Abstract

The entire range of transformations that a Laguerre–Gaussian (LG) beam with astigmatism can go through in free space is clarified. The transformations are governed by the relative phase between the astigmatic Hermite–Gaussian components. Formulas describing the behavior of this relative phase are obtained and used to classify and map the transformation patterns to initial beam parameters. The difference between an LG beam and a phase singular beam generated by a hologram under astigmatic conditions is also investigated.

© 2005 Optical Society of America

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  1. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1996).
    [CrossRef]
  2. H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beam produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
    [CrossRef]
  3. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
    [CrossRef]
  4. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  5. H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinstein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
    [CrossRef] [PubMed]
  6. M. E. J. Friese, J. Enger, H. Rubinstein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particle,” Phys. Rev. A 54, 1593–1596 (1996).
    [CrossRef] [PubMed]
  7. N. B. Simpson, L. Allen, M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
    [CrossRef] [PubMed]
  8. E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
    [CrossRef]
  9. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  10. V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 427–429 (1990).
  11. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  12. H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S961 (1992).
    [CrossRef]
  13. I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [CrossRef]
  14. J. Serna, F. Encinas-Sanz, G. Nemes, “Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens,” J. Opt. Soc. Am. A 18, 1726–1733 (2001).
    [CrossRef]
  15. A. Y. Bekshaev, M. S. Soskin, M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
    [CrossRef]
  16. J. Courtial, M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre–Gaussian laser modes,” Opt. Commun. 159, 13–18 (1999).
    [CrossRef]
  17. R. P. Singh, S. R. Chowdhury, “Trajectory of an optical vortex: canonical vs. non-canonical,” Opt. Commun. 215, 231–237 (2002).
    [CrossRef]
  18. G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
    [CrossRef]
  19. A. Wada, Y. Miyamoto, T. Ohtani, N. Nishihara, M. Takeda, “Effects of astigmatic aberration in holographic generation of Laguerre–Gaussian beam,” in International Conference on Lasers, Applications, and Technologies 2002: Advanced Lasers and Systems, G. Huber, I. A. Scherbakov, and V. Y. Panchenko, eds., Proc. SPIE5137, 177–180 (2003).
  20. M. J. Padgett, L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4, S17–S19 (2002).
    [CrossRef]
  21. L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
    [CrossRef]
  22. A. T. O’Neil, J. Countial, “Mode transformation in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35–45 (2000).
    [CrossRef]
  23. A. E. Siegman, Lasers (University Science, 1986).
  24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

2003 (1)

2002 (2)

R. P. Singh, S. R. Chowdhury, “Trajectory of an optical vortex: canonical vs. non-canonical,” Opt. Commun. 215, 231–237 (2002).
[CrossRef]

M. J. Padgett, L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4, S17–S19 (2002).
[CrossRef]

2001 (2)

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

J. Serna, F. Encinas-Sanz, G. Nemes, “Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens,” J. Opt. Soc. Am. A 18, 1726–1733 (2001).
[CrossRef]

2000 (1)

A. T. O’Neil, J. Countial, “Mode transformation in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35–45 (2000).
[CrossRef]

1999 (2)

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

J. Courtial, M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre–Gaussian laser modes,” Opt. Commun. 159, 13–18 (1999).
[CrossRef]

1997 (2)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

N. B. Simpson, L. Allen, M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[CrossRef] [PubMed]

1996 (2)

M. E. J. Friese, J. Enger, H. Rubinstein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particle,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1996).
[CrossRef]

1995 (2)

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beam produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

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

1993 (2)

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992 (3)

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S961 (1992).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1991 (1)

E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

1990 (1)

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 427–429 (1990).

Abramochkin, E.

E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Allen, L.

M. J. Padgett, L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4, S17–S19 (2002).
[CrossRef]

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

N. B. Simpson, L. Allen, M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[CrossRef] [PubMed]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Basistiy, I. V.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Bazhenov, V. Y.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 427–429 (1990).

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Bekshaev, A. Y.

Chowdhury, S. R.

R. P. Singh, S. R. Chowdhury, “Trajectory of an optical vortex: canonical vs. non-canonical,” Opt. Commun. 215, 231–237 (2002).
[CrossRef]

Countial, J.

A. T. O’Neil, J. Countial, “Mode transformation in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35–45 (2000).
[CrossRef]

Courtial, J.

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

J. Courtial, M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre–Gaussian laser modes,” Opt. Commun. 159, 13–18 (1999).
[CrossRef]

Encinas-Sanz, F.

Enger, J.

M. E. J. Friese, J. Enger, H. Rubinstein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particle,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Friese, M. E. J.

M. E. J. Friese, J. Enger, H. Rubinstein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particle,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

He, H.

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

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beam produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Heckenberg, H. R.

H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S961 (1992).
[CrossRef]

Heckenberg, N. R.

M. E. J. Friese, J. Enger, H. Rubinstein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particle,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beam produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

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

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1996).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1996).
[CrossRef]

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S961 (1992).
[CrossRef]

Miyamoto, Y.

A. Wada, Y. Miyamoto, T. Ohtani, N. Nishihara, M. Takeda, “Effects of astigmatic aberration in holographic generation of Laguerre–Gaussian beam,” in International Conference on Lasers, Applications, and Technologies 2002: Advanced Lasers and Systems, G. Huber, I. A. Scherbakov, and V. Y. Panchenko, eds., Proc. SPIE5137, 177–180 (2003).

Molina-Terriza, G.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Nemes, G.

Nishihara, N.

A. Wada, Y. Miyamoto, T. Ohtani, N. Nishihara, M. Takeda, “Effects of astigmatic aberration in holographic generation of Laguerre–Gaussian beam,” in International Conference on Lasers, Applications, and Technologies 2002: Advanced Lasers and Systems, G. Huber, I. A. Scherbakov, and V. Y. Panchenko, eds., Proc. SPIE5137, 177–180 (2003).

O’Neil, A. T.

A. T. O’Neil, J. Countial, “Mode transformation in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35–45 (2000).
[CrossRef]

Ohtani, T.

A. Wada, Y. Miyamoto, T. Ohtani, N. Nishihara, M. Takeda, “Effects of astigmatic aberration in holographic generation of Laguerre–Gaussian beam,” in International Conference on Lasers, Applications, and Technologies 2002: Advanced Lasers and Systems, G. Huber, I. A. Scherbakov, and V. Y. Panchenko, eds., Proc. SPIE5137, 177–180 (2003).

Padgett, M. J.

M. J. Padgett, L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4, S17–S19 (2002).
[CrossRef]

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

J. Courtial, M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre–Gaussian laser modes,” Opt. Commun. 159, 13–18 (1999).
[CrossRef]

N. B. Simpson, L. Allen, M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[CrossRef] [PubMed]

Recolons, J.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Rubinstein-Dunlop, H.

M. E. J. Friese, J. Enger, H. Rubinstein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particle,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

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

H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S961 (1992).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beam produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Serna, J.

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Simpson, N. B.

Singh, R. P.

R. P. Singh, S. R. Chowdhury, “Trajectory of an optical vortex: canonical vs. non-canonical,” Opt. Commun. 215, 231–237 (2002).
[CrossRef]

Smith, C. P.

H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S961 (1992).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

Soskin, M. S.

A. Y. Bekshaev, M. S. Soskin, M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 427–429 (1990).

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Takeda, M.

A. Wada, Y. Miyamoto, T. Ohtani, N. Nishihara, M. Takeda, “Effects of astigmatic aberration in holographic generation of Laguerre–Gaussian beam,” in International Conference on Lasers, Applications, and Technologies 2002: Advanced Lasers and Systems, G. Huber, I. A. Scherbakov, and V. Y. Panchenko, eds., Proc. SPIE5137, 177–180 (2003).

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vasnetsov, M. V.

A. Y. Bekshaev, M. S. Soskin, M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 427–429 (1990).

Volostnikov, V.

E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Wada, A.

A. Wada, Y. Miyamoto, T. Ohtani, N. Nishihara, M. Takeda, “Effects of astigmatic aberration in holographic generation of Laguerre–Gaussian beam,” in International Conference on Lasers, Applications, and Technologies 2002: Advanced Lasers and Systems, G. Huber, I. A. Scherbakov, and V. Y. Panchenko, eds., Proc. SPIE5137, 177–180 (2003).

Wegener, M. J.

H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S961 (1992).
[CrossRef]

White, A. G.

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Wright, E. M.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

J. Mod. Opt. (1)

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beam produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

M. J. Padgett, L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4, S17–S19 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

JETP Lett. (1)

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 427–429 (1990).

Opt. Commun. (6)

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

J. Courtial, M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre–Gaussian laser modes,” Opt. Commun. 159, 13–18 (1999).
[CrossRef]

R. P. Singh, S. R. Chowdhury, “Trajectory of an optical vortex: canonical vs. non-canonical,” Opt. Commun. 215, 231–237 (2002).
[CrossRef]

E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

A. T. O’Neil, J. Countial, “Mode transformation in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35–45 (2000).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S961 (1992).
[CrossRef]

Phys. Rev. A (2)

M. E. J. Friese, J. Enger, H. Rubinstein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particle,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

Phys. Rev. Lett. (3)

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

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Proc. IEEE (1)

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[CrossRef]

Other (3)

A. Wada, Y. Miyamoto, T. Ohtani, N. Nishihara, M. Takeda, “Effects of astigmatic aberration in holographic generation of Laguerre–Gaussian beam,” in International Conference on Lasers, Applications, and Technologies 2002: Advanced Lasers and Systems, G. Huber, I. A. Scherbakov, and V. Y. Panchenko, eds., Proc. SPIE5137, 177–180 (2003).

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J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (9)

Fig. 1
Fig. 1

Gray-scale representation of Δ θ lim versus F x , F y ( 10 < F x , F y < 10 ). Dashed curves are borders separating the region Δ θ lim > π 2 and the region Δ θ lim < π 2 .

Fig. 2
Fig. 2

Gray-scale representation of Δ θ [ z ̃ e ] versus F x , F y ( 10 < F x , F y < 10 ). Dashed lines are borders separating the region Δ θ [ z ̃ e ] > π 2 and the region Δ θ [ z ̃ e ] < π 2 . The region F x + F y 0 is not of interest and is shown in black.

Fig. 3
Fig. 3

Phase increment θ versus propagation distance z for the case where the total charge of splitting singularities is changed to m at the far field. (a) The transformation process is monotonic ( m = 3 , w = 1000 λ , R 0 x = 3 × 10 5 λ , R 0 y = 3 × 10 5 λ ). (b) The beam transformation is reversed from z ̃ = z ̃ e ( m = 3 , w = 1000 λ , R 0 x = 3 × 10 5 λ , R 0 y = 1.2 × 10 7 λ ).

Fig. 4
Fig. 4

Phase increment θ versus propagation distance z for the case where the beam profile is that of an HG beam at the far field. (a) The transformation process is monotonic ( m = 3 , w = 1000 λ , R 0 x = 3.14 × 10 6 λ , R 0 y = 3.14 × 10 6 λ ). (b) The beam transformation is reversed from z ̃ = z ̃ e ( m = 3 , w = 1000 λ , R 0 x = 7.85 × 10 5 λ , R 0 y = 1.25 × 10 7 λ ).

Fig. 5
Fig. 5

Phase increment θ versus propagation distance z for the case where the total charge of splitting phase singularities is conserved at the far field. (a) The transformation process is monotonic ( m = 3 , w = 1000 λ , R x = 2 × 10 6 λ , R y = 6 × 10 6 λ ). (b) The beam transformation is reversed from z ̃ = z ̃ e , and the total charge of splitting phase singularities is conserved throughout ( m = 3 , w = 1000 λ , R 0 x = 1 × 10 6 λ , R 0 y = 1.6 × 10 6 λ ). (c) The beam profile is that of an HG beam at the reversal point ( m = 3 , w = 1000 λ , R 0 x = 7.85 × 10 5 λ , R 0 y = 1.57 × 10 6 λ ). (d) The total charge of splitting singularities is changed to m at the reversal point ( m = 3 , w = 1000 λ , R 0 x = 3 × 10 5 λ , R 0 y = 9 × 10 5 λ ).

Fig. 6
Fig. 6

LG beam with astigmatic aberration ( m = 3 , w = 1000 λ , R 0 x = 3 × 10 5 λ , R 0 y = 3 × 10 5 λ ). The parameters correspond to those for Fig. 3a.

Fig. 7
Fig. 7

LG beam with astigmatic aberration ( m = 3 , w = 1000 λ , R 0 x = 3 × 10 5 λ , R 0 y = 9 × 10 5 λ ). The parameters correspond to those for Fig. 5d.

Fig. 8
Fig. 8

Beam generated by a hologram with astigmatic aberration ( m = 3 , w = 2000 λ , R 0 x = 3 × 10 5 λ , R 0 y = 3 × 10 5 λ ). The parameters correspond to those for Figs. 3a, 6.

Fig. 9
Fig. 9

Beam generated by a hologram with astigmatic aberration ( m = 3 , w = 2000 λ , R 0 x = 3 × 10 5 λ , R 0 y = 9 × 10 5 λ ). The parameters correspond to those for Figs. 5d, 7.

Equations (57)

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u [ x , y ; 0 ] = 1 w 0 2 p ! π ( p + m ) ! ( 2 ρ w 0 ) m L p m [ 2 ρ 2 w 0 2 ] exp [ r 2 w 0 2 + i π λ ( x 2 R 0 x + y 2 R 0 y ) ] exp [ i m ϕ ] ,
u [ x , y ; 0 ] = s = 0 N exp [ sgn [ m ] i s π 2 ] b [ p , m + p , s ] HG s t [ x , y ; 0 ]
( N = 2 p + m , t = N s ) ,
b [ m , n , k ] = ( ( m + n k ) ! k ! 2 ( m + n ) m ! n ! ) 1 2 1 k ! d k d a k { ( 1 a ) m ( 1 + a ) n } a = 0 .
HG s t [ x , y ; 0 ] = 1 w 0 2 π 2 ( s + t ) s ! t ! H s [ 2 x w 0 ] H t [ 2 y w 0 ] exp [ r 2 w 0 2 + i π λ ( x 2 R 0 x + y 2 R 0 y ) ] ,
sgn [ x ] = { 1 ( x > 0 ) 0 ( x = 0 ) 1 ( x < 0 ) } .
HG s t [ x , y ; z ] = 1 w x w y 2 π 2 ( s + t ) s ! t ! H s [ 2 x w x ] H t [ 2 y w y ] × exp [ x 2 ( 1 w x 2 i k 2 R x ) y 2 ( 1 w y 2 i k 2 R y ) i ( Ψ s t Φ s t ) ] ,
w α 2 = w w α 2 { 1 + ( 2 ( z z w α ) k w w α 2 ) 2 } ,
R α = ( z z w α ) { 1 + ( k w w α 2 2 ( z z w α ) ) 2 } ,
Ψ s t = Ψ x ( 2 s + 1 ) + Ψ y ( 2 t + 1 ) 2 ,
Ψ α = arctan [ 2 ( z z w α ) k w w α 2 ] ,
w w α 2 = w 0 2 { 1 + ( k w 0 2 2 R 0 α ) 2 } 1 ,
z w α = R 0 α { 1 + ( 2 R 0 α k w 0 2 ) 2 } 1 .
u [ x , y ; z ] = s = 0 N exp [ sgn [ m ] i s π 2 ] b [ p , m + p , s ] H G s t [ x , y ; z ] .
A s = b [ p , m + p , s ] 2 π 2 ( s + t ) s ! t ! ,
θ s t [ z ] = Φ s t Ψ s t [ z ] + sgn [ m ] s π 2 .
u [ x , y ; z ] = 1 w x w y exp [ x 2 ( 1 w x 2 i k 2 R x ) y 2 ( 1 w y 2 i k 2 R y ) ] × s = 0 N A s H s [ 2 x w x ] H t [ 2 y w y ] exp [ i θ s t ] .
θ s t [ z ] = θ 0 N [ z ] + s θ [ z ] ,
θ 0 N [ z ] = 1 2 ( Ψ x [ z ] Ψ x [ 0 ] ) + 2 N + 1 2 ( Ψ y [ z ] Ψ y [ 0 ] ) ,
θ [ z ] = Ψ x [ z ] + Ψ y [ z ] + Ψ x [ 0 ] Ψ y [ 0 ] + sgn [ m ] π 2
= arctan [ z ̃ ( 1 + F x 2 ) + F x ] + arctan [ z ̃ ( 1 + F y 2 ) + F y ] + arctan [ F x ] arctan [ F y ] + sgn [ m ] π 2 .
z ̃ = 2 z k w 0 2 ,
F x = k w 0 2 2 R 0 x ,
F y = k w 0 2 2 R 0 y .
s = 0 N A s H s [ 2 x w x ] H t [ 2 y w y ] exp [ i s θ ] = 0 .
( x , y ) = ( 0 , 0 ) , ( ± w x 6 cos θ 4 cos θ 2 , ± w y 6 cos θ 4 cos θ 2 )
( cos θ < 0 ) ,
( x , y ) = ( 0 , 0 ) , ( w x 6 cos θ 4 cos θ 2 , ± w y 6 cos θ 4 cos θ 2 )
( cos θ > 0 ) .
Δ θ [ z ̃ ] = θ [ z ̃ ] θ [ 0 ]
= arctan [ z ̃ ( 1 + F x 2 ) + F x ] + arctan [ z ̃ ( 1 + F y 2 ) + F y ] + arctan [ F x ] arctan [ F y ] .
Δ θ lim = lim z ̃ Δ θ [ z ̃ ] = arctan [ F x ] arctan [ F y ] ,
tan Δ θ lim = F x F y 1 + F x F y .
1 + F x F y = 0 .
z ̃ e = 2 F x + F y .
tan Δ θ [ z ̃ ] = tan [ θ A + θ B ] ,
θ A = arctan [ z ̃ ( 1 + F x 2 ) + F x ] arctan [ F x ] ,
θ B = arctan [ z ̃ ( 1 + F y 2 ) + F y ] arctan [ F y ] .
tan θ A = 1 z ̃ 1 + F x ,
tan θ B = 1 z ̃ 1 + F y .
tan Δ θ [ z ̃ ] = F x F y z ̃ 2 + ( F x + F y ) z ̃ 1 + F x F y + 1 .
tan Δ θ [ z ̃ e ] = 4 ( F x F y ) ( F x F y + 2 ) ( F x F y 2 ) .
F x F y ± 2 = 0 .
d Ψ α d z = 2 k w α 2 ,
d Δ θ d z = 2 k w x 2 + 2 k w y 2 .
u [ x , y ; 0 ] = 1 w H 2 π exp [ ρ 2 w H 2 + i π λ ( x 2 R 0 x + y 2 R 0 y ) ] exp [ i m ϕ ] ,
u [ x , y ; z ] = U [ f x , f y ; 0 ] H [ f x , f y ; z ] exp [ i 2 π ( f x x + f y y ) ] d f x d f y ,
H [ f x , f y ; z ] = exp [ i 2 π λ z 1 λ 2 f x 2 λ 2 f y 2 ] .
φ max = arctan [ λ f max 1 λ 2 f max 2 ] .
T z = T + 2 z tan φ max
= T + 2 z λ f max 1 λ 2 f max 2 .
u [ x , y ; z ] = k = N N 1 l = N N 1 U [ k f max N , l f max N ; 0 ] H [ k f max N , l f max N ; z ] exp [ i 2 π f max N ( k x + l y ) ] .
u [ x , y ; z ] = s = t = u [ x + s N f max , y + t N f max ; z ] .
T v = N f max ( T z N f max )
= 2 N f max T z
= 2 N f max T + 2 z λ f max 1 λ 2 f max 2 ,
N > f max 2 ( T v + T ) + z λ f max 2 1 ( λ f max ) 2 .

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