Abstract

Ghost-imaging experiments correlate the outputs from two photodetectors: a high-spatial-resolution (scanning pinhole or CCD array) detector that measures a field that has not interacted with the object to be imaged, and a bucket (single-pixel) detector that collects a field that has interacted with the object. We give a comprehensive review of ghost imaging—within a unified Gaussian-state framework—presenting detailed analyses of its resolution, field of view, image contrast, and signal-to-noise ratio behavior. We consider three classes of illumination: thermal-state (classical), biphoton-state (quantum), and classical-state phase-sensitive light. The first two have been employed in a variety of ghost-imaging demonstrations. The third is the classical Gaussian state that produces ghost images that most closely mimic those obtained from biphoton illumination. The insights we develop lead naturally to a new, single-beam approach to ghost imaging, called computational ghost imaging, in which only the bucket detector is required. We provide quantitative results while simultaneously emphasizing the underlying physics of ghost imaging. The key to developing the latter understanding lies in the coherence behavior of a pair of Gaussian-state light beams with either phase-insensitive or phase-sensitive cross correlation.

© 2010 Optical Society of America

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  1. T. B. Pittman, Y. H. Shih, D. V. Strekalov, A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
    [CrossRef] [PubMed]
  2. R. S. Bennink, S. J. Bentley, R. W. Boyd, ““Two-photon” coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
    [CrossRef]
  3. M. D’Angelo, Y. Shih, “Can quantum imaging be classically simulated?,” arXiv.org, arXiv:quant-ph0302146v2 (2003).
  4. A. Gatti, E. Brambilla, L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90, 133603 (2003).
    [CrossRef] [PubMed]
  5. R. S. Bennink, S. J. Bentley, R. W. Boyd, J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004).
    [CrossRef] [PubMed]
  6. B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
    [CrossRef]
  7. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
    [CrossRef] [PubMed]
  8. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19, 1174–1184 (2002).
    [CrossRef]
  9. A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
    [CrossRef]
  10. A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
    [CrossRef] [PubMed]
  11. Y. Cai, S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
    [CrossRef]
  12. Y. Cai, S.-Y. Zhu, “Ghost interference with partially coherent light radiation,” Opt. Lett. 29, 2716–2718 (2004).
    [CrossRef] [PubMed]
  13. A. Valencia, G. Scarcelli, M. D’Angelo, Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
    [CrossRef] [PubMed]
  14. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
    [CrossRef] [PubMed]
  15. G. Scarcelli, V. Berardi, Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96, 063602 (2006).
    [CrossRef] [PubMed]
  16. L. Basano, P. Ottonello, “A conceptual experiment on single-beam coincidence detection with pseudothermal light,” Opt. Express 19, 12386–12394 (2009).
  17. B. I. Erkmen, J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
    [CrossRef]
  18. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).
    [CrossRef]
  19. Y. Bromberg, O. Katz, Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
    [CrossRef]
  20. O. Katz, Y. Bromberg, Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 113110 (2009).
    [CrossRef]
  21. A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
    [CrossRef]
  22. B. E. A. Saleh, M. C. Teich, Noise in Classical and Quantum Photon-Correlation Imaging (SPIE, 2008), Vol. 183, Chap. 21.
  23. J. Cheng, S.-S. Han, “Theoretical analysis of quantum noise in ghost imaging,” Chin. Phys. Lett. 22, 1676–1679 (2005).
    [CrossRef]
  24. B. I. Erkmen, J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009).
    [CrossRef]
  25. M. Bache, E. Brambilla, A. Gatti, L. A. Lugiato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
    [CrossRef]
  26. A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
    [CrossRef] [PubMed]
  27. X.-H. Chen, Q. Liu, K.-H. Luo, L.-A. Wu, “Lensless ghost imaging with true thermal light,” Opt. Lett. 34, 695–697 (2009).
    [CrossRef] [PubMed]
  28. M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, L. A. Lugiato, “Coherent imaging of a pure phase object with classical incoherent light,” Phys. Rev. A 73, 053802 (2006).
    [CrossRef]
  29. M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75, 021803(R) (2007).
    [CrossRef]
  30. J. Cheng, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823 (2008).
    [CrossRef]
  31. R. Meyers, K. S. Deacon, Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008).
    [CrossRef]
  32. R. E. Meyers, K. S. Deacon, Y. Shih, “Quantum imaging of an obscured object by measurement of reflected photons,” Proc. SPIE 7092, 70920E (2008).
    [CrossRef]
  33. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916–7921 (2009).
    [CrossRef] [PubMed]
  34. L. Wang, S. Qamar, S.-Y. Zhu, M. S. Zubairy, “Hanbury Brown-Twiss effect and thermal light ghost imaging: a unified approach,” Phys. Rev. A 79, 033835 (2009).
    [CrossRef]
  35. M. H. Rubin, Y. Shih, “Resolution of ghost imaging for nondegenerate spontaneous parametric down-conversion,” Phys. Rev. A 78, 033836 (2008).
    [CrossRef]
  36. K. Chan, M. N. O’Sullivan, R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79, 033808 (2009).
    [CrossRef]
  37. Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009).
    [CrossRef]
  38. L.-H. Ou, L.-M. Kuang, “Ghost imaging with third-order correlated thermal light,” J. Phys. B 40, 1833–1844 (2007).
    [CrossRef]
  39. K. W. C. Chan, M. N. O’Sullivan, R. W. Boyd, “High-order thermal ghost imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuD100.
  40. I. Agafonov, M. Chekhova, T. S. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56, 422–431 (2009).
    [CrossRef]
  41. X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv.org, arXiv:0902.3713v1 [quant-ph] (2009).
  42. The positive-frequency electric field operator is non-Hermitian; so both its phase-insensitive and phase-sensitive correlation functions must be specified in order to fully describe a zero-mean Gaussian state. This is discussed further in Section 4.
  43. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2000).
  44. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
    [CrossRef]
  45. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
    [CrossRef]
  46. R. M. Gagliardi, S. Karp, Optical Communications (Wiley, 1976).
  47. J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547–1569 (2009).
    [CrossRef]
  48. J. H. Shapiro, “Corrections to “The Quantum Theory of Optical Communications” [Nov/Dec 09 1547-1569],” IEEE J. Sel. Top. Quantum Electron. 16, 698 (2010).
    [CrossRef]
  49. We are assuming polarized light sources and targets that do not depolarize, so that scalar-wave theory suffices. The extension of our treatment to vector-wave sources and depolarizing targets is straightforward.
  50. R. G. Gallager, Discrete Stochastic Processes (Kluwer Academic, 1996).
    [CrossRef]
  51. J. H. Shapiro, “Quantum Gaussian noise,” in Proc. SPIE 5111, 382–395 (2003).
    [CrossRef]
  52. J. W. Goodman, Statistical Optics, Classics ed. (Wiley, 2000).
  53. H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory 26, 78–92 (1980).
    [CrossRef]
  54. J. H. Shapiro, “Quantum measurement eigenkets for continuous-time direct detection,” Quantum Semiclassic. Opt. 10, 567–578 (1998).
    [CrossRef]
  55. J. H. Shapiro, K.-X. Sun, “Semiclassical versus quantum behavior in fourth-order interference,” J. Opt. Soc. Am. B 11, 1130–1141 (1994).
    [CrossRef]
  56. J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, “Optical communication with two-photon coherent states—Part II: photoemissive detection and structured receiver performance,” IEEE Trans. Inf. Theory 25, 179–192 (1979).
    [CrossRef]
  57. Strictly speaking, the quasi-monochromatic and paraxial conditions refer to the excited modes of the quantized field, i.e., all non-vacuum-state modes of the field operators ÊS(ρ,t)e−iω0t and ÊR(ρ,t)e−iω0t are confined to a temporal-frequency bandwidth much smaller than ω0 and a spatial-frequency bandwidth much smaller than ω0∕c.
  58. H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part I: quantum state propagation and quantum noise reduction,” IEEE Trans. Inf. Theory 24, 657–668 (1978).
    [CrossRef]
  59. J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, 1965).
  60. In writing Eq. (23), we have used Eq. (16) to eliminate the temporal correlation terms.
  61. Here ⋆ denotes convolution.
  62. Because Ê(ρ,t) is a baseband field operator, Ω is the frequency detuning of the plane-wave component Â(k,Ω) from the wave’s center frequency ω0.
  63. B. I. Erkmen, J. H. Shapiro, “Optical coherence theory for phase-sensitive light,” Proc. SPIE 6305, 63050G (2006).
    [CrossRef]
  64. Although it is more common to express the field of view as a solid angle, here we use the mean-square radius at the transverse target plane as our field-of-view measure.
  65. F. N. C. Wong, T. Kim, J. H. Shapiro, “Efficient generation of polarization-entangled photons in a nonlinear crystal,” Laser Phys. 16, 1517–1524 (2006).
    [CrossRef]
  66. E. Brambilla, A. Gatti, M. Bache, 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]
  67. The SPDC output field operators presented herein are derived from quantized coupled-mode equations using the typical nondepleting plane-wave pump approximation. The transverse boundary effects within the crystal have been ignored, and unimportant global phase factors have been omitted.
  68. Although the exact solution of the coupled-mode equations and the boundary conditions at the input facet of the nonlinear crystal does not lead to a Gaussian ν(k,Ω), this assumption facilitates an analytic treatment without compromising the fundamental physics we are after.
  69. Excess noise refers to fluctuations on the light illuminating the photodetectors that is transferred to the resulting photocurrents.
  70. The filter HB(Ω), including its AC coupling, will be assumed to be within the photodetector blocks shown in Fig. 1, so that ⟨ı̂m(t)⟩=0 for m=1,2 for all the field states we shall consider.
  71. Because the maximally entangled phase-sensitive cross-correlation function is not coherence separable, but rather a sum of two coherence-separable terms [see Eq. (57)], some extra steps are needed in evaluating the integrals in this case.
  72. This is the same parametric dependence reported earlier in [21], without derivation.
  73. It is necessary to utilize photon-number-resolving detectors in order to reap the advantages ascribed to this high-flux, low-brightness regime.
  74. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).
  75. Recall that Cn is defined in Eq. (24).
  76. The classical photocurrent measured by the bucket detector is a random process with shot-noise fluctuations. In deriving Eq. (84) we have used the ensemble average (mean) of the measured photocurrent, as we have done in all of the previous sections.
  77. Y. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016–1030 (2007).
    [CrossRef]
  78. J. F. Clauser, M. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
    [CrossRef]
  79. In our view, the interpretation of computational ghost imaging that suggests nonlocal interference between a physical photon in a light beam and a “virtual photon” arising from numerical computations on a computer processor—see M. H. Rubin, “Comment on ghost imaging with a single detector [arXiv0812.2633v2],” arXiv.org, arXiv:0902.1940v1 [quant-ph] (2009)—is far from being convincing, and is highly speculative.
  80. B. I. Erkmen, J. H. Shapiro, “Phase-conjugate optical coherence tomography,” Phys. Rev. A 74, 041601(R) (2006).
    [CrossRef]
  81. B. I. Erkmen, J. H. Shapiro, “Gaussian-state theory of two-photon imaging,” Phys. Rev. A 78, 023835 (2008).
    [CrossRef]

2010 (1)

J. H. Shapiro, “Corrections to “The Quantum Theory of Optical Communications” [Nov/Dec 09 1547-1569],” IEEE J. Sel. Top. Quantum Electron. 16, 698 (2010).
[CrossRef]

2009 (11)

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547–1569 (2009).
[CrossRef]

L. Basano, P. Ottonello, “A conceptual experiment on single-beam coincidence detection with pseudothermal light,” Opt. Express 19, 12386–12394 (2009).

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

O. Katz, Y. Bromberg, Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 113110 (2009).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009).
[CrossRef]

X.-H. Chen, Q. Liu, K.-H. Luo, L.-A. Wu, “Lensless ghost imaging with true thermal light,” Opt. Lett. 34, 695–697 (2009).
[CrossRef] [PubMed]

J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916–7921 (2009).
[CrossRef] [PubMed]

L. Wang, S. Qamar, S.-Y. Zhu, M. S. Zubairy, “Hanbury Brown-Twiss effect and thermal light ghost imaging: a unified approach,” Phys. Rev. A 79, 033835 (2009).
[CrossRef]

K. Chan, M. N. O’Sullivan, R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79, 033808 (2009).
[CrossRef]

Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009).
[CrossRef]

I. Agafonov, M. Chekhova, T. S. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56, 422–431 (2009).
[CrossRef]

2008 (7)

M. H. Rubin, Y. Shih, “Resolution of ghost imaging for nondegenerate spontaneous parametric down-conversion,” Phys. Rev. A 78, 033836 (2008).
[CrossRef]

J. Cheng, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823 (2008).
[CrossRef]

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

R. E. Meyers, K. S. Deacon, Y. Shih, “Quantum imaging of an obscured object by measurement of reflected photons,” Proc. SPIE 7092, 70920E (2008).
[CrossRef]

B. I. Erkmen, 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]

B. I. Erkmen, J. H. Shapiro, “Gaussian-state theory of two-photon imaging,” Phys. Rev. A 78, 023835 (2008).
[CrossRef]

2007 (3)

Y. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016–1030 (2007).
[CrossRef]

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

L.-H. Ou, L.-M. Kuang, “Ghost imaging with third-order correlated thermal light,” J. Phys. B 40, 1833–1844 (2007).
[CrossRef]

2006 (6)

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

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

B. I. Erkmen, J. H. Shapiro, “Optical coherence theory for phase-sensitive light,” Proc. SPIE 6305, 63050G (2006).
[CrossRef]

F. N. C. Wong, T. Kim, J. H. Shapiro, “Efficient generation of polarization-entangled photons in a nonlinear crystal,” Laser Phys. 16, 1517–1524 (2006).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Phase-conjugate optical coherence tomography,” Phys. Rev. A 74, 041601(R) (2006).
[CrossRef]

2005 (4)

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

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

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

J. Cheng, S.-S. Han, “Theoretical analysis of quantum noise in ghost imaging,” Chin. Phys. Lett. 22, 1676–1679 (2005).
[CrossRef]

2004 (7)

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

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[CrossRef] [PubMed]

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

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

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

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

E. Brambilla, A. Gatti, M. Bache, 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]

2003 (2)

J. H. Shapiro, “Quantum Gaussian noise,” in Proc. SPIE 5111, 382–395 (2003).
[CrossRef]

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

2002 (2)

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

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

2001 (1)

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

1998 (1)

J. H. Shapiro, “Quantum measurement eigenkets for continuous-time direct detection,” Quantum Semiclassic. Opt. 10, 567–578 (1998).
[CrossRef]

1995 (1)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

1994 (1)

J. H. Shapiro, K.-X. Sun, “Semiclassical versus quantum behavior in fourth-order interference,” J. Opt. Soc. Am. B 11, 1130–1141 (1994).
[CrossRef]

1980 (1)

H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory 26, 78–92 (1980).
[CrossRef]

1979 (1)

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, “Optical communication with two-photon coherent states—Part II: photoemissive detection and structured receiver performance,” IEEE Trans. Inf. Theory 25, 179–192 (1979).
[CrossRef]

1978 (1)

H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part I: quantum state propagation and quantum noise reduction,” IEEE Trans. Inf. Theory 24, 657–668 (1978).
[CrossRef]

1969 (1)

J. F. Clauser, M. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

1963 (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[CrossRef] [PubMed]

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

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

Agafonov, I.

I. Agafonov, M. Chekhova, T. S. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56, 422–431 (2009).
[CrossRef]

Agafonov, I. N.

X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv.org, arXiv:0902.3713v1 [quant-ph] (2009).

Bache, M.

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

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

E. Brambilla, A. Gatti, M. Bache, 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]

Basano, L.

L. Basano, P. Ottonello, “A conceptual experiment on single-beam coincidence detection with pseudothermal light,” Opt. Express 19, 12386–12394 (2009).

Bennink, R. S.

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

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

Bentley, S. J.

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

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

Berardi, V.

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

Boyd, R. W.

K. Chan, M. N. O’Sullivan, R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79, 033808 (2009).
[CrossRef]

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

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

K. W. C. Chan, M. N. O’Sullivan, R. W. Boyd, “High-order thermal ghost imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuD100.

Brambilla, E.

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

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

E. Brambilla, A. Gatti, M. Bache, 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]

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

Bromberg, Y.

O. Katz, Y. Bromberg, Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 113110 (2009).
[CrossRef]

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

Cai, Y.

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

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

Chan, K.

K. Chan, M. N. O’Sullivan, R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79, 033808 (2009).
[CrossRef]

Chan, K. W. C.

K. W. C. Chan, M. N. O’Sullivan, R. W. Boyd, “High-order thermal ghost imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuD100.

Chekhova, M.

I. Agafonov, M. Chekhova, T. S. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56, 422–431 (2009).
[CrossRef]

Chekhova, M. V.

X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv.org, arXiv:0902.3713v1 [quant-ph] (2009).

Chen, X.-H.

Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009).
[CrossRef]

X.-H. Chen, Q. Liu, K.-H. Luo, L.-A. Wu, “Lensless ghost imaging with true thermal light,” Opt. Lett. 34, 695–697 (2009).
[CrossRef] [PubMed]

X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv.org, arXiv:0902.3713v1 [quant-ph] (2009).

Cheng, J.

J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916–7921 (2009).
[CrossRef] [PubMed]

J. Cheng, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823 (2008).
[CrossRef]

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

J. Cheng, S.-S. Han, “Theoretical analysis of quantum noise in ghost imaging,” Chin. Phys. Lett. 22, 1676–1679 (2005).
[CrossRef]

Clauser, J. F.

J. F. Clauser, M. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

D’Angelo, M.

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

M. D’Angelo, Y. Shih, “Can quantum imaging be classically simulated?,” arXiv.org, arXiv:quant-ph0302146v2 (2003).

Deacon, K. S.

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

R. E. Meyers, K. S. Deacon, Y. Shih, “Quantum imaging of an obscured object by measurement of reflected photons,” Proc. SPIE 7092, 70920E (2008).
[CrossRef]

Erkmen, B. I.

B. I. Erkmen, J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009).
[CrossRef]

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

B. I. Erkmen, J. H. Shapiro, “Gaussian-state theory of two-photon imaging,” Phys. Rev. A 78, 023835 (2008).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Phase-conjugate optical coherence tomography,” Phys. Rev. A 74, 041601(R) (2006).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Optical coherence theory for phase-sensitive light,” Proc. SPIE 6305, 63050G (2006).
[CrossRef]

Ferri, F.

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

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

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, 1976).

Gallager, R. G.

R. G. Gallager, Discrete Stochastic Processes (Kluwer Academic, 1996).
[CrossRef]

Gatti, A.

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

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

E. Brambilla, A. Gatti, M. Bache, 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]

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

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics, Classics ed. (Wiley, 2000).

Han, S.

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

Han, S.-S.

J. Cheng, S.-S. Han, “Theoretical analysis of quantum noise in ghost imaging,” Chin. Phys. Lett. 22, 1676–1679 (2005).
[CrossRef]

Holt, R. A.

J. F. Clauser, M. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Horne, M.

J. F. Clauser, M. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Howell, J. C.

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

Iskhakov, T. S.

I. Agafonov, M. Chekhova, T. S. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56, 422–431 (2009).
[CrossRef]

Jacobs, I. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, 1965).

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, 1976).

Katz, O.

O. Katz, Y. Bromberg, Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 113110 (2009).
[CrossRef]

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

Kim, T.

F. N. C. Wong, T. Kim, J. H. Shapiro, “Efficient generation of polarization-entangled photons in a nonlinear crystal,” Laser Phys. 16, 1517–1524 (2006).
[CrossRef]

Kuang, L.-M.

L.-H. Ou, L.-M. Kuang, “Ghost imaging with third-order correlated thermal light,” J. Phys. B 40, 1833–1844 (2007).
[CrossRef]

Liu, H.

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

Liu, Q.

X.-H. Chen, Q. Liu, K.-H. Luo, L.-A. Wu, “Lensless ghost imaging with true thermal light,” Opt. Lett. 34, 695–697 (2009).
[CrossRef] [PubMed]

Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009).
[CrossRef]

X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv.org, arXiv:0902.3713v1 [quant-ph] (2009).

Liu, Y.

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

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2000).

Lugiato, L. A.

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

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

E. Brambilla, A. Gatti, M. Bache, 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]

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

Luo, K.-H.

X.-H. Chen, Q. Liu, K.-H. Luo, L.-A. Wu, “Lensless ghost imaging with true thermal light,” Opt. Lett. 34, 695–697 (2009).
[CrossRef] [PubMed]

Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009).
[CrossRef]

X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv.org, arXiv:0902.3713v1 [quant-ph] (2009).

Machado Mata, J. A.

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, “Optical communication with two-photon coherent states—Part II: photoemissive detection and structured receiver performance,” IEEE Trans. Inf. Theory 25, 179–192 (1979).
[CrossRef]

Magatti, D.

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

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
[CrossRef]

Meyers, R.

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

Meyers, R. E.

R. E. Meyers, K. S. Deacon, Y. Shih, “Quantum imaging of an obscured object by measurement of reflected photons,” Proc. SPIE 7092, 70920E (2008).
[CrossRef]

O’Sullivan, M. N.

K. Chan, M. N. O’Sullivan, R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79, 033808 (2009).
[CrossRef]

K. W. C. Chan, M. N. O’Sullivan, R. W. Boyd, “High-order thermal ghost imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuD100.

Ottonello, P.

L. Basano, P. Ottonello, “A conceptual experiment on single-beam coincidence detection with pseudothermal light,” Opt. Express 19, 12386–12394 (2009).

Ou, L.-H.

L.-H. Ou, L.-M. Kuang, “Ghost imaging with third-order correlated thermal light,” J. Phys. B 40, 1833–1844 (2007).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Qamar, S.

L. Wang, S. Qamar, S.-Y. Zhu, M. S. Zubairy, “Hanbury Brown-Twiss effect and thermal light ghost imaging: a unified approach,” Phys. Rev. A 79, 033835 (2009).
[CrossRef]

Rubin, M. H.

M. H. Rubin, Y. Shih, “Resolution of ghost imaging for nondegenerate spontaneous parametric down-conversion,” Phys. Rev. A 78, 033836 (2008).
[CrossRef]

In our view, the interpretation of computational ghost imaging that suggests nonlocal interference between a physical photon in a light beam and a “virtual photon” arising from numerical computations on a computer processor—see M. H. Rubin, “Comment on ghost imaging with a single detector [arXiv0812.2633v2],” arXiv.org, arXiv:0902.1940v1 [quant-ph] (2009)—is far from being convincing, and is highly speculative.

Saleh, B. E. A.

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[CrossRef] [PubMed]

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

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

B. E. A. Saleh, M. C. Teich, Noise in Classical and Quantum Photon-Correlation Imaging (SPIE, 2008), Vol. 183, Chap. 21.

Scarcelli, G.

G. Scarcelli, V. Berardi, 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. Valencia, G. Scarcelli, M. D’Angelo, Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

Sergienko, A. V.

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[CrossRef] [PubMed]

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

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, “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, 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, A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Shapiro, J. H.

J. H. Shapiro, “Corrections to “The Quantum Theory of Optical Communications” [Nov/Dec 09 1547-1569],” IEEE J. Sel. Top. Quantum Electron. 16, 698 (2010).
[CrossRef]

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547–1569 (2009).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009).
[CrossRef]

B. I. Erkmen, 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]

B. I. Erkmen, J. H. Shapiro, “Gaussian-state theory of two-photon imaging,” Phys. Rev. A 78, 023835 (2008).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Phase-conjugate optical coherence tomography,” Phys. Rev. A 74, 041601(R) (2006).
[CrossRef]

F. N. C. Wong, T. Kim, J. H. Shapiro, “Efficient generation of polarization-entangled photons in a nonlinear crystal,” Laser Phys. 16, 1517–1524 (2006).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Optical coherence theory for phase-sensitive light,” Proc. SPIE 6305, 63050G (2006).
[CrossRef]

J. H. Shapiro, “Quantum Gaussian noise,” in Proc. SPIE 5111, 382–395 (2003).
[CrossRef]

J. H. Shapiro, “Quantum measurement eigenkets for continuous-time direct detection,” Quantum Semiclassic. Opt. 10, 567–578 (1998).
[CrossRef]

J. H. Shapiro, K.-X. Sun, “Semiclassical versus quantum behavior in fourth-order interference,” J. Opt. Soc. Am. B 11, 1130–1141 (1994).
[CrossRef]

H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory 26, 78–92 (1980).
[CrossRef]

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, “Optical communication with two-photon coherent states—Part II: photoemissive detection and structured receiver performance,” IEEE Trans. Inf. Theory 25, 179–192 (1979).
[CrossRef]

H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part I: quantum state propagation and quantum noise reduction,” IEEE Trans. Inf. Theory 24, 657–668 (1978).
[CrossRef]

Shen, X.

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

Shih, Y.

M. H. Rubin, Y. Shih, “Resolution of ghost imaging for nondegenerate spontaneous parametric down-conversion,” Phys. Rev. A 78, 033836 (2008).
[CrossRef]

R. E. Meyers, K. S. Deacon, Y. Shih, “Quantum imaging of an obscured object by measurement of reflected photons,” Proc. SPIE 7092, 70920E (2008).
[CrossRef]

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

Y. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016–1030 (2007).
[CrossRef]

G. Scarcelli, V. Berardi, 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. Valencia, G. Scarcelli, M. D’Angelo, Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

M. D’Angelo, Y. Shih, “Can quantum imaging be classically simulated?,” arXiv.org, arXiv:quant-ph0302146v2 (2003).

Shih, Y. H.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Shimony, A.

J. F. Clauser, M. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Silberberg, Y.

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

O. Katz, Y. Bromberg, Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 113110 (2009).
[CrossRef]

Stone, P. R.

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[CrossRef] [PubMed]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

Sun, K.-X.

J. H. Shapiro, K.-X. Sun, “Semiclassical versus quantum behavior in fourth-order interference,” J. Opt. Soc. Am. B 11, 1130–1141 (1994).
[CrossRef]

Teich, M. C.

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[CrossRef] [PubMed]

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

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

B. E. A. Saleh, M. C. Teich, Noise in Classical and Quantum Photon-Correlation Imaging (SPIE, 2008), Vol. 183, Chap. 21.

Valencia, A.

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

Wang, L.

L. Wang, S. Qamar, S.-Y. Zhu, M. S. Zubairy, “Hanbury Brown-Twiss effect and thermal light ghost imaging: a unified approach,” Phys. Rev. A 79, 033835 (2009).
[CrossRef]

Wei, Q.

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

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
[CrossRef]

Wong, F. N. C.

F. N. C. Wong, T. Kim, J. H. Shapiro, “Efficient generation of polarization-entangled photons in a nonlinear crystal,” Laser Phys. 16, 1517–1524 (2006).
[CrossRef]

Wozencraft, J. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, 1965).

Wu, L.-A.

X.-H. Chen, Q. Liu, K.-H. Luo, L.-A. Wu, “Lensless ghost imaging with true thermal light,” Opt. Lett. 34, 695–697 (2009).
[CrossRef] [PubMed]

Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009).
[CrossRef]

I. Agafonov, M. Chekhova, T. S. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56, 422–431 (2009).
[CrossRef]

X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv.org, arXiv:0902.3713v1 [quant-ph] (2009).

Wu, W.

Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009).
[CrossRef]

Xian, R.

X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv.org, arXiv:0902.3713v1 [quant-ph] (2009).

Yuen, H. P.

H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory 26, 78–92 (1980).
[CrossRef]

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, “Optical communication with two-photon coherent states—Part II: photoemissive detection and structured receiver performance,” IEEE Trans. Inf. Theory 25, 179–192 (1979).
[CrossRef]

H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part I: quantum state propagation and quantum noise reduction,” IEEE Trans. Inf. Theory 24, 657–668 (1978).
[CrossRef]

Zhang, M.

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

Zhu, S.-Y.

L. Wang, S. Qamar, S.-Y. Zhu, M. S. Zubairy, “Hanbury Brown-Twiss effect and thermal light ghost imaging: a unified approach,” Phys. Rev. A 79, 033835 (2009).
[CrossRef]

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

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

Zubairy, M. S.

L. Wang, S. Qamar, S.-Y. Zhu, M. S. Zubairy, “Hanbury Brown-Twiss effect and thermal light ghost imaging: a unified approach,” Phys. Rev. A 79, 033835 (2009).
[CrossRef]

Appl. Phys. Lett. (1)

O. Katz, Y. Bromberg, Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 113110 (2009).
[CrossRef]

Chin. Phys. Lett. (1)

J. Cheng, S.-S. Han, “Theoretical analysis of quantum noise in ghost imaging,” Chin. Phys. Lett. 22, 1676–1679 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547–1569 (2009).
[CrossRef]

J. H. Shapiro, “Corrections to “The Quantum Theory of Optical Communications” [Nov/Dec 09 1547-1569],” IEEE J. Sel. Top. Quantum Electron. 16, 698 (2010).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016–1030 (2007).
[CrossRef]

IEEE Trans. Inf. Theory (1)

H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory 26, 78–92 (1980).
[CrossRef]

IEEE Trans. Inf. Theory (2)

H. P. Yuen, J. H. Shapiro, “Optical communication with two-photon coherent states—Part I: quantum state propagation and quantum noise reduction,” IEEE Trans. Inf. Theory 24, 657–668 (1978).
[CrossRef]

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, “Optical communication with two-photon coherent states—Part II: photoemissive detection and structured receiver performance,” IEEE Trans. Inf. Theory 25, 179–192 (1979).
[CrossRef]

J. Mod. Opt. (1)

I. Agafonov, M. Chekhova, T. S. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56, 422–431 (2009).
[CrossRef]

J. Mod. Opt. (1)

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

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

J. H. Shapiro, K.-X. Sun, “Semiclassical versus quantum behavior in fourth-order interference,” J. Opt. Soc. Am. B 11, 1130–1141 (1994).
[CrossRef]

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

J. Phys. B (1)

L.-H. Ou, L.-M. Kuang, “Ghost imaging with third-order correlated thermal light,” J. Phys. B 40, 1833–1844 (2007).
[CrossRef]

Laser Phys. (1)

F. N. C. Wong, T. Kim, J. H. Shapiro, “Efficient generation of polarization-entangled photons in a nonlinear crystal,” Laser Phys. 16, 1517–1524 (2006).
[CrossRef]

Opt. Express (2)

L. Basano, P. Ottonello, “A conceptual experiment on single-beam coincidence detection with pseudothermal light,” Opt. Express 19, 12386–12394 (2009).

J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916–7921 (2009).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. A (1)

Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, L.-A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009).
[CrossRef]

Phys. Rev. Lett. (2)

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

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

Phys. Rev. A (2)

L. Wang, S. Qamar, S.-Y. Zhu, M. S. Zubairy, “Hanbury Brown-Twiss effect and thermal light ghost imaging: a unified approach,” Phys. Rev. A 79, 033835 (2009).
[CrossRef]

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

Phys. Rev. Lett. (2)

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

J. F. Clauser, M. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Phys. Rev. (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Phys. Rev. A (16)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

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

E. Brambilla, A. Gatti, M. Bache, 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]

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

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

B. I. Erkmen, 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]

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

J. Cheng, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823 (2008).
[CrossRef]

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

M. H. Rubin, Y. Shih, “Resolution of ghost imaging for nondegenerate spontaneous parametric down-conversion,” Phys. Rev. A 78, 033836 (2008).
[CrossRef]

K. Chan, M. N. O’Sullivan, R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79, 033808 (2009).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009).
[CrossRef]

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

B. I. Erkmen, J. H. Shapiro, “Phase-conjugate optical coherence tomography,” Phys. Rev. A 74, 041601(R) (2006).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Gaussian-state theory of two-photon imaging,” Phys. Rev. A 78, 023835 (2008).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (6)

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

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

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

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

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[CrossRef] [PubMed]

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

Proc. SPIE (1)

R. E. Meyers, K. S. Deacon, Y. Shih, “Quantum imaging of an obscured object by measurement of reflected photons,” Proc. SPIE 7092, 70920E (2008).
[CrossRef]

Proc. SPIE (2)

J. H. Shapiro, “Quantum Gaussian noise,” in Proc. SPIE 5111, 382–395 (2003).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Optical coherence theory for phase-sensitive light,” Proc. SPIE 6305, 63050G (2006).
[CrossRef]

Quantum Semiclassic. Opt. (1)

J. H. Shapiro, “Quantum measurement eigenkets for continuous-time direct detection,” Quantum Semiclassic. Opt. 10, 567–578 (1998).
[CrossRef]

Other (28)

J. W. Goodman, Statistical Optics, Classics ed. (Wiley, 2000).

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, 1965).

In writing Eq. (23), we have used Eq. (16) to eliminate the temporal correlation terms.

Here ⋆ denotes convolution.

Because Ê(ρ,t) is a baseband field operator, Ω is the frequency detuning of the plane-wave component Â(k,Ω) from the wave’s center frequency ω0.

The SPDC output field operators presented herein are derived from quantized coupled-mode equations using the typical nondepleting plane-wave pump approximation. The transverse boundary effects within the crystal have been ignored, and unimportant global phase factors have been omitted.

Although the exact solution of the coupled-mode equations and the boundary conditions at the input facet of the nonlinear crystal does not lead to a Gaussian ν(k,Ω), this assumption facilitates an analytic treatment without compromising the fundamental physics we are after.

Excess noise refers to fluctuations on the light illuminating the photodetectors that is transferred to the resulting photocurrents.

The filter HB(Ω), including its AC coupling, will be assumed to be within the photodetector blocks shown in Fig. 1, so that ⟨ı̂m(t)⟩=0 for m=1,2 for all the field states we shall consider.

Because the maximally entangled phase-sensitive cross-correlation function is not coherence separable, but rather a sum of two coherence-separable terms [see Eq. (57)], some extra steps are needed in evaluating the integrals in this case.

This is the same parametric dependence reported earlier in [21], without derivation.

It is necessary to utilize photon-number-resolving detectors in order to reap the advantages ascribed to this high-flux, low-brightness regime.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

Recall that Cn is defined in Eq. (24).

The classical photocurrent measured by the bucket detector is a random process with shot-noise fluctuations. In deriving Eq. (84) we have used the ensemble average (mean) of the measured photocurrent, as we have done in all of the previous sections.

X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, L.-A. Wu, “Arbitrary-order lensless ghost imaging with thermal light,” arXiv.org, arXiv:0902.3713v1 [quant-ph] (2009).

The positive-frequency electric field operator is non-Hermitian; so both its phase-insensitive and phase-sensitive correlation functions must be specified in order to fully describe a zero-mean Gaussian state. This is discussed further in Section 4.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2000).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
[CrossRef]

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, 1976).

K. W. C. Chan, M. N. O’Sullivan, R. W. Boyd, “High-order thermal ghost imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuD100.

We are assuming polarized light sources and targets that do not depolarize, so that scalar-wave theory suffices. The extension of our treatment to vector-wave sources and depolarizing targets is straightforward.

R. G. Gallager, Discrete Stochastic Processes (Kluwer Academic, 1996).
[CrossRef]

B. E. A. Saleh, M. C. Teich, Noise in Classical and Quantum Photon-Correlation Imaging (SPIE, 2008), Vol. 183, Chap. 21.

M. D’Angelo, Y. Shih, “Can quantum imaging be classically simulated?,” arXiv.org, arXiv:quant-ph0302146v2 (2003).

Although it is more common to express the field of view as a solid angle, here we use the mean-square radius at the transverse target plane as our field-of-view measure.

Strictly speaking, the quasi-monochromatic and paraxial conditions refer to the excited modes of the quantized field, i.e., all non-vacuum-state modes of the field operators ÊS(ρ,t)e−iω0t and ÊR(ρ,t)e−iω0t are confined to a temporal-frequency bandwidth much smaller than ω0 and a spatial-frequency bandwidth much smaller than ω0∕c.

In our view, the interpretation of computational ghost imaging that suggests nonlocal interference between a physical photon in a light beam and a “virtual photon” arising from numerical computations on a computer processor—see M. H. Rubin, “Comment on ghost imaging with a single detector [arXiv0812.2633v2],” arXiv.org, arXiv:0902.1940v1 [quant-ph] (2009)—is far from being convincing, and is highly speculative.

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Figures (9)

Fig. 1
Fig. 1

Simple ghost-imaging setup. Correlated signal ( S ) and reference ( R ) fields—here shown as quantum operators that can be used to analyze all source possibilities—propagate through L-meter-long free space paths. The signal then illuminates a high-spatial-resolution detector, shown here as a scanning pinhole detector, whereas the reference illuminates a single-pixel (bucket) detector through an object transparency with field transmission T ( ρ ) . Cross correlation of the resulting photocurrents yields the ghost image of the intensity transmission | T ( ρ ) | 2 as the pinhole is scanned.

Fig. 2
Fig. 2

Coherence behavior and angular spectrum of the source-plane ( z = 0 ) baseband field operator E ̂ ( ρ , t ) with phase-insensitive correlation function given by Eq. (26). (a) The average z = L plane irradiance is appreciable only within a region of diameter 2 λ 0 L π ρ 0 (red) around the optical axis. The phase-insensitive fluctuations seen at two transverse points that are symmetrically displaced from the optical axis are correlated only when their separation is less than 2 λ 0 L π a 0 (blue). (b) Three plane-wave components are shown here as three arrows with different colors (and line styles). The plane waves (of the same frequency) with which they have phase-insensitive correlation lie within the shaded cones of the same color (and same line-style borders). Because phase-insensitive coherence is quasi-monoplanatic, the coherence cone for each plane wave is centered on its own propagation direction.

Fig. 3
Fig. 3

Coherence behavior and angular spectrum of the source-plane ( z = 0 ) baseband field operator E ̂ ( ρ , t ) with the phase-sensitive correlation function given in Eq. (31). (a) The mean-square phase-sensitive fluctuations on the z = L plane are appreciable within the diameter 2 λ 0 L π a 0 (red). The phase-sensitive fluctuations seen at two transverse points displaced in the opposite direction by an equal amount are correlated as long as the distance between the two points is less than 2 λ 0 L π ρ 0 (blue). (b) Three plane-wave components are shown here as three arrows with different colors (and line styles). The plane waves with which they have phase-sensitive correlation are shown as shaded cones having the same color (and same line-style borders). Because phase-sensitive coherence is quasi-biplanatic, the coherence cone for each plane wave component is centered around its mirror image about the optical axis.

Fig. 4
Fig. 4

Isocontours corresponding to the e 2 -attenuation levels for the phase-sensitive and phase-insensitive correlation functions in the near-field and the far-field regimes.

Fig. 5
Fig. 5

Thermal-state ghost-imaging SNR, normalized by T I T 0 , plotted as a function of source brightness I P T 0 ρ 0 2 a 0 2 = P T 0 ρ L 2 a L 2 , for a far-field configuration ( π a 0 ρ 0 λ 0 L 1 ) with | T ( ρ 1 ) | = 1 , A T ρ L 2 = 10 4 , ρ L 2 A 1 = 10 , and η = 0.9 . Various Ω B T 0 values are shown in the (a) narrowband and (b) broadband limits. Dashed–dotted lines represent low-brightness asymptotes, and dashed lines correspond to high-brightness asymptotes.

Fig. 6
Fig. 6

Nonclassical phase-sensitive Gaussian-state ghost-imaging SNR, normalized by T I T 0 , plotted versus source brightness I P T 0 ρ 0 2 a 0 2 = P T 0 ρ L 2 a L 2 for a far-field configuration ( π a 0 2 λ 0 L 1 ) with | T ( ρ 1 ) | = 1 , A T ρ L 2 = 10 4 , ρ L 2 A 1 = 10 , and η = 0.9 . Various Ω B T 0 values are shown in the (a) narrowband and (b) broadband limits. Dashed–dotted lines represent low-brightness asymptotes, and dashed lines correspond to high-brightness asymptotes.

Fig. 7
Fig. 7

Spatiotemporal speckle of partially coherent light. (a) The transverse speckle pattern generated by illuminating a sheet of paper with a cw laser beam that has been rendered spatially incoherent by transmission through a ground-glass diffuser, and (b) the temporal fluctuations seen in a single speckle cell of a cw laser beam that has been transmitted through a rotating ground-glass diffuser. T 0 denotes the coherence time, and the dashed line (red) indicates the dark baseline.

Fig. 8
Fig. 8

Ghost imaging with a cw laser and a SLM.

Fig. 9
Fig. 9

Computational ghost-imaging setup.

Equations (84)

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μ ( t ) = A d ρ | E ( ρ , t ) | 2 ,
ı ̂ ( t ) = q A d ρ E ̂ ( ρ , t ) E ̂ ( ρ , t ) .
E ̂ ( ρ , t ) | E ( ρ , t ) = E ( ρ , t ) | E ( ρ , t ) ,
[ E ̂ m ( ρ 1 , t 1 ) , E ̂ l ( ρ 2 , t 2 ) ] = 0 ,
[ E ̂ m ( ρ 1 , t 1 ) , E ̂ l ( ρ 2 , t 2 ) ] = δ m , l δ ( ρ 1 ρ 2 ) δ ( t 1 t 2 ) ,
E ̂ l ( ρ , t ) = d ρ E ̂ m ( ρ , t L c ) h L ( ρ ρ ) ,
h L ( ρ ) e i 2 π L λ 0 e i π | ρ | 2 λ 0 L i λ 0 L ,
C ̂ ( ρ 1 ) = 1 T I T I 2 T I 2 d t ı ̂ 1 ( t ) ı ̂ 2 ( t ) ,
ı ̂ m ( t ) = q d u A m d ρ E ̂ m ( ρ , u ) E ̂ m ( ρ , u ) h B ( t u ) ,
E ̂ m ( ρ , t ) { η E ̂ 1 ( ρ , t ) + 1 η E ̂ vac 1 ( ρ , t ) , for m = 1 , η T ( ρ ) E ̂ 2 ( ρ , t ) + 1 η | T ( ρ ) | 2 E ̂ vac 2 ( ρ , t ) , for m = 2 , }
C ̂ ( ρ 1 ) = ı ̂ 1 ( t ) ı ̂ 2 ( t ) = q 2 η 2 A 1 A 2 d ρ d u 1 d u 2 h B ( t u 1 ) h B ( t u 2 ) | T ( ρ ) | 2 × E ̂ 1 ( ρ 1 , u 1 ) E ̂ 2 ( ρ , u 2 ) E ̂ 1 ( ρ 1 , u 1 ) E ̂ 2 ( ρ , u 2 ) ,
E ̂ 1 ( ρ 1 , u 1 ) E ̂ 2 ( ρ , u 2 ) E ̂ 1 ( ρ 1 , u 1 ) E ̂ 2 ( ρ , u 2 ) = E ̂ 1 ( ρ 1 , u 1 ) E ̂ 1 ( ρ 1 , u 1 ) E ̂ 2 ( ρ , u 2 ) E ̂ 2 ( ρ , u 2 ) + | E ̂ 1 ( ρ 1 , u 1 ) E ̂ 2 ( ρ , u 2 ) | 2 + | E ̂ 1 ( ρ 1 , u 1 ) E ̂ 2 ( ρ , u 2 ) | 2 .
E ̂ m ( ρ 1 , t 1 ) E ̂ l ( ρ 2 , t 2 ) = K m , l ( n ) ( ρ 1 , ρ 2 ) R m , l ( n ) ( t 2 t 1 ) ,
E ̂ S ( ρ 1 , t 1 ) E ̂ R ( ρ 2 , t 2 ) = K S , R ( p ) ( ρ 1 , ρ 2 ) R S , R ( p ) ( t 2 t 1 ) ,
E ̂ m ( ρ 1 , t 1 ) E ̂ m ( ρ 2 , t 2 ) = 0 ,
R m , l ( n ) ( 0 ) = R S , R ( p ) ( 0 ) = 1 .
E ̂ m ( ρ 1 , t 1 ) E ̂ l ( ρ 2 , t 2 ) = K m , l ( n ) ( ρ 1 , ρ 2 ) R m , l ( n ) ( t 2 t 1 ) ,
E ̂ 1 ( ρ 1 , t 1 ) E ̂ 2 ( ρ 2 , t 2 ) = K 1 , 2 ( p ) ( ρ 1 , ρ 2 ) R 1 , 2 ( p ) ( t 2 t 1 ) ,
E ̂ m ( ρ 1 , t 1 ) E ̂ m ( ρ 2 , t 2 ) = 0 ,
K m , l ( n ) ( ρ 1 , ρ 2 ) = d ρ 1 d ρ 2 K m , l ( n ) ( ρ 1 , ρ 2 ) h L * ( ρ 1 ρ 1 ) h L ( ρ 2 ρ 2 ) ,
K 1 , 2 ( p ) ( ρ 1 , ρ 2 ) = d ρ 1 d ρ 2 K S , R ( p ) ( ρ 1 , ρ 2 ) h L ( ρ 1 ρ 1 ) h L ( ρ 2 ρ 2 ) ,
C ̂ ( ρ 1 ) = C 0 ( ρ 1 ) + A 2 d ρ [ C n | K 1 , 2 ( n ) ( ρ 1 , ρ ) | 2 + C p | K 1 , 2 ( p ) ( ρ 1 , ρ ) | 2 ] | T ( ρ ) | 2 ,
C 0 ( ρ 1 ) = q 2 η 2 A 1 ( d t h B ( t ) ) 2 K 1 , 1 ( n ) ( ρ 1 , ρ 1 ) A 2 d ρ K 2 , 2 ( n ) ( ρ , ρ ) | T ( ρ ) | 2
C n = q 2 η 2 A 1 [ | R 1 , 2 ( n ) ( t ) | 2 h B ( t ) h B ( t ) ] t = 0 ,
C p = q 2 η 2 A 1 [ | R 1 , 2 ( p ) ( t ) | 2 h B ( t ) h B ( t ) ] t = 0
E ̂ ( ρ 1 , t 1 ) E ̂ ( ρ 2 , t 2 ) = 2 P π a 0 2 e ( | ρ 1 | 2 + | ρ 2 | 2 ) a 0 2 | ρ 2 ρ 1 | 2 2 ρ 0 2 e ( t 2 t 1 ) 2 2 T 0 2 ,
E ̂ ( ρ , t ) = R 2 d k 2 π d Ω 2 π A ̂ ( k , Ω ) e i k ρ i Ω t ,
[ A ̂ ( k 1 , Ω 1 ) , A ̂ ( k 2 , Ω 2 ) ] = 0 ,
[ A ̂ ( k 1 , Ω 1 ) , A ̂ ( k 2 , Ω 2 ) ] = δ ( k 1 k 2 ) δ ( Ω 1 Ω 2 ) .
A ̂ ( k 1 , Ω 1 ) A ̂ ( k 2 , Ω 2 ) = P T 0 ρ 0 2 2 π e a 0 2 | k d | 2 8 ρ 0 2 | k s | 2 2 e T 0 2 Ω 2 2 2 δ ( Ω 2 Ω 1 ) ,
E ̂ ( ρ 1 , t 1 ) E ̂ ( ρ 2 , t 2 ) = 2 P s π a 0 2 e ( | ρ 1 | 2 + | ρ 2 | 2 ) a 0 2 | ρ 2 ρ 1 | 2 2 ρ 0 2 e ( t 2 t 1 ) 2 2 T 0 2 ,
P s R 2 d ρ E ̂ 2 ( ρ , t )
A ̂ ( k 1 , Ω 1 ) A ̂ ( k 2 , Ω 2 ) = P s T 0 ρ 0 2 2 π e a 0 2 | k s | 2 2 e ρ 0 2 | k d | 2 8 e T 0 2 Ω 2 2 2 δ ( Ω 2 + Ω 1 ) ,
E ̂ L ( ρ 1 , t 1 ) E ̂ L ( ρ 2 , t 2 ) = 2 π P ρ 0 2 λ 0 2 L 2 e i π ( | ρ 1 | 2 | ρ 2 | 2 ) λ 0 L e 2 π 2 ρ 0 2 | ρ s | 2 λ 0 2 L 2 e π 2 a 0 2 | ρ d | 2 2 λ 0 2 L 2 e ( t 2 t 1 ) 2 2 T 0 2 ,
E ̂ L ( ρ 1 , t 1 ) E ̂ L ( ρ 2 , t 2 ) = 2 π P s ρ 0 2 λ 0 2 L 2 e i π ( | ρ 1 | 2 + | ρ 2 | 2 ) λ 0 L e 2 π 2 a T 2 | ρ s | 2 λ 0 2 L 2 e π 2 ρ 0 2 | ρ d | 2 2 λ 0 2 L 2 e ( t 2 t 1 ) 2 2 T 0 2 .
K m , m ( n ) ( ρ 1 , ρ 2 ) R m , m ( n ) ( t 2 t 1 ) = 2 P π a 0 2 e ( | ρ 1 | 2 + | ρ 2 | 2 ) a 0 2 | ρ 2 ρ 1 | 2 2 ρ 0 2 e ( t 2 t 1 ) 2 2 T 0 2 ,
H B ( Ω ) F [ h B ( t ) ] = e 2 Ω 2 Ω B 2 ,
K S , R ( n ) ( ρ 1 , ρ 2 ) R S , R ( n ) ( t 2 t 1 ) = 2 P π a 0 2 e ( | ρ 1 | 2 + | ρ 2 | 2 ) a 0 2 | ρ 2 ρ 1 | 2 2 ρ 0 2 e ( t 2 t 1 ) 2 2 T 0 2 ,
| K 1 , 2 ( n ) ( ρ 1 , ρ 2 ) R 1 , 2 ( n ) ( t 2 t 1 ) | = 2 P π a L 2 e ( | ρ 1 | 2 + | ρ 2 | 2 ) a L 2 | ρ 2 ρ 1 | 2 2 ρ L 2 e ( t 2 t 1 ) 2 2 T 0 2 ,
C ̂ ( ρ 1 ) = C 0 ( ρ 1 ) + C n ( 2 P π a L 2 ) 2 e 2 | ρ 1 | 2 a L 2 A 2 d ρ e | ρ 1 ρ | 2 ρ L 2 e 2 | ρ | 2 a L 2 | T ( ρ ) | 2 .
C ̂ ( ρ 1 ) = q 2 η 2 A 1 ( 2 P π a L 2 ) 2 [ A 2 d ρ | T ( ρ ) | 2 + C t ( n ) A 2 d ρ e | ρ 1 ρ | 2 ρ L 2 | T ( ρ ) | 2 ] ,
C ( n ) max ρ 1 [ C ̂ ( ρ 1 ) ] min ρ 1 [ C ̂ ( ρ 1 ) ] C 0 ( 0 ) .
C ( n ) = C s ( n ) C t ( n ) ,
C s ( n ) = max ρ 1 [ I c ( ρ 1 ) ] min ρ 1 [ I c ( ρ 1 ) ] A 2 d ρ | T ( ρ ) | 2 ,
I c ( ρ 1 ) A 2 d ρ e | ρ 1 ρ | 2 ρ L 2 | T ( ρ ) | 2
C s ( n ) π ρ L 2 A T 1 ,
A T d ρ | T ( ρ ) | 2 ,
K S , R ( p ) ( ρ 1 , ρ 2 ) R S , R ( p ) ( t 2 t 1 ) = 2 P π a 0 2 e ( | ρ 1 | 2 + | ρ 2 | 2 ) a 0 2 | ρ 2 ρ 1 | 2 2 ρ 0 2 e ( t 2 t 1 ) 2 2 T 0 2 ,
| K 1 , 2 ( p ) ( ρ 1 , ρ 2 ) R 1 , 2 ( p ) ( t 2 t 1 ) | = 2 P π a L 2 e ( | ρ 1 | 2 + | ρ 2 | 2 ) a L 2 | ρ 2 + ρ 1 | 2 2 ρ L 2 e ( t 2 t 1 ) 2 2 T 0 2 ,
C ̂ ( ρ 1 ) = C 0 ( ρ 1 ) + C p ( 2 P π a L 2 ) 2 e 2 | ρ 1 | 2 a L 2 A 2 d ρ e | ρ 1 + ρ | 2 ρ L 2 e 2 | ρ | 2 a L 2 | T ( ρ ) | 2 .
C ̂ ( ρ 1 ) = q 2 η 2 A 1 ( 2 P π a L 2 ) 2 [ A 2 d ρ | T ( ρ ) | 2 + C t ( p ) A 2 d ρ e | ρ 1 + ρ | 2 ρ L 2 | T ( ρ ) | 2 ] ,
E ̂ m ( ρ , t ) = A ( ρ ) E ̂ m ( ρ , t ) + L ̂ m ( ρ , t )
A ̂ S ( k , Ω ) = μ ( k , Ω ) a ̂ S ( k , Ω ) + ν ( k , Ω ) a ̂ R ( k , Ω ) ,
A ̂ R ( k , Ω ) = μ ( k , Ω ) a ̂ R ( k , Ω ) + ν ( k , Ω ) a ̂ S ( k , Ω ) .
ν ( k , Ω ) = 2 ( 2 π ) 1 4 P T 0 ρ 0 2 a 0 2 e ρ 0 2 | k | 2 4 T 0 2 Ω 2 4 ,
A ( ρ ) = exp { | ρ | 2 a 0 2 } ,
E ̂ S ( ρ 1 , t 1 ) E ̂ R ( ρ 2 , t 2 ) = 2 P π a 0 2 e ( | ρ 1 | 2 + | ρ 2 | 2 ) a 0 2 × [ i e | ρ 2 ρ 1 | 2 2 ρ 0 2 e ( t 2 t 1 ) 2 2 T 0 2 + ( 2 π ) 1 4 a 0 2 P T 0 ρ 0 2 e | ρ 2 ρ 1 | 2 ρ 0 2 e ( t 2 t 1 ) 2 T 0 2 ] .
C ̂ ( ρ 1 ) = C 0 ( ρ 1 ) + C t ( p ) ( 2 P π a L 2 ) 2 e 2 | ρ 1 | 2 a L 2 A 2 d ρ e | ρ 1 + ρ | 2 ρ L 2 e 2 | ρ | 2 a L 2 | T ( ρ ) | 2 + 2 π a 0 2 P T 0 ρ 0 2 C t ( q ) ( P π a L 2 ) 2 e | ρ 1 | 2 a L 2 A 2 d ρ e | ρ 1 + ρ | 2 ρ L 2 e | ρ | 2 a L 2 | T ( ρ ) | 2 ,
C ̂ ( ρ 1 ) C 0 ( ρ 1 ) + C t ( p ) ( 2 P π a L 2 ) 2 e 2 | ρ 1 | 2 a L 2 A 2 d ρ e | ρ 1 + ρ | 2 ρ L 2 e 2 | ρ | 2 a L 2 | T ( ρ ) | 2 .
C ̂ ( ρ 1 ) C 0 ( ρ 1 ) + 2 π a 0 2 P T 0 ρ 0 2 C t ( q ) ( P π a L 2 ) 2 e | ρ 1 | 2 a L 2 A 2 d ρ e | ρ 1 + ρ | 2 ρ L 2 e | ρ | 2 a L 2 | T ( ρ ) | 2 .
C ̂ ( ρ 1 ) = q 2 η 2 A 1 ( 2 P π a L 2 ) 2 × [ A 2 d ρ | T ( ρ ) | 2 + 1 8 π a 0 2 P T 0 ρ 0 2 C t ( q ) A 2 d ρ e | ρ 1 + ρ | 2 ρ L 2 | T ( ρ ) | 2 ] ,
C ( q ) C t ( q ) C s ( q ) ,
C s ( q ) = 1 8 π a 0 2 P T 0 ρ 0 2 max ρ 1 [ I q ( ρ 1 ) ] min ρ 1 [ I q ( ρ 1 ) ] A 2 d ρ | T ( ρ ) | 2 ,
I q ( ρ 1 ) A 2 d ρ e | ρ 1 + ρ | 2 ρ L 2 | T ( ρ ) | 2
C s ( q ) 1 8 π × π a L 2 P T 0 A T 1 P T 0
C ( q ) Ω B P 1 ,
H B ( Ω ) = F [ h B ( t ) ] = e 2 Ω 2 Ω B 2 e 2 Ω 2 Ω N 2 ,
SNR C ̂ ( ρ 1 ) 2 Δ C ̂ 2 ( ρ 1 ) ,
Δ C ̂ 2 ( ρ 1 ) = 1 T I 2 T I 2 T I 2 d t T I 2 T I 2 d u ı ̂ 1 ( t ) ı ̂ 2 ( t ) ı ̂ 1 ( u ) ı ̂ 2 ( u ) C ̂ ( ρ 1 ) 2 ,
A T d ρ | T ( ρ ) | 4 ,
SNR = 2 π T I T 0 ρ L 2 A T | T ( ρ 1 ) | 4 ,
SNR = 16 2 π T I T 0 η P A 1 Ω B a L 2 η I | T ( ρ 1 ) | 2 .
SNR = π 2 2 Ω B T I ρ L 2 A T | T ( ρ 1 ) | 4 ,
SNR = 4 π Ω B T I η P A 1 Ω B a L 2 η I | T ( ρ 1 ) | 2 .
SNR = 2 π T I T 0 ρ L 2 A T | T ( ρ 1 ) | 4 .
SNR = 8 π T I T 0 η 2 P A 1 | T ( ρ 1 ) | 2 Ω B a L 2 .
η P A T Ω B a L 2 1 ,
SNR = π 2 2 Ω B T I ρ L 2 A T | T ( ρ 1 ) | 4 .
SNR = 1 π Ω B T I η 2 P A 1 | T ( ρ 1 ) | 2 Ω B a L 2 .
T I ( q ) T I ( c ) = π 3 x 8 Ω B a L 2 η 2 P ( q ) A 1 ρ L 2 A T | T ( ρ 1 ) | 2 ,
T I ( q ) T I ( c ) = 2 π 3 Ω B ( q ) a L 2 η 2 P ( q ) A T ρ L 2 A 1 | T ( ρ 1 ) | 2 Ω B ( q ) T 0 ( c ) .
K m , l ( n ) ( ρ 1 , ρ 2 ) = P 2 ( d 2 D λ 0 L ) 2 e i k 0 ( | ρ 2 | 2 | ρ 1 | 2 ) 2 L × ( u = x , y sin ( π d u 1 λ 0 L ) π 0 d u 1 λ 0 L sin ( π d u 2 λ 0 L ) π d u 2 λ 0 L ) ( u = x , y sin [ π D ( u 1 u 2 ) λ 0 L ] sin [ π d ( u 1 u 2 ) λ 0 L ] ) ,
C ̂ ( ρ 1 ) = q 2 η 2 A 1 ( d t h B ( t ) ) 2 K 1 , 1 ( n ) ( ρ 1 , ρ 1 ) A 2 d ρ K 2 , 2 ( n ) ( ρ , ρ ) | T ( ρ ) | 2 + C n A 2 d ρ | K 1 , 2 ( n ) ( ρ 1 , ρ ) | 2 | T ( ρ ) | 2 ,
Δ C ̃ ( ρ 1 ) d τ 1 q η A 1 Δ I ̃ 1 ( ρ 1 , t τ 1 ) h B ( τ 1 ) d τ 2 q η P 2 ( t τ 2 ) h B ( τ 2 ) T I ,

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