Abstract

The point-spread function (PSF) is fundamental importance in estimating the imaging resolution in optical imaging systems. By using the Collins formula, a analytical imaging formula for ghost imaging system is obtained. The intensity fluctuation correlation function can be viewed as the convolution of the original object and a PSF. The imaging resolution is determined by the width of PSF. Based on the optical transfer matrix theory, we present the analytical formula describing the width of the PSF, by which one can estimate imaging resolution of a new-designed imaging scheme when compared with that of the existing imaging system. Several typical ghost imaging systems are chosen to verify experimentally our theoretical results.

© 2016 Optical Society of America

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References

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2017 (1)

X. Zeng, Y. Bai, X. Shi, Y. Gao, and X. Fu, “The influence of the positive and negative defocusing on lensless ghost imaging,” Opt. Communications 382, 415 (2017).
[Crossref]

2016 (1)

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

2014 (1)

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

2013 (4)

2012 (3)

J. Du, W. Gong, and S. Han, “The influence of sparsity property of images on ghost imaging with thermal light,” Opt. Lett. 37, 1067 (2012).
[Crossref] [PubMed]

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

2011 (4)

W. L. Gong and S. S. Han, “Ghost telescope and ghost Fourier telescope with thermal light,” Phys. Lett. A 375, 990 (2011).
[Crossref]

R. E. Meyers, K. S. Deacon, and Yanhua Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011).
[Crossref]

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

W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett. 36, 394 (2011).
[Crossref] [PubMed]

2010 (5)

X. Chen, I. N. Agafonov, K. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L. Wu, “High-visibility, high-order lensless ghost imaging with thermal light,” Opt. Lett. 35, 1166(2010).
[Crossref] [PubMed]

L. Jiying, Z. Jubo, L. Chuan, and H. Shisheng, “High-quality quantum-imaging algorithm and experiment based on compressive sensing,” Opt. Lett. 35, 1206 (2010).
[Crossref] [PubMed]

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “The influence of source and object characteristics on coincidence imaging,” J. Opt. 12, 045701 (2010).
[Crossref]

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

S. Karmakar and Y. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81, 033845 (2010).
[Crossref]

2009 (5)

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

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost “pinhole” imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

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

X. Chen, Q. Liu, K. Luo, and L. Wu, “Lensless ghost imaging with true thermal light,” Opt. Lett. 34, 695 (2009).
[Crossref] [PubMed]

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

2008 (2)

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

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

2007 (2)

Y. Cai and F. Wang, “Lensless imaging with partially coherent light,” Opt. Lett. 32, 205 (2007).
[Crossref] [PubMed]

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

2006 (1)

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

2005 (6)

Y. Zhai, X. Chen, D. Zhang, and L. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A 72, 043805 (2005).
[Crossref]

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

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

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

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

2004 (3)

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

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost Imaging with Thermal Light, Comparing Entanglement and Classical Correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

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

2001 (1)

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of Entanglement in Two-Photon Imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

1996 (2)

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804 (1996).
[Crossref] [PubMed]

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can Two-Photon Interference be Considered the Interference of Two Photons,” Phys. Rev. Lett. 77, 1917 (1996).
[Crossref] [PubMed]

1995 (1)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Opticla imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, 3429 (1995).
[Crossref]

1970 (1)

Abmann, M.

M. Abmann and M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3, 1545 (2013).
[Crossref] [PubMed]

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of Entanglement in Two-Photon Imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

Agafonov, I. N.

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, “Correlated imaging, quantum and classical,” Phys. Rev. A 70, 013802 (2004).
[Crossref]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost Imaging with Thermal Light, Comparing Entanglement and Classical Correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Bai, Y.

X. Zeng, Y. Bai, X. Shi, Y. Gao, and X. Fu, “The influence of the positive and negative defocusing on lensless ghost imaging,” Opt. Communications 382, 415 (2017).
[Crossref]

Bai, Y. F.

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

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

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

Basano, L.

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

Bayer, M.

M. Abmann and M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3, 1545 (2013).
[Crossref] [PubMed]

Bisht, N. S.

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “The influence of source and object characteristics on coincidence imaging,” J. Opt. 12, 045701 (2010).
[Crossref]

Boyd, R. W.

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

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]

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

Bromberg, Y.

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

Cai, Y.

Cai, Y. J.

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

Cao, D.

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

Chan, K. W. C.

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

Chekhova, M. V.

Chen, M.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

Chen, X.

Cheng, J.

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

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

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

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

Chuan, L.

Collins, S. A.

D’Angelo, M.

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

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

Deacon, K. S.

R. E. Meyers, K. S. Deacon, and Yanhua Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011).
[Crossref]

Du, J.

Fan, C.

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]

Fu, X.

X. Zeng, Y. Bai, X. Shi, Y. Gao, and X. Fu, “The influence of the positive and negative defocusing on lensless ghost imaging,” Opt. Communications 382, 415 (2017).
[Crossref]

Fu, X. Q.

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

Gao, Y.

X. Zeng, Y. Bai, X. Shi, Y. Gao, and X. Fu, “The influence of the positive and negative defocusing on lensless ghost imaging,” Opt. Communications 382, 415 (2017).
[Crossref]

Gatti, A.

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]

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

Gong, W.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

J. Du, W. Gong, and S. Han, “The influence of sparsity property of images on ghost imaging with thermal light,” Opt. Lett. 37, 1067 (2012).
[Crossref] [PubMed]

W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett. 36, 394 (2011).
[Crossref] [PubMed]

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

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost “pinhole” imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

Gong, W. L.

W. L. Gong and S. S. Han, “Ghost telescope and ghost Fourier telescope with thermal light,” Phys. Lett. A 375, 990 (2011).
[Crossref]

Gong, Y. X.

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

Han, S.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

J. Du, W. Gong, and S. Han, “The influence of sparsity property of images on ghost imaging with thermal light,” Opt. Lett. 37, 1067 (2012).
[Crossref] [PubMed]

W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett. 36, 394 (2011).
[Crossref] [PubMed]

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

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost “pinhole” imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

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

Han, S. S.

W. L. Gong and S. S. Han, “Ghost telescope and ghost Fourier telescope with thermal light,” Phys. Lett. A 375, 990 (2011).
[Crossref]

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

Hardy, N. D.

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

Jin, H.

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

Jiying, L.

Jubo, Z.

Kandpal, H. C.

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “The influence of source and object characteristics on coincidence imaging,” J. Opt. 12, 045701 (2010).
[Crossref]

Karmakar, S.

S. Karmakar and Y. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81, 033845 (2010).
[Crossref]

Katz, O.

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

Klyshko, D. N.

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804 (1996).
[Crossref] [PubMed]

Leng, H. Y.

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

Li, E.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

Li, H.

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

Liu, Q.

Lugiato, L. A.

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

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

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost Imaging with Thermal Light, Comparing Entanglement and Classical Correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Luo, C. L.

Luo, K.

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. E.

R. E. Meyers, K. S. Deacon, and Yanhua Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011).
[Crossref]

Migdall, A.

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can Two-Photon Interference be Considered the Interference of Two Photons,” Phys. Rev. Lett. 77, 1917 (1996).
[Crossref] [PubMed]

Mu, S. Y.

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

Nie, L.

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

O’Sullivan, M. N.

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

Ottonelloa, P.

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

Pittman, T. B.

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804 (1996).
[Crossref] [PubMed]

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can Two-Photon Interference be Considered the Interference of Two Photons,” Phys. Rev. Lett. 77, 1917 (1996).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Opticla imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, 3429 (1995).
[Crossref]

Qiao, C.

Qiu, H.

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

Rubin, M. H.

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

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can Two-Photon Interference be Considered the Interference of Two Photons,” Phys. Rev. Lett. 77, 1917 (1996).
[Crossref] [PubMed]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804 (1996).
[Crossref] [PubMed]

Saleh, B. E. A.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of Entanglement in Two-Photon Imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

Scarcelli, G.

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

Sergienko, A. V.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of Entanglement in Two-Photon Imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can Two-Photon Interference be Considered the Interference of Two Photons,” Phys. Rev. Lett. 77, 1917 (1996).
[Crossref] [PubMed]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804 (1996).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Opticla imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, 3429 (1995).
[Crossref]

Shapiro, J. H.

J. H. Shapiro, D. Venkatraman, and F. N. C. Wong, “Ghost Imaging without Discord,” Sci. Rep. 3, 1849 (2013).
[Crossref] [PubMed]

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

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

Sharma, E. K.

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “The influence of source and object characteristics on coincidence imaging,” J. Opt. 12, 045701 (2010).
[Crossref]

Shen, H.

Shen, X.

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

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost “pinhole” imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

Shi, D.

Shi, X.

X. Zeng, Y. Bai, X. Shi, Y. Gao, and X. Fu, “The influence of the positive and negative defocusing on lensless ghost imaging,” Opt. Communications 382, 415 (2017).
[Crossref]

Shih, Y.

S. Karmakar and Y. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81, 033845 (2010).
[Crossref]

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

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

Shih, Y. H.

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can Two-Photon Interference be Considered the Interference of Two Photons,” Phys. Rev. Lett. 77, 1917 (1996).
[Crossref] [PubMed]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804 (1996).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Opticla imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, 3429 (1995).
[Crossref]

Shih, Yanhua

R. E. Meyers, K. S. Deacon, and Yanhua Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011).
[Crossref]

Shisheng, H.

Silberberg, Y.

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

Strekalov, D. V.

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804 (1996).
[Crossref] [PubMed]

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can Two-Photon Interference be Considered the Interference of Two Photons,” Phys. Rev. Lett. 77, 1917 (1996).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Opticla imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, 3429 (1995).
[Crossref]

Tang, Q.

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

Teich, M. C.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of Entanglement in Two-Photon Imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

Valencia, A.

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

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

Venkatraman, D.

J. H. Shapiro, D. Venkatraman, and F. N. C. Wong, “Ghost Imaging without Discord,” Sci. Rep. 3, 1849 (2013).
[Crossref] [PubMed]

Wang, F.

Wang, H.

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

Wang, K.

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

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

Wang, Y.

Wong, F. N. C.

J. H. Shapiro, D. Venkatraman, and F. N. C. Wong, “Ghost Imaging without Discord,” Sci. Rep. 3, 1849 (2013).
[Crossref] [PubMed]

Wu, L.

Wu, T.

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

Xian, R.

Xie, Z. D.

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

Xiong, J.

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

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

Xu, D.

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

Xu, P.

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

Xu, W.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

Yu, X. Q.

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

Zeng, X.

X. Zeng, Y. Bai, X. Shi, Y. Gao, and X. Fu, “The influence of the positive and negative defocusing on lensless ghost imaging,” Opt. Communications 382, 415 (2017).
[Crossref]

Zhai, Y.

Y. Zhai, X. Chen, D. Zhang, and L. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A 72, 043805 (2005).
[Crossref]

Zhang, D.

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

Y. Zhai, X. Chen, D. Zhang, and L. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A 72, 043805 (2005).
[Crossref]

Zhang, J.

Zhang, P.

D. Shi, C. Fan, P. Zhang, H. Shen, J. Zhang, C. Qiao, and Y. Wang, “Two-wavelength ghost imaging through atmospheric turbulence,” Opt. Express 21, 2050 (2013).
[Crossref] [PubMed]

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

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost “pinhole” imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

Zhao, C.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

Zhu, S. N.

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

Zhu, S. Y.

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

Zhu, Z. H.

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

Appl. Phys. Lett. (5)

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost “pinhole” imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

D. Zhang, Q. Tang, T. Wu, H. Qiu, D. Xu, H. Li, H. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014).
[Crossref]

R. E. Meyers, K. S. Deacon, and Yanhua Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011).
[Crossref]

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

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

J. Opt. (1)

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “The influence of source and object characteristics on coincidence imaging,” J. Opt. 12, 045701 (2010).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Communications (2)

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

X. Zeng, Y. Bai, X. Shi, Y. Gao, and X. Fu, “The influence of the positive and negative defocusing on lensless ghost imaging,” Opt. Communications 382, 415 (2017).
[Crossref]

Opt. Express (2)

Opt. Lett. (7)

Phys. Lett. A (1)

W. L. Gong and S. S. Han, “Ghost telescope and ghost Fourier telescope with thermal light,” Phys. Lett. A 375, 990 (2011).
[Crossref]

Phys. Rev. A (15)

P. Zhang, W. Gong, X. Shen, and 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]

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

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

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

P. Xu, H. Y. Leng, Z. H. Zhu, Y. F. Bai, H. Jin, Y. X. Gong, X. Q. Yu, Z. D. Xie, S. Y. Mu, and S. N. Zhu, “Lensless imaging by entangled photons from quadratic nonlinear photonic crystals,” Phys. Rev. A 86, 013805 (2012).
[Crossref]

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

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

Y. Zhai, X. Chen, D. Zhang, and L. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A 72, 043805 (2005).
[Crossref]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Opticla imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, 3429 (1995).
[Crossref]

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

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804 (1996).
[Crossref] [PubMed]

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

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

S. Karmakar and Y. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81, 033845 (2010).
[Crossref]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (6)

T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Can Two-Photon Interference be Considered the Interference of Two Photons,” Phys. Rev. Lett. 77, 1917 (1996).
[Crossref] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of Entanglement in Two-Photon Imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

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]

A. Valencia, G. Scarcelli, M. D’Angelo, and 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, 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]

Sci. Rep. (2)

J. H. Shapiro, D. Venkatraman, and F. N. C. Wong, “Ghost Imaging without Discord,” Sci. Rep. 3, 1849 (2013).
[Crossref] [PubMed]

M. Abmann and M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3, 1545 (2013).
[Crossref] [PubMed]

Other (1)

Y. Shih, arxiv.org/abs/0805.1166 .

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Figures (6)

Fig. 1
Fig. 1 Schematic of ghost imaging with thermal light.
Fig. 2
Fig. 2 (a) Schematic of a lensless ghost imaging scheme. (b) and (c) are ghost-images (averaged 10000 measurements) under Lt1 = 200mm and Lt1 = 500mm, respectively. The normalized horizontal sections of the images are plotted in the below pictures. Open circles display the normalized horizontal sections of the images, and solid curves show the theoretical predictions. (d) is the normalized PSF of the images in (b) and (c) as a function of xr.
Fig. 3
Fig. 3 (a) Ghost imaging system with a lens inserted into the test arm. (b) Ghost-image in a lensless ghost imaging system under Lt1 + Lt2 = Lr1 = 500mm. (c) and (d) are ghost-images of the double-slit reconstructed by the scheme in Fig. 3(a) under Lt1 = 200mm and 350mm, respectively. (e) is the normalized PSF of the images in (b)-(d) as a function of xr. The parameters are chosen as Lt1 + Lt2 = 500mm.
Fig. 4
Fig. 4 (a) Schematic for ghost imaging with a lens in the reference arm. (b) Lensless ghost-image of the double-slit under Lt1 = 200mm. Two images by the imaging scheme in Fig. 4(a) are plotted in (c) Lr1 = 350mm, Lr2 = 300mm and (d) Lr1 = 500mm, Lr2 = 150mm. (e) is the normalized PSF for the images in (b)-(d).
Fig. 5
Fig. 5 (a) Ghost telescope imaging system. (b) and (c) are two images of the double-slit in the scheme of Fig. 5(a) under Lt2 = 200mm and Lt2 = 500mm, respectively. (e) is the normalized point-spread function of the images in (b) and (c) as the function of xr.
Fig. 6
Fig. 6 The experimental results of a complicated object “GI”. (a) the PSF in lensless ghost imaging system and ghost telescope system. (b) ghost-image in the lensless ghost imaging scheme [see Fig. 2(a)]. Lr = Lt1 = 700mm. (b) the reconstructed image in ghost telescope imaging system [see Fig. 5(a)]. The other parameters are the same as those in Fig. 5(b).

Equations (22)

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E t ( x t ) = d ξ d u E s ( u ) H t 1 ( u , ξ ) H t 2 ( ξ , x t ) ,
H t 1 ( u , ξ ) = ( j λ B t 1 ) 1 / 2 e j π λ B t 1 ( A t 1 u 2 2 u ξ + D t 1 ξ 2 ) , H t 2 ( ξ , x t ) = ( j λ B t 2 ) 1 / 2 e j π λ B t 2 ( A t 2 ξ 2 2 x t ξ + D t 2 x t 2 ) ,
E r ( x r ) = d u E s ( u ) H r ( u , x r ) ,
H r ( u , X r ) = ( j λ B r ) 1 / 2 e j π λ B r ( A r u 2 2 u x r + D r x r 2 ) .
G ( x t , x r ) = I t ( x t ) I r ( x r ) I t ( x t ) I r ( x r ) .
E s ( u 1 ) E s * ( u 2 ) = e u 1 2 + u 2 2 2 ρ s 2 δ ( u 1 u 2 ) ,
G ( x t , x r ) = β r β t 1 β t 2 π 3 d u 1 d u 2 d ξ d ξ t ( ξ ) t * ( ξ ) e α u 1 2 e α u 2 2 × e j β r [ ( A r u 1 2 2 u 1 x r + D r x r 2 ) ( A r u 2 2 2 u 2 x r + D r x r 2 ) ] × e j β t 1 [ ( A t 1 u 2 2 2 u 2 ξ + D t 1 ξ 2 ) ( A t 1 u 1 2 2 u 1 ξ + D t 1 ξ 2 ) ] × e j β t 2 [ ( A t 2 ξ 2 2 x t ξ + D t 2 x t 2 ) ( A t 2 ξ 2 2 x t ξ + D t 2 x t 2 ) ] ,
G ( x t , x r ) = β r β t 1 β t 2 π 3 d ξ d ξ t ( ξ ) t * ( ξ ) h ( ξ , ξ , x r ) e j β t 1 ( D t 1 ξ 2 D t 1 ξ 2 ) × e j β t 2 [ ( A t 2 ξ 2 2 x t ξ + D t 2 x t 2 ) ( A t 2 ξ 2 2 x t ξ + D t 2 x t 2 ) ] ,
h ( ξ , ξ , x r ) = d u 1 d u 2 exp { P u 1 2 Q u 2 2 M u 1 N u 2 } ,
G ( x r ) = G ( x t , x r ) d x t = β r β t 1 β t 2 π 3 d ξ T ( ξ ) h ( ξ , x r ) ,
h ( ξ , x r ) = π 2 α β t 1 Δ P S F exp { ( ξ M x r ) 2 Δ P S F 2 } ,
Δ P S F = λ | B t 1 | 2 π ρ s 1 + π 2 ρ s 4 λ 2 ( A r B r A t 1 B t 1 ) 2 .
( A r B r C r D r ) = ( 1 L r 0 1 ) , ( A t 1 B t 1 C t 1 D t 1 ) = ( 1 L t 1 0 1 ) .
Δ P S F = λ L t 1 2 π ρ s 1 + π 2 ρ s 4 λ 2 ( 1 L t 1 1 L r ) 2 .
( A t 1 B t 1 C t 1 D t 1 ) = ( 1 L t 2 0 1 ) ( 1 0 1 / f t 1 ) ( 1 L t 1 0 1 ) = ( 1 L t 2 / f t L t 1 + L t 2 L t 1 L t 2 / f t 1 / f t 1 L t 1 / f ) .
Δ P S F = λ | L t 1 + L t 2 L t 1 L t 2 / f t | 2 π ρ s [ 1 + π 2 ρ s 4 λ 2 ( 1 L r 1 1 L t 2 / f t L t 1 + L t 2 L t 1 L t 2 / f t ) 2 ] 1 / 2 ,
1 L t 1 L r 1 + 1 L t 2 = 1 f t ,
( A r B r C r D r ) = ( 1 L r 2 0 1 ) ( 1 0 1 / f r 1 ) ( 1 L r 1 0 1 ) = ( 1 L r 2 / f r L r 1 + L r 2 L r 1 L r 2 / f r 1 / f r 1 L r 1 / f r ) .
Δ P S F = λ L t 1 2 π ρ s [ 1 + π 2 ρ s 4 λ 2 ( 1 L t 1 1 L r 2 / f r L r 1 + L r 2 L r 1 L r 2 / f r ) 2 ] 1 / 2 .
1 L r 1 L t 1 + 1 L r 2 = 1 f r ,
Δ P S F = λ f t 2 π ρ s [ 1 + π 2 ρ s 4 λ 2 ( 1 L r 2 / f r f r 1 L t 2 / f t f t ) 2 ] 1 / 2 ,
L r 2 = ( f r f t ) 2 L t 2 + f r f r 2 f t .

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