Abstract

We have studied a correlated imaging scheme using parametric downconversion stimulated with a partially coherent beam as a source. We analyze the dependence of quantum spatial correlations and image reconstruction features on the crystal length, the coherence length, and the transverse size of the seed beam in low- and high-gain regimes of parametric downconversion. We show that the visibility of ghost interference can be adjusted by varying the intensity and/or the coherence length of the seed, with improved visibility in the high-gain regime, and a visibility greater than 0.5 in the low-gain regime.

© 2009 Optical Society of America

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  1. R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27-29 (1956).
    [CrossRef]
  2. A. V. Belinsky and D. N. Klyshko, “Two-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 52, R3429-R3432 (1995).
    [CrossRef] [PubMed]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. M. D'Angelo, Y-H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
    [CrossRef] [PubMed]
  9. J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
    [CrossRef] [PubMed]
  10. 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]
  11. D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “Experimental evidence of high resolution ghost imaging and ghost diffraction with classical thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
    [CrossRef] [PubMed]
  12. G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?,” Phys. Rev. Lett. 96, 063602 (2006).
    [CrossRef] [PubMed]
  13. Y. J. Cai and S. Y. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716-2718 (2004).
    [CrossRef] [PubMed]
  14. J. Xiong, De-Zhong Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005).
    [CrossRef] [PubMed]
  15. D. Zhang, Y-H. Zhai, L-A. Wu, and X-H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. 30, 2354-2356 (2005).
    [CrossRef] [PubMed]
  16. M. D'Angelo, A. Valencia, M. H. Rubin, and Y.. Shih, “Resolution of quantum and classical ghost imaging,” Phys. Rev. A 72, 013810 (2005).
    [CrossRef]
  17. 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]
  18. M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
    [CrossRef]
  19. M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, “Coherent imaging of pure phase object with classical incoherent light,” Phys. Rev. A 73, 053802 (2006).
    [CrossRef]
  20. S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modelling of classical ghost images obtained using scattered light,” New J. Phys. 9, 285-295 (2007).
    [CrossRef]
  21. Y. Bai, H. Liu, and S. Han, “Transmission area and correlated imaging,” Opt. Express 10, 6062-6068 (2007).
    [CrossRef]
  22. A. Kolkiran and G. S. Agarwal, “Quantum interference using coherent beam stimulated parametric downconversion,” Opt. Express 10, 6479-6485 (2008).
    [CrossRef]
  23. E. Puddu, A. Andreoni, I. P. Degiovanni, M. Bondani, and S. Castelletto, “Ghost imaging with intense fields from chaotically seeded parametric downconversion,” Opt. Lett. 32, 1132-1134 (2007).
    [CrossRef] [PubMed]
  24. I. P. Degiovanni, M. Bondani, A. Andreoni, E. Puddu, and M. Paris, “Intensity correlations, entanglement properties, and ghost imaging in multimode thermal-seeded parametric down-conversion: Theory,” Phys. Rev. A 76, 062309 (2007).
    [CrossRef]
  25. S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
    [CrossRef]
  26. G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
    [CrossRef]
  27. E. Brambilla, M. Bache, A. Gatti, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A 69, 023802 (2004).
    [CrossRef]

2008 (1)

A. Kolkiran and G. S. Agarwal, “Quantum interference using coherent beam stimulated parametric downconversion,” Opt. Express 10, 6479-6485 (2008).
[CrossRef]

2007 (4)

E. Puddu, A. Andreoni, I. P. Degiovanni, M. Bondani, and S. Castelletto, “Ghost imaging with intense fields from chaotically seeded parametric downconversion,” Opt. Lett. 32, 1132-1134 (2007).
[CrossRef] [PubMed]

I. P. Degiovanni, M. Bondani, A. Andreoni, E. Puddu, and M. Paris, “Intensity correlations, entanglement properties, and ghost imaging in multimode thermal-seeded parametric down-conversion: Theory,” Phys. Rev. A 76, 062309 (2007).
[CrossRef]

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modelling of classical ghost images obtained using scattered light,” New J. Phys. 9, 285-295 (2007).
[CrossRef]

Y. Bai, H. Liu, and S. Han, “Transmission area and correlated imaging,” Opt. Express 10, 6062-6068 (2007).
[CrossRef]

2006 (2)

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

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?,” Phys. Rev. Lett. 96, 063602 (2006).
[CrossRef] [PubMed]

2005 (5)

J. Xiong, De-Zhong Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

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

M. D'Angelo, A. Valencia, M. H. Rubin, and Y.. Shih, “Resolution of quantum and classical ghost imaging,” Phys. Rev. A 72, 013810 (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]

D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “Experimental evidence of high resolution ghost imaging and ghost diffraction with classical thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

2004 (8)

S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
[CrossRef]

E. Brambilla, M. Bache, A. Gatti, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A 69, 023802 (2004).
[CrossRef]

M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[CrossRef]

Y. J. Cai and S. Y. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716-2718 (2004).
[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]

M. D'Angelo, Y-H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[CrossRef] [PubMed]

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (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]

2002 (1)

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]

2001 (2)

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]

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

2000 (1)

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]

1995 (1)

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-R3432 (1995).
[CrossRef] [PubMed]

1994 (1)

A. V. Belinsky and D. N. Klyshko, “Two-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 177, 27-29 (1956).
[CrossRef]

Abouraddy, A. F.

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]

Agarwal, G. S.

A. Kolkiran and G. S. Agarwal, “Quantum interference using coherent beam stimulated parametric downconversion,” Opt. Express 10, 6479-6485 (2008).
[CrossRef]

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Andreoni, A.

I. P. Degiovanni, M. Bondani, A. Andreoni, E. Puddu, and M. Paris, “Intensity correlations, entanglement properties, and ghost imaging in multimode thermal-seeded parametric down-conversion: Theory,” Phys. Rev. A 76, 062309 (2007).
[CrossRef]

E. Puddu, A. Andreoni, I. P. Degiovanni, M. Bondani, and S. Castelletto, “Ghost imaging with intense fields from chaotically seeded parametric downconversion,” Opt. Lett. 32, 1132-1134 (2007).
[CrossRef] [PubMed]

Aruldoss, C.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modelling of classical ghost images obtained using scattered light,” New J. Phys. 9, 285-295 (2007).
[CrossRef]

Bache, M.

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, “Coherent imaging of 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]

D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “Experimental evidence of high resolution ghost imaging and ghost diffraction with classical thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (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]

E. Brambilla, M. Bache, A. Gatti, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A 69, 023802 (2004).
[CrossRef]

Bai, Y.

Y. Bai, H. Liu, and S. Han, “Transmission area and correlated imaging,” Opt. Express 10, 6062-6068 (2007).
[CrossRef]

Belinsky, A. V.

A. V. Belinsky and D. N. Klyshko, “Two-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]

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.

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]

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]

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Berardi, V.

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?,” Phys. Rev. Lett. 96, 063602 (2006).
[CrossRef] [PubMed]

Bondani, M.

I. P. Degiovanni, M. Bondani, A. Andreoni, E. Puddu, and M. Paris, “Intensity correlations, entanglement properties, and ghost imaging in multimode thermal-seeded parametric down-conversion: Theory,” Phys. Rev. A 76, 062309 (2007).
[CrossRef]

E. Puddu, A. Andreoni, I. P. Degiovanni, M. Bondani, and S. Castelletto, “Ghost imaging with intense fields from chaotically seeded parametric downconversion,” Opt. Lett. 32, 1132-1134 (2007).
[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]

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]

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Brambilla, E.

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

D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “Experimental evidence of high resolution ghost imaging and ghost diffraction with classical thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

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]

M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (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]

E. Brambilla, M. Bache, A. Gatti, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A 69, 023802 (2004).
[CrossRef]

Cai, Y. J.

Cao, De-Zhong

J. Xiong, De-Zhong Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Castelletto, S.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modelling of classical ghost images obtained using scattered light,” New J. Phys. 9, 285-295 (2007).
[CrossRef]

E. Puddu, A. Andreoni, I. P. Degiovanni, M. Bondani, and S. Castelletto, “Ghost imaging with intense fields from chaotically seeded parametric downconversion,” Opt. Lett. 32, 1132-1134 (2007).
[CrossRef] [PubMed]

Chen, X-H.

Cheng, J.

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

Crosby, S.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modelling of classical ghost images obtained using scattered light,” New J. Phys. 9, 285-295 (2007).
[CrossRef]

D'Angelo, M.

M. D'Angelo, A. Valencia, M. H. Rubin, and Y.. Shih, “Resolution of quantum and classical ghost imaging,” Phys. Rev. A 72, 013810 (2005).
[CrossRef]

M. D'Angelo, Y-H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[CrossRef] [PubMed]

Degiovanni, I. P.

I. P. Degiovanni, M. Bondani, A. Andreoni, E. Puddu, and M. Paris, “Intensity correlations, entanglement properties, and ghost imaging in multimode thermal-seeded parametric down-conversion: Theory,” Phys. Rev. A 76, 062309 (2007).
[CrossRef]

E. Puddu, A. Andreoni, I. P. Degiovanni, M. Bondani, and S. Castelletto, “Ghost imaging with intense fields from chaotically seeded parametric downconversion,” Opt. Lett. 32, 1132-1134 (2007).
[CrossRef] [PubMed]

Ferri, F.

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, “Coherent imaging of 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]

D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “Experimental evidence of high resolution ghost imaging and ghost diffraction with classical thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

Gatti, A.

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, “Coherent imaging of 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]

D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “Experimental evidence of high resolution ghost imaging and ghost diffraction with classical 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]

M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[CrossRef]

E. Brambilla, M. Bache, A. Gatti, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A 69, 023802 (2004).
[CrossRef]

Han, S.

Y. Bai, H. Liu, and S. Han, “Transmission area and correlated imaging,” Opt. Express 10, 6062-6068 (2007).
[CrossRef]

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

Hanbury-Brown, R.

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 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]

Huang, F.

J. Xiong, De-Zhong Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Kim, Y-H.

M. D'Angelo, Y-H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[CrossRef] [PubMed]

Klyshko, D. N.

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

Kolkiran, A.

A. Kolkiran and G. S. Agarwal, “Quantum interference using coherent beam stimulated parametric downconversion,” Opt. Express 10, 6479-6485 (2008).
[CrossRef]

Kulik, S. P.

M. D'Angelo, Y-H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[CrossRef] [PubMed]

Li, H.-G.

J. Xiong, De-Zhong Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Liu, H.

Y. Bai, H. Liu, and S. Han, “Transmission area and correlated imaging,” Opt. Express 10, 6062-6068 (2007).
[CrossRef]

Lugiato, L. A.

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

D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “Experimental evidence of high resolution ghost imaging and ghost diffraction with classical thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

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]

M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (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]

E. Brambilla, M. Bache, A. Gatti, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A 69, 023802 (2004).
[CrossRef]

Magatti, D.

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, “Coherent imaging of 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]

D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “Experimental evidence of high resolution ghost imaging and ghost diffraction with classical thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

Nagasako, E. M.

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Paris, M.

I. P. Degiovanni, M. Bondani, A. Andreoni, E. Puddu, and M. Paris, “Intensity correlations, entanglement properties, and ghost imaging in multimode thermal-seeded parametric down-conversion: Theory,” Phys. Rev. A 76, 062309 (2007).
[CrossRef]

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-R3432 (1995).
[CrossRef] [PubMed]

Puddu, E.

I. P. Degiovanni, M. Bondani, A. Andreoni, E. Puddu, and M. Paris, “Intensity correlations, entanglement properties, and ghost imaging in multimode thermal-seeded parametric down-conversion: Theory,” Phys. Rev. A 76, 062309 (2007).
[CrossRef]

E. Puddu, A. Andreoni, I. P. Degiovanni, M. Bondani, and S. Castelletto, “Ghost imaging with intense fields from chaotically seeded parametric downconversion,” Opt. Lett. 32, 1132-1134 (2007).
[CrossRef] [PubMed]

Roberts, A.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modelling of classical ghost images obtained using scattered light,” New J. Phys. 9, 285-295 (2007).
[CrossRef]

Rubin, M. H.

M. D'Angelo, A. Valencia, M. H. Rubin, and Y.. Shih, “Resolution of quantum and classical ghost imaging,” Phys. Rev. A 72, 013810 (2005).
[CrossRef]

S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
[CrossRef]

Saleh, B. E. A.

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]

Scarcelli, G.

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?,” Phys. Rev. Lett. 96, 063602 (2006).
[CrossRef] [PubMed]

Scholten, R. E.

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modelling of classical ghost images obtained using scattered light,” New J. Phys. 9, 285-295 (2007).
[CrossRef]

Sergienko, A. V.

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-R3432 (1995).
[CrossRef] [PubMed]

Shih, Y.

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?,” Phys. Rev. Lett. 96, 063602 (2006).
[CrossRef] [PubMed]

M. D'Angelo, Y-H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[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-R3432 (1995).
[CrossRef] [PubMed]

Shih, Y..

M. D'Angelo, A. Valencia, M. H. Rubin, and Y.. Shih, “Resolution of quantum and classical ghost imaging,” Phys. Rev. A 72, 013810 (2005).
[CrossRef]

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-R3432 (1995).
[CrossRef] [PubMed]

Sun, X.-J.

J. Xiong, De-Zhong Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Teich, M. C.

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]

Thanvanthri, S.

S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
[CrossRef]

Twiss, R. Q.

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

Valencia, A.

M. D'Angelo, A. Valencia, M. H. Rubin, and Y.. Shih, “Resolution of quantum and classical ghost imaging,” Phys. Rev. A 72, 013810 (2005).
[CrossRef]

Wang, K.

J. Xiong, De-Zhong Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Wu, L-A.

Xiong, J.

J. Xiong, De-Zhong Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Zhai, Y-H.

Zhang, D.

Zhu, S. Y.

Nature (1)

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

New J. Phys. (1)

S. Crosby, S. Castelletto, C. Aruldoss, R. E. Scholten, and A. Roberts, “Modelling of classical ghost images obtained using scattered light,” New J. Phys. 9, 285-295 (2007).
[CrossRef]

Opt. Express (2)

Y. Bai, H. Liu, and S. Han, “Transmission area and correlated imaging,” Opt. Express 10, 6062-6068 (2007).
[CrossRef]

A. Kolkiran and G. S. Agarwal, “Quantum interference using coherent beam stimulated parametric downconversion,” Opt. Express 10, 6479-6485 (2008).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (9)

M. D'Angelo, A. Valencia, M. H. Rubin, and Y.. Shih, “Resolution of quantum and classical ghost imaging,” Phys. Rev. A 72, 013810 (2005).
[CrossRef]

M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[CrossRef]

M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, “Coherent imaging of pure phase object with classical incoherent light,” Phys. Rev. A 73, 053802 (2006).
[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-R3432 (1995).
[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]

I. P. Degiovanni, M. Bondani, A. Andreoni, E. Puddu, and M. Paris, “Intensity correlations, entanglement properties, and ghost imaging in multimode thermal-seeded parametric down-conversion: Theory,” Phys. Rev. A 76, 062309 (2007).
[CrossRef]

S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
[CrossRef]

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

E. Brambilla, M. Bache, A. Gatti, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A 69, 023802 (2004).
[CrossRef]

Phys. Rev. Lett. (10)

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]

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]

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]

M. D'Angelo, Y-H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[CrossRef] [PubMed]

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (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]

D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “Experimental evidence of high resolution ghost imaging and ghost diffraction with classical thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?,” Phys. Rev. Lett. 96, 063602 (2006).
[CrossRef] [PubMed]

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]

J. Xiong, De-Zhong Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

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

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Figures (5)

Fig. 1
Fig. 1

Schematic showing the principle of correlated imaging using either a classical or entangled source. For the classical source, a beamsplitter (BS) is used to generate twin beams from an input beam indicated as a 1 . For stimulated PDC a nonlinear crystal (NLC) replaces the BS, and the crystal is pumped by a strong input beam plus a seed, again indicated by a 1 . The vacuum state v is assumed for the other port of the BS or NLC. The input response functions of the two beams as generated after the BS or the NLC, h j ( x j , x j ) j = 1 , 2 , describe the propagation of the beams to the detection planes. The lower figure shows the imaging arrangement with object of transmission function t ( x ) and lens of focal length f.

Fig. 2
Fig. 2

Effect of l seed on the reconstruction of a diffraction pattern of a double slit of width b = 0.08 mm and distance between the slits d = 0.2 mm , showing the normalized coherence parameter Δ G PDC + seed ( x 1 = 0 , x 2 ) as scanned in the detector plane x 2 . The results for a stimulated emission source are shown for various intensities of the seed (lower intensities of the seed correspond to the curves closer to the PDC result). The diffraction pattern/visibility obtained using only an equivalent partially coherent source is also shown. The optical setup has a lens of focal length f = 500 mm and wavelength λ = 702 nm . L = 1 mm , σ p = 0.1 . (a) l seed = 40 μ m ; (b) l seed = 20 μ m .

Fig. 3
Fig. 3

Normalized coherence parameter Δ G PDC + seed ( x 1 = 0 , x 2 = 1.5 ) of a double slit of width b = 0.08 mm and distance between the slits d = 0.2 mm versus the l seed , for n seed photons per mode as shown.

Fig. 4
Fig. 4

Visibility of the diffraction pattern of a double slit of width b = 0.08 mm and distance between the slits d = 0.2 mm , at position x 2 in the detector plane, for different conditions: (a) σ p = 0.1 , l seed = 40 μ m , L = 1 mm ; (b) σ p = 0.1 , l seed = 20 μ m , L = 1 mm ; (c) σ p = 0.05 , l seed = 2 μ m , L = 0.5 mm ; (d) σ p = 0.5 , l seed = 2 μ m , L = 0.5 mm . The results for a stimulated emission source are shown for various intensities of the seed, where lower intensities of the seed correspond to curves closer to the upper (PDC) curve in each case. The diffraction visibility obtained from using only an equivalent partially coherent source is also shown (lowest curve in each case). The optical setup has a lens of focal length f = 500 mm and λ = 702 nm .

Fig. 5
Fig. 5

Maximum visibility of a double slit diffraction pattern for a stimulated PDC source versus (a) the mean number of photons of the seed for different values of the parametric gain σ p , for l seed = 6 μ m ; (b) the coherence length of the seed for different values of n seed and for a fixed parametric gain σ p = 0.1 , L = 0.5 mm , and l seed = 6 μ m ; (c) the crystal length for different values of n seed ( l seed = 6 μ m ) ; (d) the parametric gain for different values of n seed ( l seed = 6 μ m ) . The visibility with only PDC is shown as the upper curve in each case, and for an equivalent partially coherent source (curve at small V).

Equations (22)

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G ( 2 ) ( x 1 , x 2 ) = G ( 1 ) ( x 1 , x 1 ) G ( 1 ) ( x 2 , x 2 ) + G ( 1 ) ( x 1 , x 2 ) 2 .
G ( 1 ) ( x 1 , x 2 ) = 1 ( 2 π ) d q e i q ( x 1 x 2 ) n ( q ) therm .
b i ( q ) = U i ( q ) a i ( q ) + V i ( q ) a j ( q ) , i , j = 1 , 2 ; i j .
Γ 12 ( 1 ) ( x 1 , x 2 ) = 1 2 π d q e i q ( x 1 x 2 ) U 1 ( q ) V 2 ( q ) ,
Γ j j ( 1 ) ( x 1 , x 2 ) = 1 2 π d q e i q ( x 1 x 2 ) V j ( q ) 2 , j = 1 , 2 .
Γ 12 ( 1 ) ( x 1 , x 2 ) = 1 2 π d q e i q ( x 1 x 2 ) U 1 ( q ) V 2 ( q ) ( 1 + n ( q ) therm ) ,
Γ 11 ( 1 ) ( x 1 , x 2 ) = 1 2 π d q e i q ( x 1 x 2 ) [ V 1 ( q ) 2 ( 1 + n ( q ) therm ) + n ( q ) therm ] ,
Γ 22 ( 1 ) ( x 1 , x 2 ) = 1 2 π d q e i q ( x 1 x 2 ) V 2 ( q ) 2 ( 1 + n ( q ) therm ) .
G ( 1 ) ( x 1 , x 2 ) = n seed exp [ ( x 1 x 2 ) 2 2 l seed 2 ] exp [ ( x 1 2 + x 2 2 ) 4 w seed 2 ] ,
G ̃ ( 1 ) ( x 1 , x 2 , q ) exp [ ( x 1 + x 2 ) 2 8 w seed 2 ] n seed ( q ) ,
Δ G seed ( x 1 , x 2 ) = 1 2 π d x 1 d x 2 d q h 1 * ( x 1 , x 1 ) h 2 ( x 2 , x 2 ) e i q ( x 1 x 2 ) G ̃ ( 1 ) ( ( x 1 , x 2 , q ) ) 2 ,
Δ G seed ( x 1 , x 2 ) N o 2 n seed [ 2 π x 2 f λ ( l seed 2 + 4 w seed 2 l seed 2 + 2 w seed 2 ) 1 2 ] × T ̃ [ 2 π f λ { ( 2 w seed 2 l 1 2 2 w seed 2 + l 1 2 ) x 2 x 1 } ] 2 ,
Δ G ( x 1 , x 2 ) n therm ( 2 π x 2 f λ ) T ̃ [ 2 π f λ ( x 2 x 1 ) ] 2 .
Δ G PDC + seed ( x 1 , x 2 ) = 1 2 π d x 1 d x 2 d q h 1 ( x 1 , x 1 ) h 2 ( x 2 , x 2 ) e i q ( x 1 x 2 ) U 1 ( q ) V 2 ( q ) × [ 1 + n therm ( q ) ] 2 .
Δ G PDC + seed ( x 1 , x 2 ) = 2 π ( f λ ) 2 U 1 ( 2 π x 2 f λ ) V 2 ( 2 π x 2 f λ ) T ̃ [ 2 π f λ ( x 1 + x 2 ) ] × ( 1 + n therm ( 2 π x 2 f λ ) ) 2 .
Δ G PDC + seed ( x 1 , x 2 ) = 1 2 π d x 1 d x 2 d q h 1 ( x 1 , x 1 ) h 2 ( x 2 , x 2 ) e i q ( x 1 x 2 ) U 1 ( q ) V 2 ( q ) × [ 1 + G ̃ ( 1 ) ( x 1 , x 2 , q ) ] 2 .
Δ G PDC + seed ( x 1 , x 2 ) = 2 π ( f λ ) 2 U 1 ( 2 π x 2 f λ ) V 2 ( 2 π x 2 f λ ) T ̃ [ 2 π f λ ( x 1 + x 2 ) ] + g ( x 1 , x 2 ) 2 ,
g ( x 1 , x 2 ) = 2 2 π ( f λ ) 2 w seed d q U 1 ( q ) V 2 ( q ) n seed ( q ) exp ( 2 w seed 2 ( 2 π x 2 + f λ q ) 2 f 2 λ 2 ) × T ̃ [ 2 π f λ ( x 1 x 2 ) + 2 q ] .
V ( x 1 , x 2 ) = Δ G ( x 1 , x 2 ) Δ G ( x 1 , x 2 ) + n 1 ( x 1 ) n 2 ( x 2 ) ,
n j ( x j ) = d x 1 d x 2 h j ( x 1 , x j ) h j * ( x 2 , x j ) G ( 1 ) ( x 1 , x 2 ) , j = 1 , 2
n 1 ( x 1 ) = 1 2 π d q d x 1 d x 2 h 1 ( x 1 , x 1 ) h 1 * ( x 2 , x 1 ) e i q ( x 1 x 2 ) × { V 1 ( q ) 2 [ 1 + G ̃ ( 1 ) ( x 1 , x 2 , q ) ] + G ̃ ( 1 ) ( x 1 , x 2 , q ) } ,
n 2 ( x 2 ) = 1 2 π d q d x 1 d x 2 h 2 ( x 1 , x 2 ) h 2 * ( x 2 , x 2 ) e i q ( x 1 x 2 ) V 2 ( q ) 2 [ 1 + G ̃ ( 1 ) ( x 1 , x 2 , q ) ] .

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