Abstract

We propose a scheme for producing and magnifying a hollow beam, as might be desired for purposes of storing and guiding cold atoms, through the use of a simple spherically aberrating lens and a projection lens. The field is a superposition of J0 Bessel fields, so that simple (linear, circular) polarizations can be utilized. We analyze some of the beam properties through analytical approximations. Some examples of field zeros along the optical axis are given, together with some of their characteristics. Numerical calculations largely confirm the validity of the analytical expressions. For the most important zero nearly all of the beam power is contained within the first two Bessel spacings, with a resulting highly efficient trapping. Isophotes are calculated and displayed for the region surrounding this null point. They have regular shapes, for which we give an approximate expression.

© 2002 Optical Society of America

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References

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  1. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
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    [CrossRef]
  10. Y.-I. Shin, K. Kim, J-A. Kim, H-R. Noh, W. Jhe, K. Oh, U-C. Paek, “Diffraction-limited dark laser spot produced by a hollow optical fiber,” Opt. Lett. 26, 119–121 (2001).
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2001 (1)

2000 (1)

1998 (1)

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

1996 (1)

1994 (1)

1991 (1)

1987 (2)

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

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

1983 (1)

1972 (1)

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

1954 (1)

Arlt, J.

Born, M.

See M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), pp. 459–479, and references therein.

Durnin, J.

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

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

Eberly, J. H.

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

Greene, P. L.

Grimm, R.

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

Hall, D. G.

Herman, R. M.

Jhe, W.

Kim, J-A.

Kim, K.

Manek, I.

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

McLeod, J. H.

Miceli, J. J.

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

Noh, H-R.

Oh, K.

Orchinnikov, Yu. B.

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

Padgett, M. J.

Paek, U-C.

Pohl, D.

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

Self, S. A.

Shin, Y.-I.

Wiggins, T. A.

Wolf, E.

See M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), pp. 459–479, and references therein.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

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

Opt. Lett. (2)

Phys. Rev. Lett. (1)

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

Other (1)

See M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), pp. 459–479, and references therein.

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Figures (2)

Fig. 1
Fig. 1

SA lens and associated geometry. L, lens; c.r., central illuminating region; i.r., illuminating ring for axial point P(z). fc, focal point for central rays; fm, focal point for marginal rays. The coordinate system (r, z) is indicated.

Fig. 2
Fig. 2

Isophotes for the first zero in intensity, corresponding to various fractions of the peak intensity, relative to the first null position. Distances are measured in micrometers. The solid curves result from numerical calculation; the dashed curves are given by Eq. 19 Fractional intensities are as follows: a, 1%; b, 2%; c, 5%; d, 10%; e, 20%. Note expanded scale (by a factor of 10) in the radial direction.

Equations (26)

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ϕ=-kfc12rfc2-18rfc4+Drfc4.
f=fc-(fc-fm)(r/rm)2,
D=(fc-fm)fc4rm2.
ri=rmfc-zfc-fm1/2.
ϕ=kz+r221z-1fc-r4fc3D+18fcz3-1.
ϕkz+r22 fc2 (fc-z)-D r4fc3.
ϕ=kz+(fc-z)2 fc2r2-r42ri2,
E(z)axialc.r.=E0fc[(fc-z)2+zR02]1/2×expikz-iπ2+i tan-1(fc-z)zR0,
E(z)axialring=E0expikz+ik(fc-z)ri24fc2×-ik2πz02πrdr×exp-rw2-ik(fc-z)4fc2ri2 (r2-ri2)2.
Eaxialring(z)=πkri2(fc-z)1/2E0exp-riw2+ikz+(fc-z)ri24fc2-i 3π4.
Eaxial(z)=E0exp(ikz)fc[(fc-z)2+zR02]1/2×exp-iπ2+i tan-1(fc-z)zR0+πkrm2(fc-fm)1/2exp-rm2(fc-z)w2(fc-fm)×expik(fc-z)2rm24fc2(fc-fm)-i3π4.
12πk(fc-z)2rm24fc2(fc-fm)-tan-1fc-zzR0+3π4=n,
exp-rm2(fc-z)w2(fc-fm)
=fc-fmπkrm21/2fc[(fc-z)2+zR02]1/2,
E(z, ρ)=E0-ikz0rdr exp[-(r/ω)2]J0krρz×expikz+(ρ2+r2)2z-r22 fc-Dr4fc3.
n=1:min at (fc-z)
=1.313 mm(versus1.30 mm)
 w=2.800 mm(versus2.63 mm)
n=2:min at (fc-z)
=1.918 mm(versus1.93 mm)
w=3.423 mm(versus3.09 mm).
L=2 fc2(fc-fm)k(fc-z)rm2.
E=Eaxialc.r.kri2 fc2δρ2.
R=2 fc/kri.
kR2/L=2.
kr222+z2=2.66×10-2(RelativeIntensity)mm2,

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