Abstract

In this report, an efficient hybrid laser technique, nanosecond laser rear-side processing and femtosecond laser-assisted selective etching (FLSE) for the manufacturing of high-density gas capillary targets, is demonstrated. Cylindrical capillary nozzles for laser betatron X-ray sources were numerically simulated, manufactured from fused silica by 3D laser inscription and characterized using interferometry and gas density reconstruction. The dependence of gas concentration profiles on the wall roughness of cylindrical channels is presented.

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

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2018 (1)

J. Ferri, S. Corde, A. Döpp, A. Lifschitz, A. Doche, C. Thaury, K. Ta Phuoc, B. Mahieu, I. A. Andriyash, V. Malka, and X. Davoine, “High-Brilliance Betatron γ-Ray Source Powered by Laser-Accelerated Electrons,” Phys. Rev. Lett. 120(25), 254802 (2018).
[Crossref] [PubMed]

2017 (5)

M. Šmíd, I. Gallardo González, H. Ekerfelt, J. Björklund Svensson, M. Hansson, J. C. Wood, A. Persson, S. P. D. Mangles, O. Lundh, and K. Falk, “Highly efficient angularly resolving x-ray spectrometer optimized for absorption measurements with collimated sources,” Rev. Sci. Instrum. 88(6), 063102 (2017).
[Crossref] [PubMed]

K. Mishchik, K. Gaudfrin, and J. Lopez, “Drilling of through holes in sapphire using femto-second laser pulses,” J. Laser Micro Nanoeng. 12(3), 321–324 (2017).

L. Hof and J. Abou Ziki, “Micro-Hole Drilling on Glass Substrates—A Review,” Micromachines (Basel) 8(2), 53 (2017).
[Crossref]

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

R. Budriūnas, T. Stanislauskas, J. Adamonis, A. Aleknavičius, G. Veitas, D. Gadonas, S. Balickas, A. Michailovas, and A. Varanavičius, “53 W average power CEP-stabilized OPCPA system delivering 5.5 TW few cycle pulses at 1 kHz repetition rate,” Opt. Express 25(5), 5797–5806 (2017).
[Crossref] [PubMed]

2016 (4)

A. Döpp, E. Guillaume, C. Thaury, J. Gautier, K. Ta Phuoc, and V. Malka, “3D printing of gas jet nozzles for laser-plasma accelerators,” Rev. Sci. Instrum. 87(7), 073505 (2016).
[Crossref] [PubMed]

G. Lott, N. Falletto, P. Devilder, and R. Kling, “Optimizing the processing of sapphire with ultrashort laser pulses,” J. Laser Appl. 28(2), 022206 (2016).
[Crossref]

G. Golovin, S. Banerjee, S. Chen, N. Powers, C. Liu, W. Yan, J. Zhang, P. Zhang, B. Zhao, and D. Umstadter, ““Control and optimization of a staged laser-wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 830, 375–380 (2016).
[Crossref]

M. Hansson, B. Aurand, H. Ekerfelt, A. Persson, and O. Lundh, ““Injection of electrons by colliding laser pulses in a laser wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 829, 99–103 (2016).
[Crossref]

2015 (6)

C. Thaury, E. Guillaume, A. Lifschitz, K. Ta Phuoc, M. Hansson, G. Grittani, J. Gautier, J. P. Goddet, A. Tafzi, O. Lundh, and V. Malka, “Shock assisted ionization injection in laser-plasma accelerators,” Sci. Rep. 5(1), 16310 (2015).
[Crossref] [PubMed]

M. Hansson, B. Aurand, X. Davoine, H. Ekerfelt, K. Svensson, A. Persson, C. G. Wahlström, and O. Lundh, “Down-ramp injection and independently controlled acceleration of electrons in a tailored laser wakefield accelerator,” Phys. Rev. Spec. Top. Accel. Beams 18(7), 071303 (2015).
[Crossref]

A. J. Goers, G. A. Hine, L. Feder, B. Miao, F. Salehi, J. K. Wahlstrand, and H. M. Milchberg, “Multi-MeV Electron Acceleration by Subterawatt Laser Pulses,” Phys. Rev. Lett. 115(19), 194802 (2015).
[Crossref] [PubMed]

Z. K. Wang, W. L. Seow, X. C. Wang, and H. Y. Zheng, “Effect of laser beam scanning mode on material removal efficiency in laser ablation for micromachining of glass,” J. Laser Appl. 27(S2), S28004 (2015).
[Crossref]

P. Gečys, J. Dudutis, and G. Račiukaitis, “Nanosecond Laser Processing of Soda-Lime Glass,” J. Laser Micro Nanoeng. 10(3), 254–258 (2015).
[Crossref]

N. M. Bulgakova, V. P. Zhukov, A. R. Collins, D. Rostohar, T. J.-Y. Derrien, and T. Mocek, “How to optimize ultrashort pulse laser interaction with glass surfaces in cutting regimes?” Appl. Surf. Sci. 336, 364–374 (2015).
[Crossref]

2014 (2)

V. Stankevič and G. Račiukaitis, ““Free-shape 3D structure formation in bulk fused silica by irradiation with femtosecond laser pulses,” JLMN-,” J. Laser Micro Nanoeng. 9(3), 271–275 (2014).
[Crossref]

M. Hermans, J. Gottmann, and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro Nanoeng. 9(2), 126–131 (2014).
[Crossref]

2012 (2)

S. W. Jolly, Z. He, C. McGuffey, W. Schumaker, K. Krushelnick, and A. G. R. Thomas, “Stereolithography based method of creating custom gas density profile targets for high intensity laser-plasma experiments,” Rev. Sci. Instrum. 83(7), 073503 (2012).
[Crossref] [PubMed]

D. Ashkenasi, T. Kaszemeikat, N. Mueller, A. Lemke, and H. J. Eichler, “Machining of glass and quartz using nanosecond and picosecond laser pulses,” Proc. SPIE 8243, 82430M (2012).
[Crossref]

2011 (2)

X. Zhao and Y. C. Shin, “Femtosecond laser drilling of high-aspect ratio microchannels in glass,” Appl. Phys., A Mater. Sci. Process. 104(2), 713–719 (2011).
[Crossref]

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

2009 (1)

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81(3), 1229–1285 (2009).
[Crossref]

2008 (1)

K. Ta Phuoc, E. Esarey, V. Leurent, E. Cormier-Michel, C. G. R. Geddes, C. B. Schroeder, A. Rousse, and W. P. Leemans, “Betatron radiation from density tailored plasmas,” Phys. Plasmas 15(6), 063102 (2008).
[Crossref]

2005 (1)

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

2004 (2)

M. Hipp, J. Woisetschläger, P. Reiterer, and T. Neger, “Digital evaluation of interferograms,” Measurement 36(1), 53–66 (2004).
[Crossref]

D. J. Hwang, T. Y. Choi, and C. P. Grigoropoulos, “Liquid-assisted femtosecond laser drilling of straight and three-dimensional microchannels in glass,” Appl. Phys., A Mater. Sci. Process. 79(3), 605–612 (2004).
[Crossref]

2001 (2)

1999 (1)

R. Azambuja, M. Eloy, G. Figueira, and D. Neely, “Three-dimensional characterization of high-density non-cylindrical pulsed gas jets,” J. Phys. D Appl. Phys. 32(8), L35–L43 (1999).
[Crossref]

Abou Ziki, J.

L. Hof and J. Abou Ziki, “Micro-Hole Drilling on Glass Substrates—A Review,” Micromachines (Basel) 8(2), 53 (2017).
[Crossref]

Adamonis, J.

Aleknavicius, A.

Anania, M. P.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Andriyash, I. A.

J. Ferri, S. Corde, A. Döpp, A. Lifschitz, A. Doche, C. Thaury, K. Ta Phuoc, B. Mahieu, I. A. Andriyash, V. Malka, and X. Davoine, “High-Brilliance Betatron γ-Ray Source Powered by Laser-Accelerated Electrons,” Phys. Rev. Lett. 120(25), 254802 (2018).
[Crossref] [PubMed]

Ashkenasi, D.

D. Ashkenasi, T. Kaszemeikat, N. Mueller, A. Lemke, and H. J. Eichler, “Machining of glass and quartz using nanosecond and picosecond laser pulses,” Proc. SPIE 8243, 82430M (2012).
[Crossref]

Aurand, B.

M. Hansson, B. Aurand, H. Ekerfelt, A. Persson, and O. Lundh, ““Injection of electrons by colliding laser pulses in a laser wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 829, 99–103 (2016).
[Crossref]

M. Hansson, B. Aurand, X. Davoine, H. Ekerfelt, K. Svensson, A. Persson, C. G. Wahlström, and O. Lundh, “Down-ramp injection and independently controlled acceleration of electrons in a tailored laser wakefield accelerator,” Phys. Rev. Spec. Top. Accel. Beams 18(7), 071303 (2015).
[Crossref]

Azambuja, R.

R. Azambuja, M. Eloy, G. Figueira, and D. Neely, “Three-dimensional characterization of high-density non-cylindrical pulsed gas jets,” J. Phys. D Appl. Phys. 32(8), L35–L43 (1999).
[Crossref]

Balickas, S.

Banerjee, S.

G. Golovin, S. Banerjee, S. Chen, N. Powers, C. Liu, W. Yan, J. Zhang, P. Zhang, B. Zhao, and D. Umstadter, ““Control and optimization of a staged laser-wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 830, 375–380 (2016).
[Crossref]

Beaurepaire, B.

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

Bhardwaj, V. R.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

Björklund Svensson, J.

M. Šmíd, I. Gallardo González, H. Ekerfelt, J. Björklund Svensson, M. Hansson, J. C. Wood, A. Persson, S. P. D. Mangles, O. Lundh, and K. Falk, “Highly efficient angularly resolving x-ray spectrometer optimized for absorption measurements with collimated sources,” Rev. Sci. Instrum. 88(6), 063102 (2017).
[Crossref] [PubMed]

Bocoum, M.

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

Böhle, F.

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

Brunetti, E.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Budriunas, R.

Bulgakova, N. M.

N. M. Bulgakova, V. P. Zhukov, A. R. Collins, D. Rostohar, T. J.-Y. Derrien, and T. Mocek, “How to optimize ultrashort pulse laser interaction with glass surfaces in cutting regimes?” Appl. Surf. Sci. 336, 364–374 (2015).
[Crossref]

Chen, S.

G. Golovin, S. Banerjee, S. Chen, N. Powers, C. Liu, W. Yan, J. Zhang, P. Zhang, B. Zhao, and D. Umstadter, ““Control and optimization of a staged laser-wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 830, 375–380 (2016).
[Crossref]

Choi, T. Y.

D. J. Hwang, T. Y. Choi, and C. P. Grigoropoulos, “Liquid-assisted femtosecond laser drilling of straight and three-dimensional microchannels in glass,” Appl. Phys., A Mater. Sci. Process. 79(3), 605–612 (2004).
[Crossref]

Cipiccia, S.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Collins, A. R.

N. M. Bulgakova, V. P. Zhukov, A. R. Collins, D. Rostohar, T. J.-Y. Derrien, and T. Mocek, “How to optimize ultrashort pulse laser interaction with glass surfaces in cutting regimes?” Appl. Surf. Sci. 336, 364–374 (2015).
[Crossref]

Corde, S.

J. Ferri, S. Corde, A. Döpp, A. Lifschitz, A. Doche, C. Thaury, K. Ta Phuoc, B. Mahieu, I. A. Andriyash, V. Malka, and X. Davoine, “High-Brilliance Betatron γ-Ray Source Powered by Laser-Accelerated Electrons,” Phys. Rev. Lett. 120(25), 254802 (2018).
[Crossref] [PubMed]

Corkum, P. B.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

Cormier-Michel, E.

K. Ta Phuoc, E. Esarey, V. Leurent, E. Cormier-Michel, C. G. R. Geddes, C. B. Schroeder, A. Rousse, and W. P. Leemans, “Betatron radiation from density tailored plasmas,” Phys. Plasmas 15(6), 063102 (2008).
[Crossref]

Davoine, X.

J. Ferri, S. Corde, A. Döpp, A. Lifschitz, A. Doche, C. Thaury, K. Ta Phuoc, B. Mahieu, I. A. Andriyash, V. Malka, and X. Davoine, “High-Brilliance Betatron γ-Ray Source Powered by Laser-Accelerated Electrons,” Phys. Rev. Lett. 120(25), 254802 (2018).
[Crossref] [PubMed]

M. Hansson, B. Aurand, X. Davoine, H. Ekerfelt, K. Svensson, A. Persson, C. G. Wahlström, and O. Lundh, “Down-ramp injection and independently controlled acceleration of electrons in a tailored laser wakefield accelerator,” Phys. Rev. Spec. Top. Accel. Beams 18(7), 071303 (2015).
[Crossref]

Derrien, T. J.-Y.

N. M. Bulgakova, V. P. Zhukov, A. R. Collins, D. Rostohar, T. J.-Y. Derrien, and T. Mocek, “How to optimize ultrashort pulse laser interaction with glass surfaces in cutting regimes?” Appl. Surf. Sci. 336, 364–374 (2015).
[Crossref]

Devilder, P.

G. Lott, N. Falletto, P. Devilder, and R. Kling, “Optimizing the processing of sapphire with ultrashort laser pulses,” J. Laser Appl. 28(2), 022206 (2016).
[Crossref]

Dias, J. M.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Doche, A.

J. Ferri, S. Corde, A. Döpp, A. Lifschitz, A. Doche, C. Thaury, K. Ta Phuoc, B. Mahieu, I. A. Andriyash, V. Malka, and X. Davoine, “High-Brilliance Betatron γ-Ray Source Powered by Laser-Accelerated Electrons,” Phys. Rev. Lett. 120(25), 254802 (2018).
[Crossref] [PubMed]

Döpp, A.

J. Ferri, S. Corde, A. Döpp, A. Lifschitz, A. Doche, C. Thaury, K. Ta Phuoc, B. Mahieu, I. A. Andriyash, V. Malka, and X. Davoine, “High-Brilliance Betatron γ-Ray Source Powered by Laser-Accelerated Electrons,” Phys. Rev. Lett. 120(25), 254802 (2018).
[Crossref] [PubMed]

A. Döpp, E. Guillaume, C. Thaury, J. Gautier, K. Ta Phuoc, and V. Malka, “3D printing of gas jet nozzles for laser-plasma accelerators,” Rev. Sci. Instrum. 87(7), 073505 (2016).
[Crossref] [PubMed]

Dudutis, J.

P. Gečys, J. Dudutis, and G. Račiukaitis, “Nanosecond Laser Processing of Soda-Lime Glass,” J. Laser Micro Nanoeng. 10(3), 254–258 (2015).
[Crossref]

Eichler, H. J.

D. Ashkenasi, T. Kaszemeikat, N. Mueller, A. Lemke, and H. J. Eichler, “Machining of glass and quartz using nanosecond and picosecond laser pulses,” Proc. SPIE 8243, 82430M (2012).
[Crossref]

Ekerfelt, H.

M. Šmíd, I. Gallardo González, H. Ekerfelt, J. Björklund Svensson, M. Hansson, J. C. Wood, A. Persson, S. P. D. Mangles, O. Lundh, and K. Falk, “Highly efficient angularly resolving x-ray spectrometer optimized for absorption measurements with collimated sources,” Rev. Sci. Instrum. 88(6), 063102 (2017).
[Crossref] [PubMed]

M. Hansson, B. Aurand, H. Ekerfelt, A. Persson, and O. Lundh, ““Injection of electrons by colliding laser pulses in a laser wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 829, 99–103 (2016).
[Crossref]

M. Hansson, B. Aurand, X. Davoine, H. Ekerfelt, K. Svensson, A. Persson, C. G. Wahlström, and O. Lundh, “Down-ramp injection and independently controlled acceleration of electrons in a tailored laser wakefield accelerator,” Phys. Rev. Spec. Top. Accel. Beams 18(7), 071303 (2015).
[Crossref]

Eloy, M.

R. Azambuja, M. Eloy, G. Figueira, and D. Neely, “Three-dimensional characterization of high-density non-cylindrical pulsed gas jets,” J. Phys. D Appl. Phys. 32(8), L35–L43 (1999).
[Crossref]

Ersfeld, B.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Esarey, E.

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81(3), 1229–1285 (2009).
[Crossref]

K. Ta Phuoc, E. Esarey, V. Leurent, E. Cormier-Michel, C. G. R. Geddes, C. B. Schroeder, A. Rousse, and W. P. Leemans, “Betatron radiation from density tailored plasmas,” Phys. Plasmas 15(6), 063102 (2008).
[Crossref]

Falk, K.

M. Šmíd, I. Gallardo González, H. Ekerfelt, J. Björklund Svensson, M. Hansson, J. C. Wood, A. Persson, S. P. D. Mangles, O. Lundh, and K. Falk, “Highly efficient angularly resolving x-ray spectrometer optimized for absorption measurements with collimated sources,” Rev. Sci. Instrum. 88(6), 063102 (2017).
[Crossref] [PubMed]

Falletto, N.

G. Lott, N. Falletto, P. Devilder, and R. Kling, “Optimizing the processing of sapphire with ultrashort laser pulses,” J. Laser Appl. 28(2), 022206 (2016).
[Crossref]

Faure, J.

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

Feder, L.

A. J. Goers, G. A. Hine, L. Feder, B. Miao, F. Salehi, J. K. Wahlstrand, and H. M. Milchberg, “Multi-MeV Electron Acceleration by Subterawatt Laser Pulses,” Phys. Rev. Lett. 115(19), 194802 (2015).
[Crossref] [PubMed]

Ferri, J.

J. Ferri, S. Corde, A. Döpp, A. Lifschitz, A. Doche, C. Thaury, K. Ta Phuoc, B. Mahieu, I. A. Andriyash, V. Malka, and X. Davoine, “High-Brilliance Betatron γ-Ray Source Powered by Laser-Accelerated Electrons,” Phys. Rev. Lett. 120(25), 254802 (2018).
[Crossref] [PubMed]

Figueira, G.

R. Azambuja, M. Eloy, G. Figueira, and D. Neely, “Three-dimensional characterization of high-density non-cylindrical pulsed gas jets,” J. Phys. D Appl. Phys. 32(8), L35–L43 (1999).
[Crossref]

Gadonas, D.

Gallardo González, I.

M. Šmíd, I. Gallardo González, H. Ekerfelt, J. Björklund Svensson, M. Hansson, J. C. Wood, A. Persson, S. P. D. Mangles, O. Lundh, and K. Falk, “Highly efficient angularly resolving x-ray spectrometer optimized for absorption measurements with collimated sources,” Rev. Sci. Instrum. 88(6), 063102 (2017).
[Crossref] [PubMed]

Gaudfrin, K.

K. Mishchik, K. Gaudfrin, and J. Lopez, “Drilling of through holes in sapphire using femto-second laser pulses,” J. Laser Micro Nanoeng. 12(3), 321–324 (2017).

Gautier, J.

A. Döpp, E. Guillaume, C. Thaury, J. Gautier, K. Ta Phuoc, and V. Malka, “3D printing of gas jet nozzles for laser-plasma accelerators,” Rev. Sci. Instrum. 87(7), 073505 (2016).
[Crossref] [PubMed]

C. Thaury, E. Guillaume, A. Lifschitz, K. Ta Phuoc, M. Hansson, G. Grittani, J. Gautier, J. P. Goddet, A. Tafzi, O. Lundh, and V. Malka, “Shock assisted ionization injection in laser-plasma accelerators,” Sci. Rep. 5(1), 16310 (2015).
[Crossref] [PubMed]

Gecys, P.

P. Gečys, J. Dudutis, and G. Račiukaitis, “Nanosecond Laser Processing of Soda-Lime Glass,” J. Laser Micro Nanoeng. 10(3), 254–258 (2015).
[Crossref]

Geddes, C. G. R.

K. Ta Phuoc, E. Esarey, V. Leurent, E. Cormier-Michel, C. G. R. Geddes, C. B. Schroeder, A. Rousse, and W. P. Leemans, “Betatron radiation from density tailored plasmas,” Phys. Plasmas 15(6), 063102 (2008).
[Crossref]

Goddet, J. P.

C. Thaury, E. Guillaume, A. Lifschitz, K. Ta Phuoc, M. Hansson, G. Grittani, J. Gautier, J. P. Goddet, A. Tafzi, O. Lundh, and V. Malka, “Shock assisted ionization injection in laser-plasma accelerators,” Sci. Rep. 5(1), 16310 (2015).
[Crossref] [PubMed]

Goers, A. J.

A. J. Goers, G. A. Hine, L. Feder, B. Miao, F. Salehi, J. K. Wahlstrand, and H. M. Milchberg, “Multi-MeV Electron Acceleration by Subterawatt Laser Pulses,” Phys. Rev. Lett. 115(19), 194802 (2015).
[Crossref] [PubMed]

Golovin, G.

G. Golovin, S. Banerjee, S. Chen, N. Powers, C. Liu, W. Yan, J. Zhang, P. Zhang, B. Zhao, and D. Umstadter, ““Control and optimization of a staged laser-wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 830, 375–380 (2016).
[Crossref]

Gottmann, J.

M. Hermans, J. Gottmann, and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro Nanoeng. 9(2), 126–131 (2014).
[Crossref]

Grigoropoulos, C. P.

D. J. Hwang, T. Y. Choi, and C. P. Grigoropoulos, “Liquid-assisted femtosecond laser drilling of straight and three-dimensional microchannels in glass,” Appl. Phys., A Mater. Sci. Process. 79(3), 605–612 (2004).
[Crossref]

Grittani, G.

C. Thaury, E. Guillaume, A. Lifschitz, K. Ta Phuoc, M. Hansson, G. Grittani, J. Gautier, J. P. Goddet, A. Tafzi, O. Lundh, and V. Malka, “Shock assisted ionization injection in laser-plasma accelerators,” Sci. Rep. 5(1), 16310 (2015).
[Crossref] [PubMed]

Guénot, D.

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

Guillaume, E.

A. Döpp, E. Guillaume, C. Thaury, J. Gautier, K. Ta Phuoc, and V. Malka, “3D printing of gas jet nozzles for laser-plasma accelerators,” Rev. Sci. Instrum. 87(7), 073505 (2016).
[Crossref] [PubMed]

C. Thaury, E. Guillaume, A. Lifschitz, K. Ta Phuoc, M. Hansson, G. Grittani, J. Gautier, J. P. Goddet, A. Tafzi, O. Lundh, and V. Malka, “Shock assisted ionization injection in laser-plasma accelerators,” Sci. Rep. 5(1), 16310 (2015).
[Crossref] [PubMed]

Gustas, D.

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

Hamilton, D. J.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Hansson, M.

M. Šmíd, I. Gallardo González, H. Ekerfelt, J. Björklund Svensson, M. Hansson, J. C. Wood, A. Persson, S. P. D. Mangles, O. Lundh, and K. Falk, “Highly efficient angularly resolving x-ray spectrometer optimized for absorption measurements with collimated sources,” Rev. Sci. Instrum. 88(6), 063102 (2017).
[Crossref] [PubMed]

M. Hansson, B. Aurand, H. Ekerfelt, A. Persson, and O. Lundh, ““Injection of electrons by colliding laser pulses in a laser wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 829, 99–103 (2016).
[Crossref]

M. Hansson, B. Aurand, X. Davoine, H. Ekerfelt, K. Svensson, A. Persson, C. G. Wahlström, and O. Lundh, “Down-ramp injection and independently controlled acceleration of electrons in a tailored laser wakefield accelerator,” Phys. Rev. Spec. Top. Accel. Beams 18(7), 071303 (2015).
[Crossref]

C. Thaury, E. Guillaume, A. Lifschitz, K. Ta Phuoc, M. Hansson, G. Grittani, J. Gautier, J. P. Goddet, A. Tafzi, O. Lundh, and V. Malka, “Shock assisted ionization injection in laser-plasma accelerators,” Sci. Rep. 5(1), 16310 (2015).
[Crossref] [PubMed]

He, Z.

S. W. Jolly, Z. He, C. McGuffey, W. Schumaker, K. Krushelnick, and A. G. R. Thomas, “Stereolithography based method of creating custom gas density profile targets for high intensity laser-plasma experiments,” Rev. Sci. Instrum. 83(7), 073503 (2012).
[Crossref] [PubMed]

Hermans, M.

M. Hermans, J. Gottmann, and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro Nanoeng. 9(2), 126–131 (2014).
[Crossref]

Hine, G. A.

A. J. Goers, G. A. Hine, L. Feder, B. Miao, F. Salehi, J. K. Wahlstrand, and H. M. Milchberg, “Multi-MeV Electron Acceleration by Subterawatt Laser Pulses,” Phys. Rev. Lett. 115(19), 194802 (2015).
[Crossref] [PubMed]

Hipp, M.

M. Hipp, J. Woisetschläger, P. Reiterer, and T. Neger, “Digital evaluation of interferograms,” Measurement 36(1), 53–66 (2004).
[Crossref]

Hnatovsky, C.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98(1), 013517 (2005).
[Crossref]

Hoek, M.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Hof, L.

L. Hof and J. Abou Ziki, “Micro-Hole Drilling on Glass Substrates—A Review,” Micromachines (Basel) 8(2), 53 (2017).
[Crossref]

Hwang, D. J.

D. J. Hwang, T. Y. Choi, and C. P. Grigoropoulos, “Liquid-assisted femtosecond laser drilling of straight and three-dimensional microchannels in glass,” Appl. Phys., A Mater. Sci. Process. 79(3), 605–612 (2004).
[Crossref]

Islam, M. R.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Issac, R. C.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Itoh, K.

Jaroszynski, D. A.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Jiang, Y.

Jolly, S. W.

S. W. Jolly, Z. He, C. McGuffey, W. Schumaker, K. Krushelnick, and A. G. R. Thomas, “Stereolithography based method of creating custom gas density profile targets for high intensity laser-plasma experiments,” Rev. Sci. Instrum. 83(7), 073503 (2012).
[Crossref] [PubMed]

Jullien, A.

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

Juodkazis, S.

Kaszemeikat, T.

D. Ashkenasi, T. Kaszemeikat, N. Mueller, A. Lemke, and H. J. Eichler, “Machining of glass and quartz using nanosecond and picosecond laser pulses,” Proc. SPIE 8243, 82430M (2012).
[Crossref]

Kling, R.

G. Lott, N. Falletto, P. Devilder, and R. Kling, “Optimizing the processing of sapphire with ultrashort laser pulses,” J. Laser Appl. 28(2), 022206 (2016).
[Crossref]

Krushelnick, K.

S. W. Jolly, Z. He, C. McGuffey, W. Schumaker, K. Krushelnick, and A. G. R. Thomas, “Stereolithography based method of creating custom gas density profile targets for high intensity laser-plasma experiments,” Rev. Sci. Instrum. 83(7), 073503 (2012).
[Crossref] [PubMed]

Kuroda, D.

Leemans, W. P.

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81(3), 1229–1285 (2009).
[Crossref]

K. Ta Phuoc, E. Esarey, V. Leurent, E. Cormier-Michel, C. G. R. Geddes, C. B. Schroeder, A. Rousse, and W. P. Leemans, “Betatron radiation from density tailored plasmas,” Phys. Plasmas 15(6), 063102 (2008).
[Crossref]

Lemke, A.

D. Ashkenasi, T. Kaszemeikat, N. Mueller, A. Lemke, and H. J. Eichler, “Machining of glass and quartz using nanosecond and picosecond laser pulses,” Proc. SPIE 8243, 82430M (2012).
[Crossref]

Lemos, N. R. C.

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M. P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7(11), 867–871 (2011).
[Crossref]

Leurent, V.

K. Ta Phuoc, E. Esarey, V. Leurent, E. Cormier-Michel, C. G. R. Geddes, C. B. Schroeder, A. Rousse, and W. P. Leemans, “Betatron radiation from density tailored plasmas,” Phys. Plasmas 15(6), 063102 (2008).
[Crossref]

Li, Y.

Lifschitz, A.

J. Ferri, S. Corde, A. Döpp, A. Lifschitz, A. Doche, C. Thaury, K. Ta Phuoc, B. Mahieu, I. A. Andriyash, V. Malka, and X. Davoine, “High-Brilliance Betatron γ-Ray Source Powered by Laser-Accelerated Electrons,” Phys. Rev. Lett. 120(25), 254802 (2018).
[Crossref] [PubMed]

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

C. Thaury, E. Guillaume, A. Lifschitz, K. Ta Phuoc, M. Hansson, G. Grittani, J. Gautier, J. P. Goddet, A. Tafzi, O. Lundh, and V. Malka, “Shock assisted ionization injection in laser-plasma accelerators,” Sci. Rep. 5(1), 16310 (2015).
[Crossref] [PubMed]

Liu, C.

G. Golovin, S. Banerjee, S. Chen, N. Powers, C. Liu, W. Yan, J. Zhang, P. Zhang, B. Zhao, and D. Umstadter, ““Control and optimization of a staged laser-wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 830, 375–380 (2016).
[Crossref]

Lopez, J.

K. Mishchik, K. Gaudfrin, and J. Lopez, “Drilling of through holes in sapphire using femto-second laser pulses,” J. Laser Micro Nanoeng. 12(3), 321–324 (2017).

Lopez-Martens, R.

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

Lott, G.

G. Lott, N. Falletto, P. Devilder, and R. Kling, “Optimizing the processing of sapphire with ultrashort laser pulses,” J. Laser Appl. 28(2), 022206 (2016).
[Crossref]

Lozano, M.

D. Guénot, D. Gustas, A. Vernier, B. Beaurepaire, F. Böhle, M. Bocoum, M. Lozano, A. Jullien, R. Lopez-Martens, A. Lifschitz, and J. Faure, “Relativistic electron beams driven by kHz single-cycle light pulses,” Nat. Photonics 11(5), 293–296 (2017).
[Crossref]

Lundh, O.

M. Šmíd, I. Gallardo González, H. Ekerfelt, J. Björklund Svensson, M. Hansson, J. C. Wood, A. Persson, S. P. D. Mangles, O. Lundh, and K. Falk, “Highly efficient angularly resolving x-ray spectrometer optimized for absorption measurements with collimated sources,” Rev. Sci. Instrum. 88(6), 063102 (2017).
[Crossref] [PubMed]

M. Hansson, B. Aurand, H. Ekerfelt, A. Persson, and O. Lundh, ““Injection of electrons by colliding laser pulses in a laser wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 829, 99–103 (2016).
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M. Hansson, B. Aurand, X. Davoine, H. Ekerfelt, K. Svensson, A. Persson, C. G. Wahlström, and O. Lundh, “Down-ramp injection and independently controlled acceleration of electrons in a tailored laser wakefield accelerator,” Phys. Rev. Spec. Top. Accel. Beams 18(7), 071303 (2015).
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M. Šmíd, I. Gallardo González, H. Ekerfelt, J. Björklund Svensson, M. Hansson, J. C. Wood, A. Persson, S. P. D. Mangles, O. Lundh, and K. Falk, “Highly efficient angularly resolving x-ray spectrometer optimized for absorption measurements with collimated sources,” Rev. Sci. Instrum. 88(6), 063102 (2017).
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Yan, W.

G. Golovin, S. Banerjee, S. Chen, N. Powers, C. Liu, W. Yan, J. Zhang, P. Zhang, B. Zhao, and D. Umstadter, ““Control and optimization of a staged laser-wakefield accelerator,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers,” Detect. Assoc. Equip. 830, 375–380 (2016).
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X. Zhao and Y. C. Shin, “Femtosecond laser drilling of high-aspect ratio microchannels in glass,” Appl. Phys., A Mater. Sci. Process. 104(2), 713–719 (2011).
[Crossref]

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Z. K. Wang, W. L. Seow, X. C. Wang, and H. Y. Zheng, “Effect of laser beam scanning mode on material removal efficiency in laser ablation for micromachining of glass,” J. Laser Appl. 27(S2), S28004 (2015).
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Figures (11)

Fig. 1
Fig. 1 Simulated gas concentration (a), Mach number (b) of nitrogen and concentration (c) and Mach number (d) of helium at the backing pressure of 60 bar of a cylindrical nozzle with the output diameter of 300 μm.
Fig. 2
Fig. 2 The sketch of the nozzle (a). The nanosecond laser processing setup (b), which consisted of the galvanometer scanner to guide the beam in the XY plane and f-theta lens for focusing. The process was initiated from the rear side of a sample, and the positioning stage was continuously moved down.
Fig. 3
Fig. 3 Images of 6 mm-length capillaries with output diameters of 100 μm (a), 200 μm (b), 300 μm (c) and 500 μm (d), manufactured in fused silica using the nanosecond laser.
Fig. 4
Fig. 4 Tilted SEM images of an opened drilled holes array with diameters of 100 µm, 200 µm, 300 µm and 500 µm (a). The magnified image of the 100 mm-diameter channel surface (b). The surface topography of 500 µm-diameter channel (c), 100 µm-diameter capillary, connected with a 0.8 mm-diameter adapter (d).
Fig. 5
Fig. 5 Fabrication geometry of the modifications for the chemical etching (a) and the SEM picture of the channel entrance after 22 h etching in 10M KOH (b).
Fig. 6
Fig. 6 SEM images of an opened channel with a diameter of ~100 µm after 22 h etching in 10 M KOH (a). The magnified image of the transition from the drilled adapter to the etched channel surface (b). Magnified surface morphology of the etched channel (c). Drilled and etched adapter (d).
Fig. 7
Fig. 7 Surface roughness measurement of the channel, fabricated by the FLSE technology (a), and of the adapter, drilled using nanosecond laser and 22 h-etched in 10 M KOH (b).
Fig. 8
Fig. 8 The nozzle (diameter 35 mm, height 12.7 mm) of 2 mm-length capillaries with the output diameter of 100 μm, laser-machined in fused silica with the following etching in KOH.
Fig. 9
Fig. 9 Diagram of Mach-Zehnder interferometer (a), and measured phase interferogram of helium (b) and nitrogen (c) gas jet at the backing pressure of 60 bar of the cylindrical nozzle with the output diameter of 300 μm.
Fig. 10
Fig. 10 Simulated (a, c, e, g) and reconstructed (b, d, f, h) integral phase profiles of Mach-Zehnder interferometer of nitrogen gas at 60 bar of cylindrical nozzles with the output diameter of 100 μm (a, b), 200 μm (c, d), 300 μm (e, f) and 500 μm (g, h).
Fig. 11
Fig. 11 Simulated and reconstructed transversal concentration profiles (a,b) of gas at 60 bar of cylindrical nozzles at the distance of 70 μm (line 1, marker 5), 170 μm (line 2, marker 6), 270 μm (line 3, marker 7) and 370 μm (line 4, marker 8). Solid lines correspond to the concentration simulations of nitrogen and dashed of helium. Markers represent the experimental data. Simulated and reconstructed longitudinal concentration profiles (c) of gas at 60 bar of cylindrical nozzles with output diameter of 300 μm (line 1, marker 6), 200 μm (line 2, marker 7) and 100 μm (line 3). Reconstructed integral phase profiles (d, e) of Mach-Zehnder interferometer of nitrogen gas at 60 bars of cylindrical nozzles with the output diameter of 100 μm and Rz = 568 nm (d), and Rz = 8 µm (e).

Tables (1)

Tables Icon

Table 1 Material properties used in simulation [30]

Equations (4)

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ω c [ keV ] 1.1 × 10 23 γ 2 n [ cm 3 ] r β [ μm ] .
μ = A s T ( 1 + T s / T ) .
p 0 = p + 1 2 ρ U 2 ,
ρ ρ 0 = ( 1 + k 1 2 M 2 ) ( 1 k 1 ) ,