Abstract

Temporal ghost imaging with classical pulses is described as a temporal counterpart of conventional ghost imaging with thermal light. A temporal object to be imaged is located in the test arm, while the reference arm consists of some simple temporal optical elements. It is shown by illustrative examples that, when a certain condition is satisfied, the correlation between intensity fluctuations in these two arms gives basically the squared modulus of the object, but it is generally distorted by the effect of the incident pulse. The resultant temporal image depends only on the single temporal variable in the reference arm, although the light in this arm never interacts with the object. Potential applications of this system are briefly discussed.

© 2010 Optical Society of America

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  1. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging,” in Quantum Imaging, M.I.Kolobov, ed. (Springer, 2007), pp. 79–111.
    [CrossRef]
  2. A. Gatti, E. Brambilla, and L. Lugiato, “Quantum imaging,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 251–348.
    [CrossRef]
  3. 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]
  4. 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]
  5. A. F. Abouraddy, B. 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. 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]
  8. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
    [CrossRef]
  9. 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]
  10. 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]
  11. B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
    [CrossRef]
  12. Y. Cai and S. Y. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716–2718 (2004).
    [CrossRef] [PubMed]
  13. Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
    [CrossRef]
  14. 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]
  15. A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. 98, 039301 (2007).
    [CrossRef] [PubMed]
  16. G. Scarcelli, V. Berardi, and Y. H. Shih, “Scarcelli, Berardi, and Shih reply,” Phys. Rev. Lett. 98, 039302 (2007).
    [CrossRef]
  17. R. Meyers, K. S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008).
    [CrossRef]
  18. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).
    [CrossRef]
  19. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
    [CrossRef]
  20. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
    [CrossRef]
  21. M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal pulse shaping with entangled photon pairs,” Phys. Rev. Lett. 90, 043602 (2003).
    [CrossRef] [PubMed]
  22. S. Viciani, A. Zavatta, and M. Bellini, “Nonlocal modulations on the temporal and spectral profiles of an entangled photon pair,” Phys. Rev. A 69, 053801 (2004).
    [CrossRef]
  23. 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]
  24. 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]
  25. B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
    [CrossRef] [PubMed]
  26. M. Tsang and D. Psaltis, “Propagation of temporal entanglement,” Phys. Rev. A 73, 013822 (2006).
    [CrossRef]
  27. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  28. V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Temporal ghost imaging and nonlocal dispersion cancellation with classical light,” in Book of Abstracts, ICO Topical Meeting on Emerging Trends & Novel Materials in Photonics (2009), p. 98.
  29. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  30. V. Torres-Company, “Coherence in ultrashort light pulses and applications in temporal optics,” Ph.D. dissertation (University of Valencia, 2008).
  31. Since the square roots of two complex numbers z1 and z2 do not, in general, satisfy the relations z1z2=z1z2 and z1/z2=z1/z2, we do not develop the factors in front of the integrals in Eqs. further.
  32. T. Setälä, T. Shirai, and A. T. Friberg, “Fractional Fourier transform in temporal ghost imaging with classical light,” Phys. Rev. A 82, 043813 (2010).
    [CrossRef]
  33. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).
  34. C. Dorrer and I. Walmsley, “Phase space in ultrafast optics,” in Phase-Space Optics, M.Testorf, B.Hennelly, and J.Ojeda-Castañeda, eds. (McGraw-Hill, 2010), pp. 337–383.

2010 (2)

T. Setälä, T. Shirai, and A. T. Friberg, “Fractional Fourier transform in temporal ghost imaging with classical light,” Phys. Rev. A 82, 043813 (2010).
[CrossRef]

C. Dorrer and I. Walmsley, “Phase space in ultrafast optics,” in Phase-Space Optics, M.Testorf, B.Hennelly, and J.Ojeda-Castañeda, eds. (McGraw-Hill, 2010), pp. 337–383.

2009 (2)

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Temporal ghost imaging and nonlocal dispersion cancellation with classical light,” in Book of Abstracts, ICO Topical Meeting on Emerging Trends & Novel Materials in Photonics (2009), p. 98.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

2008 (6)

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]

V. Torres-Company, “Coherence in ultrashort light pulses and applications in temporal optics,” Ph.D. dissertation (University of Valencia, 2008).

A. Gatti, E. Brambilla, and L. Lugiato, “Quantum imaging,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 251–348.
[CrossRef]

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
[CrossRef]

R. Meyers, K. S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).
[CrossRef]

2007 (6)

A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. 98, 039301 (2007).
[CrossRef] [PubMed]

G. Scarcelli, V. Berardi, and Y. H. Shih, “Scarcelli, Berardi, and Shih reply,” Phys. Rev. Lett. 98, 039302 (2007).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging,” in Quantum Imaging, M.I.Kolobov, ed. (Springer, 2007), pp. 79–111.
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

2006 (3)

M. Tsang and D. Psaltis, “Propagation of temporal entanglement,” Phys. Rev. A 73, 013822 (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]

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]

2005 (3)

Y. Cai and S. Y. 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]

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[CrossRef] [PubMed]

2004 (4)

S. Viciani, A. Zavatta, and M. Bellini, “Nonlocal modulations on the temporal and spectral profiles of an entangled photon pair,” Phys. Rev. A 69, 053801 (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, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
[CrossRef]

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

2003 (1)

M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal pulse shaping with entangled photon pairs,” Phys. Rev. Lett. 90, 043602 (2003).
[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 (1)

A. F. Abouraddy, B. 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 (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 (2)

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 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

1994 (1)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy, B. 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]

Arecchi, F. T.

M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal pulse shaping with entangled photon pairs,” Phys. Rev. Lett. 90, 043602 (2003).
[CrossRef] [PubMed]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging,” in Quantum Imaging, M.I.Kolobov, ed. (Springer, 2007), pp. 79–111.
[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]

Bellini, M.

S. Viciani, A. Zavatta, and M. Bellini, “Nonlocal modulations on the temporal and spectral profiles of an entangled photon pair,” Phys. Rev. A 69, 053801 (2004).
[CrossRef]

M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal pulse shaping with entangled photon pairs,” Phys. Rev. Lett. 90, 043602 (2003).
[CrossRef] [PubMed]

Bennink, R. S.

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, and R. W. Boyd, “‘Two-photon’ coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
[CrossRef] [PubMed]

Berardi, V.

G. Scarcelli, V. Berardi, and Y. H. Shih, “Scarcelli, Berardi, and Shih reply,” Phys. Rev. Lett. 98, 039302 (2007).
[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]

Bondani, M.

A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. 98, 039301 (2007).
[CrossRef] [PubMed]

Boyd, R. W.

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]

Brambilla, E.

A. Gatti, E. Brambilla, and L. Lugiato, “Quantum imaging,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 251–348.
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging,” in Quantum Imaging, M.I.Kolobov, ed. (Springer, 2007), pp. 79–111.
[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]

Bromberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

Cai, Y.

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

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

Deacon, K. S.

R. Meyers, K. S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008).
[CrossRef]

Dorrer, C.

C. Dorrer and I. Walmsley, “Phase space in ultrafast optics,” in Phase-Space Optics, M.Testorf, B.Hennelly, and J.Ojeda-Castañeda, eds. (McGraw-Hill, 2010), pp. 337–383.

Erkmen, B. I.

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
[CrossRef]

Fabre, C.

A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. 98, 039301 (2007).
[CrossRef] [PubMed]

Ferri, F.

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]

Friberg, A. T.

T. Setälä, T. Shirai, and A. T. Friberg, “Fractional Fourier transform in temporal ghost imaging with classical light,” Phys. Rev. A 82, 043813 (2010).
[CrossRef]

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Temporal ghost imaging and nonlocal dispersion cancellation with classical light,” in Book of Abstracts, ICO Topical Meeting on Emerging Trends & Novel Materials in Photonics (2009), p. 98.

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]

Gatti, A.

A. Gatti, E. Brambilla, and L. Lugiato, “Quantum imaging,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 251–348.
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging,” in Quantum Imaging, M.I.Kolobov, ed. (Springer, 2007), pp. 79–111.
[CrossRef]

A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. 98, 039301 (2007).
[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. 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]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Katz, O.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

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]

Kolner, B. H.

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

Lajunen, H.

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Temporal ghost imaging and nonlocal dispersion cancellation with classical light,” in Book of Abstracts, ICO Topical Meeting on Emerging Trends & Novel Materials in Photonics (2009), p. 98.

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, “Temporal ghost imaging and nonlocal dispersion cancellation with classical light,” in Book of Abstracts, ICO Topical Meeting on Emerging Trends & Novel Materials in Photonics (2009), p. 98.

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]

Lugiato, L.

A. Gatti, E. Brambilla, and L. Lugiato, “Quantum imaging,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 251–348.
[CrossRef]

Lugiato, L. A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging,” in Quantum Imaging, M.I.Kolobov, ed. (Springer, 2007), pp. 79–111.
[CrossRef]

A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. 98, 039301 (2007).
[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. 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]

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]

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]

Marin, F.

M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal pulse shaping with entangled photon pairs,” Phys. Rev. Lett. 90, 043602 (2003).
[CrossRef] [PubMed]

Meyers, R.

R. Meyers, K. S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008).
[CrossRef]

Paris, M. G. A.

A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. 98, 039301 (2007).
[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 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Psaltis, D.

M. Tsang and D. Psaltis, “Propagation of temporal entanglement,” Phys. Rev. A 73, 013822 (2006).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Saleh, B. A.

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

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[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. H. Shih, “Scarcelli, Berardi, and Shih reply,” Phys. Rev. Lett. 98, 039302 (2007).
[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]

Sergienko, A. V.

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[CrossRef] [PubMed]

A. F. Abouraddy, B. 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]

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 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Setälä, T.

T. Setälä, T. Shirai, and A. T. Friberg, “Fractional Fourier transform in temporal ghost imaging with classical light,” Phys. Rev. A 82, 043813 (2010).
[CrossRef]

Shapiro, J. H.

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).
[CrossRef]

Shih, Y.

R. Meyers, K. S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008).
[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]

Shih, Y. H.

G. Scarcelli, V. Berardi, and Y. H. Shih, “Scarcelli, Berardi, and Shih reply,” Phys. Rev. Lett. 98, 039302 (2007).
[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]

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]

Shirai, T.

T. Setälä, T. Shirai, and A. T. Friberg, “Fractional Fourier transform in temporal ghost imaging with classical light,” Phys. Rev. A 82, 043813 (2010).
[CrossRef]

Silberberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[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]

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]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[CrossRef] [PubMed]

A. F. Abouraddy, B. 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]

Torres-Company, V.

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Temporal ghost imaging and nonlocal dispersion cancellation with classical light,” in Book of Abstracts, ICO Topical Meeting on Emerging Trends & Novel Materials in Photonics (2009), p. 98.

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]

V. Torres-Company, “Coherence in ultrashort light pulses and applications in temporal optics,” Ph.D. dissertation (University of Valencia, 2008).

Tsang, M.

M. Tsang and D. Psaltis, “Propagation of temporal entanglement,” Phys. Rev. A 73, 013822 (2006).
[CrossRef]

Viciani, S.

S. Viciani, A. Zavatta, and M. Bellini, “Nonlocal modulations on the temporal and spectral profiles of an entangled photon pair,” Phys. Rev. A 69, 053801 (2004).
[CrossRef]

M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal pulse shaping with entangled photon pairs,” Phys. Rev. Lett. 90, 043602 (2003).
[CrossRef] [PubMed]

Walmsley, I.

C. Dorrer and I. Walmsley, “Phase space in ultrafast optics,” in Phase-Space Optics, M.Testorf, B.Hennelly, and J.Ojeda-Castañeda, eds. (McGraw-Hill, 2010), pp. 337–383.

Wolf, E.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

Zavatta, A.

S. Viciani, A. Zavatta, and M. Bellini, “Nonlocal modulations on the temporal and spectral profiles of an entangled photon pair,” Phys. Rev. A 69, 053801 (2004).
[CrossRef]

M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal pulse shaping with entangled photon pairs,” Phys. Rev. Lett. 90, 043602 (2003).
[CrossRef] [PubMed]

Zhu, S. Y.

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

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

IEEE J. Quantum Electron. (1)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[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]

Opt. Lett. (1)

Phys. Rev. A (11)

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
[CrossRef]

R. Meyers, K. S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).
[CrossRef]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[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]

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

T. Setälä, T. Shirai, and A. T. Friberg, “Fractional Fourier transform in temporal ghost imaging with classical light,” Phys. Rev. A 82, 043813 (2010).
[CrossRef]

S. Viciani, A. Zavatta, and M. Bellini, “Nonlocal modulations on the temporal and spectral profiles of an entangled photon pair,” Phys. Rev. A 69, 053801 (2004).
[CrossRef]

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]

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. Tsang and D. Psaltis, “Propagation of temporal entanglement,” Phys. Rev. A 73, 013822 (2006).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (10)

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]

A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. 98, 039301 (2007).
[CrossRef] [PubMed]

G. Scarcelli, V. Berardi, and Y. H. Shih, “Scarcelli, Berardi, and Shih reply,” Phys. Rev. Lett. 98, 039302 (2007).
[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. F. Abouraddy, B. 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]

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]

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[CrossRef] [PubMed]

M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal pulse shaping with entangled photon pairs,” Phys. Rev. Lett. 90, 043602 (2003).
[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]

Other (9)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

C. Dorrer and I. Walmsley, “Phase space in ultrafast optics,” in Phase-Space Optics, M.Testorf, B.Hennelly, and J.Ojeda-Castañeda, eds. (McGraw-Hill, 2010), pp. 337–383.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Temporal ghost imaging and nonlocal dispersion cancellation with classical light,” in Book of Abstracts, ICO Topical Meeting on Emerging Trends & Novel Materials in Photonics (2009), p. 98.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

V. Torres-Company, “Coherence in ultrashort light pulses and applications in temporal optics,” Ph.D. dissertation (University of Valencia, 2008).

Since the square roots of two complex numbers z1 and z2 do not, in general, satisfy the relations z1z2=z1z2 and z1/z2=z1/z2, we do not develop the factors in front of the integrals in Eqs. further.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging,” in Quantum Imaging, M.I.Kolobov, ed. (Springer, 2007), pp. 79–111.
[CrossRef]

A. Gatti, E. Brambilla, and L. Lugiato, “Quantum imaging,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 251–348.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

(a) One-lens and (b) lensless geometries for temporal ghost imaging with classical non-stationary pulsed light. The reference arm consists of a temporal lens and two dispersive media in the one-lens geometry, while it consists simply of a dispersive medium in the lensless geometry. In both geometries, the test arm consists of a temporal object m ( t ) to be imaged and two dispersive media. The typical example of the dispersive medium is a single-mode optical fiber, characterized by the group delay dispersion (GDD) parameters Φ l β 2 l z l ( l = a , b , c , d ) , with β 2 l and z l being, respectively, the group velocity dispersion coefficients and the lengths of the optical fibers. The instantaneous intensities are detected at the end of both arms, and the correlation between these two intensities is examined.

Equations (36)

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Δ I 1 ( t 1 ) Δ I 2 ( t 2 ) = I 1 ( t 1 ) I 2 ( t 2 ) I 1 ( t 1 ) I 2 ( t 2 ) ,
Δ I j ( t j ) = I j ( t j ) I j ( t j ) ,     ( j = 1 , 2 ) ,
I j ( t j ) = E j ( t j ) E j ( t j ) ,     ( j = 1 , 2 ) ,
Δ I 1 ( t 1 ) Δ I 2 ( t 2 ) = | Γ ( t 1 , t 2 ) | 2 ,
Γ ( t 1 , t 2 ) = E 1 ( t 1 ) E 2 ( t 2 )
E j ( t ) = E 0 ( t ) K j ( t , t ) d t ,     ( j = 1 , 2 ) ,
K 1 ( t , t ) = 1 2 π i Φ a i Φ b exp ( i t 2 2 γ ) exp [ i ( t t ) 2 2 Φ a ] exp [ i ( t t ) 2 2 Φ b ] d t ,
K 2 ( t , t ) = 1 2 π i Φ c i Φ d m ( t ) exp [ i ( t t ) 2 2 Φ c ] exp [ i ( t t ) 2 2 Φ d ] d t ,
Γ ( t 1 , t 2 ) = Γ 0 ( t 1 , t 2 ) K 1 ( t 1 , t 1 ) K 2 ( t 2 , t 2 ) d t 1 d t 2 ,
Γ 0 ( t 1 , t 2 ) = E 0 ( t 1 ) E 0 ( t 2 )
Γ 0 ( t 1 , t 2 ) = I 0 δ ( t 2 t 1 ) ,
Γ ( t 1 , t 2 ) = I 0 K 1 ( t 1 , t 1 ) K 2 ( t 2 , t 1 ) d t 1 .
exp [ i ( t 1 t 1 ) 2 2 Φ a ] exp [ i ( t 2 t 1 ) 2 2 Φ c ] d t 1 = 2 π Φ a Φ c i ( Φ a Φ c )   exp [ i ( t 1 t 2 ) 2 2 ( Φ a Φ c ) ] ,
Γ ( t 1 , t 2 ) = ( 2 π ) 3 / 2 I 0   exp ( i ψ ) i ( Φ a Φ c ) Φ b Φ d m ( t 2 ) exp [ i ( t 2 t 2 ) 2 2 Φ d ] F 1   d t 2 ,
F 1 = exp [ i 2 ( t 1 2 Φ b + t 2 2 Φ a Φ c ) i ( t 1 Φ b + t 2 Φ a Φ c ) t 1 + i 2 ( 1 Φ b + 1 Φ a Φ c 1 γ ) t 1 2 ] d t 1 ,
1 γ = 1 Φ a Φ c + 1 Φ b ,
F 1 = 2 π | Φ a Φ c | exp [ i 2 ( t 1 2 Φ b + t 2 2 Φ a Φ c ) ] δ ( t 2 + Φ a Φ c Φ b t 1 ) ,
Γ ( t 1 , t 2 ) = I 0 i 2 π s Φ d   exp ( i ψ ) exp [ i 2 Φ b ( s 1 s ) t 1 2 ] exp [ i 2 Φ d ( t 1 s t 2 ) 2 ] m ( t 1 s ) ,
s = Φ b Φ a Φ c .
Δ I 1 ( t 1 ) Δ I 2 ( t 2 ) = I 0 2 2 π | s Φ d | | m ( t 1 s ) | 2 | m ( t 1 s ) | 2 .
Δ I 1 ( t 1 ) Δ I 2 ( t 2 ) = I 0 2 2 π | Φ d | | m ( t 1 ) | 2 | m ( t 1 ) | 2 ,
Γ 0 ( t 1 , t 2 ) = I 0   exp ( t 1 2 + t 2 2 T 2 ) δ ( t 2 t 1 ) ,
Γ ( t 1 , t 2 ) = I 0 exp ( 2 t 1 2 T 2 ) K 1 ( t 1 , t 1 ) K 2 ( t 2 , t 1 ) d t 1 .
Γ ( t 1 , t 2 ) = A m ( t 2 ) exp [ i ( t 2 t 2 ) 2 2 Φ d ] F 2   d t 2 ,
A = ( 2 π ) 3 / 2 I 0 T   exp ( i ψ ) Φ b Φ d { 4 Φ a Φ c + i ( Φ a Φ c ) T 2 } ,
F 2 = exp [ i 2 { t 1 2 Φ b ( 4 Φ a i T 2 ) t 2 2 4 Φ a Φ c + i ( Φ a Φ c ) T 2 } ] exp [ i { t 1 Φ b + i T 2 t 2 4 Φ a Φ c + i ( Φ a Φ c ) T 2 } t 1 ] exp [ i 2 { 1 Φ b + 4 Φ c + i T 2 4 Φ a Φ c + i ( Φ a Φ c ) T 2 1 γ } t 1 2 ] d t 1 .
exp ( 2 t 1 2 T 2 ) exp [ i ( t 1 t 1 ) 2 2 Φ a ] exp [ i ( t 2 t 1 ) 2 2 Φ c ] d t 1 = T 2 π Φ a Φ c 4 Φ a Φ c + i ( Φ a Φ c ) T 2   exp [ ( t 1 t 2 ) 2 T 2 + 4 i ( Φ c t 1 2 Φ a t 2 2 ) 2 { 4 Φ a Φ c + i ( Φ a Φ c ) T 2 } ] .
1 γ = 4 Φ c + i T 2 4 Φ a Φ c + i ( Φ a Φ c ) T 2 + 1 Φ b ,
1 γ = 1 Φ a Φ c T 2 T 2 4 i Φ c + 1 Φ b ,
Φ c = 0.
1 γ = 1 Φ a + 1 Φ b .
F 2 = 2 π | Φ a | exp ( 2 t 2 2 T 2 ) exp [ i 2 ( t 1 2 Φ b + t 2 2 Φ a ) ] δ ( t 2 + Φ a Φ b t 1 ) .
Γ ( t 1 , t 2 ) = I 0 i 2 π s Φ d   exp ( i ψ ) exp [ 2 T 2 ( t 1 s ) 2 ] exp [ i 2 Φ b ( s 1 s ) t 1 2 ] exp [ i 2 Φ d ( t 1 s t 2 ) 2 ] m ( t 1 s ) ,
s = Φ b Φ a .
Δ I 1 ( t 1 ) Δ I 2 ( t 2 ) = I 0 2 2 π | s Φ d | exp [ 4 T 2 ( t 1 s ) 2 ] | m ( t 1 s ) | 2 exp [ 4 T 2 ( t 1 s ) 2 ] | m ( t 1 s ) | 2 .
Δ I 1 ( t 1 ) Δ I 2 ( t 2 ) = I 0 2 2 π | Φ d | exp ( 4 T 2 t 1 2 ) | m ( t 1 ) | 2 exp ( 4 T 2 t 1 2 ) | m ( t 1 ) | 2 .

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