Abstract

We report on the fabrication of an axicon by applying a two-step manufacturing process including a 1030 nm femtosecond and a 10.6 µm CO2 laser. First, the pre-defined axicon geometry is generated by high-precision femtosecond layer-by-layer ablation. In order to meet high surface quality requirements, inevitable stipulated for optical use, the surface of the thus structured axicon is smoothened by a subsequent CO2 laser polishing process. The finalized axicon fulfills optical quality as the surface roughness Ra is significantly reduced from 0.56 µm to 34 nm. For the evaluation of the optical quality, the axicon is placed in a measurement setup including the femtosecond laser. Comparison between the calculated Bessel beam for an ideal axicon and the quasi-Bessel beam generated and measured by the fabricated axicon reveals excellent agreement, verifying our precise manufacturing method.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  24. C. Weingarten, A. Schmickler, E. Willenborg, K. Wissenbach, and R. Poprawe, “Laser polishing and laser shape correction of optical glass,” J. Laser Appl. 29(1), 011702 (2017).
    [Crossref]
  25. A. D. McLachlan and F. P. Meyer, “Temperature dependence of the extinction coefficient of fused silica for CO2 laser wavelengths,” Appl. Opt. 26(9), 1728–1731 (1987).
    [Crossref] [PubMed]
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    [Crossref]

2018 (1)

2017 (4)

2016 (5)

Q. Xie, X. Li, L. Jiang, B. Xia, X. Yan, W. Zhao, and Y. Lu, “High-aspect-ratio, high-quality microdrilling by electron density control using a femtosecond laser Bessel beam,” Appl. Phys. A 122(2), 136 (2016).
[Crossref]

H.-K. Choi, J. Ryu, C. Kim, Y.-C. Noh, I.-B. Sohn, and J.-T. Kim, “Formation of micro-lens array using femtosecond and CO2 lasers,” J. Laser Micro Nanoengin. 11(3), 341–345 (2016).
[Crossref]

T. Delgado, D. Nieto, and M. T. Flores-Arias, “Soda-lime glass microlens arrays fabricated by laser: Comparison between a nanosecond and a femtosecond IR pulsed laser,” Opt. Lasers Eng. 83, 27–37 (2016).

M. Serhatlioglu, B. Ortaç, C. Elbuken, N. Biyikli, and M. E. Solmaz, “CO2 laser polishing of microfluidic channels fabricated by femtosecond laser assisted carving,” J. Micromech. Microeng. 26 (11), 115011 (2016).
[Crossref]

J. Dudutis, P. Gečys, and Gediminas Račiukaitis, “Non-ideal axicon-generated Bessel beam application for intra-volume glass modification,” Opt. Express 24(25), 28433–28443 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (2)

A. Pan, B. Gao, T. Chen, J. Si, C. Li, F. Chen, and X. Hou, “Fabrication of concave spherical microlenses on silicon by femtosecond laser irradiation and mixed acid etching,” Opt. Express 22(12), 15245–15250 (2014).
[Crossref] [PubMed]

M. K. Bhuyan, P. K. Velpula, J. P. Colombier, T. Olivier, N. Faure, and R. Stoin, “Single-shot high aspect ratio bulk nanostructuring of fused silica using chirp-controlled ultrafast laser Bessel beams,” Appl. Phys. Lett. 104(2), 021107 (2014).
[Crossref]

2011 (2)

S. Heidrich, E. Willenborg, and A. Richmann, “Development of a laser based process chain for manufacturing freeform optics,” Phys. Proc. 12, 519–528 (2011).
[Crossref]

J. Hildebrand, K. Hecht, J. Bliedtner, and H. Müller, “Laser beam polishing of quartz glass surfaces,” Phys. Proc. 12, 452–461 (2011).
[Crossref]

2009 (1)

X.-F. Li, R. J. Winfield, S. O’Brien, and G. M. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. 255(10), 5146–5149 (2009).
[Crossref]

2008 (2)

D. F. Farson, H. W. Choi, B. Zimmerman, J. K. Steach, J. J. Chalmers, S. V. Olesik, and L. J. Lee, “Femtosecond laser micromachining of dielectric materials for biomedical applications,” J. Micromech. Microeng. 18(3), 035020 (2008).
[Crossref]

O. Brzobohatý, T. Čižmár, and P. Zemánek, , “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16(17), 12688–12700 (2008).
[Crossref] [PubMed]

2006 (3)

1987 (3)

A. D. McLachlan and F. P. Meyer, “Temperature dependence of the extinction coefficient of fused silica for CO2 laser wavelengths,” Appl. Opt. 26(9), 1728–1731 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” Opt. Soc. Am. A 4(4), 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Beuton, R.

Bhuyan, M. K.

M. K. Bhuyan, P. K. Velpula, J. P. Colombier, T. Olivier, N. Faure, and R. Stoin, “Single-shot high aspect ratio bulk nanostructuring of fused silica using chirp-controlled ultrafast laser Bessel beams,” Appl. Phys. Lett. 104(2), 021107 (2014).
[Crossref]

Billet, C.

Biyikli, N.

M. Serhatlioglu, B. Ortaç, C. Elbuken, N. Biyikli, and M. E. Solmaz, “CO2 laser polishing of microfluidic channels fabricated by femtosecond laser assisted carving,” J. Micromech. Microeng. 26 (11), 115011 (2016).
[Crossref]

Bliedtner, J.

J. Hildebrand, K. Hecht, J. Bliedtner, and H. Müller, “Laser beam polishing of quartz glass surfaces,” Phys. Proc. 12, 452–461 (2011).
[Crossref]

Bridges, D.

Brzobohatý, O.

Chalmers, J. J.

D. F. Farson, H. W. Choi, B. Zimmerman, J. K. Steach, J. J. Chalmers, S. V. Olesik, and L. J. Lee, “Femtosecond laser micromachining of dielectric materials for biomedical applications,” J. Micromech. Microeng. 18(3), 035020 (2008).
[Crossref]

Chanal, M.

Chang, T.-L.

H.-Y. Tsai, S.-W. Luo, and T.-L. Chang, “Surface forming on glass material by femtosecond laser modification with HF etching process,” CIRP Annals 64(1), 205–208 (2015).
[Crossref]

Chassagne, B.

Chen, F.

Chen, T.

Cheng, G.

Cheng, Y.

Y. Cheng, H. L. Tsai, K. Sugioka, and K. Midorikawa, “Fabrication of 3D microoptical lenses in photosensitve glass using femtosecond laser micromachining,” Appl. Phys. A 85(1), 11–14 (2006).
[Crossref]

Chimier, B.

Choi, H. W.

D. F. Farson, H. W. Choi, B. Zimmerman, J. K. Steach, J. J. Chalmers, S. V. Olesik, and L. J. Lee, “Femtosecond laser micromachining of dielectric materials for biomedical applications,” J. Micromech. Microeng. 18(3), 035020 (2008).
[Crossref]

Choi, H.-K.

H.-K. Choi, J. Ryu, C. Kim, Y.-C. Noh, I.-B. Sohn, and J.-T. Kim, “Formation of micro-lens array using femtosecond and CO2 lasers,” J. Laser Micro Nanoengin. 11(3), 341–345 (2016).
[Crossref]

Cižmár, T.

Clady, R.

Colombier, J. P.

M. K. Bhuyan, P. K. Velpula, J. P. Colombier, T. Olivier, N. Faure, and R. Stoin, “Single-shot high aspect ratio bulk nanostructuring of fused silica using chirp-controlled ultrafast laser Bessel beams,” Appl. Phys. Lett. 104(2), 021107 (2014).
[Crossref]

Courvoisier, F.

Crean, G. M.

X.-F. Li, R. J. Winfield, S. O’Brien, and G. M. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. 255(10), 5146–5149 (2009).
[Crossref]

Delgado, T.

T. Delgado, D. Nieto, and M. T. Flores-Arias, “Soda-lime glass microlens arrays fabricated by laser: Comparison between a nanosecond and a femtosecond IR pulsed laser,” Opt. Lasers Eng. 83, 27–37 (2016).

Dematteo Caulier, O.

Duchateau, G.

Dudutis, J.

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” Opt. Soc. Am. A 4(4), 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Elbuken, C.

M. Serhatlioglu, B. Ortaç, C. Elbuken, N. Biyikli, and M. E. Solmaz, “CO2 laser polishing of microfluidic channels fabricated by femtosecond laser assisted carving,” J. Micromech. Microeng. 26 (11), 115011 (2016).
[Crossref]

Farson, D. F.

D. F. Farson, H. W. Choi, B. Zimmerman, J. K. Steach, J. J. Chalmers, S. V. Olesik, and L. J. Lee, “Femtosecond laser micromachining of dielectric materials for biomedical applications,” J. Micromech. Microeng. 18(3), 035020 (2008).
[Crossref]

Faure, N.

M. K. Bhuyan, P. K. Velpula, J. P. Colombier, T. Olivier, N. Faure, and R. Stoin, “Single-shot high aspect ratio bulk nanostructuring of fused silica using chirp-controlled ultrafast laser Bessel beams,” Appl. Phys. Lett. 104(2), 021107 (2014).
[Crossref]

Flores-Arias, M. T.

T. Delgado, D. Nieto, and M. T. Flores-Arias, “Soda-lime glass microlens arrays fabricated by laser: Comparison between a nanosecond and a femtosecond IR pulsed laser,” Opt. Lasers Eng. 83, 27–37 (2016).

Furfaro, L.

Gao, B.

Gecys, P.

Giust, R.

Grojo, D.

Guo, R.

Hecht, K.

J. Hildebrand, K. Hecht, J. Bliedtner, and H. Müller, “Laser beam polishing of quartz glass surfaces,” Phys. Proc. 12, 452–461 (2011).
[Crossref]

Heidrich, S.

S. Heidrich, E. Willenborg, and A. Richmann, “Development of a laser based process chain for manufacturing freeform optics,” Phys. Proc. 12, 519–528 (2011).
[Crossref]

Hellmann, R.

S. Schwarz and R. Hellmann, “Fabrication of cylindrical lenses by combining ultrashort pulsed laser and CO2 laser,” J. Laser Micro Nanoengin. 12(2), 76–79 (2017).
[Crossref]

Hildebrand, J.

J. Hildebrand, K. Hecht, J. Bliedtner, and H. Müller, “Laser beam polishing of quartz glass surfaces,” Phys. Proc. 12, 452–461 (2011).
[Crossref]

Hönninger, C.

Hou, X.

Hu, A.

Huang, W.

Ježek, J.

Jiang, L.

Q. Xie, X. Li, L. Jiang, B. Xia, X. Yan, W. Zhao, and Y. Lu, “High-aspect-ratio, high-quality microdrilling by electron density control using a femtosecond laser Bessel beam,” Appl. Phys. A 122(2), 136 (2016).
[Crossref]

Kim, C.

H.-K. Choi, J. Ryu, C. Kim, Y.-C. Noh, I.-B. Sohn, and J.-T. Kim, “Formation of micro-lens array using femtosecond and CO2 lasers,” J. Laser Micro Nanoengin. 11(3), 341–345 (2016).
[Crossref]

Kim, J.-T.

H.-K. Choi, J. Ryu, C. Kim, Y.-C. Noh, I.-B. Sohn, and J.-T. Kim, “Formation of micro-lens array using femtosecond and CO2 lasers,” J. Laser Micro Nanoengin. 11(3), 341–345 (2016).
[Crossref]

Kling, R.

Lee, L. J.

D. F. Farson, H. W. Choi, B. Zimmerman, J. K. Steach, J. J. Chalmers, S. V. Olesik, and L. J. Lee, “Femtosecond laser micromachining of dielectric materials for biomedical applications,” J. Micromech. Microeng. 18(3), 035020 (2008).
[Crossref]

Li, C.

Li, J.

Li, R.

Li, X.

Q. Xie, X. Li, L. Jiang, B. Xia, X. Yan, W. Zhao, and Y. Lu, “High-aspect-ratio, high-quality microdrilling by electron density control using a femtosecond laser Bessel beam,” Appl. Phys. A 122(2), 136 (2016).
[Crossref]

Li, X.-F.

X.-F. Li, R. J. Winfield, S. O’Brien, and G. M. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. 255(10), 5146–5149 (2009).
[Crossref]

Lopez, J.

Lu, Y.

Q. Xie, X. Li, L. Jiang, B. Xia, X. Yan, W. Zhao, and Y. Lu, “High-aspect-ratio, high-quality microdrilling by electron density control using a femtosecond laser Bessel beam,” Appl. Phys. A 122(2), 136 (2016).
[Crossref]

Luo, S.-W.

H.-Y. Tsai, S.-W. Luo, and T.-L. Chang, “Surface forming on glass material by femtosecond laser modification with HF etching process,” CIRP Annals 64(1), 205–208 (2015).
[Crossref]

McLachlan, A. D.

Meyer, F. P.

Meyer, R.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Midorikawa, K.

Y. Cheng, H. L. Tsai, K. Sugioka, and K. Midorikawa, “Fabrication of 3D microoptical lenses in photosensitve glass using femtosecond laser micromachining,” Appl. Phys. A 85(1), 11–14 (2006).
[Crossref]

Mishchik, K.

Mitra, S.

Mottay, E.

Mouskeftaras, A.

Müller, H.

J. Hildebrand, K. Hecht, J. Bliedtner, and H. Müller, “Laser beam polishing of quartz glass surfaces,” Phys. Proc. 12, 452–461 (2011).
[Crossref]

Nedela, V.

Nieto, D.

T. Delgado, D. Nieto, and M. T. Flores-Arias, “Soda-lime glass microlens arrays fabricated by laser: Comparison between a nanosecond and a femtosecond IR pulsed laser,” Opt. Lasers Eng. 83, 27–37 (2016).

Noh, Y.-C.

H.-K. Choi, J. Ryu, C. Kim, Y.-C. Noh, I.-B. Sohn, and J.-T. Kim, “Formation of micro-lens array using femtosecond and CO2 lasers,” J. Laser Micro Nanoengin. 11(3), 341–345 (2016).
[Crossref]

O’Brien, S.

X.-F. Li, R. J. Winfield, S. O’Brien, and G. M. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. 255(10), 5146–5149 (2009).
[Crossref]

Olesik, S. V.

D. F. Farson, H. W. Choi, B. Zimmerman, J. K. Steach, J. J. Chalmers, S. V. Olesik, and L. J. Lee, “Femtosecond laser micromachining of dielectric materials for biomedical applications,” J. Micromech. Microeng. 18(3), 035020 (2008).
[Crossref]

Olivier, T.

M. K. Bhuyan, P. K. Velpula, J. P. Colombier, T. Olivier, N. Faure, and R. Stoin, “Single-shot high aspect ratio bulk nanostructuring of fused silica using chirp-controlled ultrafast laser Bessel beams,” Appl. Phys. Lett. 104(2), 021107 (2014).
[Crossref]

Ortaç, B.

M. Serhatlioglu, B. Ortaç, C. Elbuken, N. Biyikli, and M. E. Solmaz, “CO2 laser polishing of microfluidic channels fabricated by femtosecond laser assisted carving,” J. Micromech. Microeng. 26 (11), 115011 (2016).
[Crossref]

Pan, A.

Poprawe, R.

C. Weingarten, A. Schmickler, E. Willenborg, K. Wissenbach, and R. Poprawe, “Laser polishing and laser shape correction of optical glass,” J. Laser Appl. 29(1), 011702 (2017).
[Crossref]

Raciukaitis, Gediminas

Rapp, L.

Richmann, A.

S. Heidrich, E. Willenborg, and A. Richmann, “Development of a laser based process chain for manufacturing freeform optics,” Phys. Proc. 12, 519–528 (2011).
[Crossref]

Ryu, J.

H.-K. Choi, J. Ryu, C. Kim, Y.-C. Noh, I.-B. Sohn, and J.-T. Kim, “Formation of micro-lens array using femtosecond and CO2 lasers,” J. Laser Micro Nanoengin. 11(3), 341–345 (2016).
[Crossref]

Schmickler, A.

C. Weingarten, A. Schmickler, E. Willenborg, K. Wissenbach, and R. Poprawe, “Laser polishing and laser shape correction of optical glass,” J. Laser Appl. 29(1), 011702 (2017).
[Crossref]

Schwarz, S.

S. Schwarz and R. Hellmann, “Fabrication of cylindrical lenses by combining ultrashort pulsed laser and CO2 laser,” J. Laser Micro Nanoengin. 12(2), 76–79 (2017).
[Crossref]

Serhatlioglu, M.

M. Serhatlioglu, B. Ortaç, C. Elbuken, N. Biyikli, and M. E. Solmaz, “CO2 laser polishing of microfluidic channels fabricated by femtosecond laser assisted carving,” J. Micromech. Microeng. 26 (11), 115011 (2016).
[Crossref]

Si, J.

Skupin, S.

Sohn, I.-B.

H.-K. Choi, J. Ryu, C. Kim, Y.-C. Noh, I.-B. Sohn, and J.-T. Kim, “Formation of micro-lens array using femtosecond and CO2 lasers,” J. Laser Micro Nanoengin. 11(3), 341–345 (2016).
[Crossref]

Solmaz, M. E.

M. Serhatlioglu, B. Ortaç, C. Elbuken, N. Biyikli, and M. E. Solmaz, “CO2 laser polishing of microfluidic channels fabricated by femtosecond laser assisted carving,” J. Micromech. Microeng. 26 (11), 115011 (2016).
[Crossref]

Steach, J. K.

D. F. Farson, H. W. Choi, B. Zimmerman, J. K. Steach, J. J. Chalmers, S. V. Olesik, and L. J. Lee, “Femtosecond laser micromachining of dielectric materials for biomedical applications,” J. Micromech. Microeng. 18(3), 035020 (2008).
[Crossref]

Stoian, R.

Stoin, R.

M. K. Bhuyan, P. K. Velpula, J. P. Colombier, T. Olivier, N. Faure, and R. Stoin, “Single-shot high aspect ratio bulk nanostructuring of fused silica using chirp-controlled ultrafast laser Bessel beams,” Appl. Phys. Lett. 104(2), 021107 (2014).
[Crossref]

Sugioka, K.

Y. Cheng, H. L. Tsai, K. Sugioka, and K. Midorikawa, “Fabrication of 3D microoptical lenses in photosensitve glass using femtosecond laser micromachining,” Appl. Phys. A 85(1), 11–14 (2006).
[Crossref]

Tsai, H. L.

Y. Cheng, H. L. Tsai, K. Sugioka, and K. Midorikawa, “Fabrication of 3D microoptical lenses in photosensitve glass using femtosecond laser micromachining,” Appl. Phys. A 85(1), 11–14 (2006).
[Crossref]

Tsai, H.-Y.

H.-Y. Tsai, S.-W. Luo, and T.-L. Chang, “Surface forming on glass material by femtosecond laser modification with HF etching process,” CIRP Annals 64(1), 205–208 (2015).
[Crossref]

Velpula, P. K.

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Q. Xie, X. Li, L. Jiang, B. Xia, X. Yan, W. Zhao, and Y. Lu, “High-aspect-ratio, high-quality microdrilling by electron density control using a femtosecond laser Bessel beam,” Appl. Phys. A 122(2), 136 (2016).
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Q. Xie, X. Li, L. Jiang, B. Xia, X. Yan, W. Zhao, and Y. Lu, “High-aspect-ratio, high-quality microdrilling by electron density control using a femtosecond laser Bessel beam,” Appl. Phys. A 122(2), 136 (2016).
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G. Zhang, R. Stoian, W. Zhao, and G. Cheng, “Femtosecond laser Bessel beam welding of transparent to nontransparent materials with large focal-position tolerant zone,” Opt. Express 26(2), 917–926 (2018).
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Q. Xie, X. Li, L. Jiang, B. Xia, X. Yan, W. Zhao, and Y. Lu, “High-aspect-ratio, high-quality microdrilling by electron density control using a femtosecond laser Bessel beam,” Appl. Phys. A 122(2), 136 (2016).
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M. K. Bhuyan, P. K. Velpula, J. P. Colombier, T. Olivier, N. Faure, and R. Stoin, “Single-shot high aspect ratio bulk nanostructuring of fused silica using chirp-controlled ultrafast laser Bessel beams,” Appl. Phys. Lett. 104(2), 021107 (2014).
[Crossref]

Appl. Surf. Sci. (1)

X.-F. Li, R. J. Winfield, S. O’Brien, and G. M. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. 255(10), 5146–5149 (2009).
[Crossref]

CIRP Annals (1)

H.-Y. Tsai, S.-W. Luo, and T.-L. Chang, “Surface forming on glass material by femtosecond laser modification with HF etching process,” CIRP Annals 64(1), 205–208 (2015).
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C. Weingarten, A. Schmickler, E. Willenborg, K. Wissenbach, and R. Poprawe, “Laser polishing and laser shape correction of optical glass,” J. Laser Appl. 29(1), 011702 (2017).
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M. Serhatlioglu, B. Ortaç, C. Elbuken, N. Biyikli, and M. E. Solmaz, “CO2 laser polishing of microfluidic channels fabricated by femtosecond laser assisted carving,” J. Micromech. Microeng. 26 (11), 115011 (2016).
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G. Zhang, R. Stoian, W. Zhao, and G. Cheng, “Femtosecond laser Bessel beam welding of transparent to nontransparent materials with large focal-position tolerant zone,” Opt. Express 26(2), 917–926 (2018).
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T. Delgado, D. Nieto, and M. T. Flores-Arias, “Soda-lime glass microlens arrays fabricated by laser: Comparison between a nanosecond and a femtosecond IR pulsed laser,” Opt. Lasers Eng. 83, 27–37 (2016).

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S. Heidrich, E. Willenborg, and A. Richmann, “Development of a laser based process chain for manufacturing freeform optics,” Phys. Proc. 12, 519–528 (2011).
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Figures (6)

Fig. 1
Fig. 1 Transformation of a Gaussian beam into a quasi-Bessel beam after an axicon within the range of zmax.
Fig. 2
Fig. 2 3D and 2D accumulated fluence profiles for different pulse distances 18, 16, 14 and 12 µm.
Fig. 3
Fig. 3 Laser scanning microscope images of the fabricated axicon (a) and cross-sections of unpolished and polished axicon (b).
Fig. 4
Fig. 4 Enlarged area of the axicon tip with fitted circle (a) and calculated error map (b) of the polished axicon.
Fig. 5
Fig. 5 (a) Optical setup for axicon characterization with a microscopy setup including an objective and a plano-convex lens. (b) Quasi-Bessel beams behind the axicon tip for different z-distances. Colour of the intensity was adjusted automatically for each picture.
Fig. 6
Fig. 6 Calculated Bessel beam (gray) and quasi-Bessel beam profile in x- (blue) and y-direction (red) at the position 8 mm behind the fabricated axicon tip.

Equations (5)

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

I ( r , Z ) = 4 P k sin Θ ω 0 z z m a x J 0 2 ( k r sin Θ ) exp ( 2 z 2 z m a x 2 ) .
Θ = arcsin ( n n 0 cos ( α 2 ) ) + α π 2 .
z m a x = ω 0 tan Θ .
r B = 2.4048 k sin Θ .
Φ a c c ( x , y ) = i = j = Φ 0 exp ( 2 ( ( x x i ) 2 + ( y y i ) 2 ) ω 0 2 )

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