Abstract

An alternative and convenient theoretical model is proposed to describe a flexible anomalous hollow beam of elliptical symmetry with an elliptical solid core, which was observed in experiment recently (Phys. Rev. Lett, 94 (2005) 134802). In this model, the electric field of anomalous hollow beam is expressed as a finite sum of elliptical Gaussian modes. Flat-topped beams, dark hollow beams and Gaussian beams are special cases of our model. Analytical propagation formulae for coherent and partially coherent anomalous hollow beams passing through astigmatic ABCD optical systems are derived. Some numerical examples are calculated to show the propagation and focusing properties of coherent and partially coherent anomalous hollow beams.

©2008 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
  4. Y. Li, “Light beam with flat-topped profiles,” Opt. Lett.  27, 1007–1009 (2002).
    [Crossref]
  5. Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profile,” J. Opt. A: Pure Appl. Opt. 6, 390–395 (2004).
    [Crossref]
  6. Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B 23, 1398–1407 (2006).
    [Crossref]
  7. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353–1367 (2006).
    [Crossref] [PubMed]
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    [Crossref]
  9. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [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. D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44, 7187–7190 (2005).
    [Crossref] [PubMed]
  18. D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202–202 (2008).
    [Crossref]
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  21. Y. Zhang, “Generation of thin and hollow beams by the axicon with a large open angle,” Opt. Commun.  281, 508–514 (2008).
    [Crossref]
  22. R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934(2008).
    [Crossref]
  23. G. Wu, Q. Lou, and J. Zhou, “Analytical vectorial structure of hollow Gaussian beams in the far field,” Opt. Express 16, 6417–6424 (2008).
    [Crossref] [PubMed]
  24. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
    [Crossref]
  25. Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in a weak turbulent atmosphere,” J. Opt. Soc. Am. A 25, 1497–1503 (2008).
    [Crossref]
  26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett.  94, 134802 (2005).
    [Crossref] [PubMed]
  27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett.  32, 3179–3181 (2007).
    [Crossref] [PubMed]
  28. Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372, 4654–4660 (2008).
  29. J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, 1973), pp. 247–304.
  30. 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 85, 67 (1990).
  31. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [Crossref]
  32. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964).
  33. 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.
  34. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  35. 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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
    [Crossref]
  36. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
    [Crossref]
  37. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Lett. 27, 216–218 (2002).
    [Crossref]
  38. Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
    [Crossref]
  39. Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716–2718 (2004).
    [Crossref] [PubMed]
  40. Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
    [Crossref]
  41. Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798–1804 (2005).
    [Crossref]
  42. F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14, 6999–7004 (2006).
    [Crossref] [PubMed]
  43. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett.  89, 041117 (2006).
    [Crossref]
  44. Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett.  31, 685–687 (2006).
    [Crossref] [PubMed]
  45. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15, 15480–15492 (2007).
    [Crossref] [PubMed]
  46. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937–1944 (2007).
    [Crossref]
  47. X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369, 157–166 (2007).
    [Crossref]
  48. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33,1389–1391(2008).
    [Crossref] [PubMed]
  49. F. V. Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575–581 (2008).
    [Crossref]
  50. F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett.  33, 1795–1797 (2008).
    [Crossref] [PubMed]

2008 (12)

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
[Crossref]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202–202 (2008).
[Crossref]

Y. Zhang, “Generation of thin and hollow beams by the axicon with a large open angle,” Opt. Commun.  281, 508–514 (2008).
[Crossref]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934(2008).
[Crossref]

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
[Crossref]

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372, 4654–4660 (2008).

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett.  33, 1795–1797 (2008).
[Crossref] [PubMed]

D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83–87 (2008).
[Crossref]

F. V. Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575–581 (2008).
[Crossref]

G. Wu, Q. Lou, and J. Zhou, “Analytical vectorial structure of hollow Gaussian beams in the far field,” Opt. Express 16, 6417–6424 (2008).
[Crossref] [PubMed]

Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in a weak turbulent atmosphere,” J. Opt. Soc. Am. A 25, 1497–1503 (2008).
[Crossref]

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33,1389–1391(2008).
[Crossref] [PubMed]

2007 (5)

2006 (8)

2005 (5)

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798–1804 (2005).
[Crossref]

D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44, 7187–7190 (2005).
[Crossref] [PubMed]

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[Crossref]

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
[Crossref]

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett.  94, 134802 (2005).
[Crossref] [PubMed]

2004 (3)

2003 (1)

2002 (3)

2000 (2)

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev.  7, 216–220 (2000).
[Crossref]

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

1998 (1)

1997 (2)

C. Palma, “Decentered Gaussian beams, ray bundles and Bessel-Gaussian beams,” Appl. Opt. 36, 1116–1120 (1997).
[Crossref] [PubMed]

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]

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 85, 67 (1990).

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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

1970 (1)

Ahmad, M. A.

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 85, 67 (1990).

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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

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, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, 1973), pp. 247–304.

Arpali, C.

Arpali, S. A.

Baykal, Y.

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 85, 67 (1990).

Cai, Y.

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372, 4654–4660 (2008).

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett.  33, 1795–1797 (2008).
[Crossref] [PubMed]

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33,1389–1391(2008).
[Crossref] [PubMed]

Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in a weak turbulent atmosphere,” J. Opt. Soc. Am. A 25, 1497–1503 (2008).
[Crossref]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15, 15480–15492 (2007).
[Crossref] [PubMed]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937–1944 (2007).
[Crossref]

X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369, 157–166 (2007).
[Crossref]

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett.  32, 3179–3181 (2007).
[Crossref] [PubMed]

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett.  31, 685–687 (2006).
[Crossref] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett.  89, 041117 (2006).
[Crossref]

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B 23, 1398–1407 (2006).
[Crossref]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353–1367 (2006).
[Crossref] [PubMed]

F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14, 6999–7004 (2006).
[Crossref] [PubMed]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798–1804 (2005).
[Crossref]

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[Crossref]

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
[Crossref]

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profile,” J. Opt. A: Pure Appl. Opt. 6, 390–395 (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 and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716–2718 (2004).
[Crossref] [PubMed]

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]

Capjack,, C. E.

Chen, Z.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
[Crossref]

Collins, S. A.

Davidson, F. M.

Deng, D.

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]

Dijk, F. V.

Eyyuboglu, H. T.

Fan, Z.

Fu, X.

Gao, W.

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

Gbur, G.

He, S.

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]

Hu, L.

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett.  31, 685–687 (2006).
[Crossref] [PubMed]

Jitsuno, T.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev.  7, 216–220 (2000).
[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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

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]

Li, J.

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett.  94, 134802 (2005).
[Crossref] [PubMed]

Li, Y.

Y. Li, “Light beam with flat-topped profiles,” Opt. Lett.  27, 1007–1009 (2002).
[Crossref]

Lin, J.

Lin, Q.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
[Crossref]

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profile,” J. Opt. A: Pure Appl. Opt. 6, 390–395 (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]

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

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 85, 67 (1990).

Liu, J.

Liu, S.

Liu, Z.

Lou, Q.

Lu, X.

Lü, X.

X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369, 157–166 (2007).
[Crossref]

Luneburg, R. K.

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

Mandel, L.

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

Matsuoka,, S.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev.  7, 216–220 (2000).
[Crossref]

Mei, Z.

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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Miyanaga, N.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev.  7, 216–220 (2000).
[Crossref]

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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Nakatsuka, M.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev.  7, 216–220 (2000).
[Crossref]

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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Nishi, N.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev.  7, 216–220 (2000).
[Crossref]

Palma, C.

Peschel, U.

Pu, J.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
[Crossref]

Ricklin, J. C.

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.

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).
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Seguin, H. J. J.

Senthilkumaran, P.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934(2008).
[Crossref]

Shao, J.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202–202 (2008).
[Crossref]

D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44, 7187–7190 (2005).
[Crossref] [PubMed]

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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]

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934(2008).
[Crossref]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934(2008).
[Crossref]

Strohschein, J. D.

Tian, G.

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]

Tsubakimoto, K.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev.  7, 216–220 (2000).
[Crossref]

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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.

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[Crossref]

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Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett.  94, 134802 (2005).
[Crossref] [PubMed]

Wu, Y. K.

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett.  94, 134802 (2005).
[Crossref] [PubMed]

Xu, S.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202–202 (2008).
[Crossref]

D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83–87 (2008).
[Crossref]

Yamanaka, C.

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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Yazicioglu, C.

Yin, J.

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

Yu, H.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202–202 (2008).
[Crossref]

D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83–87 (2008).
[Crossref]

Zhang, L.

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Y. Zhang, “Generation of thin and hollow beams by the axicon with a large open angle,” Opt. Commun.  281, 508–514 (2008).
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Zhao, D.

Zhao, H.

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Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
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Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
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Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716–2718 (2004).
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J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E. Wolf, ed., (North-Holland, Amsterdam, 2003), pp.119–204.

Appl. Opt. (3)

Appl. Phy. Lett. (1)

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
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Appl. Phys. Lett (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett.  89, 041117 (2006).
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J. Opt. A: Pure Appl. Opt. (1)

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profile,” J. Opt. A: Pure Appl. Opt. 6, 390–395 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

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

Opt. Commun (1)

Y. Zhang, “Generation of thin and hollow beams by the axicon with a large open angle,” Opt. Commun.  281, 508–514 (2008).
[Crossref]

Opt. Commun. (3)

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934(2008).
[Crossref]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202–202 (2008).
[Crossref]

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

Opt. Eng. (1)

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
[Crossref]

Opt. Express (5)

Opt. Laser Technol. (1)

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
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Y. Li, “Light beam with flat-topped profiles,” Opt. Lett.  27, 1007–1009 (2002).
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Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett.  32, 3179–3181 (2007).
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Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett.  31, 685–687 (2006).
[Crossref] [PubMed]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett.  33, 1795–1797 (2008).
[Crossref] [PubMed]

Opt. Lett. (5)

Opt. Rev (1)

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev.  7, 216–220 (2000).
[Crossref]

Optik (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 85, 67 (1990).

Phys. Lett. A (2)

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372, 4654–4660 (2008).

X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369, 157–166 (2007).
[Crossref]

Phys. Rev. E (1)

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[Crossref]

Phys. Rev. Lett (1)

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett.  94, 134802 (2005).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

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 suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[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]

Other (5)

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

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

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.

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

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

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Figures (5)

Fig. 1.
Fig. 1. Normalized intensity (contour graph) of an anomalous hollow beam and corresponding cross line (y = 0) for different values of θ, ϕ, N, p, α and β with w 0x = 2mm, w 0y = 1mm, w 1x = 3mm, w 1y = 1mm (a) θ = ϕ = π/4, N = 5, p = 0.8, α = 0.4, β = 0.5, (b) θ = ϕ = π/4, N = 10, p = 0.8, α = 0.4, β = 0.5, (c) θ = 0, ϕ = π/10, N = 10, p = 0.8,α = 0.4, β = 0.5, (d) θ = 0, ϕ = π/10, N = 10, p = 0.5, α = 0.4, β = 0.5, (e) θ = 0, ϕ = π/10, N = 10, p = 0.8, α0.35, β = 0.5, (f) θ = 0, ϕ = π/10, N = 10, p = 0.8, α = 0.4, β = 0.2

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Fig. 2.
Fig. 2. Normalized intensity (contour graph) of an anomalous hollow beam and corresponding cross line (y = 0) for different values of w 1x and w 1y with w 0x = 2mm w 0y = 1mm, θ = 0, ϕ = 0, N = 10, p = 0.8, α = 0.35 and β = 0.5 (a) w 1x = 2mm and w 1y = 1mm, (b) w 1x = 3mm, and w 1y = 1mm, (c) w 1x = 10mm and w 1y = 1mm

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Fig. 3.
Fig. 3. Normalized 3D-intensity distribution of an anomalous hollow beam and cross line (y = 0) in free space at several different propagation distances (a) z = 0, (b) z = 0.3m, (c) z = 1m, (d) z = 2m, (e) z = 5m, (f) z = 15m

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Fig. 4.
Fig. 4. Normalized 3D-intensity distribution of a partially coherent anomalous hollow beam and cross line (y = 0) in free space at several different propagation distances (a) z = 0, (b) z = 0.3m, (c) z = 1m, (d) z = 2m, (e) z = 5m, (f) z = 15m

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Fig. 5.
Fig. 5. Normalized 3D-intensity distribution of an anomalous hollow beam and cross line (y = 0) at geometrical focal plane for different values of the coherence width (a) σ g = 0 (coherent case), (b) σ g = 2mm, (c) σ g = 0.5mm, (d) σ g = 0.1mm, (e) cross lines

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Equations (35)

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E N ( x , y , 0 ) = ∑ n = 1 N ( − 1 ) n − 1 N ( N n ) [ exp ( − x 2 w 0 x x n 2 − 2 xy w 0 x y n 2 − y 2 w 0 y y n 2 )
− exp ( − x 2 w 0 x x n p 2 − 2 x y w 0 x y n p 2 − y 2 w 0 y y n p 2 ) − α exp ( − x 2 w 1 x x β 2 − 2 x y w 1 x y β 2 − y 2 w 1 y y β 2 ) ] ,
1 w 0 x x n 2 = n cos ⁡ 2 θ w 0 x 2 + n sin ⁡ 2 θ ⁡ w 0 y 2 , 1 w 0 y y n 2 = n cos ⁡ 2 θ w 0 y 2 + n sin ⁡ 2 θ w 0 x 2 , 1 w 0 x y n 2 = n sin ⁡ 2 θ 2 w 0 x 2 − n sin ⁡ 2 θ 2 w 0 y 2 , 1 w 0 x x n p 2 = n cos ⁡ 2 θ p 2 w 0 x 2 + n sin ⁡ 2 θ p 2 w 0 y 2 , 1 w 0 y y n p 2 = n cos ⁡ 2 θ p 2 w 0 y 2 + n sin ⁡ 2 θ p 2 w 0 x 2 , 1 w 0 x y n p 2 = n sin ⁡ 2 θ 2 p 2 w 0 x 2 − n sin ⁡ 2 θ 2 p 2 w 0 y 2 , 1 w 1 x x β 2 = cos ⁡ 2 ϕ β 2 w 1 x 2 + sin ⁡ 2 ϕ β 2 w 1 y 2 , 1 w 1 y y β 2 = cos ⁡ 2 ϕ β 2 w 1 y 2 + sin ⁡ 2 ϕ β 2 w 1 x 2 , 1 w 1 x y β 2 = sin ⁡ 2 ϕ 2 β 2 w 1 x 2 − sin ⁡ 2 ϕ 2 β 2 w 1 y 2 ,
E N ( r 1 , 0 ) = ∑ n = 1 N ( − 1 ) n − 1 N ( N n ) [ exp ( − i k 2 r 1 T Q 1 n − 1 r 1 ) − exp ( − i k 2 r 1 T Q 1 n p − 1 r 1 ) − α exp ( − i k 2 r 1 T Q 1 β − 1 r 1 ) ] ,
Q 1 n − 1 = ( 2 i k w 0 x x n 2 2 i k w 0 x y n 2 2 i k w 0 x y n 2 2 i k w 0 y y n 2 ) , Q 1 n p − 1 = ( 2 i k w 0 x x n p 2 2 i k w 0 x y n p 2 2 i k w 0 x y n p 2 2 i k w 0 y y n p 2 ) , Q 1 β − 1 = ( 2 i k w 1 x x β 2 2 i k w 1 x y β 2 2 i k w 1 x y β 2 2 i k w 1 y y β 2 ) .
E ( ρ 1 , l ) = − i λ [ det ( B ) ] 1 2 ∫ − ∞ ∞ ∫ − ∞ ∞ E ( r 1 , 0 ) exp [ − i k 2 ( r 1 T B − 1 A r 1 − 2 r 1 T B − 1 ρ 1 + ρ 1 T D B − 1 ρ 1 ) ] d r 1 ,
( B − 1 A ) T = B − 1 A , ( − B − 1 ) T = ( C − DB − 1 A ) , ( DB − 1 ) T = DB − 1 .
E N ( ρ 1 , l ) = ∑ n = 1 N ( − 1 ) n − 1 N ( N n ) [ 1 [ det ( A + B Q 1 n − 1 ) ] 1 2 exp ( − i k 2 ρ 1 T Q 2 n − 1 ρ 1 )
− 1 [ det ( A + B Q 1 n p − 1 ) ] 1 2 exp ( − i k 2 ρ 1 T Q 2 n p − 1 ρ 1 ) − α 1 [ det ( A + B Q 1 β − 1 ) ] 1 2 exp ( − i k 2 ρ 1 T Q 2 β − 1 ρ 1 ) ,
Q 2 n − 1 = ( C + D Q 1 n − 1 ) ( A + B Q 1 n − 1 ) − 1 , Q 2 n p − 1 = ( C + D Q 1 n p − 1 ) ( A + B Q 1 n p − 1 ) − 1 ,
Q 2 β − 1 = ( C + D Q 1 β − 1 ) ( A + B Q 1 β − 1 ) − 1 .
Γ ( 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 2 σ g 0 2 − ( y 1 − y 2 ) 2 2 σ g 0 2 ] ,
Γ N ( r ˜ , 0 ) = ∑ n = 1 N ∑ m = 1 N ( − 1 ) n + m N 2 ( N n ) ( N m ) [ exp ( − i k 2 r ˜ T M 11 − 1 r ˜ ) − exp ( − i k 2 r ˜ T M 12 − 1 r ˜ ) − α exp ( − i k 2 r ˜ T M 13 − 1 r ˜ )
− exp ( − i k 2 r ˜ T M 14 − 1 r ˜ ) + exp ( − i k 2 r ˜ T M 15 − 1 r ˜ ) + α exp ( − i k 2 r ˜ T M 16 − 1 r ˜ ) − α exp ( − i k 2 r ˜ T M 17 − 1 r ˜ )
+ α exp ( − i k 2 r ˜ T M 18 − 1 r ˜ ) + α 2 exp ( − i k 2 r ˜ T M 19 − 1 r ˜ ) ] ,
M 11 − 1 = ( − 2 i k w 0 x x n 2 − i k σ g 0 2 − 2 i k w 0 x y n 2 i k σ g 0 2 0 − 2 i k w 0 x y n 2 − 2 i k w 0 y y n 2 − i k σ g 0 2 0 i k σ g 0 2 i k σ g 0 2 0 − 2 i k w 0 x x m 2 − i k σ g 0 2 − 2 i k w 0 x y m 2 0 i k σ g 0 2 − 2 i k w 0 x y m 2 − 2 i k w 0 y y m 2 − i k σ g 0 2 ) ,
M 12 − 1 = ( − 2 i k w 0 x x n 2 − i k σ g 0 2 − 2 i k w 0 x y n 2 i k σ g 0 2 0 − 2 i k w 0 x y n 2 − 2 i k w 0 yyn 2 − i k σ g 0 2 0 i k σ g 0 2 i k σ g 0 2 0 − 2 i k w 0 x x m p 2 − i k σ g 0 2 − 2 i k w 0 x y m p 2 0 i k σ g 0 2 − 2 i k w 0 x y m p 2 − 2 i k w 0 y y m p 2 − i k σ g 0 2 ) ,
M 13 − 1 = ( − 2 i k w 0 x x n 2 − i k σ g 0 2 − 2 i k w 0 x y n 2 i k σ g 0 2 0 − 2 i k w 0 x y n 2 − 2 i k w 0 yy n 2 − i k σ g 0 2 0 i k σ g 0 2 i k σ g 0 2 0 − 2 i k w 1 x x β 2 − i k σ g 0 2 − 2 i k w 1 x y β 2 0 i k σ g 0 2 − 2 i k w 1 x y β 2 − 2 i k w 1 y y β 2 − i k σ g 0 2 ) ,
M 14 − 1 = ( − 2 i k w 0 x x n p 2 − i k σ g 0 2 − 2 i k w 0 x y n p 2 i k σ g 0 2 0 − 2 i k w 0 x y n p 2 − 2 i k w 0 yynp 2 − i k σ g 0 2 0 i k σ g 0 2 i k σ g 0 2 0 − 2 i k w 0 x x m 2 − i k σ g 0 2 − 2 i k w 0 x y m 2 0 i k σ g 0 2 − 2 i k w 0 x y m 2 − 2 i k w 0 y y m 2 − i k σ g 0 2 ) ,
M 15 − 1 = ( − 2 i k w 0 x x n p 2 − i k σ g 0 2 − 2 i k w 0 x y n p 2 i k σ g 0 2 0 − 2 i k w 0 x y n p 2 − 2 i k w 0 yynp 2 − i k σ g 0 2 0 i k σ g 0 2 i k σ g 0 2 0 − 2 i k w 0 x x m p 2 − i k σ g 0 2 − 2 i k w 0 x y m p 2 0 i k σ g 0 2 − 2 i k w 0 x y m p 2 − 2 i k w 0 y y m p 2 − i k σ g 0 2 ) ,
M 16 − 1 = ( − 2 i k w 0 x x n p 2 − i k σ g 0 2 − 2 i k w 0 x y n p 2 i k σ g 0 2 0 − 2 i k w 0 x y n p 2 − 2 i k w 0 yynp 2 − i k σ g 0 2 0 i k σ g 0 2 i k σ g 0 2 0 − 2 i k w 1 x x β 2 − i k σ g 0 2 − 2 i k w 1 x y β 2 0 i k σ g 0 2 − 2 i k w 1 x y β 2 − 2 i k w 1 y y β 2 − i k σ g 0 2 ) ,
M 17 − 1 = ( − 2 i k w 1 x x β 2 − i k σ g 0 2 − 2 i k w 1 x y β 2 i k σ g 0 2 0 − 2 i k w 1 x y β 2 − 2 i k w 1 yy β 2 − i k σ g 0 2 0 i k σ g 0 2 i k σ g 0 2 0 − 2 i k w 0 x x m 2 − i k σ g 0 2 − 2 i k w 0 x y m 2 0 i k σ g 0 2 − 2 i k w 0 x y m 2 − 2 i k w 0 y y m 2 − i k σ g 0 2 ) ,
M 18 − 1 = ( − 2 i k w 1 x x β 2 − i k σ g 0 2 − 2 i k w 1 x y β 2 i k σ g 0 2 0 − 2 i k w 1 x y β 2 − 2 i k w 1 yy β 2 − i k σ g 0 2 0 i k σ g 0 2 i k σ g 0 2 0 − 2 i k w 0 x x m p 2 − i k σ g 0 2 − 2 i k w 0 x y m p 2 0 i k σ g 0 2 − 2 i k w 0 x y m p 2 − 2 i k w 0 y y m p 2 − i k σ g 0 2 ) ,
M 19 − 1 = ( − 2 i k w 1 x x β 2 − i k σ g 0 2 − 2 i k w 1 x y β 2 i k σ g 0 2 0 − 2 i k w 1 x y β 2 − 2 i k w 1 yy β 2 − i k σ g 0 2 0 i k σ g 0 2 i k σ g 0 2 0 − 2 i k w 1 x x β 2 − i k σ g 0 2 − 2 i k w 1 x y β 2 0 i k σ g 0 2 − 2 i k w 1 x y β 2 − 2 i k w 1 y y β 2 − i k σ g 0 2 ) ,
Γ ( ρ ˜ , l ) = k 2 4 π 2 [ det ( B ˜ ) ] 1 2 ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ Γ ( r ˜ , 0 ) exp [ − i k 2 ( r ˜ T B ˜ − 1 A ˜ r ˜ − 2 r ˜ T B ˜ − 1 ρ ˜ + ρ ˜ T D ˜ B ˜ − 1 ρ ˜ ) ] d r ˜ ,
A ˜ = ( A 0 I 0 I A * ) , B ˜ = ( B 0 I 0 I − B * ) , C ˜ = ( C 0 I 0 I − C * ) , D ˜ = ( D 0 I 0 I D * ) ,
( B ˜ − 1 A ˜ ) T = B ˜ − 1 A ˜ , ( − B ˜ − 1 ) T = ( C ˜ − D ˜ B ˜ − 1 A ˜ ) , ( D ˜ B ˜ − 1 ) T = D ˜ B ˜ − 1 .
Γ N ( ρ ˜ , l ) = ∑ n = 1 N ∑ m = 1 N ( − 1 ) n + m N 2 ( N n ) ( N m ) [ S 1 exp ( − i k 2 ρ ˜ T M 21 − 1 ρ ˜ ) − S 2 exp ( − i k 2 ρ ˜ T M 22 − 1 ρ ˜ ) − α S 3 exp ( − i k 2 ρ ˜ T M 23 − 1 ρ ˜ )
− S 4 exp ( − i k 2 ρ ˜ T M 24 − 1 ρ ˜ ) + S 5 exp ( − i k 2 ρ ˜ T M 25 − 1 ρ ˜ ) + α S 6 exp ( − i k 2 ρ ˜ T M 26 − 1 ρ ˜ ) − α S 7 exp ( − i k 2 ρ ˜ T M 27 − 1 ρ ˜ )
+ α S 8 exp ( − i k 2 ρ ˜ T M 28 − 1 ρ ˜ ) + α 2 S 9 exp ( − i k 2 ρ ˜ T M 29 − 1 ρ ˜ ) ] ,
S i = 1 [ det ( A ˜ + B ˜ M 1 i − 1 ) ] 1 2 , ( i = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 )
M 2 i − 1 = ( C ˜ + D ˜ M 1 i − 1 ) ( A ˜ + B ˜ M 1 i − 1 ) − 1 , ( i = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 )
A ˜ = ( I 0 I 0 I I ) , B = ( z I 0 I 0 I − z I ) , C = ( 0 I 0 I 0 I 0 I ) , D = ( I 0 I 0 I I ) .
A ˜ = ( 0 I 0 I 0 I 0 I ) , B = ( f I 0 I 0 I − f I ) , C = ( ( − 1 f ) I 0 I 0 I ( 1 f ) I ) , D = ( I 0 I 0 I I ) .

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