Abstract

We investigate experimentally and numerically the propagation characteristics of laser beams formed by imparting an azimuthal phase ϕ to a Gaussian beam, where is an integer. We find that when high- beams of a finite extent are focused through a lens, the beams achieve peak intensity and are most sharply defined before and after the focal plane. Additionally, in these regions of highest intensity the effect of aberrations on the beam quality is greatly reduced, which we also demonstrate experimentally and numerically. We present a simple geometrical picture that provides excellent estimates of the beam radius and propagation distance to the plane of peak intensity.

© 2007 Optical Society of America

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References

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  1. D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
    [CrossRef] [PubMed]
  2. T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
    [CrossRef]
  3. F. K. Fatemi, M. Bashkansky, and Z. Dutton, "Dynamic, high-speed spatial manipulation of cold atoms using acousto-optic and spatial light modulation," Opt. Express 15, 3589-3596 (2007).
    [CrossRef] [PubMed]
  4. N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill, "Laser beams with embedded vortices: tools for atom optics," J. Opt. Soc. Am. B 23, 94-103 (2006).
    [CrossRef]
  5. D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Opt. 53, 547-556 (2006).
    [CrossRef]
  6. K. T. Gahagan and J. G. A. Swartzlander, "Trapping of low-index microparticles in an optical vortex," J. Opt. Soc. Am. B 15, 524-534 (1998).
    [CrossRef]
  7. J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
    [CrossRef]
  8. M. Schiffer, M. Rauner, S. Kuppens, M. Zinner, K. Sengstock, and W. Ertmer, "Guiding, focusing, and cooling of atoms in a strong dipole potential," Appl. Phys. B 67, 705-708 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  11. Y. Roichman and D. G. Grier, "Three-dimensional holographic ring traps," Proc. SPIE 6483, 64830F (2007).
    [CrossRef]
  12. G. Machavariani, N. Davidson, E. Hasman, S. Blit, A. Ishaaya, and A. A. Friesem, "Efficient conversion of a Gaussian beam to a high purity helical beam," Opt. Commun. 209, 265-271 (2002).
    [CrossRef]
  13. A. A. Ishaaya, N. Davidson, and A. A. Friesem, "Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser," Opt. Express 13, 4952-4962 (2005).
    [CrossRef] [PubMed]
  14. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, and C. Tamm, "Transverse laser patterns. I. phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
    [CrossRef] [PubMed]
  15. 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]
  16. M. A. Bandres and J. C. Gutierrez-Vega, "Ince-Gaussian beams," Opt. Lett. 29, 144-146 (2004).
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  17. F. K. Fatemi and M. Bashkansky, "Cold atom guidance using a binary spatial light modulator," Opt. Express 14, 1368-1375 (2006).
    [CrossRef] [PubMed]
  18. A. E. Siegman, Lasers (University Science Books, 1986).
  19. Y. Li and E. Wolf, "Focal shift in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
    [CrossRef]
  20. S. Chavez-Cerda and G. H. C. New, "Evolution of focused Hankel waves and Bessel beams," Opt. Commun. 181, 369-377 (2000).
    [CrossRef]
  21. S. Sundbeck, I. Gruzberg, and D. G. Grier, "Structure and scaling of helical modes of light," Opt. Lett. 30, 477-479 (2005).
    [CrossRef] [PubMed]
  22. Y. Roichman, A. Waldon, E. Gardel, and D. G. Grier, "Optical traps with geometric aberrations," Appl. Opt. 45, 3425-3429 (2006).
    [CrossRef] [PubMed]
  23. A. Wada, T. Ohtani, Y. Miyamoto, and M. Takeda, "Propagation analysis of the Laguerre-Gaussian beam with astigmatism," J. Opt. Soc. Am. A 22, 2746-2755 (2005).
    [CrossRef]

2007 (2)

2006 (4)

2005 (3)

2004 (1)

2003 (2)

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

2002 (2)

G. Machavariani, N. Davidson, E. Hasman, S. Blit, A. Ishaaya, and A. A. Friesem, "Efficient conversion of a Gaussian beam to a high purity helical beam," Opt. Commun. 209, 265-271 (2002).
[CrossRef]

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)

S. Chavez-Cerda and G. H. C. New, "Evolution of focused Hankel waves and Bessel beams," Opt. Commun. 181, 369-377 (2000).
[CrossRef]

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

1998 (2)

M. Schiffer, M. Rauner, S. Kuppens, M. Zinner, K. Sengstock, and W. Ertmer, "Guiding, focusing, and cooling of atoms in a strong dipole potential," Appl. Phys. B 67, 705-708 (1998).
[CrossRef]

K. T. Gahagan and J. G. A. Swartzlander, "Trapping of low-index microparticles in an optical vortex," J. Opt. Soc. Am. B 15, 524-534 (1998).
[CrossRef]

1997 (1)

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

1996 (1)

N. B. Simpson, L. Allen, and M. J. Padgett, "Optical tweezers and optical spanners with Laguerre-Gaussian modes," J. Mod. Opt. 43, 2485-2491 (1996).
[CrossRef]

1991 (1)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, and C. Tamm, "Transverse laser patterns. I. phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

1981 (1)

Y. Li and E. Wolf, "Focal shift in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
[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]

Allen, L.

N. B. Simpson, L. Allen, and M. J. Padgett, "Optical tweezers and optical spanners with Laguerre-Gaussian modes," J. Mod. Opt. 43, 2485-2491 (1996).
[CrossRef]

Arakelyan, I. V.

Arlt, J.

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

Bandres, M. A.

Bashkansky, M.

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, and C. Tamm, "Transverse laser patterns. I. phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

Blit, S.

G. Machavariani, N. Davidson, E. Hasman, S. Blit, A. Ishaaya, and A. A. Friesem, "Efficient conversion of a Gaussian beam to a high purity helical beam," Opt. Commun. 209, 265-271 (2002).
[CrossRef]

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, and C. Tamm, "Transverse laser patterns. I. phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

Chattrapiban, N.

Chavez-Cerda, S.

S. Chavez-Cerda and G. H. C. New, "Evolution of focused Hankel waves and Bessel beams," Opt. Commun. 181, 369-377 (2000).
[CrossRef]

Curtis, J. E.

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

Davidson, N.

A. A. Ishaaya, N. Davidson, and A. A. Friesem, "Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser," Opt. Express 13, 4952-4962 (2005).
[CrossRef] [PubMed]

G. Machavariani, N. Davidson, E. Hasman, S. Blit, A. Ishaaya, and A. A. Friesem, "Efficient conversion of a Gaussian beam to a high purity helical beam," Opt. Commun. 209, 265-271 (2002).
[CrossRef]

Dholakia, K.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

Dutton, Z.

Ertmer, W.

M. Schiffer, M. Rauner, S. Kuppens, M. Zinner, K. Sengstock, and W. Ertmer, "Guiding, focusing, and cooling of atoms in a strong dipole potential," Appl. Phys. B 67, 705-708 (1998).
[CrossRef]

Fatemi, F. K.

Freegarde, T.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Friesem, A. A.

A. A. Ishaaya, N. Davidson, and A. A. Friesem, "Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser," Opt. Express 13, 4952-4962 (2005).
[CrossRef] [PubMed]

G. Machavariani, N. Davidson, E. Hasman, S. Blit, A. Ishaaya, and A. A. Friesem, "Efficient conversion of a Gaussian beam to a high purity helical beam," Opt. Commun. 209, 265-271 (2002).
[CrossRef]

Gahagan, K. T.

Gardel, E.

Gherardi, D. M.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Grier, D. G.

Y. Roichman and D. G. Grier, "Three-dimensional holographic ring traps," Proc. SPIE 6483, 64830F (2007).
[CrossRef]

Y. Roichman, A. Waldon, E. Gardel, and D. G. Grier, "Optical traps with geometric aberrations," Appl. Opt. 45, 3425-3429 (2006).
[CrossRef] [PubMed]

S. Sundbeck, I. Gruzberg, and D. G. Grier, "Structure and scaling of helical modes of light," Opt. Lett. 30, 477-479 (2005).
[CrossRef] [PubMed]

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

Gruzberg, I.

Gutierrez-Vega, J. C.

Hasman, E.

G. Machavariani, N. Davidson, E. Hasman, S. Blit, A. Ishaaya, and A. A. Friesem, "Efficient conversion of a Gaussian beam to a high purity helical beam," Opt. Commun. 209, 265-271 (2002).
[CrossRef]

Hill, W. T.

Hirano, T.

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

Hitomi, T.

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

Ishaaya, A.

G. Machavariani, N. Davidson, E. Hasman, S. Blit, A. Ishaaya, and A. A. Friesem, "Efficient conversion of a Gaussian beam to a high purity helical beam," Opt. Commun. 209, 265-271 (2002).
[CrossRef]

Ishaaya, A. 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]

Kuga, T.

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

Kuppens, S.

M. Schiffer, M. Rauner, S. Kuppens, M. Zinner, K. Sengstock, and W. Ertmer, "Guiding, focusing, and cooling of atoms in a strong dipole potential," Appl. Phys. B 67, 705-708 (1998).
[CrossRef]

Li, Y.

Y. Li and E. Wolf, "Focal shift in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Livesey, J.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Lugiato, L. A.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, and C. Tamm, "Transverse laser patterns. I. phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

Machavariani, G.

G. Machavariani, N. Davidson, E. Hasman, S. Blit, A. Ishaaya, and A. A. Friesem, "Efficient conversion of a Gaussian beam to a high purity helical beam," Opt. Commun. 209, 265-271 (2002).
[CrossRef]

McGloin, D.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Melville, H.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Miyamoto, Y.

New, G. H. C.

S. Chavez-Cerda and G. H. C. New, "Evolution of focused Hankel waves and Bessel beams," Opt. Commun. 181, 369-377 (2000).
[CrossRef]

Ohtani, T.

Padgett, M. J.

N. B. Simpson, L. Allen, and M. J. Padgett, "Optical tweezers and optical spanners with Laguerre-Gaussian modes," J. Mod. Opt. 43, 2485-2491 (1996).
[CrossRef]

Penna, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, and C. Tamm, "Transverse laser patterns. I. phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

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]

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, and C. Tamm, "Transverse laser patterns. I. phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

Rauner, M.

M. Schiffer, M. Rauner, S. Kuppens, M. Zinner, K. Sengstock, and W. Ertmer, "Guiding, focusing, and cooling of atoms in a strong dipole potential," Appl. Phys. B 67, 705-708 (1998).
[CrossRef]

Rhodes, D. P.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Rogers, E. A.

Roichman, Y.

Y. Roichman and D. G. Grier, "Three-dimensional holographic ring traps," Proc. SPIE 6483, 64830F (2007).
[CrossRef]

Y. Roichman, A. Waldon, E. Gardel, and D. G. Grier, "Optical traps with geometric aberrations," Appl. Opt. 45, 3425-3429 (2006).
[CrossRef] [PubMed]

Roy, R.

Schiffer, M.

M. Schiffer, M. Rauner, S. Kuppens, M. Zinner, K. Sengstock, and W. Ertmer, "Guiding, focusing, and cooling of atoms in a strong dipole potential," Appl. Phys. B 67, 705-708 (1998).
[CrossRef]

Sengstock, K.

M. Schiffer, M. Rauner, S. Kuppens, M. Zinner, K. Sengstock, and W. Ertmer, "Guiding, focusing, and cooling of atoms in a strong dipole potential," Appl. Phys. B 67, 705-708 (1998).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, and 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 Books, 1986).

Simpson, N. B.

N. B. Simpson, L. Allen, and M. J. Padgett, "Optical tweezers and optical spanners with Laguerre-Gaussian modes," J. Mod. Opt. 43, 2485-2491 (1996).
[CrossRef]

Sundbeck, S.

Swartzlander, J. G. 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]

Takeda, M.

Tamm, C.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, and C. Tamm, "Transverse laser patterns. I. phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

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]

Torii, Y.

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

Wada, A.

Waldon, A.

Wolf, E.

Y. Li and E. Wolf, "Focal shift in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Zinner, M.

M. Schiffer, M. Rauner, S. Kuppens, M. Zinner, K. Sengstock, and W. Ertmer, "Guiding, focusing, and cooling of atoms in a strong dipole potential," Appl. Phys. B 67, 705-708 (1998).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (2)

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

M. Schiffer, M. Rauner, S. Kuppens, M. Zinner, K. Sengstock, and W. Ertmer, "Guiding, focusing, and cooling of atoms in a strong dipole potential," Appl. Phys. B 67, 705-708 (1998).
[CrossRef]

J. Mod. Opt. (2)

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

N. B. Simpson, L. Allen, and M. J. Padgett, "Optical tweezers and optical spanners with Laguerre-Gaussian modes," J. Mod. Opt. 43, 2485-2491 (1996).
[CrossRef]

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

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

Nature (1)

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

Opt. Commun. (3)

G. Machavariani, N. Davidson, E. Hasman, S. Blit, A. Ishaaya, and A. A. Friesem, "Efficient conversion of a Gaussian beam to a high purity helical beam," Opt. Commun. 209, 265-271 (2002).
[CrossRef]

Y. Li and E. Wolf, "Focal shift in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
[CrossRef]

S. Chavez-Cerda and G. H. C. New, "Evolution of focused Hankel waves and Bessel beams," Opt. Commun. 181, 369-377 (2000).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (2)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, and C. Tamm, "Transverse laser patterns. I. phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

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. (2)

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

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

Proc. SPIE (1)

Y. Roichman and D. G. Grier, "Three-dimensional holographic ring traps," Proc. SPIE 6483, 64830F (2007).
[CrossRef]

Other (1)

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

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Figures (8)

Fig. 1
Fig. 1

Overlap integral | C p , 0 | as a function of radial index p for the value of ω L G that maximizes | C 0, 0 | . The spread of contributing radial indices increases with .

Fig. 2
Fig. 2

Left: Typical phase profile applied to the SLM ( = 16 ) . Right: Optical layout for characterizing the hollow laser beams. BS = 50 : 50 beam splitter, λ / 2 = half-wave   plate .

Fig. 3
Fig. 3

Beam profiles for = 4 [left: (a), (b), and (e)] and = 16 [right; (c), (d), and (f)]. (a)–(d): Cross-sectional intensity profiles through the focus as a function of propagation distance. (e) Radial intensity profile for three different planes for = 4 and (f) = 16 . Inset to (e) and (f) are images of the | E | profile (square root of images has been taken to enhance detail).

Fig. 4
Fig. 4

(a) Peak intensity as a function of propagation distance for = 4 and = 16 , using f = 80   mm , and (b) using f = 8   mm for = 16 . The tighter focusing (larger NA) in (b) brings the planes of peak intensity closer to the focal plane. Numerical simulations (curves) and experimental results (symbols) are shown. Error bars represent range of intensities Δ I around the ring due to aberrations.

Fig. 5
Fig. 5

(a) Width of primary intensity ring for = 4 (dashed) and = 16 (solid). For = 16 , the beam is most clearly defined before and after the focal plane. (b) Beam radius as a function of z for f = 80   mm , = 4 and = 16 (as indicated). Numerical simulations (curves) and experimental results (symbols) are shown. Error bars are ranges due to aberrations. (c) Same as (b) but f = 8   mm , = 16 .

Fig. 6
Fig. 6

Definition of terms for ray-optics picture. An incident ray at distance R is deflected by angle α. In a plane z , its radius is R . This plane is imaged to a plane a distance L away from the focal plane, where the beam radius is magnified to R . Note that | z | may be larger than the focal length f, but is drawn before the lens for visual clarity. For the same reason, only one of the two solutions for z and for L have been drawn.

Fig. 7
Fig. 7

Comparison of full numerical simulation of propagation (symbols) with geometric optics estimates (curves) for (a) beam radius, and (b) shift of peak intensity (L), using = 4 , 8, 16, and 32.

Fig. 8
Fig. 8

Effect of aberrations. (a) Images of the hollow beam in three planes close to the focus (top row) without applied aberrations, (middle row) with astigmatism, and (bottom row) with coma. Aberration function is shown in leftmost column. (b) Filled circles: I p ( z ) for the case of coma. Diamonds: azimuthal intensity variation Δ I p ( z ) due to coma. In the plane of peak intensity, Δ I p with coma is no different than if no aberration function (squares) was applied to the SLM. (c) Product of I p and beam quality Q ( η = I p Q ) .

Equations (4)

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u p exp ( r 2 ω L G 2 ) exp ( i ϕ ) ( 1 ) p ( 2 r 2 ω L G 2 ) / 2 L p ( 2 r 2 ω L G 2 ) ,
C p , 0 = u p ( ω L G ) | u 0 0 ( ω 0 ) exp ( i ϕ ) ,
R / ω f = / 2 ( 1 / 2 ) ,
L = ( π / 2 λ ) ω f 2 .

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