Abstract

In recent years, considerable attention has been devoted to laser beams with specific intensity profile, i.e., non-Gaussian. In this work, we present a novel technique to generate high-radial-order Laguerre–Gaussian beams LGp0 based on the use of a binary phase diffractive optical element (BPDOE). The latter is a phase plate made up of annular zones introducing alternatively a phase shift equal to 0 or π modeled on positions which do not coincide with the position of the zeros of the desired LGp0 beam. The LGp0 beams are obtained by transforming a fundamental Gaussian beam through an appropriate BPDOE. The design of the latter is based on the calculation of the Fresnel–Kirchhoff integral, and the diffracted intensity at the focus plane of a lens has been modeled analytically for the first time. The numerical simulations and experiment demonstrate a good beam quality transformation. Obtained LGp0 are suitable for atom trap and pumping solid state laser applications.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. S. Ngcobo, K. Aït-Ameur, N. Passilly, A. Hasnaoui, and A. Forbes, “Exciting higher-order radial Laguerre–Gaussian modes in a diode-pumped solid-state laser resonator,” Appl. Opt. 52, 2093–2101 (2013).
    [CrossRef]
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2013 (3)

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre–Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

S. Ngcobo, K. Aït-Ameur, N. Passilly, A. Hasnaoui, and A. Forbes, “Exciting higher-order radial Laguerre–Gaussian modes in a diode-pumped solid-state laser resonator,” Appl. Opt. 52, 2093–2101 (2013).
[CrossRef]

2012 (1)

2011 (3)

A. Hasnaoui, A. Bencheikh, and K. Ait-Ameur, “Tailored TEMp0 beams for large size 3-D laser prototyping,” Opt. Lasers Eng. 49, 248–251 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

E. Cagniot, M. Fromager, T. Godin, N. Passilly, M. Brunel, and K. Ait-Ameur, “A variant of the method of Fox & Li dedicated to intracavity laser beam shaping,” J. Opt. Soc. Am. A 28, 489–495 (2011).
[CrossRef]

2010 (1)

2009 (1)

2007 (1)

2006 (3)

2005 (1)

2004 (2)

2003 (1)

A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A. Friesem, “Efficient selection of high-order Laguerre–Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003).
[CrossRef]

2002 (1)

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre–Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

2000 (2)

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical beam,” Opt. Lett. 25, 191–193 (2000).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

1998 (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

1997 (1)

R. Bourouis, K. Ait Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

1996 (1)

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Abraham, E. R. I.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre–Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Ait Ameur, K.

R. Bourouis, K. Ait Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

Ait-Ameur, K.

Aït-Ameur, K.

S. Ngcobo, K. Aït-Ameur, N. Passilly, A. Hasnaoui, and A. Forbes, “Exciting higher-order radial Laguerre–Gaussian modes in a diode-pumped solid-state laser resonator,” Appl. Opt. 52, 2093–2101 (2013).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

R. de Saint Denis, N. Passilly, and K. Aït-Ameur, “Laser beam brightness of apertured optical resonators,” Opt. Commun. 264, 193–202 (2006).
[CrossRef]

Allen, L.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Ando, T.

Arlt, J.

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical beam,” Opt. Lett. 25, 191–193 (2000).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

Bashkansky, M.

Bencheikh, A.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, and K. Ait-Ameur, “Tailored TEMp0 beams for large size 3-D laser prototyping,” Opt. Lasers Eng. 49, 248–251 (2011).
[CrossRef]

Bernet, S.

Bogan, C.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre–Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Bond, C.

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

Bourouis, R.

R. Bourouis, K. Ait Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

Brown, D.

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

Brueckner, F.

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

Brunel, M.

Cagniot, E.

E. Cagniot, M. Fromager, T. Godin, N. Passilly, M. Brunel, and K. Ait-Ameur, “A variant of the method of Fox & Li dedicated to intracavity laser beam shaping,” J. Opt. Soc. Am. A 28, 489–495 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

Carbone, L.

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre–Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Chen, C. H.

Davidson, N.

A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A. Friesem, “Efficient selection of high-order Laguerre–Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse Mode Shaping and Selection in Laser Resonators,” Vol. 42 of Progress in Optics Series (Elsevier, 2001), pp. 325–386.

de Saint Denis, R.

R. de Saint Denis, N. Passilly, and K. Aït-Ameur, “Laser beam brightness of apertured optical resonators,” Opt. Commun. 264, 193–202 (2006).
[CrossRef]

R. de Saint Denis, N. Passilly, M. Laroche, T. Mohamed-Brahim, and K. Ait-Ameur, “Beam shaping longitudinal range of a binary diffractive element,” Appl. Opt. 45, 8136–8141 (2006).
[CrossRef]

Dholakia, K.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

Fatemi, F.

Forbes, A.

Freise, A.

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre–Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Friesem, A.

A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A. Friesem, “Efficient selection of high-order Laguerre–Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003).
[CrossRef]

Friesem, A. A.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse Mode Shaping and Selection in Laser Resonators,” Vol. 42 of Progress in Optics Series (Elsevier, 2001), pp. 325–386.

Fromager, M.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

E. Cagniot, M. Fromager, T. Godin, N. Passilly, M. Brunel, and K. Ait-Ameur, “A variant of the method of Fox & Li dedicated to intracavity laser beam shaping,” J. Opt. Soc. Am. A 28, 489–495 (2011).
[CrossRef]

Fuckuchi, N.

Fukuchi, N.

Fulda, P.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre–Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

Fürhaper, S.

Glebov, L. B.

Godin, T.

Hara, T.

Hasman, E.

A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A. Friesem, “Efficient selection of high-order Laguerre–Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse Mode Shaping and Selection in Laser Resonators,” Vol. 42 of Progress in Optics Series (Elsevier, 2001), pp. 325–386.

Hasnaoui, A.

S. Ngcobo, K. Aït-Ameur, N. Passilly, A. Hasnaoui, and A. Forbes, “Exciting higher-order radial Laguerre–Gaussian modes in a diode-pumped solid-state laser resonator,” Appl. Opt. 52, 2093–2101 (2013).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, and K. Ait-Ameur, “Tailored TEMp0 beams for large size 3-D laser prototyping,” Opt. Lasers Eng. 49, 248–251 (2011).
[CrossRef]

Hsieh, W. F.

Inoue, T.

Ishaaya, A.

A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A. Friesem, “Efficient selection of high-order Laguerre–Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003).
[CrossRef]

Ito, H.

Jesacher, A.

Kennedy, S. A.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre–Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Kurihara, K.

Laakonen, P.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Ladjouze, H.

R. Bourouis, K. Ait Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

Laroche, M.

Lodhia, D.

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

Lumeau, J.

Machavariani, G.

A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A. Friesem, “Efficient selection of high-order Laguerre–Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003).
[CrossRef]

Matsumoto, N.

Miyaji, G.

Miyanaga, N.

Mohamed-Brahim, T.

Mokhov, S.

Ngcobo, S.

Ohtake, Y.

Oron, R.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse Mode Shaping and Selection in Laser Resonators,” Vol. 42 of Progress in Optics Series (Elsevier, 2001), pp. 325–386.

Otomo, A.

Padgett, M. J.

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical beam,” Opt. Lett. 25, 191–193 (2000).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Palmer, R.

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

Passilly, N.

Porterfield, J. Z.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre–Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Ritsch-Marte, M.

Robertson, D. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Rotar, V.

Saito, S.

SeGall, M.

Skidanov, R. V.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Smith, G. M.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Sueda, K.

Syouji, A.

Szabo, M. J.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre–Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Tai, P. T.

Teslow, H.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre–Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Turnbull, G. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Turunen, J.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Wang, M.

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

Willke, B.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre–Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Wolf, E.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse Mode Shaping and Selection in Laser Resonators,” Vol. 42 of Progress in Optics Series (Elsevier, 2001), pp. 325–386.

Zeldovich, B.

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A. Friesem, “Efficient selection of high-order Laguerre–Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003).
[CrossRef]

J. Mod. Opt. (2)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

R. Bourouis, K. Ait Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Visualized Exp. (1)

L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013).
[CrossRef]

Opt. Commun. (4)

R. de Saint Denis, N. Passilly, and K. Aït-Ameur, “Laser beam brightness of apertured optical resonators,” Opt. Commun. 264, 193–202 (2006).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (1)

A. Hasnaoui, A. Bencheikh, and K. Ait-Ameur, “Tailored TEMp0 beams for large size 3-D laser prototyping,” Opt. Lasers Eng. 49, 248–251 (2011).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A (1)

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre–Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre–Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Other (2)

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse Mode Shaping and Selection in Laser Resonators,” Vol. 42 of Progress in Optics Series (Elsevier, 2001), pp. 325–386.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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Figures (11)

Fig. 1.
Fig. 1.

Radial intensity distribution of the pure LG10, LG20, and LG30 beams.

Fig. 2.
Fig. 2.

Optical layout.

Fig. 3.
Fig. 3.

Designed BPDOE containing five phase discontinuities.

Fig. 4.
Fig. 4.

Obtained Laguerre–Gaussian beams LG10 and LG40, respectively.

Fig. 5.
Fig. 5.

Transverse distribution of the intensity for LG10 and LG40 at different z positions along the propagation axis.

Fig. 6.
Fig. 6.

Variation of the parameter R2 as function of propagating distance z for calculated LG beams

Fig. 7.
Fig. 7.

(a) Experimental setup for investigating the transformation of a fundamental Gaussian beam into a Laguerre–Gaussian beam. (b) BPDOE patterns. (a,b) Phase patterns of the designed BPDOE without grating used for generating LG10 and LG40 beams, respectively. (c,d) Corresponding phase patterns of the designed (BPDOE + grating).

Fig. 8.
Fig. 8.

Generated Laguerre–Gaussian beam in the focal plane of a lens where, from left to right, we show increasing radial orders from p=1 to p=5.

Fig. 9.
Fig. 9.

Fits of generated LG10, LG20, and LG40 beams.

Fig. 10.
Fig. 10.

How the generated Laguerre–Gaussian beams LG10 and LG40 keep their shapes for several values of ZR

Fig. 11.
Fig. 11.

Transverse distribution of the intensity for generated LG10 and LG40 at different z positions along the propagation axis.

Tables (6)

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Table 1. Roots of Laguerre Polynomials Lp(ρi/W)

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Table 2. Empirical Phase Discontinuity Positions

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Table 3. Characteristics of the Quality of the Transformation at Distance z after the Focusing Lens

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Table 4. Calculated M2 of the Obtained Laguerre–Gaussian Beam Based on the Second-Order Moments of the Intensity Distribution

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Table 5. Quality of the Transformation of Generated LG Beams in the Focal Plane of Lens L5

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Table 6. Characteristics of the Quality of Generated LG10 and LG40 Beams at Distance z after the Focusing Lens

Equations (21)

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Upl(ρ,ϕ)=2PπW2p!(l+p)!(1)pexp[ilϕ]×exp[ρ2W2](2ρW)lLpl(2ρ2W2).
Up0(ρ,ϕ)=U0exp[ρ2W2]×Lp(2ρ2W2),
Ein(ρ)=E0exp[ρ2W2].
E(r,z=f)=2πλfDOEτ(ρ)Ein(ρ)J0(2πλfrρ)ρdρ,
E(r)=2πλf{k=0Nm=0Ni=1p(1)k+m+i+1a2mRi2(k+m+1)4mW2k(m!)2k!(2k+2m+2)r2m+(1)p12exp[a2W24r2]},
I(r)=(2πλf)2|k=0Nm=0Ni=1p(1)k+m+i+1(πλf)2mRi2(k+m+1)W2k(m!)2k!(2k+2m+2)r2m+(1)p12exp[(πWλf)2r2]|2,
g(k,m,i)=(1)k+m+i+1(πλf)2m2(m!)2k!(k+m+1).
I(r)=(2πλf)2|k=0Nm=0Ni=1pg(k,m,i)Ri2(k+m+1)W2kr2m+(1)p2exp[(πWλf)2r2]|2,
E(r,z=f)=2πλf{0Rexp[ρ2W2]J0(2πλfrρ)ρdρRexp[ρ2W2]J0(2πλfrρ)ρdρ}.
Rexp[ρ2W2]J0(arρ)rdr=0exp[ρ2W2]J0(arρ)ρdρ0Rexp[ρ2W2]J0(arρ)ρdρ,
E(ρ,z=f)=2πλf{20RτEOD(r)exp[r2W2k]J0(arρ)rdr0exp[r2W2k]J0(arρ)rdr},
E1(ρ)=0exp[r2W2]J0(arρ)rdr
E2(ρ)=20RτEOD(r)exp[r2W2k]J0(arρ)rdr.
0exp[bx]J0(ax)dx=exp[a24b]b.
E1(ρ)=12W2exp[W24(aρ)2].
J0(arρ)=m=0(1)m(arρ/2)2m(m!)2=m=0(1)mρ2m(ar)2m22m(m!)2
exp[r2W2]=k=0(1)kr2kW2kk!,
E2(ρ)=0RτEOD(r)k=0N(1)kr2kW2kk!×m=0N(1)ma2mρ2mr2m4m(m!)2rdr.
E2(ρ,z=f)=k=0Nm=0N(1)k+ma2mρ2m4mW2k(m!)2k!×0RτEOD(r)r2k+2m+1dr.
E2total(ρ)=k=0Nm=0Ni=1p(1)k+m(1)i+1a2mρ2m4mW2k(m!)2k!0Rir2k+2m+1dr.
E2total(ρ)=k=0Nm=0Ni=1p(1)k+m+i+1a2mρ2mRi2(k+m+1)4mW2k(m!)2k!(2k+2m+2).

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