Abstract

A theoretical model is proposed to describe coherent dark hollow beams (DHBs) with rectangular symmetry. The electric field of a coherent rectangular DHB is expressed as a superposition of a series of the electric field of a finite series of fundamental Gaussian beams. Analytical propagation formulas for a coherent rectangular DHB passing through paraxial optical systems are derived in a tensor form. Furthermore, for the more general case, we propose a theoretical model to describe a partially coherent rectangular DHB. Analytical propagation formulas for a partially coherent rectangular DHB passing through paraxial optical systems are derived. The beam propagation factor (M2 factor) for both coherent and partially coherent rectangular DHBs are studied. Numerical examples are given by using the derived formulas. Our models and method provide an effective way to describe and treat the propagation of coherent and partially coherent rectangular DHBs.

© 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,E.Wolf, ed. (North-Holland, 2003), Vol. 44, 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. L. Zhang, X. Lu, X. Chen, and S. He, "Generation of a dark hollow beam inside a cavity," Chin. Phys. Lett. 21, 298-301 (2004).
    [CrossRef]
  5. 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]
  6. J. Arlt and K. Dholakia, "Generation of high-order Bessel beamsby use of an axicon," Opt. Commun. 177, 297-301 (2000).
    [CrossRef]
  7. Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beam and its propagation," Opt. Lett. 28, 1084-1086 (2003).
    [CrossRef] [PubMed]
  8. 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]
  9. A. E. Siegman, Lasers (University Science, 1986).
  10. Y. Li, "Flat-topped light beam with non-circular cross-sections," J. Mod. Opt. 50, 1957-1966 (2003).
    [CrossRef]
  11. B. D. Stone and T. J. Bruegge, "Practical considerations for simulating beam propagation: a comparison of three approaches," in International Optical Design Conference 2002, P. K. Manhart and J. M. Sasian, eds, Proc. SPIE 4832, 359-377 (2002).
    [CrossRef]
  12. T. J. Bruegge, M. P. Rimmer, and J. D. Targove, "Software for free-space beam propagation," in Optical Design and Analysis Software, R. C. Juergens, ed., Proc. SPIE 3780, 14-22 (1999).
    [CrossRef]
  13. M. M. Popov, "A new method of computation of wave fields using Gaussian beams," Wave Motion 4, 85-97 (1982).
    [CrossRef]
  14. M. A. Alonso and G. W. Forbes, "Stable aggregates of flexible elements give a stronger link between rays and waves," Opt. Express 10, 728-739 (2002).
    [PubMed]
  15. R. Simon and G. S. Agarwal, "Wigner representation of Laguerre-Gaussian beams," Opt. Lett. 25, 1313-1315 (2000).
    [CrossRef]
  16. A. T. Friberg and J. Turnnen, "Algebraic and graphical propagation methods for Gaussian Schell-model beams," Opt. Eng. 25, 857-864 (1986).
  17. J. A. Arnaud, "Hamiltonian theory of beam mode propagation," in Progress in Optics, E.Wolf, ed. (North-Holland, 1973) Vol. XI, pp. 247-304.
    [CrossRef]
  18. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into a symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).
  19. Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002).
    [CrossRef]
  20. A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  21. R. Martinez-Herrero and P. M. Mejias, "Second-order spatial characterization of hard-edge diffracted beams," Opt. Lett. 18, 1669-1671 (1993).
    [CrossRef] [PubMed]
  22. S. Wang and L. Ronchi, "Principles and design of optical array," in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1988), Vol. 25, p. 279.
    [CrossRef]
  23. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.
  24. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppession," Phys. Rev. Lett. 53, 1057-1060 (1984).
    [CrossRef]
  25. A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver halide noise gratings," Opt. Commun. 98, 236-240 (1993).
    [CrossRef]
  26. F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
    [CrossRef]
  27. Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian-Schell model beams through misaligned optical system," Opt. Commun. 211, 1-8 (2002).
    [CrossRef]
  28. J. R. Leger, D. Chen, and G. Mowry, "Design and performance of diffractive optics for custom laser resonators," Appl. Opt. 34, 2498-2509 (1995).
    [CrossRef] [PubMed]
  29. J. Yin, W. Gao, H. Wang, Q. Long, and Y. Wang, "Generations of dark hollow beams and their applications in laser cooling of atoms and all-optical-type Bose-Einstein condensation," Chin. Phys. 11, 1157-1169 (2002).
    [CrossRef]
  30. N. G. Parker, N. P. Proukakis, M. Leadbeater, and C. S. Adams, "Soliton-sound interactions in quasi-one-dimensional Bose-Einstein condensates," Phys. Rev. Lett. 90, 220401 (2003).
    [CrossRef] [PubMed]
  31. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).
    [CrossRef]
  32. A. Ottl, S. Ritter, M. Kohl, and T. Esslinger, "Correlations and counting statistics of an atom laser," Phys. Rev. Lett. 95, 090404 (2005).
    [CrossRef] [PubMed]

2005 (1)

A. Ottl, S. Ritter, M. Kohl, and T. Esslinger, "Correlations and counting statistics of an atom laser," Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef] [PubMed]

2004 (2)

2003 (3)

Y. Li, "Flat-topped light beam with non-circular cross-sections," J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

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

N. G. Parker, N. P. Proukakis, M. Leadbeater, and C. S. Adams, "Soliton-sound interactions in quasi-one-dimensional Bose-Einstein condensates," Phys. Rev. Lett. 90, 220401 (2003).
[CrossRef] [PubMed]

2002 (5)

Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002).
[CrossRef]

M. A. Alonso and G. W. Forbes, "Stable aggregates of flexible elements give a stronger link between rays and waves," Opt. Express 10, 728-739 (2002).
[PubMed]

B. D. Stone and T. J. Bruegge, "Practical considerations for simulating beam propagation: a comparison of three approaches," in International Optical Design Conference 2002, P. K. Manhart and J. M. Sasian, eds, Proc. SPIE 4832, 359-377 (2002).
[CrossRef]

Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian-Schell model beams through misaligned optical system," Opt. Commun. 211, 1-8 (2002).
[CrossRef]

J. Yin, W. Gao, H. Wang, Q. Long, and Y. Wang, "Generations of dark hollow beams and their applications in laser cooling of atoms and all-optical-type Bose-Einstein condensation," Chin. Phys. 11, 1157-1169 (2002).
[CrossRef]

2000 (2)

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

R. Simon and G. S. Agarwal, "Wigner representation of Laguerre-Gaussian beams," Opt. Lett. 25, 1313-1315 (2000).
[CrossRef]

1999 (1)

T. J. Bruegge, M. P. Rimmer, and J. D. Targove, "Software for free-space beam propagation," in Optical Design and Analysis Software, R. C. Juergens, ed., Proc. SPIE 3780, 14-22 (1999).
[CrossRef]

1997 (2)

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]

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

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]

J. R. Leger, D. Chen, and G. Mowry, "Design and performance of diffractive optics for custom laser resonators," Appl. Opt. 34, 2498-2509 (1995).
[CrossRef] [PubMed]

1993 (2)

R. Martinez-Herrero and P. M. Mejias, "Second-order spatial characterization of hard-edge diffracted beams," Opt. Lett. 18, 1669-1671 (1993).
[CrossRef] [PubMed]

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

1991 (1)

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

1990 (2)

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

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

1986 (1)

A. T. Friberg and J. Turnnen, "Algebraic and graphical propagation methods for Gaussian Schell-model beams," Opt. Eng. 25, 857-864 (1986).

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppession," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

1982 (1)

M. M. Popov, "A new method of computation of wave fields using Gaussian beams," Wave Motion 4, 85-97 (1982).
[CrossRef]

Adams, C. S.

N. G. Parker, N. P. Proukakis, M. Leadbeater, and C. S. Adams, "Soliton-sound interactions in quasi-one-dimensional Bose-Einstein condensates," Phys. Rev. Lett. 90, 220401 (2003).
[CrossRef] [PubMed]

Agarwal, G. S.

Alda, J.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into a 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]

Alonso, M. A.

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppession," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Arlt, J.

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

Arnaud, J. A.

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

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]

Belendez, A.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Bernabeu, E.

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

Bruegge, T. J.

B. D. Stone and T. J. Bruegge, "Practical considerations for simulating beam propagation: a comparison of three approaches," in International Optical Design Conference 2002, P. K. Manhart and J. M. Sasian, eds, Proc. SPIE 4832, 359-377 (2002).
[CrossRef]

T. J. Bruegge, M. P. Rimmer, and J. D. Targove, "Software for free-space beam propagation," in Optical Design and Analysis Software, R. C. Juergens, ed., Proc. SPIE 3780, 14-22 (1999).
[CrossRef]

Cai, Y.

Carretero, L.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Chen, D.

Chen, X.

L. Zhang, X. Lu, X. Chen, and S. He, "Generation of a dark hollow beam inside a cavity," Chin. Phys. Lett. 21, 298-301 (2004).
[CrossRef]

Dholakia, K.

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

Esslinger, T.

A. Ottl, S. Ritter, M. Kohl, and T. Esslinger, "Correlations and counting statistics of an atom laser," Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef] [PubMed]

Fimia, A.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Forbes, G. W.

Friberg, A. T.

A. T. Friberg and J. Turnnen, "Algebraic and graphical propagation methods for Gaussian Schell-model beams," Opt. Eng. 25, 857-864 (1986).

Gao, W.

J. Yin, W. Gao, H. Wang, Q. Long, and Y. Wang, "Generations of dark hollow beams and their applications in laser cooling of atoms and all-optical-type Bose-Einstein condensation," Chin. Phys. 11, 1157-1169 (2002).
[CrossRef]

J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics,E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

Gori, F.

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

He, S.

L. Zhang, X. Lu, X. Chen, and S. He, "Generation of a dark hollow beam inside a cavity," Chin. Phys. Lett. 21, 298-301 (2004).
[CrossRef]

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]

Jhe, W.

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

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppession," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Kawano, K.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).
[CrossRef]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppession," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Kitoh, T.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).
[CrossRef]

Kohl, M.

A. Ottl, S. Ritter, M. Kohl, and T. Esslinger, "Correlations and counting statistics of an atom laser," Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef] [PubMed]

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]

Leadbeater, M.

N. G. Parker, N. P. Proukakis, M. Leadbeater, and C. S. Adams, "Soliton-sound interactions in quasi-one-dimensional Bose-Einstein condensates," Phys. Rev. Lett. 90, 220401 (2003).
[CrossRef] [PubMed]

Leger, J. R.

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]

Li, Y.

Y. Li, "Flat-topped light beam with non-circular cross-sections," J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

Lin, Q.

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]

Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002).
[CrossRef]

Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian-Schell model beams through misaligned optical system," Opt. Commun. 211, 1-8 (2002).
[CrossRef]

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

Long, Q.

J. Yin, W. Gao, H. Wang, Q. Long, and Y. Wang, "Generations of dark hollow beams and their applications in laser cooling of atoms and all-optical-type Bose-Einstein condensation," Chin. Phys. 11, 1157-1169 (2002).
[CrossRef]

Lu, X.

L. Zhang, X. Lu, X. Chen, and S. He, "Generation of a dark hollow beam inside a cavity," Chin. Phys. Lett. 21, 298-301 (2004).
[CrossRef]

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

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.

Martinez-Herrero, R.

Mejias, P. M.

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppession," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppession," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Mowry, G.

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppession," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Ohtsu, M.

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

Ottl, A.

A. Ottl, S. Ritter, M. Kohl, and T. Esslinger, "Correlations and counting statistics of an atom laser," Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef] [PubMed]

Parker, N. G.

N. G. Parker, N. P. Proukakis, M. Leadbeater, and C. S. Adams, "Soliton-sound interactions in quasi-one-dimensional Bose-Einstein condensates," Phys. Rev. Lett. 90, 220401 (2003).
[CrossRef] [PubMed]

Popov, M. M.

M. M. Popov, "A new method of computation of wave fields using Gaussian beams," Wave Motion 4, 85-97 (1982).
[CrossRef]

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]

Proukakis, N. P.

N. G. Parker, N. P. Proukakis, M. Leadbeater, and C. S. Adams, "Soliton-sound interactions in quasi-one-dimensional Bose-Einstein condensates," Phys. Rev. Lett. 90, 220401 (2003).
[CrossRef] [PubMed]

Rimmer, M. P.

T. J. Bruegge, M. P. Rimmer, and J. D. Targove, "Software for free-space beam propagation," in Optical Design and Analysis Software, R. C. Juergens, ed., Proc. SPIE 3780, 14-22 (1999).
[CrossRef]

Ritter, S.

A. Ottl, S. Ritter, M. Kohl, and T. Esslinger, "Correlations and counting statistics of an atom laser," Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef] [PubMed]

Ronchi, L.

S. Wang and L. Ronchi, "Principles and design of optical array," in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1988), Vol. 25, p. 279.
[CrossRef]

Sakaki, K.

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

Santarsiero, M.

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

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]

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.

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

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

Simon, R.

Stone, B. D.

B. D. Stone and T. J. Bruegge, "Practical considerations for simulating beam propagation: a comparison of three approaches," in International Optical Design Conference 2002, P. K. Manhart and J. M. Sasian, eds, Proc. SPIE 4832, 359-377 (2002).
[CrossRef]

Targove, J. D.

T. J. Bruegge, M. P. Rimmer, and J. D. Targove, "Software for free-space beam propagation," in Optical Design and Analysis Software, R. C. Juergens, ed., Proc. SPIE 3780, 14-22 (1999).
[CrossRef]

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]

Turnnen, J.

A. T. Friberg and J. Turnnen, "Algebraic and graphical propagation methods for Gaussian Schell-model beams," Opt. Eng. 25, 857-864 (1986).

Wang, H.

J. Yin, W. Gao, H. Wang, Q. Long, and Y. Wang, "Generations of dark hollow beams and their applications in laser cooling of atoms and all-optical-type Bose-Einstein condensation," Chin. Phys. 11, 1157-1169 (2002).
[CrossRef]

Wang, S.

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

S. Wang and L. Ronchi, "Principles and design of optical array," in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1988), Vol. 25, p. 279.
[CrossRef]

Wang, Y.

J. Yin, W. Gao, H. Wang, Q. Long, and Y. Wang, "Generations of dark hollow beams and their applications in laser cooling of atoms and all-optical-type Bose-Einstein condensation," Chin. Phys. 11, 1157-1169 (2002).
[CrossRef]

Wolf, E.

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Figures (5)

Fig. 1
Fig. 1

Contour graphs of normalized irradiance distribution of a coherent DHB for different M and N with p = 0.9 , w 0 x = 1 , and w 0 y = 1.5 mm ; (a) M = 3 , N = 5 ; (b) M = 6 , N = 10 .

Fig. 2
Fig. 2

Dependences of the M 2 factor of a coherent DHB on (a) p, (b) M and N.

Fig. 3
Fig. 3

Dependence of the M 2 factor of a partially coherent DHB on σ g .

Fig. 4
Fig. 4

Three-dimensional normalized irradiance distribution of a coherent rectangular DHB in free space at different propagation distances: (a) z = 0 , (b) z = 1500 mm , (c) z = 5000 mm , (d) z = 20 , 000 mm .

Fig. 5
Fig. 5

Cross lines ( y = 0 ) of the normalized irradiance distribution of a partially coherent rectangular DHB at different propagation distances for different values of σ g .

Equations (66)

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E M N ( x , y ; 0 ) = m = 1 M n = 1 N C m C n [ E m n ( x , y ; 0 ) E m n p ( x , y ; 0 ) ] = m = 1 M n = 1 N ( 1 ) m + n 2 M N [ M m ] [ N n ] [ exp ( m x 2 w 0 x 2 n y 2 w 0 y 2 ) exp ( m x 2 p w 0 x 2 n y 2 p w 0 y 2 ) ] .
C m = ( 1 ) m 1 M [ M m ] , C n = ( 1 ) n 1 M [ N n ] , and [ M m ] and [ N n ]
M x 2 = 4 π Δ x Δ p x , M y 2 = 4 π Δ y Δ p y ,
Δ x = 1 I ( x x ¯ ) 2 E ( x , y ; 0 ) 2 d x d y ,
Δ y = 1 I ( y y ¯ ) 2 E ( x , y ; 0 ) 2 d x d y ,
Δ p x = 1 I ( p x p ¯ x ) 2 E ̃ ( p x , p y ; 0 ) 2 d p x d p y ,
Δ p y = 1 I ( p y p ¯ y ) 2 E ̃ ( p x , p y ; 0 ) 2 d p x d p y .
E ̃ ( p x , p y ; 0 ) = E ( x , y ; 0 ) exp ( 2 π i p x x 2 π i p y y ) d x d y .
x ¯ = 1 I x E ( x , y ; 0 ) 2 d x d y ,
y ¯ = 1 I y E ( x , y ; 0 ) 2 d x d y ,
p ¯ x = 1 I p x E ̃ ( p x , p y ; 0 ) 2 d p x d p y ,
p ¯ y = 1 I p y E ̃ ( p x , p y ; 0 ) 2 d p x d p y ,
I = E ( x , y ; 0 ) 2 d x d y = E ̃ ( p x , p y ; 0 ) 2 d p x d p y .
M x 2 = 4 { m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l M 2 N 2 2 [ M m ] [ N n ] [ M h ] [ N l ] ( α m h 3 2 α n l 1 2 α m h p 3 2 α n l p 1 2 α p m h 3 2 α p n l 1 2 + α p m h p 3 2 α p n l p 1 2 ) } 1 2 × { m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l m h M 2 N 2 [ M m ] [ N n ] [ M h ] [ N l ] ( α m h 3 2 α n l 1 2 α m h p 3 2 α n l p 1 2 α p m h 3 2 α p n l 1 2 + α p m h p 3 2 α p n l p 1 2 ) } 1 2 × 1 m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l M 2 N 2 [ M m ] [ N n ] [ M h ] [ N l ] ( α m h 1 2 α n l 1 2 α m h p 1 2 α n l p 1 2 α p m h 1 2 α p n l 1 2 + α p m h p 1 2 α p n l p 1 2 ) ,
m + h = α m h , n + l = α n l , ( m + h p ) = α m h p , ( n + l p ) = α n l p .
( m p + h ) = α p m h , ( n p + l ) = α p n l , ( m + h ) p = α p m h p , ( n + l ) p = α p n l p .
E M N ( r 1 ; 0 ) = m = 1 M n = 1 N ( 1 ) m + n 2 M N [ M m ] [ N n ] [ exp ( i k 2 r 1 T Q 1 m n 1 r 1 ) exp ( i k 2 r 1 T Q 1 m n p 1 r 1 ) ] ,
Q 1 m n 1 = 2 i k [ m w 0 x 2 0 0 n w 0 y 2 ] , Q 1 m n p 1 = 1 p Q 1 m n 1 .
E 2 ( r 2 ; z ) = i k 2 π [ det ( B ) ] 1 2 exp ( i k l 0 ) E 1 ( r 1 ; 0 ) exp [ i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ) ] d r 1 ,
E M N ( r 2 ; z ) = m = 1 M n = 1 N ( 1 ) m + n 2 M N [ M m ] [ N n ] exp [ i k l 0 ] × { [ det ( A + BQ 1 m n 1 ) ] 1 2 exp ( i k 2 r 2 T Q 2 m n 1 r 2 ) [ det ( A + BQ 1 m n p 1 ) ] 1 2 exp ( i k 2 r 2 T Q 2 m n p 1 r 2 ) } ,
Q 2 m n 1 = ( C + DQ 1 m n 1 ) ( A + BQ 1 m n 1 ) 1 ,
Q 2 m n p 1 = ( C + DQ 1 m n p 1 ) ( A + BQ 1 m n p 1 ) 1 .
E 1 2 ( r 2 ; z ) = i k 2 π [ det ( B ) ] 1 2 exp ( i k l 0 ) E ( r 1 ; 0 ) exp [ i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ) ] exp [ i k 2 ( r 1 T B 1 e f + r 2 T B 1 g h ) ] d r 1 ,
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 M N ( r 2 ; z ) = m = 1 M n = 1 N ( 1 ) m + n 2 M N [ M m ] [ N n ] exp ( i k l 0 ) ( i k 2 r 2 T B 1 g h ) { [ det ( A + BQ 1 m n 1 ) ] 1 2 exp [ i k 2 r 2 T Q 2 m n 1 r 2 i k 2 r 2 T B 1 T ( A + BQ 1 m n 1 ) 1 e f + i k 8 e f T B 1 T ( A + BQ 1 m n 1 ) 1 e f ] [ det ( A + BQ 1 m n p 1 ) ] 1 2 exp [ i k 2 r 2 T Q 2 m n p 1 r 2 i k 2 r 2 T B 1 T ( A + BQ 1 m n p 1 ) 1 e f + i k 8 e f T B 1 T ( A + BQ 1 m n p 1 ) 1 e f ] } ,
W ( x 1 , y 1 , x 2 , y 2 ; 0 ) = I ( x 1 , y 1 ; 0 ) I ( x 2 , y 2 ; 0 ) g ( x 1 x 2 ; y 1 y 2 ) ,
g ( x 1 x 2 ; y 1 y 2 ) = exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 2 ] .
W M N ( x 1 , y 1 , x 2 , y 2 ; 0 ) = m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l M 2 N 2 [ M m ] [ N n ] [ M h ] [ N l ] [ exp ( m x 1 2 w 0 x 2 h x 2 2 w 0 x 2 n y 1 2 w 0 y 2 l y 2 2 w 0 y 2 ) ) exp ( m x 1 2 w 0 x 2 h x 2 2 p w 0 x 2 n y 1 2 w 0 y 2 l y 2 2 p w 0 y 2 ) exp ( m x 1 2 p w 0 x 2 h x 2 2 w 0 x 2 n y 1 2 p w 0 y 2 l y 2 2 w 0 y 2 ) [ + exp ( m x 1 2 p w 0 x 2 h x 2 2 p w 0 x 2 n y 1 2 p w 0 y 2 l y 2 2 p w 0 y 2 ) ] exp [ ( x 1 x 2 ) 2 2 σ g 2 ( y 1 y 2 ) 2 2 σ g 2 ] .
M x 2 = 4 π Δ x Δ p x , M y 2 = 4 π Δ y Δ p y ,
Δ x = 1 I ( x x ¯ ) 2 W ( x , x , y , y ) d x d y ,
Δ y = 1 I ( y y ¯ ) 2 W ( x , x , y , y ) d x d y ,
Δ p x = 1 I ( p x p ¯ x ) 2 W ̃ ( p x , p x , p y , p y ) d p x d p y ,
Δ p y = 1 I ( p y p ¯ y ) 2 W ̃ ( p x , p x , p y , p y ) d p x d p y ,
I = W ( x , y , x , y ) d x d y = W ̃ ( p x , p y , p x , p y ) d p x d p y ,
M x 2 = { m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l 2 M 2 N 2 [ M m ] [ N n ] [ M h ] [ N l ] ( α m h 3 2 α n l 1 2 α m h p 3 2 α n l p 1 2 α p m h 3 2 α p n l 1 2 + α p m h p 3 2 α p n l p 1 2 ) } 1 2 { m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l M 2 N 2 [ M m ] [ N n ] [ M h ] [ N l ] ( w 0 x 2 σ g 2 α m h 1 2 + 4 m h α m h 3 2 ) α n l 1 2 ( w 0 x 2 σ g 2 α m h p 1 2 + 4 m h p α m h p 3 2 ) α n l p 1 2 ) { [ ( w 0 x 2 σ g 2 α p m h 1 2 + 4 m h p α p m h 3 2 ) α p n l 1 2 + ( w 0 x 2 σ g 2 α p m h p 1 2 + 4 m h p 2 α p m h p 3 2 ) α p n l p 1 2 ] } 1 2 × 1 m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l M 2 N 2 [ M m ] [ N n ] [ M h ] [ N l ] ( α m h 1 2 α n l 1 2 α m h p 1 2 α n l p 1 2 α p m h 1 2 α p n l 1 2 + α p m h p 1 2 α p n l p 1 2 ) .
W M N ( r ̃ ; 0 ) = m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l M 2 N 2 [ M m ] [ N n ] [ M h ] [ N l ] [ exp ( i k 2 r ̃ T M i 1 m h n l 1 r ̃ ) ) [ exp ( i k 2 r ̃ T M i 2 m h n l 1 r ̃ ) exp ( i k 2 r ̃ T M i 3 m h n l 1 r ̃ ) + exp ( i k 2 r ̃ T M i 4 m h n l 1 r ̃ ) ] ,
M i 1 m h n l 1 = i k [ m m n m g m g m h l ] , M i 2 m h n l 1 = i k [ m m n m g m g m h l p ] ,
M i 3 m h n l 1 = i k [ m m n p m g m g m h l ] ,
M i 4 m h n l 1 = i k [ m m n p m g m g m h l p ] ,
m g = [ 1 σ g 2 0 0 1 σ g 2 ] ,
m m n = [ 2 m w 0 x 2 1 σ g 2 0 0 2 n w 0 y 2 1 σ g 2 ] ,
m h l = [ 2 h w 0 x 2 1 σ g 2 0 0 2 l w 0 y 2 1 σ g 2 ] ,
m m n p = [ 2 m p w 0 x 2 1 σ g 2 0 0 2 n p w 0 y 2 1 σ g 2 ] ,
m h l p = [ 2 h p w 0 x 2 1 σ g 2 0 0 2 l p w 0 y 2 1 σ g 2 ] .
W ( ρ ̃ ; z ) = 1 λ 2 [ det ( B ¯ ) ] 1 2 W ( r ̃ ; 0 ) exp [ i π λ ( r ̃ T B ¯ 1 A ¯ r ̃ 2 r ̃ T B ¯ 1 ρ ̃ + ρ ̃ T D ¯ B ¯ 1 ρ ̃ ) ] d r ̃ ,
A ¯ = [ A 0 0 A ] , B ¯ = [ B 0 0 B ] ,
C ¯ = [ C 0 0 C ] , D ¯ = [ D 0 0 D ] .
W ( ρ ̃ ; z ) = m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l M 2 N 2 [ M m ] [ N n ] [ M h ] [ N l ] { [ det ( A ¯ + B ¯ M i 1 m n h l 1 ) ] 1 2 exp ( i k 2 ρ ̃ T M o 1 m n h l 1 ρ ̃ ) [ det ( A ¯ + B ¯ M i 2 m n h j 1 ) ] 1 2 exp ( i k 2 ρ ̃ T M o 2 m n h l 1 ρ ̃ ) [ det ( A ¯ + B ¯ M i 3 m n h l 1 ) ] 1 2 exp ( i k 2 ρ ̃ T M o 3 m n h l 1 ρ ̃ ) + [ det ( A ¯ + B ¯ M i 4 m n h j 1 ) ] 1 2 exp ( i k 2 ρ ̃ T M o 4 m n h l 1 ρ ̃ ) } ,
M o j m n h l 1 = ( C ¯ + D ¯ M i j m n h l ) ( A ¯ + B ¯ M i j m n h l ) , ( j = 1 , 2 , 3 , 4 ) .
W ( ρ ; z ) = 1 λ 2 [ det ( B ¯ ) ] 1 2 W ( r ̃ ; 0 ) exp [ i π λ ( r ̃ T B ¯ 1 A ¯ r ̃ 2 r ̃ T B ¯ 1 ρ ̃ + ρ ̃ T D ¯ B ¯ 1 ρ ̃ ) ] exp [ i π λ ( r ̃ T B ¯ 1 e ¯ f + ρ ̃ T B ¯ 1 g ¯ h ) ] d r ̃ ,
A ¯ = [ a I 0 0 a I ] , B ¯ = [ b I 0 0 b I ] ,
C ¯ = [ c I 0 0 c I ] , D ¯ = [ d I 0 0 d I ] .
W ( ρ ̃ ; z ) = m = 1 M n = 1 N h = 1 M l = 1 N ( 1 ) m + n + h + l M 2 N 2 [ M m ] [ N n ] [ M h ] [ N l ] exp ( i k 2 ρ ̃ T B ¯ 1 g ¯ h ) { [ det ( A ¯ + B ¯ M i 1 m n h l 1 ) ] 1 2 exp [ i k 2 ρ ̃ T M i 1 m n h l 1 ρ ̃ i k 2 ρ ̃ T B ¯ 1 T ( A ¯ + B ¯ M i 1 m n h l 1 ) 1 e ¯ f + i k 8 e ¯ f T B ¯ 1 T ( A ¯ + B ¯ M i 1 m n h l 1 ) 1 e ¯ f ] ] [ det ( A ¯ + B ¯ M i 2 m n h l 1 ) ] 1 2 exp [ i k 2 ρ ̃ T M o 2 m n h l 1 ρ ̃ i k 2 ρ ̃ T B ¯ 1 T ( A ¯ + B ¯ M i 2 m n h l 1 ) 1 e ¯ f + i k 8 e ¯ f T B ¯ 1 T ( A ¯ + B ¯ M i 2 m n h l 1 ) 1 e ¯ f ] [ det ( A ¯ + B ¯ M i 3 m n h l 1 ) ] 1 2 exp [ i k 2 ρ ̃ T M o 3 m n h l 1 ρ ̃ i k 2 ρ ̃ T B ¯ 1 T ( A ¯ + B ¯ M i 3 m n h l 1 ) 1 e ¯ f + i k 8 e ¯ f T B ¯ 1 T ( A ¯ + B ¯ M i 3 m n h l 1 ) 1 e ¯ f ] { [ det ( A ¯ + B ¯ M i 4 m n h l 1 ) ] 1 2 exp [ i k 2 ρ ̃ T M o 4 m n h l 1 ρ ̃ i k 2 ρ ̃ T B ¯ 1 T ( A ¯ + B ¯ M i 4 m n h l 1 ) 1 e ¯ f + i k 8 e ¯ T B ¯ 1 T ( A ¯ + B ¯ M i 4 m n h l 1 ) 1 e ¯ f ] } ,
E M N ( r 2 ; z ) = i k 2 π [ det ( B ) ] 1 2 exp ( i k l 0 ) m = 1 M n = 1 N ( 1 ) m + n 2 M N [ M m ] [ N n ] { exp [ i k 2 r 2 T DB 1 r 2 + i k 2 r 2 T B 1 T ( Q 1 m n 1 + B 1 A ) 1 B 1 r 2 ] ] exp [ i k 2 ( Q 1 m n 1 + B 1 A ) 1 2 r 1 ( Q 1 m n 1 + B 1 A ) 1 2 B 1 r 2 2 ] d r 1 exp [ i k 2 r 2 T DB 1 r 2 + i k 2 r 2 T B 1 T ( Q 1 m n p 1 + B 1 A ) 1 B 1 r 2 ] × exp { [ i k 2 ( Q 1 m n p 1 + B 1 A ) 1 2 r 1 ( Q 1 m n p 1 + B 1 A ) 1 2 B 1 r 2 2 ] d r 1 } .
E M N ( r 2 ; z ) = exp ( i k l 0 ) [ det ( B ) ] 1 2 m = 1 M n = 1 N ( 1 ) m + n 2 M N [ M m ] [ N n ] { [ det ( Q 1 m n 1 + B 1 A ) ] 1 2 exp [ i k 2 r 2 T DB 1 r 2 + i k 2 r 2 T B 1 T ( Q 1 m n 1 + B 1 A ) 1 B 1 r 2 ] [ det ( Q 1 m n 1 + B 1 A ) ] 1 2 exp [ i k 2 r 2 T DB 1 r 2 + i k 2 r 2 T B 1 T ( Q 1 m n p 1 + B 1 A ) 1 B 1 r 2 ] } .
[ det ( B ) ] 1 2 [ det ( Q 1 m n 1 + B 1 A ) ] 1 2 = [ det ( A + BQ 1 m n 1 ) ] 1 2 ,
[ det ( B ) ] 1 2 [ det ( Q 1 m n p 1 + B 1 A ) ] 1 2 = [ det ( A + BQ 1 m n p 1 ) ] 1 2 ,
DB 1 B 1 T ( Q 1 m n 1 + B 1 A ) 1 B 1 = DB 1 B 1 T ( BQ 1 m n 1 + A ) 1 = ( DQ 1 m n 1 + DB 1 A B 1 T ) ( BQ 1 m n 1 + A ) 1 = ( C + DQ 1 m n 1 ) ( BQ 1 m n 1 + A ) 1 ,
DB 1 B 1 T ( Q 1 m n p 1 + B 1 A ) 1 B 1 = DB 1 B 1 T ( BQ 1 m n p 1 + A ) 1 = ( DQ 1 m n p 1 + DB 1 A B 1 T ) ( BQ 1 m n p 1 + A ) 1 = ( C + DQ 1 m n p 1 ) ( BQ 1 m n p 1 + A ) 1 ,
Q 2 m n 1 = ( C + DQ 1 m n 1 ) ( A + BQ 1 m n 1 ) 1 ,
Q 2 m n p 1 = ( C + DQ 1 m n p 1 ) ( A + BQ 1 m n p 1 ) 1 ,
( B ¯ 1 A ¯ ) T = B ¯ 1 A ¯ , ( B ¯ 1 ) T = ( C ¯ D ¯ B ¯ 1 A ¯ ) ,
( D ¯ B ¯ 1 ) T = D ¯ B ¯ 1 .

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