Abstract

On the basis of the generalized Collins formula and the expansion of the hard-aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula for a hollow Gaussian beam propagating through an apertured paraxial stigmatic (ST) ABCD optical system is derived. Some numerical examples are given. Furthermore, by using a tensor method, we derive approximate analytical formulas for a hollow elliptical Gaussian beam propagating through an apertured paraxial general astigmatic ABCD optical system and an apertured paraxial misaligned ST ABCD optical system. Our results provide a convenient way for studying the propagation and transformation of a hollow Gaussian beam and a hollow elliptical Gaussian beam through an apertured general optical system.

© 2006 Optical Society of America

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  1. J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics, Vol. XLIV, E.Wolf, ed. (North-Holland, 2003), pp. 119-204.
    [Crossref]
  2. H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, "Atomic funnel with evanescent light," Phys. Rev. A 56, 712-718 (1997).
    [Crossref]
  3. W. L. Power, L. Allen, M. Babiker, and V. E. Lembessis, "Atomic motion in light beams possessing orbital angular momentum," Phys. Rev. A 52, 479-488 (1995).
    [Crossref] [PubMed]
  4. H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
    [Crossref] [PubMed]
  5. X. Wang and M. G. Littman, "Laser cavity for generation of variable-radius rings of light," Opt. Lett. 18, 767-768 (1993).
    [Crossref] [PubMed]
  6. J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, "Generation of a dark hollow beam by a small hollow fiber," Opt. Commun. 138, 287-292 (1997).
    [Crossref]
  7. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, "Generation of optical phase singularities by computer-generated holograms," Opt. Lett. 17, 221-223 (1992).
    [Crossref] [PubMed]
  8. A. V. Mamaev, M. Saffman, and A. A. Zozulya, "Vortex evolution and bound pair formation in anisotropic nonlinear optical media," Phys. Rev. E 77, 4544-4547 (1996).
  9. S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
    [Crossref] [PubMed]
  10. M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
    [Crossref] [PubMed]
  11. M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, "Evanescent-wave guiding of atoms in hollow optical fibers," Phys. Rev. A 53, R648-R651 (1996).
    [Crossref] [PubMed]
  12. H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500-4503 (1996).
    [Crossref] [PubMed]
  13. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
    [Crossref]
  14. J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000).
    [Crossref]
  15. Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beam and its propagation," Opt. Lett. 28, 1084-1086 (2003).
    [Crossref] [PubMed]
  16. Y. Cai and Q. Lin, "Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems," J. Opt. Soc. Am. A 21, 1058-1065 (2004).
    [Crossref]
  17. Z. Wang, Q. Lin, and Y. Wang, "Control of atomic rotation by elliptical hollow beam carrying zero angular momentum," Opt. Commun. 240, 357-362 (2004).
    [Crossref]
  18. Z. Wang, Y. Dong, and Q. Lin, "Atomic trapping and guiding by quasi-dark hollow beams," J. Opt. A, Pure Appl. Opt. 7, 147-153 (2005).
    [Crossref]
  19. J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
    [Crossref]
  20. D. Ding and X. Liu, "Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture," J. Opt. Soc. Am. A 16, 1286-1293 (1999).
    [Crossref]
  21. S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [Crossref]
  22. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  23. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. 1.
  24. J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt. 8, 1687-1693 (1969).
    [Crossref] [PubMed]
  25. J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).
  26. J. A. Arnaud, "Hamiltonian theory of beam mode propagation," in Progress in Optics, Vol. XI, E.Wolf, ed. (North-Holland, 1973), pp. 247-304.
    [Crossref]
  27. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).
  28. G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am. A 11, 2257-2264 (1994).
    [Crossref]
  29. Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).
  30. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964), Chap. 4.
  31. S. Wang and L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E.Wolf, ed. (North-Holland, 1988), pp. 279.

2005 (1)

Z. Wang, Y. Dong, and Q. Lin, "Atomic trapping and guiding by quasi-dark hollow beams," J. Opt. A, Pure Appl. Opt. 7, 147-153 (2005).
[Crossref]

2004 (2)

Z. Wang, Q. Lin, and Y. Wang, "Control of atomic rotation by elliptical hollow beam carrying zero angular momentum," Opt. Commun. 240, 357-362 (2004).
[Crossref]

Y. Cai and Q. Lin, "Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems," J. Opt. Soc. Am. A 21, 1058-1065 (2004).
[Crossref]

2003 (1)

2002 (1)

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).

2000 (1)

J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000).
[Crossref]

1999 (1)

1997 (3)

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, "Atomic funnel with evanescent light," Phys. Rev. A 56, 712-718 (1997).
[Crossref]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, "Generation of a dark hollow beam by a small hollow fiber," Opt. Commun. 138, 287-292 (1997).
[Crossref]

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

1996 (3)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, "Vortex evolution and bound pair formation in anisotropic nonlinear optical media," Phys. Rev. E 77, 4544-4547 (1996).

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, "Evanescent-wave guiding of atoms in hollow optical fibers," Phys. Rev. A 53, R648-R651 (1996).
[Crossref] [PubMed]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500-4503 (1996).
[Crossref] [PubMed]

1995 (2)

W. L. Power, L. Allen, M. Babiker, and V. E. Lembessis, "Atomic motion in light beams possessing orbital angular momentum," Phys. Rev. A 52, 479-488 (1995).
[Crossref] [PubMed]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[Crossref] [PubMed]

1994 (3)

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[Crossref] [PubMed]

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[Crossref] [PubMed]

G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am. A 11, 2257-2264 (1994).
[Crossref]

1993 (1)

1992 (1)

1990 (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

1988 (1)

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[Crossref]

1970 (2)

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).

S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
[Crossref]

1969 (1)

Alda, J.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

Allen, L.

W. L. Power, L. Allen, M. Babiker, and V. E. Lembessis, "Atomic motion in light beams possessing orbital angular momentum," Phys. Rev. A 52, 479-488 (1995).
[Crossref] [PubMed]

Anderson, D. Z.

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, "Evanescent-wave guiding of atoms in hollow optical fibers," Phys. Rev. A 53, R648-R651 (1996).
[Crossref] [PubMed]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[Crossref] [PubMed]

Arlt, J.

J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000).
[Crossref]

Arnaud, J. A.

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).

J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt. 8, 1687-1693 (1969).
[Crossref] [PubMed]

J. A. Arnaud, "Hamiltonian theory of beam mode propagation," in Progress in Optics, Vol. XI, E.Wolf, ed. (North-Holland, 1973), pp. 247-304.
[Crossref]

Atewart, B. W.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[Crossref] [PubMed]

Babiker, M.

W. L. Power, L. Allen, M. Babiker, and V. E. Lembessis, "Atomic motion in light beams possessing orbital angular momentum," Phys. Rev. A 52, 479-488 (1995).
[Crossref] [PubMed]

Bernabeu, E.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[Crossref]

Cai, Y.

Choi, K.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[Crossref] [PubMed]

Collins, S. A.

Cornell, E. A.

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, "Evanescent-wave guiding of atoms in hollow optical fibers," Phys. Rev. A 53, R648-R651 (1996).
[Crossref] [PubMed]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[Crossref] [PubMed]

Dholakia, K.

J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000).
[Crossref]

Ding, D.

Dong, Y.

Z. Wang, Y. Dong, and Q. Lin, "Atomic trapping and guiding by quasi-dark hollow beams," J. Opt. A, Pure Appl. Opt. 7, 147-153 (2005).
[Crossref]

Donley, E. A.

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, "Evanescent-wave guiding of atoms in hollow optical fibers," Phys. Rev. A 53, R648-R651 (1996).
[Crossref] [PubMed]

Erdelyi, A.

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

Fenichel, H.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[Crossref] [PubMed]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. 1.

Gao, W.

J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics, Vol. XLIV, E.Wolf, ed. (North-Holland, 2003), pp. 119-204.
[Crossref]

Heckenberg, N. R.

Hirano, T.

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

Ito, H.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, "Atomic funnel with evanescent light," Phys. Rev. A 56, 712-718 (1997).
[Crossref]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500-4503 (1996).
[Crossref] [PubMed]

Jhe, W.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, "Atomic funnel with evanescent light," Phys. Rev. A 56, 712-718 (1997).
[Crossref]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, "Generation of a dark hollow beam by a small hollow fiber," Opt. Commun. 138, 287-292 (1997).
[Crossref]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500-4503 (1996).
[Crossref] [PubMed]

Kim, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, "Generation of a dark hollow beam by a small hollow fiber," Opt. Commun. 138, 287-292 (1997).
[Crossref]

Kogelnik, H.

Kuga, T.

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

Lee, H. S.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[Crossref] [PubMed]

Lee, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, "Generation of a dark hollow beam by a small hollow fiber," Opt. Commun. 138, 287-292 (1997).
[Crossref]

Lee, K. I.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500-4503 (1996).
[Crossref] [PubMed]

Lembessis, V. E.

W. L. Power, L. Allen, M. Babiker, and V. E. Lembessis, "Atomic motion in light beams possessing orbital angular momentum," Phys. Rev. A 52, 479-488 (1995).
[Crossref] [PubMed]

Lin, Q.

Z. Wang, Y. Dong, and Q. Lin, "Atomic trapping and guiding by quasi-dark hollow beams," J. Opt. A, Pure Appl. Opt. 7, 147-153 (2005).
[Crossref]

Z. Wang, Q. Lin, and Y. Wang, "Control of atomic rotation by elliptical hollow beam carrying zero angular momentum," Opt. Commun. 240, 357-362 (2004).
[Crossref]

Y. Cai and Q. Lin, "Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems," J. Opt. Soc. Am. A 21, 1058-1065 (2004).
[Crossref]

Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beam and its propagation," Opt. Lett. 28, 1084-1086 (2003).
[Crossref] [PubMed]

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

Littman, M. G.

Liu, X.

Lu, X.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964), Chap. 4.

Magnus, W.

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

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, "Vortex evolution and bound pair formation in anisotropic nonlinear optical media," Phys. Rev. E 77, 4544-4547 (1996).

Marksteiner, S.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[Crossref] [PubMed]

McDuff, R.

Montgomery, D.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[Crossref] [PubMed]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. 1.

Nakata, T.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500-4503 (1996).
[Crossref] [PubMed]

Nemes, G.

Noh, H.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, "Generation of a dark hollow beam by a small hollow fiber," Opt. Commun. 138, 287-292 (1997).
[Crossref]

Oberhettinger, F.

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

Ohtsu, M.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, "Atomic funnel with evanescent light," Phys. Rev. A 56, 712-718 (1997).
[Crossref]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500-4503 (1996).
[Crossref] [PubMed]

Power, W. L.

W. L. Power, L. Allen, M. Babiker, and V. E. Lembessis, "Atomic motion in light beams possessing orbital angular momentum," Phys. Rev. A 52, 479-488 (1995).
[Crossref] [PubMed]

Renn, M. J.

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, "Evanescent-wave guiding of atoms in hollow optical fibers," Phys. Rev. A 53, R648-R651 (1996).
[Crossref] [PubMed]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[Crossref] [PubMed]

Rolston, S.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[Crossref] [PubMed]

Ronchi, L.

S. Wang and L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E.Wolf, ed. (North-Holland, 1988), pp. 279.

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, "Vortex evolution and bound pair formation in anisotropic nonlinear optical media," Phys. Rev. E 77, 4544-4547 (1996).

Sakaki, K.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, "Atomic funnel with evanescent light," Phys. Rev. A 56, 712-718 (1997).
[Crossref]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500-4503 (1996).
[Crossref] [PubMed]

Sasada, H.

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

Savage, C. M.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[Crossref] [PubMed]

Shimizu, Y.

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

Siegman, A. E.

Smith, C. P.

Torii, Y.

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

Vdovin, O.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[Crossref] [PubMed]

Wang, S.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

S. Wang and L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E.Wolf, ed. (North-Holland, 1988), pp. 279.

Wang, X.

Wang, Y.

Z. Wang, Q. Lin, and Y. Wang, "Control of atomic rotation by elliptical hollow beam carrying zero angular momentum," Opt. Commun. 240, 357-362 (2004).
[Crossref]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, "Generation of a dark hollow beam by a small hollow fiber," Opt. Commun. 138, 287-292 (1997).
[Crossref]

Wang, Z.

Z. Wang, Y. Dong, and Q. Lin, "Atomic trapping and guiding by quasi-dark hollow beams," J. Opt. A, Pure Appl. Opt. 7, 147-153 (2005).
[Crossref]

Z. Wang, Q. Lin, and Y. Wang, "Control of atomic rotation by elliptical hollow beam carrying zero angular momentum," Opt. Commun. 240, 357-362 (2004).
[Crossref]

Wen, J. J.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[Crossref]

White, A. G.

Wieman, C. E.

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[Crossref]

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[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Normalized irradiance distribution of a HGB at several different propagation distances after passing through an aperture of radius 1 mm . (a) z = 5 mm , (b) z = 50 mm , (c) z = 200 mm , (d) z = 500 mm .

Fig. 2
Fig. 2

Normalized irradiance distribution of a HGB at distance z = 200 mm after passing through an aperture with different values of radius a 1 . (a) a 1 = 1 mm , (b) a 1 = 1.2 mm , (c) a 1 = 2 mm , (d) a 1 = 5 mm .

Fig. 3
Fig. 3

Normalized 3D irradiance distribution of a HEGB and its cross-irradiance distribution ( y = 0 ) at several different propagation distances after passing through a circular aperture of radius 0.8 mm . (a) z = 50 mm , (b) z = 200 mm .

Fig. 4
Fig. 4

Normalized 3D irradiance distribution of a HEGB at distance z = 50 mm after passing through a circular aperture with different values of radius a 1 . (a) a 1 = 0.3 mm , (b) a 1 = 0.5 mm , (c) a 1 = 0.8 mm , (d) a 1 = 5 mm .

Equations (45)

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E n ( r , 0 ) = G 0 ( r 2 w 0 2 ) n exp ( r 2 w 0 2 ) , n = 0 , 1 , 2 , ,
E ( r , z ) = i λ B exp [ i k l 0 ( z ) ] 0 2 π 0 a 1 E 0 ( r 1 , 0 ) exp { i k 2 B [ A r 1 2 2 r 1 r cos ( θ θ 1 ) + D r 2 ] } r 1 d r 1 d θ 1 ,
H ( r 1 ) = { 1 , r 1 a 1 0 , r 1 > a 1 }
J 0 ( x ) = 1 2 π 0 2 π exp ( i x cos θ ) d θ ,
E ( r , z ) = 2 π i λ B exp ( i k l 0 ) exp ( i k D r 2 2 B ) 0 E n ( r 1 , 0 ) H ( r 1 ) exp ( i k A r 1 2 2 B ) J 0 ( k r r 1 B ) r 1 d r 1 .
H ( r ) = m = 1 M A m exp ( B m a 1 2 r 2 ) ,
0 exp ( p t ) t v 2 + n J v ( 2 α 1 2 t 1 2 ) d t = n ! a v 2 p ( n + v + 1 ) exp ( a p ) L n v ( a p )
E n ( r , z ) = i k G 0 n ! 2 B w 0 2 n exp ( i k l 0 ) exp ( i k D r 2 2 B ) m = 1 M A m ( 1 w 0 2 + i k A 2 B + B m a 1 2 ) n 1 exp [ ( k r 2 B ) 2 ( 1 w 0 2 + i k A 2 B + B m a 1 2 ) ] L n [ ( k r 2 B ) 2 ( 1 w 0 2 + i k A 2 B + B m a 1 2 ) ] .
x 2 n = n ! 2 n p = 0 n ( 1 ) p ( n p ) L p ( 2 x 2 ) , n = 0 , 1 , ,
E n ( r , 0 ) = G 0 n ! 2 n p = 0 n ( 1 ) p ( n p ) L p ( 2 r 2 w 0 2 ) exp ( r 2 w 0 2 ) .
0 x v + 1 2 exp ( β x 2 ) L n v ( a x 2 ) J v ( x y ) ( x y ) 1 2 d x = 2 v 1 β v n 1 ( β α ) n y v + 1 2 × exp ( y 2 4 β ) L n v [ α y 2 4 β ( α β ) ] ( Re β > 0 , Re v > 0 ) ,
E n ( r , z ) = G 0 n ! 2 n exp ( i k l 0 ) p = 0 n m = 1 M ( 1 ) p ( n p ) A m ( 2 B m B i k a 1 2 + A B q 0 ) n ( 2 B m B i k a 1 2 + A + B q 0 ) n + 1 exp [ i k ( D q 0 + C + 2 B m D i k a 1 2 ) r 2 2 ( B q 0 + A + 2 B m B i k a 1 2 ) ] L p [ k 2 r 2 2 B 2 w 0 2 2 w 0 2 B 2 ( B m a 1 2 + i k A 2 B ) 2 ] ,
E n ( r , 0 ) = G 0 ( i k 2 r T Q 1 r ) n exp [ i k 2 r T Q 1 r ] , n = 0 , 1 , 2 , 3 ,
Q 1 = [ q 0 x x 1 q 0 x y 1 q 0 x y 1 q 0 y y 1 ] ,
( r T Q 1 r ) n = ( 2 n ) ! 2 3 n m = 0 n 1 ( n m ) ! ( 2 m ) ! H 2 m [ 2 Q 1 2 r ] ,
E n ( r , 0 ) = G 0 ( 2 n ) ! 2 3 n exp [ i k 2 r T Q 1 r ] m = 0 n 1 ( n m ) ! ( 2 m ) ! H 2 m [ ( i k Q 1 ) 1 2 r ] , n = 0 , 1 , 2 , 3 .
E 2 ( r 2 , z ) = i n 1 λ [ det ( B ) ] 1 2 exp ( i k l 0 ) H ( r 1 ) E 1 ( r 1 , 0 ) exp ( i k l 1 ) d r 1 ,
l 1 = 1 2 ( r 1 r 2 ) T [ n 1 B 1 A n 1 B 1 n 2 ( C D B 1 A ) n 2 D B 1 ] ( r 1 r 2 ) ,
( r 2 r 2 ) = [ A B C D ] ( r 1 r 1 ) .
H ( r ) = m = 1 M A m exp ( i k 2 r T R m r ) ,
R m = 2 B m i k a 1 2 [ 1 0 0 1 ] .
E 2 p ( r 2 , z ) = i λ [ det ( B ) ] 1 2 exp ( i k l 0 ) G 0 ( 2 n ) ! 2 3 n m = 1 M p = 0 n A m ( n p ) ! ( 2 p ) ! exp [ i k 2 r 1 T ( Q 1 1 + R m ) r 1 ] H 2 p [ i k r 1 T Q 1 1 r 1 ] exp [ i k 2 ( r 1 T B 1 A r 1 2 r 1 T B 1 r 2 + r 2 T D B 1 r 2 ) ] d r 1 .
E n ( r 2 , z ) = G 0 exp ( i k l 0 ) ( 2 n ) ! 2 3 n m = 1 M p = 0 n A m ( n p ) ! ( 2 p ) ! [ det ( A + B Q 1 1 + B R m ) ] 1 2 exp [ i k 2 r 2 T Q 2 m 1 r 2 ] × [ 1 2 det ( B 1 A Q 1 + R m Q 1 + I ) ] p H 2 p ( [ 1 2 det ( B 1 A Q 1 + R m Q 1 + I ) ] 1 2 { i k r 2 T [ A + B ( Q 1 1 + R m ) ] 1 T [ A Q 1 + B ( I + R m Q 1 ) ] 1 r 2 } 1 2 ) ,
Q 2 m 1 = [ C + D ( Q 1 1 + R m ) ] [ A + B ( Q 1 1 + R m ) ] 1 .
0 exp [ ( a 1 2 x a 1 2 b ) 2 2 ] H p ( x ) d x = 2 π a ( 1 2 a ) p 2 H p [ b 1 2 a ]
( B 1 A ) T = B 1 A , ( B 1 ) T = ( C D B 1 A ) , ( D B 1 ) T = D B 1 .
E ( r 2 , z ) = i k 2 π [ det ( B ¯ ) ] 1 2 exp ( i k l 0 ) E ( r 1 , 0 ) H ( r 1 ) exp [ i k 2 ( r 1 T B ¯ 1 A ¯ r 1 2 r 1 T B ¯ 1 r 2 + r 2 T D ¯ B ¯ 1 r 2 ) ] × exp [ i k 2 ( r 1 T B ¯ 1 e f + r 2 T B ¯ 1 g h ) ] d r 1 ,
A ¯ = [ a 0 0 a ] , B ¯ = [ b 0 0 b ] , C ¯ = [ c 0 0 c ] , D ¯ = [ d 0 0 d ] .
e = 2 ( α T ϵ x + β T ϵ x ) ,
f = 2 ( α T ϵ y + β T ϵ y ) ,
g = 2 ( b γ T d α T ) ϵ x + 2 ( b δ T d β T ) ϵ x ,
h = 2 ( b γ T d α T ) ϵ y + 2 ( b δ T d β T ) ϵ y ,
α T = 1 a , β T = l b , γ T = c , δ T = ± 1 d .
E n ( r 2 , z ) = G 0 ( 2 n ) ! 2 3 n exp ( i k l 0 ) m = 1 M p = 0 n A m ( n p ) ! ( 2 p ) ! [ det ( A ¯ + B ¯ Q 1 1 + B ¯ R m ) ] 1 2 exp [ i k 2 r 2 T B ¯ 1 g g ] exp [ i k 2 r 2 T Q 2 m 1 r 2 ] exp [ i k 2 r 2 T B ¯ 1 T ( A ¯ + B ¯ Q 1 1 + B ¯ R m ) 1 e f ] exp [ i k 8 e f T B ¯ 1 T ( A ¯ + B ¯ Q 1 1 + B ¯ R m ) e f ] [ 1 2 det ( B ¯ 1 A ¯ Q 1 + R m Q 1 + I ) ] p H 2 p ( [ 1 2 det ( B ¯ 1 A ¯ Q 1 + R m Q 1 + I ) ] 1 2 { i k ( r 2 e f 2 ) T [ A ¯ + B ¯ ( Q 1 1 + R m ) ] 1 T [ A ¯ Q 1 + B ¯ ( I + R m Q 1 ) ] 1 ( r 2 e f 2 ) } 1 2 ) ,
Q 2 m 1 = [ C ¯ + D ¯ ( Q 1 1 + R m ) ] [ A ¯ + B ¯ ( Q 1 1 + R m ) ] 1 .
( B ¯ 1 A ¯ ) T = B ¯ 1 A ¯ , ( B ¯ 1 ) T = ( C ¯ D ¯ B ¯ 1 A ¯ ) , ( D ¯ B ¯ 1 ) T = D ¯ B ¯ 1 .
H ( x , y ) = { 1 , x a 1 , y b 1 0 , x > a 1 , y > b 1 } ,
H ( x , y ) = m = 1 M A m exp ( B m a 1 2 x 2 ) j = 1 J A j exp ( B j b 1 2 y 2 ) ,
H ( r ) = m = 1 M j = 1 J A m A j exp ( i k 2 r T R m j r ) ,
R m j = [ 2 B m i k a 1 2 0 0 2 B j i k b 1 2 ] .
E n ( r 2 , z ) = G 0 ( 2 n ) ! 2 3 n exp ( i k l 0 ) m = 1 M j = 1 J p = 0 n A m A j ( n p ) ! ( 2 p ) ! [ det ( A + B Q 1 1 + B R m j ) ] 1 2 exp [ i k 2 r 2 T Q 2 m j 1 r 2 ] [ 1 2 det ( B 1 A Q 1 + R m j Q 1 + I ) ] p H 2 p ( [ 1 2 det ( B 1 A Q 1 + R m j Q 1 + I ) ] 1 2 { i k r 2 T [ A + B ( Q 1 1 + R m j ) ] 1 T [ A Q 1 + B ( I + R m j Q 1 ) ] 1 r 2 } 1 2 ) ,
Q 2 m j 1 = [ C + D ( Q 1 1 + R m j ) ] [ A + B ( Q 1 1 + R m j ) ] 1 .
E n ( r 2 , z ) = G 0 ( 2 n ) ! 2 3 n exp ( i k l 0 ) m = 1 M j = 1 J p = 0 n A m A j ( n p ) ! ( 2 p ) ! [ det ( A ¯ + B ¯ Q 1 1 + B ¯ R m j ) ] 1 2 exp [ i k 2 r 2 T B ¯ 1 g g ] exp [ i k 2 r 2 T Q 2 m j 1 r 2 ] exp [ i k 2 r 2 T B ¯ 1 T ( A ¯ + B ¯ Q 1 1 + B ¯ R m j ) 1 e f ] exp [ i k 8 e f T B ¯ 1 T ( A ¯ + B ¯ Q 1 1 + B ¯ R m j ) e f ] + [ 1 2 det ( B ¯ 1 A ¯ Q 1 + R m j Q 1 + I ) ] p H 2 p ( [ 1 2 det ( B 1 A ¯ Q 1 + R m j Q 1 + I ) ] 1 2 { i k ( r 2 e f 2 ) T [ A ¯ + B ¯ ( Q 1 1 + R m j ) ] 1 T [ A ¯ Q 1 + B ¯ ( I + R m j Q 1 ) ] 1 ( r 2 e f 2 ) } 1 2 ) ,
Q 2 m j 1 = [ C ¯ + D ¯ ( Q 1 1 + R m j ) ] [ A ¯ + B ¯ ( Q 1 1 + R m j ) ] 1 .
A = [ 1 0 0 1 ] , B = [ z 0 0 z ] , C = [ 0 0 0 0 ] , D = [ 1 0 0 1 ] ,

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