Abstract

A new mathematical model, described as hollow Gaussian beams (HGBs), is proposed to describe a dark hollow laser beam (DHB). The area of the dark region across the HGBs can easily be controlled by proper choice of the beam parameters. Based on the Collins integral, an analytical propagation formula for the HGBs through a paraxial optical system is derived. The HGBs also can be expressed as a superposition of a series of Lagurerre–Gaussian modes by use of a polynomial expansion. As a numerical example, the propagation properties of a DHB in free space are illustrated graphically. The HGBs provide a convenient and powerful way to describe and treat the propagation of DHBs and can be used conveniently to analyze atoms manipulated with a DHB.

© 2003 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. Y. Song, D. Milam, and W. T. Hill, Opt. Lett. 24, 1805 (1999).
    [CrossRef]
  4. J. Soding, R. Grimm, and Yu. B. Ovchinnikov, Opt. Commun. 119, 652 (1995).
    [CrossRef]
  5. J. Yin, Y. Zhu, W. Jhe, and Y. Wang, Phys. Rev. A 58, 509 (1998).
    [CrossRef]
  6. X. Xu, Y. Wang, and W. Jhe, J. Opt. Soc. Am. B 17, 1039 (2000).
    [CrossRef]
  7. J. Yin, W. Gao, H. Wang, Q. Long, and Y. Wang, Chin. Phys. 11, 1157 (2002).
    [CrossRef]
  8. X. Wang and M. G. Littman, Opt. Lett. 18, 767 (1993).
    [CrossRef] [PubMed]
  9. R. M. Herman and T. A. Wiggins, J. Opt. Soc. Am. A 8, 932 (1991).
    [CrossRef]
  10. H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, Phys. Rev. A 49, 4922 (1994).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).
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  17. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Erdelyi, A.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Magnus, W.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Oberhettinger, F.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Other (17)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. 79, 2225 (1997).
[CrossRef]

Y. Song, D. Milam, and W. T. Hill, Opt. Lett. 24, 1805 (1999).
[CrossRef]

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, Opt. Commun. 119, 652 (1995).
[CrossRef]

J. Yin, Y. Zhu, W. Jhe, and Y. Wang, Phys. Rev. A 58, 509 (1998).
[CrossRef]

X. Xu, Y. Wang, and W. Jhe, J. Opt. Soc. Am. B 17, 1039 (2000).
[CrossRef]

J. Yin, W. Gao, H. Wang, Q. Long, and Y. Wang, Chin. Phys. 11, 1157 (2002).
[CrossRef]

X. Wang and M. G. Littman, Opt. Lett. 18, 767 (1993).
[CrossRef] [PubMed]

R. M. Herman and T. A. Wiggins, J. Opt. Soc. Am. A 8, 932 (1991).
[CrossRef]

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, Phys. Rev. A 49, 4922 (1994).
[CrossRef] [PubMed]

C. Paterson and R. Smith, Opt. Commun. 124, 121 (1996).
[CrossRef]

S. Marksteiner, C. M. Savage, and P. Zoller, S. Rolston, Phys. Rev. A 50, 2680 (1994).
[CrossRef] [PubMed]

J. Arlt and K. Dholakia, Opt. Commun. 177, 297 (2000).
[CrossRef]

V. I. Balykin V. S. Letokhov, Opt. Commun. 64, 151 (1987).
[CrossRef]

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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Figures (2)

Fig. 1
Fig. 1

Normalized two-dimensional intensity distributions of HGBs for various r and n values: (a) n=3, (b) n=5, (c) n=8, (d) n=15.

Fig. 2
Fig. 2

Normalized three-dimensional-intensity distributions of a HGB in free-space propagation at two propagation distances: (a) z=zR, (b) z=10zR.

Equations (13)

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Enr,0=G0r2ω02nexp-r2ω02,    n=0,1,2,,
Er,z=iλBexp-ikz02π0E0r,0×exp-ik2BAr2-2rr cosθ-θ+Dr2rdrdθ,
J0x=12π02πexpix cos θdθ,
Er,z=iλBexp-ikzexp-ikDr22B0Enr,0×exp-ikAr22BJ0krrBrdr.
0exp-pttν/2+nJν2α1/2t1/2dt=n!aν/2p-n+ν+1exp-a/pLnνa/p,
Er,z=ikAG0n!2Bω02n1ω02+ikA2B-n-1×exp-ikzexp-ikDr22B×exp-kr/2B21/ω02+ikA/2B×Lnkr/2B21/ω02+ikA/2B.
x2n=n!2nm=0n-1mnmLm2x2,    n=0,1,,
Enr,0=G0n!2nm=0n-1mnmLm2r2ω02exp-r2ω02.
ψpr,0=exp-r2ω02Lp2r2ω02.
Enr,z=G0n!2nm=0n-1mnmA-B/q0mA+B/q0m+1×Lm2r2ω2zexp-ikr22qexp-ikz,
1q=C+D/q0A+B/q0.
ω2z=ω02A2+Bλπω022.
ABCD=1z01.

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