Abstract

We compare the performance of high-order thermal ghost imaging with that of conventional (that is, lowest-order) thermal ghost imaging for different data processing methods. Particular attention is given to high-order thermal ghost imaging with background normalization and conventional ghost imaging with background subtraction. The contrast-to-noise ratio (CNR) of the ghost image is used as the figure of merit for the comparison. We find analytically that the CNR of the normalized high-order ghost image is inversely proportional to the square root of the number of transmitting pixels of the object. This scaling law is independent of the exponents used in calculating the high-order correlation and is the same as that of conventional ghost imaging with background subtraction. We find that no data processing procedure performs better than lowest-order ghost imaging with background subtraction. Our results are found to be able to explain the observations of a recent experiment [Chen et al., arXiv:0902.3713v3 [quant-ph]].

© 2010 Optical Society of America

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  1. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
    [CrossRef] [PubMed]
  2. 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]
  3. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).
    [CrossRef]
  4. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, "Two-Photon Imaging with Thermal Light," Phys. Rev. Lett. 94, 063601 (2005).
    [CrossRef] [PubMed]
  5. Y. Bromberg, O. Katz, and Y. Silberberg, "Ghost imaging with a single detector," Phys. Rev. A 79, 053840 (2009).
    [CrossRef]
  6. 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]
  7. 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]
  8. B. I. Erkmen and J. H. Shapiro, "Unified theory of ghost imaging with Gaussian-state light," Phys. Rev. A 77, 043809 (2008).
    [CrossRef]
  9. L.-G. Wang, S. Qamar, S.-Y. Zhu, and M. S. Zubairy, "Hanbury Brown-Twiss effect and thermal light ghost imaging: A unified approach," Phys. Rev. A 79, 033835 (2009).
    [CrossRef]
  10. J. H. Shapiro, "Computational ghost imaging," Phys. Rev. A 78, 061802 (2008).
    [CrossRef]
  11. R. Meyers, K. S. Deacon, and Y. H. Shih, "Ghost-imaging experiment by measuring reflected photons," Phys. Rev. A 77, 041801 (2008).
    [CrossRef]
  12. J. Cheng and S. Han, "Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction," Phys. Rev. Lett. 92, 093903 (2004).
    [CrossRef] [PubMed]
  13. G. Scarcelli, V. Berardi, and Y. Shih, "Phase-conjugate mirror via two-photon thermal light imaging," Appl. Phys. Lett. 88, 061106 (2006).
    [CrossRef]
  14. L. Basano and P. Ottonello, "Experiment in lensless ghost imaging with thermal light," Appl. Phys. Lett. 89, 091109 (2006)
    [CrossRef]
  15. 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]
  16. L. Basano and P. Ottonello, "Use of an intensity threshold to improve the visibility of ghost images produced by incoherent light," Appl. Opt. 46, 6291-6296 (2007).
    [CrossRef] [PubMed]
  17. Y. Bai and S. Han, "Ghost imaging with thermal light by third-order correlation," Phys. Rev. A. 76, 043828 (2007).
    [CrossRef]
  18. L.-H. Ou and L.-M. Kuang, "Ghost imaging with third-order correlated thermal light," J. Phys. B: At. Mol. Opt. Phys. 40, 1833-1844 (2007).
    [CrossRef]
  19. D.-Z. Cao, J. Xiong, S.-H. Zhang, L.-F. Lin, L. Gao, and K. Wang, "Enhancing visibility and resolution in Nth order intensity correlation of thermal light," Appl. Phys. Lett. 92, 201102 (2008).
    [CrossRef]
  20. I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, "High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light," Phys. Rev. A 77, 053801 (2008).
    [CrossRef]
  21. Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, and L.-A. Wu, "Role of multiphoton bunching in high-order ghost imaging with thermal light sources," Phys. Rev. A 79, 053844 (2009).
    [CrossRef]
  22. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, "High-Order Thermal Ghost Imaging," Opt. Lett. 34, 3343-3345 (2009).
    [CrossRef] [PubMed]
  23. 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:0902.3713v3 [quant-ph].
  24. O. Katz, Y. Bromberg, and Y. Silberberg, "Compressive ghost imaging," Appl. Phys. Lett. 95, 131110 (2009).
    [CrossRef]
  25. L. Basano and P. Ottonello, "A conceptual experiment on single-beam coincidence detection with pseudothermal light," Opt. Express 15, 12386-12394 (2007).
    [CrossRef] [PubMed]
  26. D. Cao, J. Xiong, and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005).
    [CrossRef]
  27. Y. Cai and F. Wang, "Lensless imaging with partially coherent light," Opt. Lett. 32, 205-207 (2007).
    [CrossRef] [PubMed]
  28. B. I. Erkmen and J. H. Shapiro, "Signal-to-noise ratio of Gaussian-state ghost imaging," Phys. Rev. A 79, 023833 (2009).
    [CrossRef]
  29. D. Zhang, Y.-H. Zhai, L.-A. Wu, and X.-H. Chen, "Correlated two-photon imaging with true thermal light," Opt. Lett. 30, 2354-2356 (2005).
    [CrossRef] [PubMed]
  30. D. V. Hinkley, "On the Ratio of Two Correlated Normal Random Variables," Biometrika 56, 635-639 (1969).
    [CrossRef]
  31. A. Cedilnik, K. Košmelj, and A. Blejec, "Ratio of Two Random Variables: A Note on the Existence of its Moments," Metodološki zvezki 3, 1-7 (2006).
  32. R. C. Geary, "The Frequency Distribution of the Quotient of Two Normal Variates," J. Roy. Statistical Society 93, 442-446 (1930).
    [CrossRef]
  33. K. N. Boyadzhiev, "Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals," Abstract and Applied Analysis 2009, 168672 (2009).
    [CrossRef]
  34. S. Roman, The Umbral Calculus (Academic Press, New York, 1984), pp. 63-67 and 82-87.

2009 (7)

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

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

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

O. Katz, Y. Bromberg, and Y. Silberberg, "Compressive ghost imaging," Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

K. N. Boyadzhiev, "Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals," Abstract and Applied Analysis 2009, 168672 (2009).
[CrossRef]

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

K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, "High-Order Thermal Ghost Imaging," Opt. Lett. 34, 3343-3345 (2009).
[CrossRef] [PubMed]

2008 (5)

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

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, "High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light," Phys. Rev. A 77, 053801 (2008).
[CrossRef]

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

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

B. I. Erkmen and J. H. Shapiro, "Unified theory of ghost imaging with Gaussian-state light," Phys. Rev. A 77, 043809 (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]

Y. Bai and S. Han, "Ghost imaging with thermal light by third-order correlation," Phys. Rev. A. 76, 043828 (2007).
[CrossRef]

L.-H. Ou and L.-M. Kuang, "Ghost imaging with third-order correlated thermal light," J. Phys. B: At. Mol. Opt. Phys. 40, 1833-1844 (2007).
[CrossRef]

Y. Cai and F. Wang, "Lensless imaging with partially coherent light," Opt. Lett. 32, 205-207 (2007).
[CrossRef] [PubMed]

L. Basano and P. Ottonello, "Use of an intensity threshold to improve the visibility of ghost images produced by incoherent light," Appl. Opt. 46, 6291-6296 (2007).
[CrossRef] [PubMed]

L. Basano and P. Ottonello, "A conceptual experiment on single-beam coincidence detection with pseudothermal light," Opt. Express 15, 12386-12394 (2007).
[CrossRef] [PubMed]

2006 (4)

A. Cedilnik, K. Košmelj, and A. Blejec, "Ratio of Two Random Variables: A Note on the Existence of its Moments," Metodološki zvezki 3, 1-7 (2006).

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

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

G. Scarcelli, V. Berardi, and Y. Shih "Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?" Phys. Rev. Lett. 96, 063602 (2006).
[CrossRef] [PubMed]

2005 (4)

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. 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, and L. A. Lugiato, "High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

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

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

2004 (3)

J. Cheng and S. Han, "Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction," Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation," Phys. Rev. Lett. 93, 093602 (2004);
[CrossRef] [PubMed]

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

1995 (1)

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

1969 (1)

D. V. Hinkley, "On the Ratio of Two Correlated Normal Random Variables," Biometrika 56, 635-639 (1969).
[CrossRef]

1930 (1)

R. C. Geary, "The Frequency Distribution of the Quotient of Two Normal Variates," J. Roy. Statistical Society 93, 442-446 (1930).
[CrossRef]

Agafonov, I. N.

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, "High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light," Phys. Rev. A 77, 053801 (2008).
[CrossRef]

Bache, M.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation," Phys. Rev. Lett. 93, 093602 (2004);
[CrossRef] [PubMed]

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

Bai, Y.

Y. Bai and S. Han, "Ghost imaging with thermal light by third-order correlation," Phys. Rev. A. 76, 043828 (2007).
[CrossRef]

Basano, L.

Berardi, V.

G. Scarcelli, V. Berardi, and Y. Shih "Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?" Phys. Rev. Lett. 96, 063602 (2006).
[CrossRef] [PubMed]

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

Blejec, A.

A. Cedilnik, K. Košmelj, and A. Blejec, "Ratio of Two Random Variables: A Note on the Existence of its Moments," Metodološki zvezki 3, 1-7 (2006).

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]

Boyadzhiev, K. N.

K. N. Boyadzhiev, "Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals," Abstract and Applied Analysis 2009, 168672 (2009).
[CrossRef]

Boyd, R. W.

Brambilla, E.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation," Phys. Rev. Lett. 93, 093602 (2004);
[CrossRef] [PubMed]

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

Bromberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, "Compressive ghost imaging," Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

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

Cai, Y.

Cao, D.

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

Cao, D.-Z.

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

Cedilnik, A.

A. Cedilnik, K. Košmelj, and A. Blejec, "Ratio of Two Random Variables: A Note on the Existence of its Moments," Metodološki zvezki 3, 1-7 (2006).

Chan, K. W. C.

Chekhova, M. V.

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, "High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light," Phys. Rev. A 77, 053801 (2008).
[CrossRef]

Chen, X.-H.

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

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

Cheng, J.

J. Cheng and S. Han, "Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction," Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef] [PubMed]

D’Angelo, M.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, "Two-Photon Imaging with Thermal Light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

Deacon, K. S.

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

Erkmen, B. I.

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

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.

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]

Gao, L.

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

Gatti, 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]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation," Phys. Rev. Lett. 93, 093602 (2004);
[CrossRef] [PubMed]

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

Geary, R. C.

R. C. Geary, "The Frequency Distribution of the Quotient of Two Normal Variates," J. Roy. Statistical Society 93, 442-446 (1930).
[CrossRef]

Han, S.

Y. Bai and S. Han, "Ghost imaging with thermal light by third-order correlation," Phys. Rev. A. 76, 043828 (2007).
[CrossRef]

J. Cheng and S. Han, "Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction," Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef] [PubMed]

Hinkley, D. V.

D. V. Hinkley, "On the Ratio of Two Correlated Normal Random Variables," Biometrika 56, 635-639 (1969).
[CrossRef]

Iskhakov, T. Sh.

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, "High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light," Phys. Rev. A 77, 053801 (2008).
[CrossRef]

Katz, O.

O. Katz, Y. Bromberg, and Y. Silberberg, "Compressive ghost imaging," Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

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

Košmelj, K.

A. Cedilnik, K. Košmelj, and A. Blejec, "Ratio of Two Random Variables: A Note on the Existence of its Moments," Metodološki zvezki 3, 1-7 (2006).

Kuang, L.-M.

L.-H. Ou and L.-M. Kuang, "Ghost imaging with third-order correlated thermal light," J. Phys. B: At. Mol. Opt. Phys. 40, 1833-1844 (2007).
[CrossRef]

Lin, L.-F.

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

Liu, Q.

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

Lugiato, L. 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]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation," Phys. Rev. Lett. 93, 093602 (2004);
[CrossRef] [PubMed]

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

Luo, K.-H.

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

Magatti, D.

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]

Meyers, R.

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

O’Sullivan, M. N.

Ottonello, P.

Ou, L.-H.

L.-H. Ou and L.-M. Kuang, "Ghost imaging with third-order correlated thermal light," J. Phys. B: At. Mol. Opt. Phys. 40, 1833-1844 (2007).
[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]

Penin, A. N.

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, "High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light," Phys. Rev. A 77, 053801 (2008).
[CrossRef]

Pittman, T. B.

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

Qamar, S.

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

Scarcelli, G.

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

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. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, "Two-Photon Imaging with Thermal Light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

Sergienko, A. 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]

Shapiro, J. H.

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

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

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

Shih, Y.

G. Scarcelli, V. Berardi, and Y. Shih "Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?" Phys. Rev. Lett. 96, 063602 (2006).
[CrossRef] [PubMed]

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

Shih, Y. H.

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

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, "Two-Photon Imaging with Thermal Light," Phys. Rev. Lett. 94, 063601 (2005).
[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]

Silberberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, "Compressive ghost imaging," Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

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]

Valencia, A.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, "Two-Photon Imaging with Thermal Light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

Wang, F.

Wang, K.

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

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

Wang, L.-G.

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

Wu, L.-A.

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

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

Wu, W.

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

Xiong, J.

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

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

Zhai, Y.-H.

Zhang, D.

Zhang, S.-H.

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

Zhu, S.-Y.

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

Zubairy, M. S.

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

Abstract and Applied Analysis (1)

K. N. Boyadzhiev, "Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals," Abstract and Applied Analysis 2009, 168672 (2009).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

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

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

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

O. Katz, Y. Bromberg, and Y. Silberberg, "Compressive ghost imaging," Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Biometrika (1)

D. V. Hinkley, "On the Ratio of Two Correlated Normal Random Variables," Biometrika 56, 635-639 (1969).
[CrossRef]

J. Phys. B: At. Mol. Opt. Phys. (1)

L.-H. Ou and L.-M. Kuang, "Ghost imaging with third-order correlated thermal light," J. Phys. B: At. Mol. Opt. Phys. 40, 1833-1844 (2007).
[CrossRef]

J. Roy. Statistical Society (1)

R. C. Geary, "The Frequency Distribution of the Quotient of Two Normal Variates," J. Roy. Statistical Society 93, 442-446 (1930).
[CrossRef]

Metodološki zvezki (1)

A. Cedilnik, K. Košmelj, and A. Blejec, "Ratio of Two Random Variables: A Note on the Existence of its Moments," Metodološki zvezki 3, 1-7 (2006).

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (11)

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

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

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

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, "High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light," Phys. Rev. A 77, 053801 (2008).
[CrossRef]

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

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

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

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

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

Phys. Rev. A. (1)

Y. Bai and S. Han, "Ghost imaging with thermal light by third-order correlation," Phys. Rev. A. 76, 043828 (2007).
[CrossRef]

Phys. Rev. Lett. (6)

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]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, "High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light," Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef] [PubMed]

J. Cheng and S. Han, "Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction," Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef] [PubMed]

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, "Two-Photon Imaging with Thermal Light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation," Phys. Rev. Lett. 93, 093602 (2004);
[CrossRef] [PubMed]

Other (2)

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:0902.3713v3 [quant-ph].

S. Roman, The Umbral Calculus (Academic Press, New York, 1984), pp. 63-67 and 82-87.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematic of the lenless thermal ghost imaging setup. (b) The simplified object model for analysis. The object plane (left) and the reference detector plane (right) are discretized by pixels of finite sizes. BS: beam-splitter.

Fig. 2.
Fig. 2.

Numerical simulations. (a) The object mask (50 × 50 pixels). (b) The averaged speckle pattern M[Ir (x⃗)]. (c) The conventional ghost image g 1,1 (x⃗). (d) The optimal high-order ghost image g opt(x⃗) = g 20,2(x⃗). (e) The normalized conventional ghost image g1,1(x⃗). (f) The normalized high-order ghost image g20,2(x⃗). The number of transmitting pixels is T = 410 and the number of samplings is N = 150000. The latter is chosen to be large to make the ghost image in (c) visible.

Fig. 3.
Fig. 3.

The probability density functions (pdf) of gmn and gmn ′ inside and outside the transmitting regions of the object with Pi (gmn ) ≡P(gmn (x⃗i )), i = in or out. (a) The pdf for the conventional ghost image Pi (g 1,1). (b) The pdf for the optimal high-order ghost image Pi (g 20,2). (c) The pdf for the normalized conventional ghost image Pi (g1,1). (d) The pdf for the normalized high-order ghost image Pi (g20,2). The number of transmitting pixels is T =410 and the number of samplings is N = 150000. These give ⟨g 1,1 (x⃗ in)⟩ = ⟨g 1,1′ (x⃗ in)⟩ = 1.0024 and ⟨g 20,2(x⃗)⟩ = ⟨g20,2(x⃗ in)) = 1.0998. The data points are accumulated by repeating the simulations for 500 times. The solid curves (overlapping with the numerical data) are the theoretical results using Eq. (20).

Fig. 4.
Fig. 4.

Plots of the contrast-to-noise ratio CNR(gmn ′) (normalized by √N) against the number of transmitting pixels T for various values of m and n. The theoretical curves are plotted using Eq. (29). The data points are from simulations using N = 20000. Also shown are CNR(G) from Eq. (8), CNR(G′) from Eq. (10) and CNR(g opt) from Ref. [22]. We note that under all circumstances the function G′ produces the best image, with procedures g1,1 and g2,2 performing nearly as well.

Equations (47)

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G ( x ) = M [ I o I r ( x ) ] ,
M [ X ] 1 N s = 1 N X ( s )
I o = 1 Δ beam area d 2 y O ( y ) I ( y ) y 𝓣 I ( y )
I r ( x ) = 1 Δ area of pixel at x d 2 yI ( y ) I ( x )
CNR ( G ) G ( x in ) G ( x out ) 1 2 [ Δ 2 G ( x in ) + Δ 2 G ( x out ) ] ,
Δ 2 G ( x ) = 1 N [ I o 2 I r 2 ( x ) I o I r ( x ) 2 ] .
I o m I r n ( x ) = { Γ ( T + m + n ) n ! Γ ( T + n ) μ m + n , for x = x in , Γ ( T + m ) n ! Γ ( T ) μ m + n , for x = x out ,
CNR ( G ) = N T 2 + 5 T + 11 / 2 .
G ( x ) = M [ ( I o M [ I o ] ) ( I r ( x ) M [ I r ( x ) ] ) ]
= M [ I o I r ( x ) ] M [ I o ] M [ I r ( x ) ] .
CNR ( G ) = N 1 T + 7 / 2 3 / N .
g 1,1 ( x ) = M [ I o I r ( x ) ] M [ I o ] M [ I r ( x ) ] .
CNR ( g 1,1 ) N 2 T + 3 .
CNR ( G ) N T + 7 / 2 .
g mn ( full ) ( x ) = M [ I o m I r n ( x ) ] M [ I o m ] M [ I r n ( x ) ] .
g mn ( x ) = 1 C m B n M [ I o m I r n ( x ) ] ,
g mn ( x ) = 1 C m M [ I o m I r n ( x ) ] M [ I r n ( x ) ] .
A mn ( x ) = I o m I r n ( x ) , B n = I r n ( x ) ,
α mn 2 ( x ) = Δ 2 [ I o m I r n ( x ) ] N , β n 2 = Δ 2 [ I r n ( x ) ] N ,
t mn ( x ) = B n C m g mn ( x ) A mn ( x ) β n 2 C m 2 g mn 2 ( x ) 2 ρ mn ( x ) C m g mn ( x ) + α mn 2 ( x )
P ( t mn ( x ) ) = 1 2 π exp [ t mn 2 ( x ) 2 ] .
A mn ( x ) = { Γ ( T + m + n ) n ! Γ ( T + n ) μ m + n , for x = x in , Γ ( T + m ) n ! Γ ( T ) μ m + n , for x = x out ,
B n = n ! μ n , C m = Γ ( T + m ) Γ ( T ) μ m ,
α mn 2 ( x ) = A 2 m 2 n ( x ) A mn 2 ( x ) N , β n 2 = B 2 n B n 2 N ,
ρ mn ( x ) = A m 2 n ( x ) A mn ( x ) B n N .
P ( g mn ( x ) ) 1 2 π Δ mn ( x ) exp [ [ g mn ( x ) g mn ( x ) ] 2 2 Δ mn 2 ( x ) ] .
g mn ( x ) = A mn ( x ) C m B n ( x ) = { Γ ( T ) Γ ( T + m + n ) Γ ( T + m ) Γ ( T + n ) , for x = x in , 1 , for x = x out,
Δ mn 2 ( x in ) = ( 2 n ) ! N ( n ! ) 2 [ Γ ( T + 2 m + 2 n ) Γ ( T ) 2 Γ ( T + m ) 2 Γ ( T + 2 n ) + Γ ( T + m + n ) 2 Γ ( T ) 2 Γ ( T + m ) 2 Γ ( T + n ) 2
2 Γ ( T + m + n ) Γ ( T + m + 2 n ) Γ ( T ) 2 Γ ( T + m ) 2 Γ ( T + n ) Γ ( T + 2 n ) ] ,
Δ mn 2 ( x out ) = ( 2 n ) ! N ( n ! ) 2 [ Γ ( T + 2 m ) Γ ( T ) Γ ( T + m ) 2 1 ] .
CNR ( g mn ) = g mn ( x in ) g mn ( x out ) 1 2 [ Δ mn 2 ( x in ) + Δ mn 2 ( x out ) ] .
g ' mn ( x in ) ~ 1 + mn T , g mn ( x out ) = 1 ,
Δ mn 2 ( x in ) Δ mn 2 ( x out ) ~ m 2 ( 2 n ) ! NT ( n ! ) 2 .
CNR ( g mn ) ~ N T [ n ( n ! ) ( 2 n ) ! ] .
V g mn ( x in ) g mn ( x out ) g mn ( x in ) + g mn ( x out ) tanh ( mn 2 T ) .
i a = e n Δ I a μ ,
n Δ = ημ Δ τ hv ,
i a n shot noise = e n ϕ n ( n Δ μ I a ) e n ( n Δ μ ) n { I a } n ,
I o m I r n ( x ) { I o } m { I r ( x ) } n
= ( μ n Δ ) m + n ϕ m ( n Δ μ I o ) ϕ n ( n Δ μ I r ( x ) ) ,
( μ n Δ ) m + n ϕ m ( n Δ μ I o ) ϕ n ( n Δ μ I r ( x ) ) = ( μ n Δ ) m + n i = 0 m j = 0 n S m i S n j I o i I r j ( x ) μ i + j n Δ i + j ,
G ( sn ) ' ( x ) = ( μ n Δ ) 2 ϕ 1 ( n Δ μ I o n Δ μ I o ) ϕ 1 ( n Δ μ I r ( x ) n Δ μ I r ( x ) )
Δ 2 G ( sn ) ( x ) = 1 N { ( μ n Δ ) 4 ϕ 2 ( n Δ μ I o n Δ μ I o ) ϕ 2 ( n Δ μ I r ( x ) n Δ μ I r ( x ) )
[ G ( sn ) ( x ) ] 2 } .
G ( sn ) ( x ) ΔG ( sn ) ( x ) = NO ( x ) [ T + 7 O ( x ) ] + O ( x ) [ 4 n Δ 1 + n Δ 2 ] ,
CNR ( G ( sn ) ) = N ( T + 7 / 2 ) + ( 4 n Δ 1 + n Δ 2 ) / 2 .
n Δ = ημ Δ τ hc / λ = 10 9 / ( 5 × 10 6 ) × ( 10 × 10 12 ) ( 6.63 × 10 34 ) × ( 3 × 10 8 ) / ( 780 × 10 9 ) ~ 7800

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