Abstract

The far-field intensity distribution of hollow Gaussian beams was investigated based on scalar diffraction theory. An analytical expression of the M2 factor of the beams was derived on the basis of the second-order moments. Moreover, numerical examples to illustrate our analytical results are given.

© 2005 Optical Society of America

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References

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  1. R. Borghi, M. Santarsiero, “M2 factor of Bessel Gauss beams,” Opt. Lett. 22, 262–264 (1997).
    [CrossRef] [PubMed]
  2. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509–513 (1998).
    [CrossRef]
  7. X. Xu, Y. Wang, W. Jhe, “Theory of atom guidance in a hollow laser beam: dressed-atom approach,” J. Opt. Soc. Am. B 17, 1039–1050 (2000).
    [CrossRef]
  8. V. I. Balykin, V. S. Letokhov, “The possibility of deep laser focusing of an atomic beam into the AA-region,” Opt. Commun. 64, 151–156 (1987).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. K. Zhu, H. Tang, X. Wang, T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Weimar) 113, 222–226 (2002).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Hollems, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  16. I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1980).
  17. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  18. A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

2004 (1)

2003 (1)

2002 (2)

K. Zhu, H. Tang, X. Sun, X. Wang, T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).
[CrossRef]

K. Zhu, H. Tang, X. Wang, T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Weimar) 113, 222–226 (2002).
[CrossRef]

2001 (1)

2000 (2)

J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

X. Xu, Y. Wang, W. Jhe, “Theory of atom guidance in a hollow laser beam: dressed-atom approach,” J. Opt. Soc. Am. B 17, 1039–1050 (2000).
[CrossRef]

1999 (1)

1998 (1)

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509–513 (1998).
[CrossRef]

1997 (3)

R. Borghi, M. Santarsiero, “M2 factor of Bessel Gauss beams,” Opt. Lett. 22, 262–264 (1997).
[CrossRef] [PubMed]

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

Yu. B. Ovchinnikov, I. Manek, R. Grimm, “Surface trap for Cs atoms based on evanescent-wave cooling,” Phys. Rev. Lett. 79, 2225–2228 (1997).
[CrossRef]

1995 (1)

J. Soding, R. Grimm, Yu. B. Ovchinnikov, “Gravitational laser trap for atoms with evanescent-wave cooling,” Opt. Commun. 119, 652–662 (1995).
[CrossRef]

1987 (1)

V. I. Balykin, V. S. Letokhov, “The possibility of deep laser focusing of an atomic beam into the AA-region,” Opt. Commun. 64, 151–156 (1987).
[CrossRef]

Arlt, J.

J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

Balykin, V. I.

V. I. Balykin, V. S. Letokhov, “The possibility of deep laser focusing of an atomic beam into the AA-region,” Opt. Commun. 64, 151–156 (1987).
[CrossRef]

Borghi, R.

Cai, Y.

Dholakia, K.

J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

Erdelyi, A.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1980).

Grimm, R.

Yu. B. Ovchinnikov, I. Manek, R. Grimm, “Surface trap for Cs atoms based on evanescent-wave cooling,” Phys. Rev. Lett. 79, 2225–2228 (1997).
[CrossRef]

J. Soding, R. Grimm, Yu. B. Ovchinnikov, “Gravitational laser trap for atoms with evanescent-wave cooling,” Opt. Commun. 119, 652–662 (1995).
[CrossRef]

Hill, W. T.

Hirano, T.

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

Jhe, W.

X. Xu, Y. Wang, W. Jhe, “Theory of atom guidance in a hollow laser beam: dressed-atom approach,” J. Opt. Soc. Am. B 17, 1039–1050 (2000).
[CrossRef]

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509–513 (1998).
[CrossRef]

Kuga, T.

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

Letokhov, V. S.

V. I. Balykin, V. S. Letokhov, “The possibility of deep laser focusing of an atomic beam into the AA-region,” Opt. Commun. 64, 151–156 (1987).
[CrossRef]

Lin, Q.

Liu, T.

K. Zhu, H. Tang, X. Wang, T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Weimar) 113, 222–226 (2002).
[CrossRef]

K. Zhu, H. Tang, X. Sun, X. Wang, T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).
[CrossRef]

Lu, X.

Magnus, W.

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

Manek, I.

Yu. B. Ovchinnikov, I. Manek, R. Grimm, “Surface trap for Cs atoms based on evanescent-wave cooling,” Phys. Rev. Lett. 79, 2225–2228 (1997).
[CrossRef]

Milam, D.

Oberhettinger, F.

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

Ovchinnikov, Yu. B.

Yu. B. Ovchinnikov, I. Manek, R. Grimm, “Surface trap for Cs atoms based on evanescent-wave cooling,” Phys. Rev. Lett. 79, 2225–2228 (1997).
[CrossRef]

J. Soding, R. Grimm, Yu. B. Ovchinnikov, “Gravitational laser trap for atoms with evanescent-wave cooling,” Opt. Commun. 119, 652–662 (1995).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1980).

Santarsiero, M.

Sasada, H.

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

Shimizu, Y.

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

Shiokawa, N.

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

Siegman, A.

A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Hollems, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Soding, J.

J. Soding, R. Grimm, Yu. B. Ovchinnikov, “Gravitational laser trap for atoms with evanescent-wave cooling,” Opt. Commun. 119, 652–662 (1995).
[CrossRef]

Song, Y.

Sun, X.

K. Zhu, H. Tang, X. Sun, X. Wang, T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).
[CrossRef]

Tang, H.

K. Zhu, H. Tang, X. Sun, X. Wang, T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).
[CrossRef]

K. Zhu, H. Tang, X. Wang, T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Weimar) 113, 222–226 (2002).
[CrossRef]

Torii, Y.

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

Tovar, A. A.

Wang, X.

K. Zhu, H. Tang, X. Wang, T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Weimar) 113, 222–226 (2002).
[CrossRef]

K. Zhu, H. Tang, X. Sun, X. Wang, T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).
[CrossRef]

Wang, Y.

X. Xu, Y. Wang, W. Jhe, “Theory of atom guidance in a hollow laser beam: dressed-atom approach,” J. Opt. Soc. Am. B 17, 1039–1050 (2000).
[CrossRef]

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509–513 (1998).
[CrossRef]

Xu, X.

Yin, J.

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509–513 (1998).
[CrossRef]

Zhu, K.

K. Zhu, H. Tang, X. Sun, X. Wang, T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).
[CrossRef]

K. Zhu, H. Tang, X. Wang, T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Weimar) 113, 222–226 (2002).
[CrossRef]

Zhu, Y.

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509–513 (1998).
[CrossRef]

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

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

Opt. Commun. (4)

V. I. Balykin, V. S. Letokhov, “The possibility of deep laser focusing of an atomic beam into the AA-region,” Opt. Commun. 64, 151–156 (1987).
[CrossRef]

J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

J. Soding, R. Grimm, Yu. B. Ovchinnikov, “Gravitational laser trap for atoms with evanescent-wave cooling,” Opt. Commun. 119, 652–662 (1995).
[CrossRef]

K. Zhu, H. Tang, X. Sun, X. Wang, T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).
[CrossRef]

Opt. Lett. (3)

Optik (Weimar) (1)

K. Zhu, H. Tang, X. Wang, T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Weimar) 113, 222–226 (2002).
[CrossRef]

Phys. Rev. A (1)

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509–513 (1998).
[CrossRef]

Phys. Rev. Lett. (2)

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

Yu. B. Ovchinnikov, I. Manek, R. Grimm, “Surface trap for Cs atoms based on evanescent-wave cooling,” Phys. Rev. Lett. 79, 2225–2228 (1997).
[CrossRef]

Other (4)

A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Hollems, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1980).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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

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Figures (2)

Fig. 1
Fig. 1

Normalized far-field intensity distributions of HGBs of order n versus parameter πω0f.

Fig. 2
Fig. 2

M2 factor of HGBs of order n versus parameter β.

Equations (19)

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M 2 = 2 π σ 0 σ ,
E n ( r , 0 ) = G 0 ( r 2 ω 0 2 ) n exp ( r 2 ω 0 2 ) , n = 0 , 1 , 2 , ,
σ 0 2 = 0 r 3 | E n ( r , 0 ) | 2 d r 0 r | E n ( r , 0 ) | 2 d r .
0 exp ( s t ) t n d t = n ! s n 1 ,
σ 0 = ( 2 n + 1 ) ω 0 2 2 .
E n ( r 1 , z ) = exp ( i k z ) i λ z 0 2 π 0 E n ( r , 0 ) × exp [ i k z r r 1 cos ( θ θ 1 ) ] r d r d θ ,
J 0 ( x ) = 1 2 π 0 2 π exp [ i x cos ( θ θ 1 ) ] d θ ,
E n ( r 1 , z ) = 2 π i λ z exp ( i k z ) 0 E n ( r , 0 ) J 0 ( k r r 1 z ) r d r .
0 exp ( p t ) t υ / 2 + n J υ ( 2 a 1 / 2 t 1 / 2 ) d t = n ! a υ / 2 p ( n + υ + 1 ) exp ( a / p ) L n υ ( a / p ) ,
E n ( f , z ) = π G 0 exp ( i k z ) i λ z n ! × ω 0 2 × exp ( π 2 ω 0 2 f 2 ) L n ( π 2 ω 0 2 f 2 ) .
I n ( f , z ) = E n ( f , z ) E n * ( f , z ) = I n ( 0 , z ) exp ( 2 π 2 ω 0 2 f 2 ) [ L n ( π 2 ω 0 2 f 2 ) ] 2 ,
σ 2 = 0 f 3 | E n ( f , z ) | 2 d f 0 f | E n ( f , z ) | 2 d f .
F ( α , q ) = 0 exp ( 2 β x ) [ L n ( β x ) ] 2 x q d x .
σ 2 = [ F ( β , 1 ) / F ( β , 0 ) ] β = π 2 ω 0 2 .
L n ( x ) = m = 0 n ( 1 ) m ( n n m ) x m m !
F ( β , q ) = m = 0 n m = 0 n ( 1 ) m + m m ! m ! ( n n m ) ( n n m ) × 0 x m + m + p exp ( 2 β x ) d x ,
( n n m )
α 2 = 1 2 β m = 0 n m = 0 n ( 1 ) m + m m ! m ! ( n n m ) ( n n m ) ( 2 β ) m m ( m + m + 1 ) ! m = 0 n m = 0 n ( 1 ) m + m m ! m ! ( n n m ) ( n n m ) ( 2 β ) m m ( m + m ) ! .
M 2 = { ( 2 n + 1 ) m = 0 n m = 0 n ( 1 ) m + m m ! m ! ( n n m ) ( n n m ) ( 2 β ) m m ( m + m + 1 ) ! m = 0 n m = 0 n ( 1 ) m + m m ! m ! ( n n m ) ( n n m ) ( 2 β ) m m ( m + m ) ! } 1 / 2 .

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