Abstract

We demonstrate for the first time (to our knowledge) that a high-quality image can still be obtained in atmospheric turbulence by applying adaptive optical ghost imaging (AOGI) system even when conventional ghost imaging system fails to produce an image. The performance of AOGI under different strength of atmospheric turbulence is investigated by simulation. The influence of adaptive optics system with different numbers of adaptive mirror elements on obtained image quality is also studied.

© 2012 OSA

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  1. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett.93(9), 093602 (2004).
    [CrossRef] [PubMed]
  2. 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(18), 183602 (2005).
    [CrossRef] [PubMed]
  3. 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(6), 063602 (2006).
    [CrossRef] [PubMed]
  4. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104(25), 253603 (2010).
    [CrossRef] [PubMed]
  5. J. Shapiro, and R. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. DOI 10.1007 (2012).
  6. R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett.98(11), 111115 (2011).
    [CrossRef]
  7. R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett.100(6), 061126 (2012).
    [CrossRef]
  8. R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett.100(13), 131114 (2012).
    [CrossRef]
  9. C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through Fturbulent atmosphere,” Appl. Phys. B99(3), 599–604 (2010).
    [CrossRef]
  10. B. I. Erkmen, “Computational ghost imaging for remote sensing,” J. Opt. Soc. Am. A29(5), 782–789 (2012).
    [CrossRef] [PubMed]
  11. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express17(10), 7916–7921 (2009).
    [CrossRef] [PubMed]
  12. N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A84(6), 063824 (2011).
    [CrossRef]
  13. P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A82(3), 033817 (2010).
    [CrossRef]
  14. W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett.36(3), 394–396 (2011).
    [CrossRef] [PubMed]
  15. G. Wenlin, Z. Pengli, S. Xia, and H. Shensheng, “Ghost imaging in scattering media,” arXiv:0806.3543, (2008).
  16. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett.95(13), 131110 (2009).
    [CrossRef]
  17. K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A79(3), 033808 (2009).
    [CrossRef]
  18. K. W. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett.34(21), 3343–3345 (2009).
    [CrossRef] [PubMed]
  19. A. Yariv and T. L. Koch, “One-way coherent imaging through a distorting medium using four-wave mixing,” Opt. Lett.7(3), 113–115 (1982).
    [CrossRef] [PubMed]
  20. J. Feinberg, “Imaging through a distorting medium with and without phase conjugation,” Appl. Phys. Lett.42(1), 30–32 (1983).
    [CrossRef]
  21. F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).
  22. D. S. Simon and A. V. Sergienko, “Turbulence mitigation in phase-conjugated two-photon imaging,” arXiv:1105.0128v2, (2011).
  23. A. K. Majumdar and J. C. Rickloin, Free-Space Laser Communications: Principles and Advances (Springer, 2008).

2012 (3)

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett.100(6), 061126 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett.100(13), 131114 (2012).
[CrossRef]

B. I. Erkmen, “Computational ghost imaging for remote sensing,” J. Opt. Soc. Am. A29(5), 782–789 (2012).
[CrossRef] [PubMed]

2011 (3)

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett.98(11), 111115 (2011).
[CrossRef]

W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett.36(3), 394–396 (2011).
[CrossRef] [PubMed]

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A84(6), 063824 (2011).
[CrossRef]

2010 (3)

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A82(3), 033817 (2010).
[CrossRef]

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104(25), 253603 (2010).
[CrossRef] [PubMed]

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through Fturbulent atmosphere,” Appl. Phys. B99(3), 599–604 (2010).
[CrossRef]

2009 (4)

J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express17(10), 7916–7921 (2009).
[CrossRef] [PubMed]

K. W. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett.34(21), 3343–3345 (2009).
[CrossRef] [PubMed]

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett.95(13), 131110 (2009).
[CrossRef]

K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A79(3), 033808 (2009).
[CrossRef]

2006 (1)

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(6), 063602 (2006).
[CrossRef] [PubMed]

2005 (1)

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(18), 183602 (2005).
[CrossRef] [PubMed]

2004 (1)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett.93(9), 093602 (2004).
[CrossRef] [PubMed]

1983 (1)

J. Feinberg, “Imaging through a distorting medium with and without phase conjugation,” Appl. Phys. Lett.42(1), 30–32 (1983).
[CrossRef]

1982 (1)

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(18), 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(9), 093602 (2004).
[CrossRef] [PubMed]

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(6), 063602 (2006).
[CrossRef] [PubMed]

Boyd, R. W.

K. W. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett.34(21), 3343–3345 (2009).
[CrossRef] [PubMed]

K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A79(3), 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(18), 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(9), 093602 (2004).
[CrossRef] [PubMed]

Bromberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett.95(13), 131110 (2009).
[CrossRef]

Chan, K. W.

Chan, K. W. C.

K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A79(3), 033808 (2009).
[CrossRef]

Cheng, J.

Deacon, K. S.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett.100(6), 061126 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett.100(13), 131114 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett.98(11), 111115 (2011).
[CrossRef]

Erkmen, B. I.

Feinberg, J.

J. Feinberg, “Imaging through a distorting medium with and without phase conjugation,” Appl. Phys. Lett.42(1), 30–32 (1983).
[CrossRef]

Ferri, F.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104(25), 253603 (2010).
[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(18), 183602 (2005).
[CrossRef] [PubMed]

Gatti, A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104(25), 253603 (2010).
[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(18), 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(9), 093602 (2004).
[CrossRef] [PubMed]

Gong, W.

W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett.36(3), 394–396 (2011).
[CrossRef] [PubMed]

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A82(3), 033817 (2010).
[CrossRef]

Han, S.

W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett.36(3), 394–396 (2011).
[CrossRef] [PubMed]

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A82(3), 033817 (2010).
[CrossRef]

Hardy, N. D.

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A84(6), 063824 (2011).
[CrossRef]

Katz, O.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett.95(13), 131110 (2009).
[CrossRef]

Koch, T. L.

Li, C.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through Fturbulent atmosphere,” Appl. Phys. B99(3), 599–604 (2010).
[CrossRef]

Lugiato, L. A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104(25), 253603 (2010).
[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(18), 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(9), 093602 (2004).
[CrossRef] [PubMed]

Magatti, D.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104(25), 253603 (2010).
[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(18), 183602 (2005).
[CrossRef] [PubMed]

Meyers, R. E.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett.100(13), 131114 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett.100(6), 061126 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett.98(11), 111115 (2011).
[CrossRef]

O’Sullivan, M. N.

O'Sullivan, M. N.

K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A79(3), 033808 (2009).
[CrossRef]

Pu, J.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through Fturbulent atmosphere,” Appl. Phys. B99(3), 599–604 (2010).
[CrossRef]

Rao, R.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through Fturbulent atmosphere,” Appl. Phys. B99(3), 599–604 (2010).
[CrossRef]

Scarcelli, G.

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(6), 063602 (2006).
[CrossRef] [PubMed]

Shapiro, J. H.

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A84(6), 063824 (2011).
[CrossRef]

Shen, X.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A82(3), 033817 (2010).
[CrossRef]

Shih, Y.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett.100(13), 131114 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett.100(6), 061126 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett.98(11), 111115 (2011).
[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(6), 063602 (2006).
[CrossRef] [PubMed]

Silberberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett.95(13), 131110 (2009).
[CrossRef]

Tunick, A. D.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett.100(6), 061126 (2012).
[CrossRef]

Wang, T.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through Fturbulent atmosphere,” Appl. Phys. B99(3), 599–604 (2010).
[CrossRef]

Yariv, A.

Zhang, P.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A82(3), 033817 (2010).
[CrossRef]

Zhu, W.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through Fturbulent atmosphere,” Appl. Phys. B99(3), 599–604 (2010).
[CrossRef]

Appl. Phys. B (1)

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through Fturbulent atmosphere,” Appl. Phys. B99(3), 599–604 (2010).
[CrossRef]

Appl. Phys. Lett. (5)

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett.98(11), 111115 (2011).
[CrossRef]

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett.100(6), 061126 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett.100(13), 131114 (2012).
[CrossRef]

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett.95(13), 131110 (2009).
[CrossRef]

J. Feinberg, “Imaging through a distorting medium with and without phase conjugation,” Appl. Phys. Lett.42(1), 30–32 (1983).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (3)

K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A79(3), 033808 (2009).
[CrossRef]

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A84(6), 063824 (2011).
[CrossRef]

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A82(3), 033817 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett.93(9), 093602 (2004).
[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(18), 183602 (2005).
[CrossRef] [PubMed]

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(6), 063602 (2006).
[CrossRef] [PubMed]

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett.104(25), 253603 (2010).
[CrossRef] [PubMed]

Other (5)

J. Shapiro, and R. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. DOI 10.1007 (2012).

G. Wenlin, Z. Pengli, S. Xia, and H. Shensheng, “Ghost imaging in scattering media,” arXiv:0806.3543, (2008).

F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).

D. S. Simon and A. V. Sergienko, “Turbulence mitigation in phase-conjugated two-photon imaging,” arXiv:1105.0128v2, (2011).

A. K. Majumdar and J. C. Rickloin, Free-Space Laser Communications: Principles and Advances (Springer, 2008).

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Figures (4)

Fig. 1
Fig. 1

The schematic of AOGI system. A thermal light is split into two beams by the beam splitter (BS). Z0, Z1 and Z2 are the distances from the Light Source to the Object, from the Light Source to the CCD Detector and from the Object to the Single Pixel Bucket Detector, respectively. The u, y, xr, xt are the coordinators at the Light Source plane, object plane, CCD Detector plane and Single Pixel Bucket Detector plane, respectively. DM, WFS, AM, GS are the Dichroic Mirror, the Wave-front Sensing, the adaptive mirror and the Guide Star, respectively.

Fig. 2
Fig. 2

The obtained images of the double slit in atmospheric turbulence. (a) corresponds to the situation II; (b) corresponds to the situation I; (c) corresponds to the realistic AOGI. The normalized horizontal sections of the images are plotted in (d,e) where red lines show the simulated data and blue lines correspond to the theoretical prediction from Eqs. (9) and (8). The blue line, green line and red line in (F) represent the normalized horizontal sections of the images (a,b,c), respectively.

Fig. 3
Fig. 3

The obtained images of letter ‘A’ under different strength atmospheric turbulence. (a,b,c) are the images of convention ghost imaging under different strength turbulence through multiplying HV21 model by different coefficient 2.0, 3.2 and 6.8, respectively. (d,e,f) are the images of AOGI under different strength turbulence through multiplying HV21 model by different coefficient 2.0, 3.2 and 6.8, respectively. Statistical over 10000 samples.

Fig. 4
Fig. 4

The obtained images of letter ‘H’ for ghost imaging via different AO systems under the strength turbulence through multiplying HV21 model by coefficient 3.2. (a) is the images of ghost imaging for the situation in which atmospheric turbulence is purely compensated. (b, c, d) are the images of ghost imaging with AO systems of PCM, 127 elements AM and 61 elements AM, respectively. Statistical over 10000 samples.

Equations (9)

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G( x t , x r )= I t ( x t ) I r ( x r ) I t ( x t ) I r ( x r ) = E t * ( x t ) E t ( x t ) E r * ( x r ) E r ( x r ) E t * ( x t ) E t ( x t ) E r * ( x r ) E r ( x r ) ,
E t ( x t )= exp( 2jπ( z 0 + z 2 ) λ ) λ 2 z 0 z 2 dyduE(u)exp[ jπ λ z 0 (yu) 2 ] exp[ ψ 0 (y,u)i ϕ AM (u) ] ×t(y)exp[ jπ λ z 2 ( x t y) 2 ]exp[ ψ 2 ( x t ,y) ],
E r ( x r )= jexp( 2jπ z 1 λ ) λ z 1 duE(u)exp[ jπ λ z 1 ( x r u) 2 ] .
E(u) E * ( u ) = I 0 exp( u 2 / ρ s 2 )δ(u u ),
exp[ i ϕ i (x,y)i ϕ i ( x , y ) ] =exp{ (x x ) 2 +(x x )(y y )+ (y y ) 2 ρ i 2 },
G( x t , x r )= I 0 2 λ 6 z 0 2 z 1 2 z 2 2 d u 1 d u 2 dyd y t(y) t * ( y )exp( u 1 2 u 2 2 ρ s 2 ) exp[ i ϕ 2 (y, x t )i ϕ 2 ( y , x t ) ] × exp[ i ϕ 0 ( u 1 ,y)i ϕ 0 ( u 2 , y )i ϕ AM ( u 1 )+i ϕ AM ( u 2 ) ] exp{ jπ λ 2 z 1 [ ( x r u 2 ) 2 ( x r u 1 ) 2 ] } ×exp{ jπ λ 1 z 0 [ (y u 1 ) 2 ( y u 2 ) 2 ] }exp{ jπ λ 1 z 2 [ ( x t y) 2 ( x t y ) 2 ] }.
G( x r )= | t(y) | 2 exp( y 2 / R 1Z 2 ),
R 1Z = λz 2 π ρ s 1+ 2 ρ s 2 ρ 0 2 .
R 2Z = λz 2 π ρ s ,

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