv. Mater. (1)

S. Juodkazis and H. Misawa, “Laser processing of sapphire by strongly focused femtosecond pulses,” Appl. Phys. Adv. Mater. 93(4), 857–861 (2008).

Appl. Phys. B-Lasers O. (1)

G. A. Torchia, C. Mendez, I. Arias, L. Roso, A. Rodenas, and D. Jaque, “Laser gain in femtosecond microstructured Nd:MgO:LiNbO3crystals,” Appl. Phys. B-Lasers O. 83(4), 559–563 (2006).
[Crossref]

Appl. Phys. Lett. (8)

J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82(15), 2371–2373 (2003).
[Crossref]

B. McMillen, K. Chen, H. An, S. Fleming, V. Hartwell, and D. Snoke, “Waveguiding and nonlinear optical properties of three-dimensional waveguides in LiTaO3 written by high-repetition rate ultrafast laser,” Appl. Phys. Lett. 93(11), 111106 (2008).
[Crossref]

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

Y. Bellouard, M. Dugan, A. A. Said, and P. Bado, “Thermal conductivity contrast measurement of fused silica exposed to low-energy femtosecond laser pulses,” Appl. Phys. Lett. 89(16), 161911 (2006).
[Crossref]

P. S. Salter and M. J. Booth, “Dynamic control of directional asymmetry observed in ultrafast laser direct writing,” Appl. Phys. Lett. 101(14), 141109 (2012).
[Crossref]

P. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. 90(15), 151120 (2007).
[Crossref]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011).
[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]

Electron. Lett. (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]

Int. J. Optomechatronics (1)

S. Hasegawa and Y. Hayasaki, “Holographic Vector Wave Femtosecond Laser Processing,” Int. J. Optomechatronics 8(2), 73–88 (2014).
[Crossref]

J. Am. Ceram. Soc. (1)

M. Tomozawa and T. Takamori, “Relation of Surface Structure of Glass to HF Acid Attack and Stress State,” J. Am. Ceram. Soc. 62(7–8), 370–373 (1979).
[Crossref]

J. Appl. Phys. (1)

P. Yang, G. R. Burns, J. Guo, T. S. Luk, and G. A. Vawter, “Femtosecond laser-pulse-induced birefringence in optically isotropic glass,” J. Appl. Phys. 95(10), 5280–5283 (2004).
[Crossref]

J. Lightwave Technol. (1)

J. Non-Cryst. Solids (1)

Y. Tokuda, M. Saito, M. Takahashi, and K. Yamada, “Waveguide formation in niobium tellurite glasses by pico- and femtosecond laser pulses,” J. Non-Cryst. Solids 326–327, 472–475 (2003).
[Crossref]

J. Phys. D Appl. Phys. (1)

T. N. Vasudevan and R. S. Krishnan, “Dispersion of the stress-optic coefficient in glasses,” J. Phys. D Appl. Phys. 5(12), 2283–2287 (1972).
[Crossref]

Laser Photon. Rev. (1)

K. Sugioka, Y. Hanada, and K. Midorikawa, “Three-dimensional femtosecond laser micromachining of photosensitive glass for biomicrochips,” Laser Photon. Rev. 4(3), 386–400 (2010).
[Crossref]

Nat. Photonics (1)

W. Yang, P. G. Kazansky, and Y. P. Svirko, “Non-reciprocal ultrafast laser writing,” Nat. Photonics 2(2), 99–104 (2008).
[Crossref]

Opt. Commun. (1)

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. 191(3–6), 333–339 (2001).
[Crossref]

Opt. Express (12)

G. Zhou, A. Jesacher, M. Booth, T. Wilson, A. Ródenas, D. Jaque, and M. Gu, “Axial birefringence induced focus splitting in lithium niobate,” Opt. Express 17(20), 17970–17975 (2009).
[Crossref] [PubMed]

B. Poumellec, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Femtosecond laser irradiation stress induced in pure silica,” Opt. Express 11(9), 1070–1079 (2003).
[Crossref] [PubMed]

A. Champion, M. Beresna, P. Kazansky, and Y. Bellouard, “Stress distribution around femtosecond laser affected zones: effect of nanogratings orientation,” Opt. Express 21(21), 24942–24951 (2013).
[Crossref] [PubMed]

F. Madani-Grasset and Y. Bellouard, “Femtosecond laser micromachining of fused silica molds,” Opt. Express 18(21), 21826–21840 (2010).
[Crossref] [PubMed]

Y. Bellouard, T. Colomb, C. Depeursinge, M. Dugan, A. A. Said, and P. Bado, “Nanoindentation and birefringence measurements on fused silica specimen exposed to low-energy femtosecond pulses,” Opt. Express 14(18), 8360–8366 (2006).
[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]

M. Ams, G. D. Marshall, and M. J. Withford, “Study of the influence of femtosecond laser polarisation on direct writing of waveguides,” Opt. Express 14(26), 13158–13163 (2006).
[Crossref] [PubMed]

K. Minoshima, A. Kowalevicz, E. Ippen, and J. Fujimoto, “Fabrication of coupled mode photonic devices in glass by nonlinear femtosecond laser materials processing,” Opt. Express 10(15), 645–652 (2002).
[Crossref] [PubMed]

G. A. Torchia, P. F. Meilán, A. Ródenas, D. Jaque, C. Mendez, and L. Roso, “Femtosecond laser written surface waveguides fabricated in Nd:YAG ceramics,” Opt. Express 15(20), 13266–13271 (2007).
[Crossref] [PubMed]

P. Nandi, G. Jose, C. Jayakrishnan, S. Debbarma, K. Chalapathi, K. Alti, A. K. Dharmadhikari, J. A. Dharmadhikari, and D. Mathur, “Femtosecond laser written channel waveguides in tellurite glass,” Opt. Express 14(25), 12145–12150 (2006).
[Crossref] [PubMed]

J. Bai, G. Cheng, X. Long, Y. Wang, W. Zhao, G. Chen, R. Stoian, and R. Hui, “Polarization behavior of femtosecond laser written optical waveguides in Ti:Sapphire,” Opt. Express 20(14), 15035–15044 (2012).
[Crossref] [PubMed]

F. Liang, R. Vallée, and S. L. Chin, “Mechanism of nanograting formation on the surface of fused silica,” Opt. Express 20(4), 4389–4396 (2012).
[Crossref] [PubMed]

Opt. Lett. (6)

Opt. Mater. (1)

A. Champion and Y. Bellouard, “Direct volume variation measurements in fused silica specimens exposed to femtosecond laser,” Opt. Mater. 2(6), 789–798 (2012).
[Crossref]

Phys. Rev. Lett. (3)

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-Induced Microexplosion Confined in the Bulk of a Sapphire Crystal: Evidence of Multimegabar Pressures,” Phys. Rev. Lett. 96(16), 166101 (2006).
[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]

V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically Produced Arrays of Planar Nanostructures inside Fused Silica,” Phys. Rev. Lett. 96(5), 057404 (2006).
[Crossref] [PubMed]

Proc. SPIE (1)

A. Champion, Y. Bellouard, G. Mindaugas, M. Beresna, and P. G. Kazansky, “Role of stress in the chemical etching of fused silica exposed to low-energy femtosecond laser pulses,” Proc. SPIE 7925, 79250Z (2011).
[Crossref]

Prog. Quantum Electron. (1)

R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: Recent advances,” Prog. Quantum Electron. 38(3), 119–156 (2014).
[Crossref]

Other (1)

J. F. Nye, Physical Properties of Crystals (Oxford University, 1985).

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Figures (9)

Fig. 1
Fig. 1 Illustration of the process used for fabricating tube-based stressors. A tube is composed of individual rings (A), fabricated by translating the substrate in a circular motion (1) at a fixed velocity. Rings are then stacked (B) by translating the focus along the z-direction (2) and the process repeated (3), forming a tube. Focal-plane separation (z-distance between rings) was fixed at 5 um.
Fig. 2
Fig. 2 Illustrative examples of laser-induced stress anisotropy in (a) borosilicate glass, (b) fused silica, and (c) crystalline quartz. Although the three patterns have similar geometry, written under the same polarization and focusing conditions (but with different laser fluence adapted for each material), the stress-induced retardance profiles are noticeably different. These simple examples demonstrate how stress induced birefringence resulting from symmetric laser-written patterns may be used to reveal anisotropy in the laser-writing process. This anisotropy may be due to the presence of self-organized nanostructure, anisotropy present the writing beam, or the structure of the material itself. In (a), the green arrow indicates the starting position and direction of writing for each successive tube layer (for all materials). The polarization of the writing beam was linear, and fixed parallel to the vertical axis of the figure.
Fig. 3
Fig. 3 – Demonstration of the dependence of anisotropic stress field orientation on writing polarization. Retardance maps (a-c) and corresponding SEM images (d-f) of nanograting structure are shown for 50 μm diameter tubes fabricated with linear polarization orientations of 90°, 45°, and 0° respectively. The green arrows indicate the starting position and writing direction for successive layers in each tube. The pulse energy was with a writing speed of 1 mm/s and a fixed z-spacing of 5 μm.
Fig. 4
Fig. 4 Examples of isotropic retardance plots for (d) circular, (e) radial, and (f) azimuthal polarization. Circular polarization induces an anisotropic stress, while radial and azimuthal are nearly isotropic. Irregularities are found in the nanograting formation as shown in the corresponding SEM images above and below each field map (a-c, g-i). Writing speed was fixed at 1 mm/s with a pulse energy of 200 nJ for circular polarization and 1 uJ for radial and azimuthal polarization.
Fig. 5
Fig. 5 Nanograting formation in the case of radial polarization manifests asymmetric (anisotropic) geometries that are not entirely apparent from the intensity of the retardance alone, which appears to be nearly isotropic. The grating structure itself, though well formed, has a preferential orientation (concave-up) around the radius of each tube, and is mirrored left-to-right (a). Additional irregularities are also present at the ‘poles’ of the feature. The stress produced by this configuration is oriented tangentially to the tube structure, generating a retardance pattern whose slow-axis orientation is also anisotropic (b).
Fig. 6
Fig. 6 – Retardance images for 25 μm diameter tubes written in crystalline quartz with linear writing polarizations of 90, 45, and 0 degrees respectively (a-c). In (d), a tube written with radial polarization is given as a comparative example of a stress state that is free of anisotropy induced by directionally dependent nanostructure formation. For the linearly polarized features (a-c), note the visible deformation of the stress field, which is most visible in (b), confirming the formation of laser-induced nanostructure formation as shown in the SEM images (e-h). The triad in (d) indicates the orientation of the crystallographic axes of the substrate.
Fig. 7
Fig. 7 Quantitative analysis of the lattice distortion in β-quartz due to anisotropic stress. In (a-d), the intensity retardance maps displayed in Fig. 6 have been processed to produce a set of contours at fixed values of retardance. The largest, lowest-value contour (shown in dark-blue) was extracted for analysis. The plots shown in (e-h) represent the angles between a linear fit of each of the facet point groups shown in (a-d) for each respective writing polarization. The reference location of each angle in relation to the hexagonal contour is given in (i). For reference, the structure of β-quartz is also displayed in (j). This method of analysis clearly reveals the presence of anisotropic stress due to the formation of nanogratings within the laser-modified zone. As with the previous analysis in fused silica, radial polarization is used as a reference case for isotropic stress. This is reflected in the plot shown in (h), where the angles between each face are approximately equal. At locations 2 and 5, the angle is slightly smaller. This may be attributed to two factors: the slightly anisotropic distribution of stress at the ‘poles’ of the tube structure due to formation irregularities in the nanogratings as well as a different Young’s modulus along the y-direction of the crystal (when compared to the x-direction) [42]. If this case is compared to that shown for 90° polarization (h), we find that the angles at locations 2 and 5 are smaller still, which may be correlated to the orientation of the writing polarization and the maximum anisotropic stress component. Similarly, the plots for 45° and 0° (f-g) show similar trends with regard to angle location and stress orientation. Note that, for the 0° case (c), the pattern is not distorted perfectly along the x-direction, due to slight misalignment of the sample with respect to the motion stages during fabrication.
Fig. 8
Fig. 8 Proposed model describing anisotropic stress loading in tube structures resulting from nanograting formation when using a linearly polarized writing beam. In this model, each grating lamella contributes to the overall deformation of the ring structure, as shown in (b), where only deformations along the + y axis are considered for simplicity. The deformation curve, shown in red, may be calculated by taking the chord length along the + y direction (perpendicular to the grating lamella), between the inner and outer radius, with respect to distance along the + x direction. This model is further developed in (c) for both principal directions. Here the values of σi and σo represent the inner and outer stress profiles respectively. These values are computed with respect to the radius of the tube ρ, and the thickness of the laser modified zone w, as a function of angle around the tube radius. The indicated zones (1, 2, and 3) represent geometrically discontinuous regions which must be taken into account when calculating the stress profile for the outer radius, σo.
Fig. 9
Fig. 9 Comparison between the stress calculated from a measured retardance pattern (a) and a simulated tube structure with a diameter of 100 µm (b). The simulated peak stress was scaled to a value of 2 GPa, with a ratio between the y and x components of the stress of ~2.5 [27].

Equations (13)

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B 0 ( x 1 2 + x 2 2 + x 3 2 )=1,
Δ B ij = z ijk E k +π i jkl σ j .
σ 1 σ 2 = R T( C 1 C 1 ) ,
C=( n 3 2 )( π 11 π 12 ).
σ ox { θ 0 θ max wcos θ 0 +ρsin θ 0 [ sin 1 { ( ρ+w ρ )sin θ 0 } θ 0 ] η Lxo θ 0 θ max ( ρ+w )cos θ 0 η Lxo ,
σ oy { θ 0 θ min ( ρ+w )sin θ 0 η Lyo θ 0 θ min wsin θ 0 +ρcos θ 0 [ θ 0 cos 1 { ( ρ+w ρ )cos θ 0 } ] η Lyo ,
η Lxo =( ρ+w )cos θ max ,
η Lyo =( ρ+w )sin θ max .
θ min = cos 1 ( ρ ρ+w ),
θ max = sin 1 ( ρ ρ+w ).
σ ix wcos θ i +( ρ+w )sin θ i [ θ i sin 1 { ( ρ+w ρ )sin θ i } ] η Lxi ,
σ iy wsin θ i +( ρ+w )cos θ i [ cos 1 { ( ρ ρ+w )cos θ i } θ i ] η Lyi .
η Lxi , η Lyi =(ρ+w) cos 1 ( ρ ρ+w ).

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