Abstract

The relationship between transmission area of an object imaged and the visibility of correlated imaging is investigated in a lensless system. We show that they are not in simple inverse proportion, as usually depicted. The changes of the visibility will be quite different when the transmission area is varied by different manners, which may motivate people to seek a new understanding about the influence factors of the visibility.

© 2007 Optical Society of America

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  1. P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, "Controlling the degree of visibility of Young’s fringers with photon coincidence measurements," Phys. Rev. A 49, 4176-4179 (1994).
    [CrossRef] [PubMed]
  2. A. V. Belinsky and D. N. Klyshko, "2-photon optics-diffraction, holography and transformation of 2-dimensional signals," Sov. Phys. JETP 78, 259-262 (1994).
  3. 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 53, R3429-R3432 (1995).
    [CrossRef]
  4. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ‘ghost’ interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
    [CrossRef] [PubMed]
  5. A. F. Abouraddy, B. E. Saleh, A. V. Sergienko, and M. C. Teich, "Entangled-photon Fourier optics," J. Opt. Soc. Am. B 19, 1174-1184 (2002).
    [CrossRef]
  6. A. Gatti, E. Brambilla, and L. A. Lugiatio, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).
    [CrossRef]
  7. G. A. Barbosa, "Quantum images in double-slit experiments with spontaneous down-conversion light," Phys. Rev. A 54, 4473-4478 (1996).
    [CrossRef] [PubMed]
  8. A. F. Abouraddy, B. E. Saleh, A. V. Sergienko, and M. C. Teich, "Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
    [CrossRef] [PubMed]
  9. R. S. Bennink, S. J. Bentley, R.W. Boyd, and J. C. Howell, "Quantum and classical coincidence imaging," Phys. Rev. Lett. 92, 033601 (2004).
    [CrossRef] [PubMed]
  10. Jing Cheng and Shensheng Han, "Incoherent coincidence imaging and its applicability in X-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004).
    [CrossRef] [PubMed]
  11. Da Zhang, Yanhua Zhai, Lingan Wu, and Xihao Chen, "Correlated two-photon imaging with true thermal light," Opt. Lett. 30, 2354-2356 (2005).
    [CrossRef] [PubMed]
  12. Yangjian Cai and Shiyao Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E 71, 056607 (2005).
    [CrossRef]
  13. Yangjian Cai, Qiang Lin, and Shiyao Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phys. Lett. 86, 021112 (2005).
    [CrossRef]
  14. 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]
  15. B. E. 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]
  16. J. W. Goodman, Statictical Optics (Wiley, New York, 1985).
  17. R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).
    [CrossRef]
  18. M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, Y. Bai, and S. Han, "Sub-wavelength Fourier-transform imaging of a pure-phase object with thermal light," Phys. Lett. A will be published; arXiv.org e-Print archive, arXiv. quant-ph/0612060, (2006).
  19. 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]

2006 (2)

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]

2005 (3)

Da Zhang, Yanhua Zhai, Lingan Wu, and Xihao Chen, "Correlated two-photon imaging with true thermal light," Opt. Lett. 30, 2354-2356 (2005).
[CrossRef] [PubMed]

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

Yangjian Cai, Qiang Lin, and Shiyao Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

2004 (3)

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

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

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

2002 (1)

2001 (1)

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

2000 (1)

B. E. 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]

1996 (1)

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

1995 (2)

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 53, R3429-R3432 (1995).
[CrossRef]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ‘ghost’ interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

1994 (2)

P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, "Controlling the degree of visibility of Young’s fringers with photon coincidence measurements," Phys. Rev. A 49, 4176-4179 (1994).
[CrossRef] [PubMed]

A. V. Belinsky and D. N. Klyshko, "2-photon optics-diffraction, holography and transformation of 2-dimensional signals," Sov. Phys. JETP 78, 259-262 (1994).

1956 (1)

R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy, B. E. 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. 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. 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]

Bache, M.

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]

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]

P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, "Controlling the degree of visibility of Young’s fringers with photon coincidence measurements," Phys. Rev. A 49, 4176-4179 (1994).
[CrossRef] [PubMed]

Belinsky, A. V.

A. V. Belinsky and D. N. Klyshko, "2-photon optics-diffraction, holography and transformation of 2-dimensional signals," Sov. Phys. JETP 78, 259-262 (1994).

Bennink, R. S.

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

Bentley, S. J.

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

Boyd, R.W.

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

Brambilla, E.

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]

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

Ferri, F.

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]

Gatti, A.

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]

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

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).
[CrossRef]

Howell, J. C.

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

Klyshko, D. N.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ‘ghost’ interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

A. V. Belinsky and D. N. Klyshko, "2-photon optics-diffraction, holography and transformation of 2-dimensional signals," Sov. Phys. JETP 78, 259-262 (1994).

Lugiatio, L. A.

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

Machado da Silva, J. C.

P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, "Controlling the degree of visibility of Young’s fringers with photon coincidence measurements," Phys. Rev. A 49, 4176-4179 (1994).
[CrossRef] [PubMed]

Magatti, D.

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]

Padua, S.

P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, "Controlling the degree of visibility of Young’s fringers with photon coincidence measurements," Phys. Rev. A 49, 4176-4179 (1994).
[CrossRef] [PubMed]

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 53, R3429-R3432 (1995).
[CrossRef]

Ribeiro, P. H. S.

P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, "Controlling the degree of visibility of Young’s fringers with photon coincidence measurements," Phys. Rev. A 49, 4176-4179 (1994).
[CrossRef] [PubMed]

Saleh, B. E.

A. F. Abouraddy, B. E. 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. 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. 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]

Sergienko, A. V.

A. F. Abouraddy, B. E. 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. 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. 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]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ‘ghost’ interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

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 53, R3429-R3432 (1995).
[CrossRef]

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 53, R3429-R3432 (1995).
[CrossRef]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ‘ghost’ interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

Strekalov, D. V.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ‘ghost’ interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

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 53, R3429-R3432 (1995).
[CrossRef]

Teich, M. C.

A. F. Abouraddy, B. E. 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. 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. 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]

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).
[CrossRef]

Appl. Phys. Lett. (1)

Yangjian Cai, Qiang Lin, and Shiyao Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

J. Mod. Opt. (1)

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]

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

Nature (London) (1)

R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (6)

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]

B. E. 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]

P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, "Controlling the degree of visibility of Young’s fringers with photon coincidence measurements," Phys. Rev. A 49, 4176-4179 (1994).
[CrossRef] [PubMed]

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 53, R3429-R3432 (1995).
[CrossRef]

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

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

Phys. Rev. E (1)

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

Phys. Rev. Lett. (4)

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

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

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

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ‘ghost’ interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

A. V. Belinsky and D. N. Klyshko, "2-photon optics-diffraction, holography and transformation of 2-dimensional signals," Sov. Phys. JETP 78, 259-262 (1994).

Other (2)

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, Y. Bai, and S. Han, "Sub-wavelength Fourier-transform imaging of a pure-phase object with thermal light," Phys. Lett. A will be published; arXiv.org e-Print archive, arXiv. quant-ph/0612060, (2006).

J. W. Goodman, Statictical Optics (Wiley, New York, 1985).

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Figures (4)

Fig. 1.
Fig. 1.

Correlated imaging with incoherent light.

Fig. 2.
Fig. 2.

(a) The normalized conditional intensity correlation G (2)(u 1 = 0,u 2) versus u 2 with incoherent light source for different number of slits. (b) Dependence of the corresponding visibility on the number of slits n.

Fig. 3.
Fig. 3.

(a) The normalized G (2)(u 1 = 0,u 2) as the function of u 2 for different ratios of the slit widthto the slit distance. (b) The corresponding visibility versus ω/d.

Fig. 4.
Fig. 4.

Normalized ΔG (2)(u 2) vs the reference detector position for an object with different transmission. (a) Intensity fluctuation correlations for different number of slits. (b) The curves for different slit width.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

E i ( u i ) = E ( x ) h i ( x , u i ) dx , i = 1,2 .
G ( 2 ) ( u 1 , u 2 ) = E ( u 1 ) E ( u 2 ) E * ( u 2 ) E * ( u 1 ) ,
G ( 2 ) ( u 1 , u 2 ) = E ( x 1 ) E ( x 2 ) E * ( x 2 ) E * ( x 2 )
× h 1 ( x 1 , u 1 ) h 2 ( x 2 , u 2 ) h 2 * ( x 2 , u 2 )
× h 1 * ( x 1 , u 1 ) dx 1 dx 2 dx 2 dx 1 ,
G ( 2 ) ( x 1 , x 2 , x 2 , x 1 ) = G ( 1 ) ( x 1 , x 1 ) G ( 1 ) ( x 2 , x 2 )
+ G ( 1 ) ( x 1 , x 2 ) G ( 1 ) ( x 2 , x 1 ) ,
G ( 1 ) ( x i , x j ) = E ( x i ) E * ( x j ) ,
G ( 2 ) ( u 1 , u 2 ) = E ( x 1 ) E * ( x 1 ) h 1 ( x 1 , u 1 ) h 1 * ( x 1 , u 1 ) dx 1 dx 1
× E ( x 2 ) E * ( x 2 ) h 2 ( x 2 , u 2 ) h 2 * ( x 2 , u 2 ) dx 2 dx 2
+ E ( x 1 ) E * ( x 2 ) h 1 ( x 1 , u 1 ) h 2 * ( x 2 , u 2 ) dx 1 dx 2
× E ( x 2 ) E * ( x 1 ) h 2 ( x 2 , u 2 ) h 1 * ( x 1 , u 1 ) dx 2 dx 1
= I 1 ( u 1 ) I 2 ( u 2 ) + E ( x 1 ) E * ( x 2 ) h 1 ( x 1 , u 1 ) h 2 * ( x 2 , u 2 ) dx 1 dx 2 2
= I 1 ( u 1 ) I 2 ( u 2 ) + Δ G ( 2 ) ( u 1 , u 2 ) .
Δ G ( 2 ) ( u 1 , u 2 ) = I 1 ( u 1 ) I 2 ( u 2 ) I 1 ( u 1 ) I 2 ( u 2 ) .
V = Δ G max ( 2 ) I 1 I 2 max = Δ G max ( 2 ) [ I 1 I 2 + Δ G ( 2 ) ] max .
E s ( x 1 ) E s * ( x 2 ) = G 0 exp ( x 1 2 + x 2 2 4 a 2 ) δ ( x 1 x 2 ) ,
h 1 ( x 1 , u 1 ) = e i k z 1 iλz 1 exp [ i π λ z 1 ( x x 1 ) 2 ] t ( x )
× e i k z 2 i λ z 2 exp [ i π λ z 2 ( u 1 x ) 2 ] dx .
h 2 ( x 2 , u 2 ) = e i k z i λ z exp [ i π λ z ( u 2 x 2 ) 2 ] .

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