Abstract

We show how quantum entangled biphoton states can be used to realize ghost scattering, a nonlocal scheme to obtain scattering information of an unknown object through the correlation measurement of the scattering photons in two different optical paths. We present a framework to describe the biphoton ghost scattering process from the T-matrix formula of the scattering theory. We find the scattering information of a test object can be retrieved from either the test arm or the reference arm. By adjusting the biphoton states, the ghost scattering patterns may be varied from the scattering pattern of the object in the test arm to the object in the reference arm.

© 2016 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (7)

L. Nie, Y. Bai, and X. Fu, “Ghost telescope imaging system from the perspective of coherent-mode representation,” Opt. Commun. 358, 88–91 (2016).
[Crossref]

H. Yu, R. H. Lu, S. S. Han, H. L. Xie, G. H. Du, T. Q. Xiao, and D. M. Zhu, “Fourier-transform ghost imaging with hard x-rays,” Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganini, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117, 113902 (2016).
[Crossref]

L. Wang and S. M. Zhao, “Fast reconstructed and high-quality ghost imaging with fast Walsh–Hadamard transform,” Photon. Res. 4, 240–244 (2016).
[Crossref]

J. Cheng, “Theory of ghost scattering with incoherent light sources,” Phys. Rev. A 93, 043808 (2016).
[Crossref]

I. Starshynov, J. Bertolotti, and J. Anders, “Quantum correlation of light scattered by disordered media,” Opt. Express 24, 4662–4671 (2016).
[Crossref]

J. C. Schotland, A. Caze, and T. B. Norris, “Scattering of entangled two-photon states,” Opt. Lett. 41, 444–447 (2016).
[Crossref]

2015 (6)

2014 (2)

2013 (4)

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, “Time correspondence differential ghost imaging,” Phys. Rev. A 87, 033813 (2013).
[Crossref]

C. L. Luo and J. Cheng, “Ghost imaging with shaped incoherent sources,” Opt. Lett. 38, 5381–5384 (2013).
[Crossref]

J. Cheng and J. Lin, “Unified theory of thermal ghost imaging and ghost diffraction through turbulent atmosphere,” Phys. Rev. A 87, 043810 (2013).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

2012 (2)

R. E. Meyers, K. S. Deacon, and Y. H. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100, 131114 (2012).
[Crossref]

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

2011 (3)

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[Crossref]

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

2010 (3)

P. Zhang, W. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010), and references therein.
[Crossref]

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref]

2009 (3)

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

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

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

2008 (3)

J. Cheng, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823 (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 (2)

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

J. Cheng and S. S. Han, “Classical correlated imaging from the perspective of coherent-mode representation,” Phys. Rev. A 76, 023824 (2007).
[Crossref]

2006 (2)

G. Scarcelli, V. Berardi, and Y. H. 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]

2005 (1)

2004 (2)

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

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

2002 (1)

S. Savasta, O. D. Stefano, and R. Girlanda, “Light quantization for arbitrary scattering systems,” Phys. Rev. A 65, 043801 (2002).
[Crossref]

1995 (2)

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[Crossref]

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]

Akhlaghi, M. I.

Anders, J.

Andersen, U. L.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

Bache, M.

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]

Bai, Y.

L. Nie, Y. Bai, and X. Fu, “Ghost telescope imaging system from the perspective of coherent-mode representation,” Opt. Commun. 358, 88–91 (2016).
[Crossref]

Basano, L.

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

Berardi, V.

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

Bertolotti, J.

Body, R. W.

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

Bowman, A.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Bowman, R.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Brambilla, E.

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]

Bromberg, Y.

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

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

Cantelli, V.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganini, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117, 113902 (2016).
[Crossref]

Caze, A.

Chan, K. W.

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Chen, X. H.

Cheng, J.

J. Cheng, “Theory of ghost scattering with incoherent light sources,” Phys. Rev. A 93, 043808 (2016).
[Crossref]

C. L. Luo, H. H. Xu, and J. Cheng, “High-resolution ghost imaging experiments with Cosh–Gaussian modulated incoherent sources,” J. Opt. Soc. Am. A 32, 482–485 (2015).
[Crossref]

C. L. Luo and J. Cheng, “Ghost imaging with shaped incoherent sources,” Opt. Lett. 38, 5381–5384 (2013).
[Crossref]

J. Cheng and J. Lin, “Unified theory of thermal ghost imaging and ghost diffraction through turbulent atmosphere,” Phys. Rev. A 87, 043810 (2013).
[Crossref]

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

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

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

J. Cheng and S. S. Han, “Classical correlated imaging from the perspective of coherent-mode representation,” Phys. Rev. A 76, 023824 (2007).
[Crossref]

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

Deacon, K. S.

R. E. Meyers, K. S. Deacon, and Y. H. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100, 131114 (2012).
[Crossref]

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

Dixon, P. B.

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

Dogariu, A.

Du, G. H.

H. Yu, R. H. Lu, S. S. Han, H. L. Xie, G. H. Du, T. Q. Xiao, and D. M. Zhu, “Fourier-transform ghost imaging with hard x-rays,” Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Duan, D.

Eberly, J. H.

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Edgar, M. P.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Elsasser, W.

Erkmen, B. I.

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010), and references therein.
[Crossref]

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

Fan, H.

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, “Time correspondence differential ghost imaging,” Phys. Rev. A 87, 033813 (2013).
[Crossref]

Fu, X.

L. Nie, Y. Bai, and X. Fu, “Ghost telescope imaging system from the perspective of coherent-mode representation,” Opt. Commun. 358, 88–91 (2016).
[Crossref]

Gatti, A.

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]

Girlanda, R.

S. Savasta, O. D. Stefano, and R. Girlanda, “Light quantization for arbitrary scattering systems,” Phys. Rev. A 65, 043801 (2002).
[Crossref]

Gong, W.

P. Zhang, W. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Gong, W. L.

Han, S. S.

H. Yu, R. H. Lu, S. S. Han, H. L. Xie, G. H. Du, T. Q. Xiao, and D. M. Zhu, “Fourier-transform ghost imaging with hard x-rays,” Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

H. Yu, E. R. Li, W. L. Gong, and S. S. Han, “Structured image reconstruction for three-dimensional ghost imaging lidar,” Opt. Express 23, 14541–14551 (2015).
[Crossref]

P. Zhang, W. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

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

J. Cheng and S. S. Han, “Classical correlated imaging from the perspective of coherent-mode representation,” Phys. Rev. A 76, 023824 (2007).
[Crossref]

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

Hardy, N. D.

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[Crossref]

Hartmann, S.

Howell, J. C.

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Howland, G.

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

Huck, A.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

Katz, O.

Y. Bromberg, O. Katz, 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]

Klyshko, D. N.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[Crossref]

Li, E. R.

Li, H. G.

D. J. Zhang, H. G. Li, Q. L. Zhao, S. Wang, H. B. Wang, J. Xiong, and K. G. Wang, “Wavelength-multiplexing ghost imaging,” Phys. Rev. A 92, 013823 (2015).
[Crossref]

Li, M. F.

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, “Time correspondence differential ghost imaging,” Phys. Rev. A 87, 033813 (2013).
[Crossref]

Lin, J.

J. Cheng and J. Lin, “Unified theory of thermal ghost imaging and ghost diffraction through turbulent atmosphere,” Phys. Rev. A 87, 043810 (2013).
[Crossref]

Liu, H. L.

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

Liu, X. F.

Liu, Y. F.

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

Lodahl, P.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

Lu, R. H.

H. Yu, R. H. Lu, S. S. Han, H. L. Xie, G. H. Du, T. Q. Xiao, and D. M. Zhu, “Fourier-transform ghost imaging with hard x-rays,” Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Lugiato, L. A.

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]

Luo, C. L.

Luo, K. H.

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, “Time correspondence differential ghost imaging,” Phys. Rev. A 87, 033813 (2013).
[Crossref]

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]

Meyers, R. E.

R. E. Meyers, K. S. Deacon, and Y. H. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100, 131114 (2012).
[Crossref]

Moerman, J. J. D.

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref]

Molitor, A.

Nie, L.

L. Nie, Y. Bai, and X. Fu, “Ghost telescope imaging system from the perspective of coherent-mode representation,” Opt. Commun. 358, 88–91 (2016).
[Crossref]

Norris, T. B.

Ott, J. R.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

Ottonello, P.

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

Padgett, M. J.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Paganini, D. M.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganini, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117, 113902 (2016).
[Crossref]

Peeters, W. H.

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref]

Pelliccia, D.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganini, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117, 113902 (2016).
[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]

Rack, A.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganini, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117, 113902 (2016).
[Crossref]

Rodenburg, B.

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

Savasta, S.

S. Savasta, O. D. Stefano, and R. Girlanda, “Light quantization for arbitrary scattering systems,” Phys. Rev. A 65, 043801 (2002).
[Crossref]

Scarcelli, G.

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

Scheel, M.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganini, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117, 113902 (2016).
[Crossref]

Schotland, J. C.

Sergienko, A. V.

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[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]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[Crossref]

Shapiro, J. H.

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[Crossref]

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010), and references therein.
[Crossref]

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

Shaprio, J. H.

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Shen, X.

P. Zhang, W. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

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

Shih, Y. H.

R. E. Meyers, K. S. Deacon, and Y. H. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100, 131114 (2012).
[Crossref]

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

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

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

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[Crossref]

Silberberg, Y.

Y. Bromberg, O. Katz, 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]

Simon, D. S.

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Smolka, S.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

Starshynov, I.

Stefano, O. D.

S. Savasta, O. D. Stefano, and R. Girlanda, “Light quantization for arbitrary scattering systems,” Phys. Rev. A 65, 043801 (2002).
[Crossref]

Strekalov, D. V.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[Crossref]

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]

Sullivan, M. N. O.

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Sullivan-Hale, M. N. O.

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

Sun, B.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

van Exter, M. P.

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref]

Vittert, L. E.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Wang, H. B.

D. J. Zhang, H. G. Li, Q. L. Zhao, S. Wang, H. B. Wang, J. Xiong, and K. G. Wang, “Wavelength-multiplexing ghost imaging,” Phys. Rev. A 92, 013823 (2015).
[Crossref]

Wang, K. G.

D. J. Zhang, H. G. Li, Q. L. Zhao, S. Wang, H. B. Wang, J. Xiong, and K. G. Wang, “Wavelength-multiplexing ghost imaging,” Phys. Rev. A 92, 013823 (2015).
[Crossref]

Wang, L.

Wang, S.

D. J. Zhang, H. G. Li, Q. L. Zhao, S. Wang, H. B. Wang, J. Xiong, and K. G. Wang, “Wavelength-multiplexing ghost imaging,” Phys. Rev. A 92, 013823 (2015).
[Crossref]

Wei, Q.

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

Welsh, S.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Wu, L. A.

Xia, Y. J.

Xiao, T. Q.

H. Yu, R. H. Lu, S. S. Han, H. L. Xie, G. H. Du, T. Q. Xiao, and D. M. Zhu, “Fourier-transform ghost imaging with hard x-rays,” Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Xie, H. L.

H. Yu, R. H. Lu, S. S. Han, H. L. Xie, G. H. Du, T. Q. Xiao, and D. M. Zhu, “Fourier-transform ghost imaging with hard x-rays,” Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Xiong, J.

D. J. Zhang, H. G. Li, Q. L. Zhao, S. Wang, H. B. Wang, J. Xiong, and K. G. Wang, “Wavelength-multiplexing ghost imaging,” Phys. Rev. A 92, 013823 (2015).
[Crossref]

Xu, H. H.

Yao, X. R.

Yu, H.

H. Yu, R. H. Lu, S. S. Han, H. L. Xie, G. H. Du, T. Q. Xiao, and D. M. Zhu, “Fourier-transform ghost imaging with hard x-rays,” Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

H. Yu, E. R. Li, W. L. Gong, and S. S. Han, “Structured image reconstruction for three-dimensional ghost imaging lidar,” Opt. Express 23, 14541–14551 (2015).
[Crossref]

Yu, W. K.

Zhai, G. J.

Zhai, Y. H.

Zhang, D.

Zhang, D. J.

D. J. Zhang, H. G. Li, Q. L. Zhao, S. Wang, H. B. Wang, J. Xiong, and K. G. Wang, “Wavelength-multiplexing ghost imaging,” Phys. Rev. A 92, 013823 (2015).
[Crossref]

Zhang, M. H.

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

Zhang, P.

P. Zhang, W. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Zhang, Y. R.

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, “Time correspondence differential ghost imaging,” Phys. Rev. A 87, 033813 (2013).
[Crossref]

Zhao, Q. L.

D. J. Zhang, H. G. Li, Q. L. Zhao, S. Wang, H. B. Wang, J. Xiong, and K. G. Wang, “Wavelength-multiplexing ghost imaging,” Phys. Rev. A 92, 013823 (2015).
[Crossref]

Zhao, S. M.

Zhu, D. M.

H. Yu, R. H. Lu, S. S. Han, H. L. Xie, G. H. Du, T. Q. Xiao, and D. M. Zhu, “Fourier-transform ghost imaging with hard x-rays,” Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (4)

G. Scarcelli, V. Berardi, and Y. H. 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]

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

R. E. Meyers, K. S. Deacon, and Y. H. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100, 131114 (2012).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

L. Nie, Y. Bai, and X. Fu, “Ghost telescope imaging system from the perspective of coherent-mode representation,” Opt. Commun. 358, 88–91 (2016).
[Crossref]

Opt. Express (2)

Opt. Express. (1)

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

Opt. Lett. (6)

Photon. Res. (2)

Phys. Rev. A (17)

J. Cheng and J. Lin, “Unified theory of thermal ghost imaging and ghost diffraction through turbulent atmosphere,” Phys. Rev. A 87, 043810 (2013).
[Crossref]

J. Cheng, “Theory of ghost scattering with incoherent light sources,” Phys. Rev. A 93, 043808 (2016).
[Crossref]

S. Savasta, O. D. Stefano, and R. Girlanda, “Light quantization for arbitrary scattering systems,” Phys. Rev. A 65, 043801 (2002).
[Crossref]

D. J. Zhang, H. G. Li, Q. L. Zhao, S. Wang, H. B. Wang, J. Xiong, and K. G. Wang, “Wavelength-multiplexing ghost imaging,” Phys. Rev. A 92, 013823 (2015).
[Crossref]

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

P. Zhang, W. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[Crossref]

P. B. Dixon, G. Howland, K. W. Chan, M. N. O. Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shaprio, D. S. Simon, A. V. Sergienko, R. W. Body, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83, 051803 (2011).
[Crossref]

K. W. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shaprio, P. B. Dixon, G. Howland, J. C. Howell, J. H. Eberly, M. N. O. Sullivan, B. Rodenburg, and R. W. Body, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, “Time correspondence differential ghost imaging,” Phys. Rev. A 87, 033813 (2013).
[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]

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

J. Cheng and S. S. Han, “Classical correlated imaging from the perspective of coherent-mode representation,” Phys. Rev. A 76, 023824 (2007).
[Crossref]

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

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

Phys. Rev. Lett. (6)

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

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

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[Crossref]

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref]

H. Yu, R. H. Lu, S. S. Han, H. L. Xie, G. H. Du, T. Q. Xiao, and D. M. Zhu, “Fourier-transform ghost imaging with hard x-rays,” Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganini, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117, 113902 (2016).
[Crossref]

Science (1)

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

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Figures (2)

Fig. 1.
Fig. 1. Geometry of a biphoton ghost scattering system. A pump beam illuminates a nonlinear crystal (NLC) to generate the biphoton quantum states. The biphoton state is then split by a beam splitter, leading to the separation of the test and reference photons (with wave vectors k 1 , 2 ). The test photon illuminates an unknown object and is scattered into the far field, measured by the test detector located at x t . The other reference photon is scattered by a known reference object and collected by the reference detector D r at x r . Corresponding wave vectors of the scattered photons are k t and k r . The second-order coherence function can be measured by the correlator.
Fig. 2.
Fig. 2. (a) T-matrix of the test object | T ( k , k i ) | 2 . (b) T-matrix of the reference object | R ( k , k i ) | 2 . (c), (e), and (g) Direct scattering patterns G t ( k ) for the small-, medium-, and large-sized incident states. (d), (f), and (h) Ghost scattering patterns G r ( k ) for the small-, medium-, and large-sized incident states. Detailed explanations and parameters are given in the main text.

Equations (12)

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| ψ i = d k d k A i ( k , k ) a ^ k a ^ k | 0 = d x d x A i ( x , ω ; x , ω ) a ^ x a ^ x | 0 ,
A s ( x , ω ; x , ω ) = d 3 x 1 d 3 x 2 d 3 x 1 d 3 x 2 G t ( x , x 1 ) T t ( x 1 , x 2 ; ω ) G r ( x , x 1 ) T r ( x 1 , x 2 ; ω ) A i ( x 2 , ω ; x 2 , ω ) ,
T s ( x 1 , x 2 ) = k s 2 η s ( x 1 ) δ ( x 1 x 2 ) + k s 2 η s ( x 1 ) d 3 x G s ( x 1 , x ) T s ( x , x 2 ) ,
A s ( x , ω ; x , ω ) = e j k t x x e j k r x x d 3 x 1 d 3 x 2 d 3 x 1 d 3 x 2 e j k t x ^ · x 1 T t ( x 1 , x 2 ; ω ) e j k r x ^ · x 1 T r ( x 1 , x 2 ; ω ) A i ( x 2 , ω ; x 2 , ω ) = e j k t x x e j k r x x d k d k T ( k t , k ) R ( k r , k ) A i ( k , k ) = e j k t x x e j k r x x A s ( k t , k r ) ,
T ( k t , k ) = d 3 x 1 d 3 x 2 e i ( k t · x 1 k · x 2 ) T t ( x 1 , x 2 ; ω )
R ( k r , k ) = d 3 x 1 d 3 x 2 e i ( k r · x 1 k · x 2 ) T r ( x 1 , x 2 ; ω )
A s ( k t , k r ) = d k d k T ( k t , k ) R ( k r , k ) A i ( k , k )
G ( k t , k r ) = | d k d k T ( k t , k ) R ( k r , k ) A i ( k , k ) | 2 .
A i ( k , k ) = U ( k ) δ ( k k ) ,
G t ( k ) = d k i T ( k , k i ) U ( k i ) R ( k 0 , k i ) ,
G r ( k ) = d k i T ( k 0 , k i ) U ( k i ) R ( k , k i ) ,
U ( k ) = { 1 , k x 2 + k y 2 < α k 0 , other

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