Abstract

Laser beams with ring-shaped intensity distributions have attracted much attention for a wide variety of applications in science and technology. In these applications, commonly used techniques for generating ring-shaped beams involve use of optical elements such as spiral phase plates or axicons. However, the ring-shaped beams formed by these methods have non-diffraction-limited resolution because the practical numerical aperture of a ring-shaped beam is smaller than that of a Gaussian beam. This is due to a reduction of the effective numerical aperture of the propagating beam after passing through the element. To overcome the issue, we propose an optical element composed of a radial grating to obtain a diffraction-limited ring-shaped beam. We found that a key point for optimizing the diffraction-limited ring-shaped beam is fine-tuning of the phase distribution in the ring-shaped beam. In an experiment for femtosecond laser processing with a ring-shaped beam, we demonstrated that the debris on a morphological structure fabricated by single-shot irradiation was considerably reduced because the subsequent pulse did not destroy the melt-solidification structure. Furthermore, we demonstrated the fabrication of an annular structure by single-shot ring-shaped beam irradiation in water without any inhibitory influence due to laser-induced bubble generation.

© 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]

2014 (2)

A. Sabatyan and B. Meshginqalam, “Generation of annular beam by a novel class of Fresnel zone plate,” Appl. Opt. 53(26), 5995–6000 (2014).
[Crossref] [PubMed]

Z. Kuang, W. Perrie, S. P. Edwardson, E. Fearon, and G. Dearden, “Ultrafast laser parallel microdrilling using multiple annular beams generated by a spatial light modulator,” J. Phys. D Appl. Phys. 47(11), 115501 (2014).
[Crossref]

2012 (1)

Y. Hayasaki, M. Nishitani, H. Takahashi, H. Yamamoto, A. Takita, D. Suzuki, and S. Hasegawa, “Experimental investigation of the closest parallel pulses in holographic femtosecond laser processing,” Appl. Phys., A Mater. Sci. Process. 107(2), 357–362 (2012).
[Crossref]

2010 (1)

2007 (1)

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, and B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[Crossref]

2006 (2)

2005 (1)

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87(3), 031101 (2005).
[Crossref]

2002 (1)

2001 (2)

J. C. Kim, D. J. Kim, D. S. Kim, S. W. Kim, and O. Y. Troitsky, “One-level, two-point method for estimation of thermal diffusivity by the converging thermal-wave technique,” Int. J. Thermophys. 22(3), 933–942 (2001).
[Crossref]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

2000 (1)

1999 (1)

R. B. Charters, B. Luther-Davies, and F. Ladouceur, “Improved performance of laser written channel waveguides using a TEM*01 beam,” IEEE Photon. Technol. Lett. 11(12), 1617–1619 (1999).
[Crossref]

1998 (1)

I. Manek, Yu. B. Ovchinnicov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1), 67–70 (1998).
[Crossref]

1997 (1)

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4), 287–292 (1997).
[Crossref]

1996 (2)

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124(1), 121–130 (1996).
[Crossref]

X. Wang, M. G. Littman, J. B. McManus, M. Tadi, Y. S. Kim, A. Askar, and H. Rabitz, “Focused bulk ultrasonic waves generated by ring-shaped laser illumination and application to flaw detection,” J. Appl. Phys. 80(8), 4274–4281 (1996).
[Crossref]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

1994 (2)

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5), 321–327 (1994).
[Crossref]

S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994).
[Crossref] [PubMed]

1990 (1)

Q. Ren and R. Birngruber, “A new laser beam delivery system for corneal surgery,” IEEE J. Quantum Electron. 26(12), 2305–2308 (1990).
[Crossref]

1978 (3)

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

Askar, A.

X. Wang, M. G. Littman, J. B. McManus, M. Tadi, Y. S. Kim, A. Askar, and H. Rabitz, “Focused bulk ultrasonic waves generated by ring-shaped laser illumination and application to flaw detection,” J. Appl. Phys. 80(8), 4274–4281 (1996).
[Crossref]

Beijersbergen, M.

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5), 321–327 (1994).
[Crossref]

Bélanger, P. A.

Berns, M. W.

Birngruber, R.

Q. Ren and R. Birngruber, “A new laser beam delivery system for corneal surgery,” IEEE J. Quantum Electron. 26(12), 2305–2308 (1990).
[Crossref]

Botvinick, E. L.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

Charters, R. B.

R. B. Charters, B. Luther-Davies, and F. Ladouceur, “Improved performance of laser written channel waveguides using a TEM*01 beam,” IEEE Photon. Technol. Lett. 11(12), 1617–1619 (1999).
[Crossref]

Chujo, K.

Coerwinkel, R.

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5), 321–327 (1994).
[Crossref]

Davidson, N.

Dearden, G.

Z. Kuang, W. Perrie, S. P. Edwardson, E. Fearon, and G. Dearden, “Ultrafast laser parallel microdrilling using multiple annular beams generated by a spatial light modulator,” J. Phys. D Appl. Phys. 47(11), 115501 (2014).
[Crossref]

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

Edwardson, S. P.

Z. Kuang, W. Perrie, S. P. Edwardson, E. Fearon, and G. Dearden, “Ultrafast laser parallel microdrilling using multiple annular beams generated by a spatial light modulator,” J. Phys. D Appl. Phys. 47(11), 115501 (2014).
[Crossref]

Esener, S. C.

Fearon, E.

Z. Kuang, W. Perrie, S. P. Edwardson, E. Fearon, and G. Dearden, “Ultrafast laser parallel microdrilling using multiple annular beams generated by a spatial light modulator,” J. Phys. D Appl. Phys. 47(11), 115501 (2014).
[Crossref]

Friedman, N.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Gan, X.

Ganic, D.

Grimm, R.

I. Manek, Yu. B. Ovchinnicov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1), 67–70 (1998).
[Crossref]

Gu, M.

Guan, J. F.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, and B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[Crossref]

Hain, M.

Hamazaki, J.

Hasegawa, S.

Y. Hayasaki, M. Nishitani, H. Takahashi, H. Yamamoto, A. Takita, D. Suzuki, and S. Hasegawa, “Experimental investigation of the closest parallel pulses in holographic femtosecond laser processing,” Appl. Phys., A Mater. Sci. Process. 107(2), 357–362 (2012).
[Crossref]

S. Hasegawa, Y. Hayasaki, and N. Nishida, “Holographic femtosecond laser processing with multiplexed phase Fresnel lenses,” Opt. Lett. 31(11), 1705–1707 (2006).
[Crossref] [PubMed]

Hayasaki, Y.

Y. Hayasaki, M. Nishitani, H. Takahashi, H. Yamamoto, A. Takita, D. Suzuki, and S. Hasegawa, “Experimental investigation of the closest parallel pulses in holographic femtosecond laser processing,” Appl. Phys., A Mater. Sci. Process. 107(2), 357–362 (2012).
[Crossref]

S. Hasegawa, Y. Hayasaki, and N. Nishida, “Holographic femtosecond laser processing with multiplexed phase Fresnel lenses,” Opt. Lett. 31(11), 1705–1707 (2006).
[Crossref] [PubMed]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87(3), 031101 (2005).
[Crossref]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Hell, S. W.

Imbert, C.

G. Roosen and C. Imbert, “The TEM*01 mode laser beam-A powerful tool for optical levitation of various types of spheres,” Opt. Commun. 26(3), 432–436 (1978).
[Crossref]

Jhe, W.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4), 287–292 (1997).
[Crossref]

Khaykovich, L.

Kim, D. J.

J. C. Kim, D. J. Kim, D. S. Kim, S. W. Kim, and O. Y. Troitsky, “One-level, two-point method for estimation of thermal diffusivity by the converging thermal-wave technique,” Int. J. Thermophys. 22(3), 933–942 (2001).
[Crossref]

Kim, D. S.

J. C. Kim, D. J. Kim, D. S. Kim, S. W. Kim, and O. Y. Troitsky, “One-level, two-point method for estimation of thermal diffusivity by the converging thermal-wave technique,” Int. J. Thermophys. 22(3), 933–942 (2001).
[Crossref]

Kim, J. C.

J. C. Kim, D. J. Kim, D. S. Kim, S. W. Kim, and O. Y. Troitsky, “One-level, two-point method for estimation of thermal diffusivity by the converging thermal-wave technique,” Int. J. Thermophys. 22(3), 933–942 (2001).
[Crossref]

Kim, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4), 287–292 (1997).
[Crossref]

Kim, S. W.

J. C. Kim, D. J. Kim, D. S. Kim, S. W. Kim, and O. Y. Troitsky, “One-level, two-point method for estimation of thermal diffusivity by the converging thermal-wave technique,” Int. J. Thermophys. 22(3), 933–942 (2001).
[Crossref]

Kim, Y. S.

X. Wang, M. G. Littman, J. B. McManus, M. Tadi, Y. S. Kim, A. Askar, and H. Rabitz, “Focused bulk ultrasonic waves generated by ring-shaped laser illumination and application to flaw detection,” J. Appl. Phys. 80(8), 4274–4281 (1996).
[Crossref]

Kobayashi, Y.

Kristensen, M.

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5), 321–327 (1994).
[Crossref]

Kuang, Z.

Z. Kuang, W. Perrie, S. P. Edwardson, E. Fearon, and G. Dearden, “Ultrafast laser parallel microdrilling using multiple annular beams generated by a spatial light modulator,” J. Phys. D Appl. Phys. 47(11), 115501 (2014).
[Crossref]

Ladouceur, F.

R. B. Charters, B. Luther-Davies, and F. Ladouceur, “Improved performance of laser written channel waveguides using a TEM*01 beam,” IEEE Photon. Technol. Lett. 11(12), 1617–1619 (1999).
[Crossref]

Lee, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4), 287–292 (1997).
[Crossref]

Littman, M. G.

X. Wang, M. G. Littman, J. B. McManus, M. Tadi, Y. S. Kim, A. Askar, and H. Rabitz, “Focused bulk ultrasonic waves generated by ring-shaped laser illumination and application to flaw detection,” J. Appl. Phys. 80(8), 4274–4281 (1996).
[Crossref]

Lu, J.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, and B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[Crossref]

Luther-Davies, B.

R. B. Charters, B. Luther-Davies, and F. Ladouceur, “Improved performance of laser written channel waveguides using a TEM*01 beam,” IEEE Photon. Technol. Lett. 11(12), 1617–1619 (1999).
[Crossref]

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

Manek, I.

I. Manek, Yu. B. Ovchinnicov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1), 67–70 (1998).
[Crossref]

McManus, J. B.

X. Wang, M. G. Littman, J. B. McManus, M. Tadi, Y. S. Kim, A. Askar, and H. Rabitz, “Focused bulk ultrasonic waves generated by ring-shaped laser illumination and application to flaw detection,” J. Appl. Phys. 80(8), 4274–4281 (1996).
[Crossref]

Meshginqalam, B.

Morita, R.

Nascimento, J. M.

Ni, X. W.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, and B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[Crossref]

Nishida, N.

S. Hasegawa, Y. Hayasaki, and N. Nishida, “Holographic femtosecond laser processing with multiplexed phase Fresnel lenses,” Opt. Lett. 31(11), 1705–1707 (2006).
[Crossref] [PubMed]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87(3), 031101 (2005).
[Crossref]

Nishitani, M.

Y. Hayasaki, M. Nishitani, H. Takahashi, H. Yamamoto, A. Takita, D. Suzuki, and S. Hasegawa, “Experimental investigation of the closest parallel pulses in holographic femtosecond laser processing,” Appl. Phys., A Mater. Sci. Process. 107(2), 357–362 (2012).
[Crossref]

Noh, H.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4), 287–292 (1997).
[Crossref]

Omatsu, T.

Ovchinnicov, Yu. B.

I. Manek, Yu. B. Ovchinnicov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1), 67–70 (1998).
[Crossref]

Ozeri, R.

Paterson, C.

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124(1), 121–130 (1996).
[Crossref]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

Perrie, W.

Z. Kuang, W. Perrie, S. P. Edwardson, E. Fearon, and G. Dearden, “Ultrafast laser parallel microdrilling using multiple annular beams generated by a spatial light modulator,” J. Phys. D Appl. Phys. 47(11), 115501 (2014).
[Crossref]

Rabitz, H.

X. Wang, M. G. Littman, J. B. McManus, M. Tadi, Y. S. Kim, A. Askar, and H. Rabitz, “Focused bulk ultrasonic waves generated by ring-shaped laser illumination and application to flaw detection,” J. Appl. Phys. 80(8), 4274–4281 (1996).
[Crossref]

Ren, Q.

Q. Ren and R. Birngruber, “A new laser beam delivery system for corneal surgery,” IEEE J. Quantum Electron. 26(12), 2305–2308 (1990).
[Crossref]

Rioux, M.

Roosen, G.

G. Roosen and C. Imbert, “The TEM*01 mode laser beam-A powerful tool for optical levitation of various types of spheres,” Opt. Commun. 26(3), 432–436 (1978).
[Crossref]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Sabatyan, A.

Shao, B.

Shen, Z. H.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, and B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[Crossref]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

Smith, R.

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124(1), 121–130 (1996).
[Crossref]

Somalingam, S.

Stankovic, S.

Sugimoto, T.

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87(3), 031101 (2005).
[Crossref]

Suzuki, D.

Y. Hayasaki, M. Nishitani, H. Takahashi, H. Yamamoto, A. Takita, D. Suzuki, and S. Hasegawa, “Experimental investigation of the closest parallel pulses in holographic femtosecond laser processing,” Appl. Phys., A Mater. Sci. Process. 107(2), 357–362 (2012).
[Crossref]

Tadi, M.

X. Wang, M. G. Littman, J. B. McManus, M. Tadi, Y. S. Kim, A. Askar, and H. Rabitz, “Focused bulk ultrasonic waves generated by ring-shaped laser illumination and application to flaw detection,” J. Appl. Phys. 80(8), 4274–4281 (1996).
[Crossref]

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Y. Hayasaki, M. Nishitani, H. Takahashi, H. Yamamoto, A. Takita, D. Suzuki, and S. Hasegawa, “Experimental investigation of the closest parallel pulses in holographic femtosecond laser processing,” Appl. Phys., A Mater. Sci. Process. 107(2), 357–362 (2012).
[Crossref]

Takita, A.

Y. Hayasaki, M. Nishitani, H. Takahashi, H. Yamamoto, A. Takita, D. Suzuki, and S. Hasegawa, “Experimental investigation of the closest parallel pulses in holographic femtosecond laser processing,” Appl. Phys., A Mater. Sci. Process. 107(2), 357–362 (2012).
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[Crossref]

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Tremblay, R.

Troitsky, O. Y.

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Tschudi, T.

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J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, and B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
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X. Wang, M. G. Littman, J. B. McManus, M. Tadi, Y. S. Kim, A. Askar, and H. Rabitz, “Focused bulk ultrasonic waves generated by ring-shaped laser illumination and application to flaw detection,” J. Appl. Phys. 80(8), 4274–4281 (1996).
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[Crossref]

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Y. Hayasaki, M. Nishitani, H. Takahashi, H. Yamamoto, A. Takita, D. Suzuki, and S. Hasegawa, “Experimental investigation of the closest parallel pulses in holographic femtosecond laser processing,” Appl. Phys., A Mater. Sci. Process. 107(2), 357–362 (2012).
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J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4), 287–292 (1997).
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Figures (10)

Fig. 1
Fig. 1 Phase patterns behind each axicon and their reconstructed patterns. (a) Conventional axicon. (b) Positive diffractive axicon. (c) Negative diffractive axicon. (d) Radial grating.
Fig. 2
Fig. 2 Radial gratings and their reconstructions versus the phase difference θd. θd was set to (a) 0.00π, (b) 0.35π, (c) 0.72π, (d) 1.09π and (e) 1.44π.
Fig. 3
Fig. 3 Change of sidelobe intensity in the ring beam versus the phase difference θd. Each color corresponds to the intensity profiles in the case of each θd. The dashed line indicates the profile in the case of positive diffractive axicon.
Fig. 4
Fig. 4 Change of the intensity profile of the ring-shaped beam versus the phase difference θd. (b) Peak intensity of the side lobe PIside and peak intensity of the main lobe PImain versus θd.
Fig. 5
Fig. 5 Intensity profile of the ring beam versus the distance on the z-axis in the case of (a) the positive axicon, (b) the negative axicon and (c) the radial grating. (d) Peak intensities in the profiles versus the distance on the z-axis.
Fig. 6
Fig. 6 Radial gratings and their computer reconstructions with ring diameter D versus various small spatial frequencies ν of the grating: (a) ν = 0.13 lp/mm, (b) ν = 0.17 lp/mm, and (c) ν = 0.19 lp/mm.
Fig. 7
Fig. 7 Femtosecond laser processing system using the ring-shaped beam.
Fig. 8
Fig. 8 Optical reconstructions of the ring-shaped beams and optical microscope images of the structures fabricated by their reconstruction versus the phase difference θd. θd was set to (a) 0.00, (b) 0.35, and (c) 1.44π.
Fig. 9
Fig. 9 Intensity and height images obtained with the confocal microscope and top view and perspective view images of the SEM for the structures fabricated by (a) the circularly scanned spot beam and (b) the ring-shaped beam.
Fig. 10
Fig. 10 In situ image of laser processing in water and intensity and height images obtained with the confocal microscope for the structures fabricated by (a) the circularly scanned spot beam and (b) the ring-shaped beam.

Equations (8)

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ϕ 0 (r)= c 0 r R + θ 0 , U 0 (r)=A(r)expi[ ϕ 0 (r)],
I 0 (ρ)= | F[ U 0 (r)] | 2 ,
ϕ 1 (r)=mod( c 0 r R + θ 1 , 2π ), U 1 (r)=A(r)expi[ ϕ 1 (r)],
ϕ 2 (r)=mod( c 0 r R + θ 2 , 2π ), U 2 (r)=A(r)expi[ ϕ 2 (r)],
ϕ 3 (r)=arg[ U 1 (r)+ U 2 (r)], U 3 (r)=A(r)expi[ ϕ 3 (r)],
P I r = P I rg D E rg P I ax D E ax , P I rg =P 2 π ω rg 2 ,P I ax =P 2 π ( 2 ω rg ) 2 ,
I(r,z)= | F[U(r)]h(r,z) | 2               = | F 1 ( F{ F[U(r)] }F[h(r,z)] ) | 2 ,
D=2r=2fvλ,