Abstract

Based on the classical optical coherence theory, ghost imaging with twisted Gaussian Schell-model (GSM) beams is analyzed. It is found that the twist phase of the GSM beam has strong influence on ghost imaging. As the absolute value of the twist factor increases, the ghost image disappears gradually, but its visibility increases. This phenomenon is caused by the fact that the twist phase enhances the transverse spatial coherence of the twisted GSM beam on propagation.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
  31. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Entangled-photon Fourier optics," J. Opt. Soc. Am. B 19, 1174-1184 (2002).
    [CrossRef]
  32. A. Gatti, E. Brambilla, and L. A. Lugiato, "Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm," Phys. Rev. Lett. 90, 133603 (2003).
    [CrossRef] [PubMed]
  33. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
    [CrossRef] [PubMed]
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  35. R. S. Bennink, S. J. Bentley, and R. W. Boyd, "Quantum and classical coincidence imaging," Phys. Rev. Lett. 92, 033601 (2004).
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  37. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).
    [CrossRef]
  38. J. Cheng and S. Han, "Incoherent coincidence imaging and its applicability in x-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004).
    [CrossRef] [PubMed]
  39. D. Cao, J. Xiong and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005).
    [CrossRef]
  40. Y.  Cai and S.  Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E  71, 056607 (2005).
    [CrossRef]
  41. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-resolution ghost image and ghost diffraction experiments with thermal light," Phys. Rev. Lett. 94, 183602 (2005).
    [CrossRef] [PubMed]
  42. A. Valencia, G. Scarcelli, M. D'Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
    [CrossRef] [PubMed]
  43. A.  Gatti, M.  Bache, D.  Magatti, E.  Brambilla, F.  Ferri, and L. A.  Lugiato, "Coherent imaging with pseudo-thermal incoherent light," J. Mod. Opt.  53, 739-760 (2006).
    [CrossRef]
  44. D. Zhang, Y. Zhai, L. Wu, and X. Chen, "Correlated two-photon imaging with true thermal light," Opt. Lett.  30, 2354-2356 (2005).
    [CrossRef] [PubMed]
  45. G. Scarcelli, V. Berardi, and Y. Shih, "Phase-conjugate mirror via two-photon thermal light imaging," Appl. Phys. Lett. 88, 061106 (2006).
    [CrossRef]
  46. L. Basano and P. Ottonello, "Experiment in lensless ghost imaging with thermal light," Appl. Phys. Lett. 89, 091109 (2006).
    [CrossRef]
  47. M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, "Coherent imaging of a pure phase object with classical incoherent light," Phys. Rev. A 73, 053802 (2006)
    [CrossRef]
  48. R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
    [CrossRef] [PubMed]
  49. M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, "Lensless Fourier-transform ghost imaging with classical incoherent light," Phys. Rev. A 75, 021803 (2007).
    [CrossRef]
  50. Y. Bai, H. Liu, and S. Han, "Transmission area and correlated imaging," Opt. Express 15, 6062-6068 (2007).
    [CrossRef] [PubMed]
  51. J. Chen and S. Han, "Classical correlated imaging from the perspective of coherent-mode representation," Phys. Rev. A 76, 023824 (2007).
    [CrossRef]
  52. S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, "Modelling of classical ghost images obtained using scattering light," New J. Phys. 9, 285 (2007).
    [CrossRef]
  53. V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, "Ghost interference with classical partially coherent light pulses," Phys. Rev. A 77, 043811(2008).
    [CrossRef]
  54. D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, "Enhancing visibility and resolution in Nth-order intensity correlation of thermal light," Appl. Phys. Lett. 92, 201102 (2008).
    [CrossRef]
  55. H. Liu and S. Han, "Spatial longitudinal coherence length of a thermal source and its influence on lensless ghost imaging," Opt. Lett. 33, 824-826 (2008).
    [CrossRef] [PubMed]
  56. F. Ferri, D. Magatti, V. G. Sala, and A. Gatti, "Longitudinal coherence in thermal ghost imaging," Appl. Phys. Lett. 92, 261109 (2008).
    [CrossRef]
  57. A. Dogariu and E. Wolf, "Coherence theory of pairs of correlated wave fields," J. Mod. Opt. 50, 1791-1796 (2003).
    [CrossRef]
  58. M. Santarsiero and R. Borghi, "Correspondence between super-Gaussian and flattened Gaussian beams," J. Opt. Soc. Am. A 16, 188-190 (1999).
    [CrossRef]
  59. F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
    [CrossRef]
  60. Y. Li, "Light beam with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
    [CrossRef]
  61. A. A. Tovar, "Propagation of flat-topped multi-Gaussian laser beams," J. Opt. Soc. Am. A 18, 1897-1904 (2001).
    [CrossRef]
  62. 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]

2008

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, "Ghost interference with classical partially coherent light pulses," Phys. Rev. A 77, 043811(2008).
[CrossRef]

D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, "Enhancing visibility and resolution in Nth-order intensity correlation of thermal light," Appl. Phys. Lett. 92, 201102 (2008).
[CrossRef]

H. Liu and S. Han, "Spatial longitudinal coherence length of a thermal source and its influence on lensless ghost imaging," Opt. Lett. 33, 824-826 (2008).
[CrossRef] [PubMed]

F. Ferri, D. Magatti, V. G. Sala, and A. Gatti, "Longitudinal coherence in thermal ghost imaging," Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

2007

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, "Lensless Fourier-transform ghost imaging with classical incoherent light," Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Y. Bai, H. Liu, and S. Han, "Transmission area and correlated imaging," Opt. Express 15, 6062-6068 (2007).
[CrossRef] [PubMed]

J. Chen and S. Han, "Classical correlated imaging from the perspective of coherent-mode representation," Phys. Rev. A 76, 023824 (2007).
[CrossRef]

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, "Modelling of classical ghost images obtained using scattering light," New J. Phys. 9, 285 (2007).
[CrossRef]

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]

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

2006

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]

G. Scarcelli, V. Berardi, and Y. Shih, "Phase-conjugate mirror via two-photon thermal light imaging," Appl. Phys. Lett. 88, 061106 (2006).
[CrossRef]

L. Basano and P. Ottonello, "Experiment in lensless ghost imaging with thermal light," Appl. Phys. Lett. 89, 091109 (2006).
[CrossRef]

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, "Coherent imaging of a pure phase object with classical incoherent light," Phys. Rev. A 73, 053802 (2006)
[CrossRef]

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

A.  Gatti, M.  Bache, D.  Magatti, E.  Brambilla, F.  Ferri, and L. A.  Lugiato, "Coherent imaging with pseudo-thermal incoherent light," J. Mod. Opt.  53, 739-760 (2006).
[CrossRef]

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]

2005

D. Zhang, Y. Zhai, L. Wu, and X. Chen, "Correlated two-photon imaging with true thermal light," Opt. Lett.  30, 2354-2356 (2005).
[CrossRef] [PubMed]

D. Cao, J. Xiong and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005).
[CrossRef]

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

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-resolution ghost image and ghost diffraction experiments with thermal light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

A. Valencia, G. Scarcelli, M. D'Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

2004

R. S. Bennink, S. J. Bentley, and R. W. Boyd, "Quantum and classical coincidence imaging," Phys. Rev. Lett. 92, 033601 (2004).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).
[CrossRef]

J. Cheng and S. Han, "Incoherent coincidence imaging and its applicability in x-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef] [PubMed]

2003

A. Gatti, E. Brambilla, and L. A. Lugiato, "Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm," Phys. Rev. Lett. 90, 133603 (2003).
[CrossRef] [PubMed]

A. Dogariu and E. Wolf, "Coherence theory of pairs of correlated wave fields," J. Mod. Opt. 50, 1791-1796 (2003).
[CrossRef]

2002

2001

S. A. Ponomarenko, "Twisted Gaussian Schell-mode solitons," Phys. Rev. E 64, 036618 (2001).
[CrossRef]

J. Serna and J. M. Movilla, "Orbital angular momentum of partially coherent beams," Opt. Lett. 26, 405-406 (2001).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A.V. Sergienko, M. C. Teich, "Quantum holography," Opt Express 9, 498-505 (2001).
[CrossRef] [PubMed]

A. A. Tovar, "Propagation of flat-topped multi-Gaussian laser beams," J. Opt. Soc. Am. A 18, 1897-1904 (2001).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

2000

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, "Duality between partial coherence and partial entanglement," Phys. Rev. A 62, 043816 (2000).
[CrossRef]

M. J. Bastiaans. Wigner-distribution function applied to twisted Gaussian light propagating in first-order optical systems.J. Opt. Soc. Am. A 17, 2475-2480 (2000).
[CrossRef]

1999

1998

1996

R. Simon, A. T. Friberg, and E. Wolf, "Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems," Pure Appl. Opt. 5, 331-343 (1996).
[CrossRef]

G. A. Barbosa, "Quantum images in double-slit experiments with spontaneous down-conversion light," Phys. Rev. A 54, 4473-4478 (1996).
[CrossRef] [PubMed]

1995

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429 (1995).
[CrossRef] [PubMed]

1994

A. T. Friberg, E. Tervonen, and J. Turunen, "Interpretation and experimental demonstration of twisted Gaussian Schell-model beams," J. Opt. Soc. Am. A 11, 1818-1826 (1994).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1993

1992

E. Tervonen, A. T. Friberg, and J. Turunen. Gaussian Schell-model beams generated with synthetic acousto-optic holograms.J. Opt. Soc. Am. A. 9, 796-803 (1992).
[CrossRef]

1982

A. T. Friberg, and R. J. Sudol. Propagation parameters of Gaussian Schell-model beams.Opt. Commun 41, 383-387 (1982).
[CrossRef]

1979

P. de Santis, F. Gori, G. Guattari, and C. Palma. "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

1978

E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?," Opt. Lett. 2, 27-29 (1978)
[CrossRef] [PubMed]

E. Wolf and E. Collett, "Partially coherent sources which produce same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
[CrossRef]

F. Gori, "Collet-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1978).
[CrossRef]

1976

J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1976).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Entangled-photon Fourier optics," J. Opt. Soc. Am. B 19, 1174-1184 (2002).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A.V. Sergienko, M. C. Teich, "Quantum holography," Opt Express 9, 498-505 (2001).
[CrossRef] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, "Duality between partial coherence and partial entanglement," Phys. Rev. A 62, 043816 (2000).
[CrossRef]

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

Aruldoss, C.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, "Modelling of classical ghost images obtained using scattering light," New J. Phys. 9, 285 (2007).
[CrossRef]

Bache, M.

A.  Gatti, M.  Bache, D.  Magatti, E.  Brambilla, F.  Ferri, and L. A.  Lugiato, "Coherent imaging with pseudo-thermal incoherent light," J. Mod. Opt.  53, 739-760 (2006).
[CrossRef]

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, "Coherent imaging of a pure phase object with classical incoherent light," Phys. Rev. A 73, 053802 (2006)
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-resolution ghost image and ghost diffraction experiments with thermal light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

Bai, Y.

Barbosa, G. A.

G. A. Barbosa, "Quantum images in double-slit experiments with spontaneous down-conversion light," Phys. Rev. A 54, 4473-4478 (1996).
[CrossRef] [PubMed]

Basano, L.

L. Basano and P. Ottonello, "Experiment in lensless ghost imaging with thermal light," Appl. Phys. Lett. 89, 091109 (2006).
[CrossRef]

Bastiaans, M. J.

Bennink, R. S.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, "Quantum and classical coincidence imaging," Phys. Rev. Lett. 92, 033601 (2004).
[CrossRef] [PubMed]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, "Two-photon coincidence imaging with a classical source," Phys. Rev. Lett. 89, 113601 (2002).
[CrossRef] [PubMed]

Bentley, S. J.

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G. Scarcelli, V. Berardi, and Y. Shih, "Phase-conjugate mirror via two-photon thermal light imaging," Appl. Phys. Lett. 88, 061106 (2006).
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Borghi, R.

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
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M. Santarsiero and R. Borghi, "Correspondence between super-Gaussian and flattened Gaussian beams," J. Opt. Soc. Am. A 16, 188-190 (1999).
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Boyd, R. W.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, "Quantum and classical coincidence imaging," Phys. Rev. Lett. 92, 033601 (2004).
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M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, "Coherent imaging of a pure phase object with classical incoherent light," Phys. Rev. A 73, 053802 (2006)
[CrossRef]

A.  Gatti, M.  Bache, D.  Magatti, E.  Brambilla, F.  Ferri, and L. A.  Lugiato, "Coherent imaging with pseudo-thermal incoherent light," J. Mod. Opt.  53, 739-760 (2006).
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-resolution ghost image and ghost diffraction experiments with thermal light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
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A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).
[CrossRef]

A. Gatti, E. Brambilla, and L. A. Lugiato, "Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm," Phys. Rev. Lett. 90, 133603 (2003).
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A. Gatti, E. Brambilla, L. A. Lugiato and M. I. Kolobov, "Quantum entangled images," Phys. Rev. Lett. 83, 1763-1766 (1999).
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Cai, Y.

Cao, D.

D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, "Enhancing visibility and resolution in Nth-order intensity correlation of thermal light," Appl. Phys. Lett. 92, 201102 (2008).
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D. Cao, J. Xiong and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005).
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Castelletto, S.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, "Modelling of classical ghost images obtained using scattering light," New J. Phys. 9, 285 (2007).
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Chen, J.

J. Chen and S. Han, "Classical correlated imaging from the perspective of coherent-mode representation," Phys. Rev. A 76, 023824 (2007).
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Chen, X.

Cheng, J.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, "Lensless Fourier-transform ghost imaging with classical incoherent light," Phys. Rev. A 75, 021803 (2007).
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J. Cheng and S. Han, "Incoherent coincidence imaging and its applicability in x-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004).
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S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, "Modelling of classical ghost images obtained using scattering light," New J. Phys. 9, 285 (2007).
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D'Angelo, M.

A. Valencia, G. Scarcelli, M. D'Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
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P. de Santis, F. Gori, G. Guattari, and C. Palma. "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
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Dogariu, A.

A. Dogariu and E. Wolf, "Coherence theory of pairs of correlated wave fields," J. Mod. Opt. 50, 1791-1796 (2003).
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Ferri, F.

F. Ferri, D. Magatti, V. G. Sala, and A. Gatti, "Longitudinal coherence in thermal ghost imaging," Appl. Phys. Lett. 92, 261109 (2008).
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A.  Gatti, M.  Bache, D.  Magatti, E.  Brambilla, F.  Ferri, and L. A.  Lugiato, "Coherent imaging with pseudo-thermal incoherent light," J. Mod. Opt.  53, 739-760 (2006).
[CrossRef]

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, "Coherent imaging of a pure phase object with classical incoherent light," Phys. Rev. A 73, 053802 (2006)
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-resolution ghost image and ghost diffraction experiments with thermal light," Phys. Rev. Lett. 94, 183602 (2005).
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J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1976).
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V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, "Ghost interference with classical partially coherent light pulses," Phys. Rev. A 77, 043811(2008).
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R. Simon, A. T. Friberg, and E. Wolf, "Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems," Pure Appl. Opt. 5, 331-343 (1996).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, "Interpretation and experimental demonstration of twisted Gaussian Schell-model beams," J. Opt. Soc. Am. A 11, 1818-1826 (1994).
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E. Tervonen, A. T. Friberg, and J. Turunen. Gaussian Schell-model beams generated with synthetic acousto-optic holograms.J. Opt. Soc. Am. A. 9, 796-803 (1992).
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A. T. Friberg, and R. J. Sudol. Propagation parameters of Gaussian Schell-model beams.Opt. Commun 41, 383-387 (1982).
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Gao, L.

D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, "Enhancing visibility and resolution in Nth-order intensity correlation of thermal light," Appl. Phys. Lett. 92, 201102 (2008).
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Gatti, A.

F. Ferri, D. Magatti, V. G. Sala, and A. Gatti, "Longitudinal coherence in thermal ghost imaging," Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, "Coherent imaging of a pure phase object with classical incoherent light," Phys. Rev. A 73, 053802 (2006)
[CrossRef]

A.  Gatti, M.  Bache, D.  Magatti, E.  Brambilla, F.  Ferri, and L. A.  Lugiato, "Coherent imaging with pseudo-thermal incoherent light," J. Mod. Opt.  53, 739-760 (2006).
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-resolution ghost image and ghost diffraction experiments with thermal light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, "Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm," Phys. Rev. Lett. 90, 133603 (2003).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, L. A. Lugiato and M. I. Kolobov, "Quantum entangled images," Phys. Rev. Lett. 83, 1763-1766 (1999).
[CrossRef]

Gori, F.

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 1391-1399 (1994).
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F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
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P. de Santis, F. Gori, G. Guattari, and C. Palma. "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
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F. Gori, "Collet-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1978).
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Guattari, G.

P. de Santis, F. Gori, G. Guattari, and C. Palma. "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
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Han, S.

H. Liu and S. Han, "Spatial longitudinal coherence length of a thermal source and its influence on lensless ghost imaging," Opt. Lett. 33, 824-826 (2008).
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J. Chen and S. Han, "Classical correlated imaging from the perspective of coherent-mode representation," Phys. Rev. A 76, 023824 (2007).
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Y. Bai, H. Liu, and S. Han, "Transmission area and correlated imaging," Opt. Express 15, 6062-6068 (2007).
[CrossRef] [PubMed]

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, "Lensless Fourier-transform ghost imaging with classical incoherent light," Phys. Rev. A 75, 021803 (2007).
[CrossRef]

J. Cheng and S. Han, "Incoherent coincidence imaging and its applicability in x-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004).
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He, S.

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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Hu, L.

Kolobov, M. I.

A. Gatti, E. Brambilla, L. A. Lugiato and M. I. Kolobov, "Quantum entangled images," Phys. Rev. Lett. 83, 1763-1766 (1999).
[CrossRef]

Lajunen, H.

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, "Ghost interference with classical partially coherent light pulses," Phys. Rev. A 77, 043811(2008).
[CrossRef]

Lancis, J.

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, "Ghost interference with classical partially coherent light pulses," Phys. Rev. A 77, 043811(2008).
[CrossRef]

Li, Y.

Lin, L.

D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, "Enhancing visibility and resolution in Nth-order intensity correlation of thermal light," Appl. Phys. Lett. 92, 201102 (2008).
[CrossRef]

Lin, Q.

Liu, H.

Liu, Y.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, "Lensless Fourier-transform ghost imaging with classical incoherent light," Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Lugiato, L. A.

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, "Coherent imaging of a pure phase object with classical incoherent light," Phys. Rev. A 73, 053802 (2006)
[CrossRef]

A.  Gatti, M.  Bache, D.  Magatti, E.  Brambilla, F.  Ferri, and L. A.  Lugiato, "Coherent imaging with pseudo-thermal incoherent light," J. Mod. Opt.  53, 739-760 (2006).
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-resolution ghost image and ghost diffraction experiments with thermal light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, "Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm," Phys. Rev. Lett. 90, 133603 (2003).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, L. A. Lugiato and M. I. Kolobov, "Quantum entangled images," Phys. Rev. Lett. 83, 1763-1766 (1999).
[CrossRef]

Magatti, D.

F. Ferri, D. Magatti, V. G. Sala, and A. Gatti, "Longitudinal coherence in thermal ghost imaging," Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

A.  Gatti, M.  Bache, D.  Magatti, E.  Brambilla, F.  Ferri, and L. A.  Lugiato, "Coherent imaging with pseudo-thermal incoherent light," J. Mod. Opt.  53, 739-760 (2006).
[CrossRef]

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, "Coherent imaging of a pure phase object with classical incoherent light," Phys. Rev. A 73, 053802 (2006)
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-resolution ghost image and ghost diffraction experiments with thermal light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

Movilla, J. M.

Mukunda, N.

Ottonello, P.

L. Basano and P. Ottonello, "Experiment in lensless ghost imaging with thermal light," Appl. Phys. Lett. 89, 091109 (2006).
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Palma, C.

P. de Santis, F. Gori, G. Guattari, and C. Palma. "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Peschel, U.

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429 (1995).
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Ponomarenko, S. A.

S. A. Ponomarenko, "Twisted Gaussian Schell-mode solitons," Phys. Rev. E 64, 036618 (2001).
[CrossRef]

Roberts, A.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, "Modelling of classical ghost images obtained using scattering light," New J. Phys. 9, 285 (2007).
[CrossRef]

Sala, V. G.

F. Ferri, D. Magatti, V. G. Sala, and A. Gatti, "Longitudinal coherence in thermal ghost imaging," Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

Saleh, B. E. A.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Entangled-photon Fourier optics," J. Opt. Soc. Am. B 19, 1174-1184 (2002).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A.V. Sergienko, M. C. Teich, "Quantum holography," Opt Express 9, 498-505 (2001).
[CrossRef] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, "Duality between partial coherence and partial entanglement," Phys. Rev. A 62, 043816 (2000).
[CrossRef]

Santarsiero, M.

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

M. Santarsiero and R. Borghi, "Correspondence between super-Gaussian and flattened Gaussian beams," J. Opt. Soc. Am. A 16, 188-190 (1999).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

Scarcelli, G.

G. Scarcelli, V. Berardi, and Y. Shih, "Phase-conjugate mirror via two-photon thermal light imaging," Appl. Phys. Lett. 88, 061106 (2006).
[CrossRef]

A. Valencia, G. Scarcelli, M. D'Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

Scholten, R. E.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, "Modelling of classical ghost images obtained using scattering light," New J. Phys. 9, 285 (2007).
[CrossRef]

Sergienko, A. V.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Entangled-photon Fourier optics," J. Opt. Soc. Am. B 19, 1174-1184 (2002).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, "Duality between partial coherence and partial entanglement," Phys. Rev. A 62, 043816 (2000).
[CrossRef]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429 (1995).
[CrossRef] [PubMed]

Sergienko, A.V.

A. F. Abouraddy, B. E. A. Saleh, A.V. Sergienko, M. C. Teich, "Quantum holography," Opt Express 9, 498-505 (2001).
[CrossRef] [PubMed]

Serna, J.

Shen, X.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, "Lensless Fourier-transform ghost imaging with classical incoherent light," Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Shih, Y.

G. Scarcelli, V. Berardi, and Y. Shih, "Phase-conjugate mirror via two-photon thermal light imaging," Appl. Phys. Lett. 88, 061106 (2006).
[CrossRef]

A. Valencia, G. Scarcelli, M. D'Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

Shih, Y. H.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429 (1995).
[CrossRef] [PubMed]

Simon, R.

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429 (1995).
[CrossRef] [PubMed]

Sudol, R. J.

A. T. Friberg, and R. J. Sudol. Propagation parameters of Gaussian Schell-model beams.Opt. Commun 41, 383-387 (1982).
[CrossRef]

Sundar, K.

Teich, M. C.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Entangled-photon Fourier optics," J. Opt. Soc. Am. B 19, 1174-1184 (2002).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A.V. Sergienko, M. C. Teich, "Quantum holography," Opt Express 9, 498-505 (2001).
[CrossRef] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, "Duality between partial coherence and partial entanglement," Phys. Rev. A 62, 043816 (2000).
[CrossRef]

Tervonen, E.

A. T. Friberg, E. Tervonen, and J. Turunen, "Interpretation and experimental demonstration of twisted Gaussian Schell-model beams," J. Opt. Soc. Am. A 11, 1818-1826 (1994).
[CrossRef]

E. Tervonen, A. T. Friberg, and J. Turunen. Gaussian Schell-model beams generated with synthetic acousto-optic holograms.J. Opt. Soc. Am. A. 9, 796-803 (1992).
[CrossRef]

Torres-Company, V.

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, "Ghost interference with classical partially coherent light pulses," Phys. Rev. A 77, 043811(2008).
[CrossRef]

Tovar, A. A.

Turunen, J.

A. T. Friberg, E. Tervonen, and J. Turunen, "Interpretation and experimental demonstration of twisted Gaussian Schell-model beams," J. Opt. Soc. Am. A 11, 1818-1826 (1994).
[CrossRef]

E. Tervonen, A. T. Friberg, and J. Turunen. Gaussian Schell-model beams generated with synthetic acousto-optic holograms.J. Opt. Soc. Am. A. 9, 796-803 (1992).
[CrossRef]

Valencia, A.

A. Valencia, G. Scarcelli, M. D'Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

Wang, F.

Wang, K.

D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, "Enhancing visibility and resolution in Nth-order intensity correlation of thermal light," Appl. Phys. Lett. 92, 201102 (2008).
[CrossRef]

D. Cao, J. Xiong and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005).
[CrossRef]

Wei, Q.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, "Lensless Fourier-transform ghost imaging with classical incoherent light," Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Wolf, E.

A. Dogariu and E. Wolf, "Coherence theory of pairs of correlated wave fields," J. Mod. Opt. 50, 1791-1796 (2003).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Scheme for lensless ghost imaging with a twisted GSM beam

Fig. 2.
Fig. 2.

3D-distribution and corresponding contour graph of the square of the transmission function (∣H(v 1)∣2) of a soft circular aperture versus v 1x and v 1y with a 1 = 0.1mm and N=30

Fig. 3.
Fig. 3.

Normalized ghost image ∣W(u 1=0,u 2)∣2 / ∣W(u 1=0,u 2=0)∣2 of a soft circular aperture formed with twisted GSM beam for different values of the initial transverse coherence width σ g 0

Fig. 4.
Fig. 4.

Normalized ghost image ∣W(u 1=0,u 2)∣2 / ∣W(u 1=0,u 2=0)∣2 of a soft circular aperture formed with twisted GSM beam for different values of the initial twist factor μ 0

Fig. 5.
Fig. 5.

Normalized ghost image ∣W(u 1=0,u 2)∣2 / ∣W(u 1=0,u 2=0)∣2 of a soft circular aperture formed with twisted GSM beam for different values of Z

Fig. 6.
Fig. 6.

Dependence of the transverse coherence length σgz of the twisted GSM beam on the propagation distance z for different values of the initial twist factor μ 0

Equations (32)

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W ( r 1 , r 2 ) = U ( r 1 ) U * ( r 2 ) = exp [ r 1 2 + r 2 2 4 σ I 0 2 ( r 1 r 2 ) 2 2 σ g 0 2 i k μ 0 2 ( r 1 r 2 ) T J ( r 1 + r 2 ) ] ,
J = ( 0 1 1 0 ) .
W ( r ̃ ) = exp ( i k 2 r ̃ T M 0 1 r ̃ ) ,
M 0 1 = ( ( i 2 k σ I 0 2 i k σ g 0 2 ) I i k σ g 0 2 I + μ 0 J i k σ g 0 2 I + μ 0 J T ( i 2 k σ I 0 2 i k σ g 0 2 ) I ) ,
W ( u ̃ ) = 1 [ Det ( A ̃ + B ̃ M 0 1 ) ] 1 / 2 exp ( i k 2 u ̃ T M 1 u ̃ ) ,
M 1 = ( C ̃ + D ̃ M 0 1 ) ( A ̃ + B ̃ M 0 1 ) 1 ,
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 * ) ,
G ( 2 ) = ( u 1 = 0 , u 2 ) = < I ( u 1 = 0 ) I ( u 2 ) > = < I ( u 1 = 0 ) > < I ( u 2 ) > + W ( u 1 = 0 , u 2 ) 2 ,
W ( u 1 , u 2 ) = U ( r 1 ) U * ( r 2 ) h 1 ( u 1 , r 2 ) h 2 * ( u 2 , r 2 ) d r 1 d r 2 ,
< I ( u i ) > = h i ( u i , r 1 ) h i * ( u i , r 2 ) U ( r 1 ) U * ( r 2 ) d r 1 d r 2 , i = 1,2
h 1 ( u 1 , r 1 ) = 1 λ 2 z z 1 exp [ i k 2 z ( r 1 v 1 ) 2 i k 2 z 1 ( v 1 u 1 ) 2 ] H ( v 1 ) d v 1 ,
h 2 ( u 2 , r 2 ) = i λ z exp [ i k 2 z ( r 2 u 2 ) 2 ] ,
W ( u 1 = 0 , u 2 ) = i λ 3 z 1 [ Det ( B ̃ ) ] 1 / 2 exp [ i k 2 r ̃ T M 0 1 r ̃ i k 2 ( r ̃ T B ̃ 1 r ̃ T 2 r ̃ T B ̃ 1 v ̃ u + v ̃ u T B ̃ 1 v ̃ u ) ]
× exp [ i k 2 z 1 v 1 2 ] H ( v 1 ) d r ̃ d v 1 ,
B ̃ = ( z I 0 I 0 I z I ) .
W ( u 1 = 0 , u 2 ) = i λ z 1 [ Det ( I ̃ + B ̃ M 0 1 ) ] 1 / 2 exp [ i k 2 v ̃ u T M 1 v ̃ u i k 2 z 1 v 1 2 ] H ( v 1 ) d v 1 ,
M 1 = ( M 0 + B ̃ ) 1 = ( M 11 1 M 12 1 ( M 12 1 ) T M 22 1 ) .
exp ( i k 2 v ̃ u T M 1 v ̃ u ) = exp ( i k 2 v 1 T M 11 1 v 1 i k v 1 T M 12 1 u 2 i k 2 u 2 T M 22 1 u 2 ) .
W ( u 1 = 0 , u 2 ) = i λ [ Det ( B 1 ) ] 1 / 2 [ Det ( I ̃ + B ̃ M 0 1 ) ] 1 / 2 exp ( i k 2 u 2 T M 22 1 u 2 )
× exp [ i k 2 v 1 T M 11 1 v 1 i k 2 v 1 T B 1 1 v 1 i k v 1 T M 12 1 u 2 ] H ( v 1 ) d v 1 ,
H ( v 1 ) = 1 C 0 n 1 N ( 1 ) n 1 N N n exp ( n v 1 2 a 1 2 ) .
H ( v 1 ) = 1 C 0 n = 1 N ( 1 ) n 1 N N n exp ( i k 2 v 1 T H n v 1 ) ,
H n = 2 n i k a 1 2 I .
W ( u 1 = 0 , u 2 ) = 1 C 0 n = 1 N ( 1 ) n 1 N N n exp 1 [ Det ( B 1 M 11 1 + B 1 H n + I ) Det ( I ̃ + B ̃ M 1 1 ) ] 1 / 2
× exp ( i k 2 u 2 T M 22 1 u 2 + i k 2 u 2 T ( M 22 1 ) T ( M 11 1 + B 1 1 + H n ) 1 M 12 1 u 2 ) .
< I ( u 1 = 0 ) > = 1 C 0 2 n = 1 N m = 1 N ( 1 ) n + m N 2 N n N m 1 [ Det ( B ̃ 1 M 1 + B ̃ 1 H n m + I ̃ ) Det ( B ̃ M 0 1 + I ̃ ) ] 1 / 2 ,
< I ( u 2 ) > = 1 [ Det ( I ̃ + B ̃ M 0 1 ) ] 1 / 2 exp [ i k 2 u ̃ 2 T M 1 u ̃ 2 ] ,
B ̃ 1 = ( z 1 I 0 I 0 I z 1 I ) , H n m = 2 i k a 1 2 ( n I 0 I 0 I m I ) .
W ( u ̃ ) = 1 [ Det ( I ̃ + B ̃ M 0 1 ) ] 1 / 2 exp ( i k 2 u ̃ T M 1 u ̃ ) ,
M 1 = ( M 0 + B ̃ ) 1 = ( m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ) ,
σ g z = ( i k m 13 ) 1 / 2 = ( i k m 24 ) 1 / 2 = ( λ 2 z 2 4 π 2 σ I 0 2 + σ g 0 2 + σ g 0 2 μ 0 2 z 2 + λ 2 z 2 σ g 0 2 16 π 2 σ I 0 4 ) 1 / 2 .
V = W ( u 1 = 0 , u 2 ) max 2 G ( 2 ) ( u 1 = 0 , u 2 ) max .

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