Abstract

Ghost imaging through turbulent atmospheres are theoretically studied. Based on the extended Huygens-Fresnel integral, we obtain an analytical imaging formula. The ghost image can be viewed as the convolution of the original object and a point-spread function (PSF). The imaging quality is determined by the size of the PSF. Increasing the turbulence strength and propagation distance, or decreasing the source size, will increase the size of the PSF, and lead to the degradation of the imaging quality.

© 2009 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2009 (1)

2008 (5)

Y.B. Zhu, D.M. Zhao, and X.Y. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008).
[Crossref] [PubMed]

K.C. Zhu, G.Q. Zhou, X.G. Li, X.J. Zheng, and H.Q. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16, 21315–21320 (2008).
[Crossref] [PubMed]

J. Cheng, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823:1–5 (2008).
[Crossref]

R. Meyers, K.S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A,  77, 041801:1–4 (2008).
[Crossref]

B.I. Erkmen and J.H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A,  77, 043809:1–13 (2008).
[Crossref]

2007 (2)

J. Cheng and S. Han, “Classical correlated imaging from the perspective of coherent-mode representation,” Phys. Rev. A 76, 023824:1–5 (2007).
[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, 021803:1–4 (2007).
[Crossref]

2006 (5)

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106:1–3 (2006).
[Crossref]

L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. 89, 091109:1–3 (2006).
[Crossref]

G. Scarcelli, V. Berardi, and Shih Y., “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations,” Phys. Rev. Lett. 96, 063602:1–4 (2006).
[Crossref]

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

H.T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profilesafter propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[Crossref]

2005 (4)

A. Valencia, Scarcelli G., M. D’Angelo, and Y. Shih, “Two-Photon Imaging with Thermal Light,” Phys. Rev. Lett. 94, 063601:1–4 (2005).
[Crossref]

F. Ferri, D. Magatti, Gatti A., M. Bache, E. Brambilla, and Lugiato L. A., “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602:1–4 (2005).
[Crossref]

D.Z. Cao, J. Xiong, and K.G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801:1–5 (2005).
[Crossref]

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

2004 (3)

J. Cheng and S. Han, “Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction,” Phys. Rev. Lett. 92, 093903:1–4 (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, 093602:1–4 (2004).
[Crossref]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).
[Crossref]

2002 (1)

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] [PubMed]

T.B. Pittman, Y.H. Shih, D.V. Strekalov, and A.V. Sergienko, “Optical imaging by meansof two-photon quantum entanglement,” Phys. Rev. A 52, 3429–3432 (1995).
[Crossref]

1972 (1)

A., Gatti

F. Ferri, D. Magatti, Gatti A., M. Bache, E. Brambilla, and Lugiato L. A., “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602:1–4 (2005).
[Crossref]

Bache, M.

F. Ferri, D. Magatti, Gatti A., M. Bache, E. Brambilla, and Lugiato L. A., “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602:1–4 (2005).
[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, 093602:1–4 (2004).
[Crossref]

Basano, L.

L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. 89, 091109:1–3 (2006).
[Crossref]

Baykal, Y.

H.T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profilesafter propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[Crossref]

Berardi, V.

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106:1–3 (2006).
[Crossref]

G. Scarcelli, V. Berardi, and Shih Y., “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations,” Phys. Rev. Lett. 96, 063602:1–4 (2006).
[Crossref]

Boyd, R.W.

Brambilla, E.

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

F. Ferri, D. Magatti, Gatti A., M. Bache, E. Brambilla, and Lugiato L. A., “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602:1–4 (2005).
[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, 093602:1–4 (2004).
[Crossref]

Cao, D.Z.

D.Z. Cao, J. Xiong, and K.G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801:1–5 (2005).
[Crossref]

Chen, X.H.

Cheng, J.

J. Cheng, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823:1–5 (2008).
[Crossref]

J. Cheng and S. Han, “Classical correlated imaging from the perspective of coherent-mode representation,” Phys. Rev. A 76, 023824:1–5 (2007).
[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, 021803:1–4 (2007).
[Crossref]

J. Cheng and S. Han, “Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction,” Phys. Rev. Lett. 92, 093903:1–4 (2004).
[Crossref]

D’Angelo, M.

A. Valencia, Scarcelli G., M. D’Angelo, and Y. Shih, “Two-Photon Imaging with Thermal Light,” Phys. Rev. Lett. 94, 063601:1–4 (2005).
[Crossref]

Davidson, F.M.

Deacon, K.S.

R. Meyers, K.S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A,  77, 041801:1–4 (2008).
[Crossref]

Dogariu, A.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).
[Crossref]

Du, X.Y.

Erkmen, B.I.

B.I. Erkmen and J.H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A,  77, 043809:1–13 (2008).
[Crossref]

Eyyuboglu, H.T.

H.T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profilesafter propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[Crossref]

Ferri, F.

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

F. Ferri, D. Magatti, Gatti A., M. Bache, E. Brambilla, and Lugiato L. A., “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602:1–4 (2005).
[Crossref]

G., Scarcelli

A. Valencia, Scarcelli G., M. D’Angelo, and Y. Shih, “Two-Photon Imaging with Thermal Light,” Phys. Rev. Lett. 94, 063601:1–4 (2005).
[Crossref]

Gatti, A.

A. Gatti, Bache M., D. Magatti, E. Brambilla, F. Ferri, and L.A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006), and references therein.
[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, 093602:1–4 (2004).
[Crossref]

Han, S.

J. Cheng and S. Han, “Classical correlated imaging from the perspective of coherent-mode representation,” Phys. Rev. A 76, 023824:1–5 (2007).
[Crossref]

J. Cheng and S. Han, “Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction,” Phys. Rev. Lett. 92, 093903:1–4 (2004).
[Crossref]

Han, S.S.

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, 021803:1–4 (2007).
[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] [PubMed]

Korotkova, O.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).
[Crossref]

L. A., Lugiato

F. Ferri, D. Magatti, Gatti A., M. Bache, E. Brambilla, and Lugiato L. A., “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602:1–4 (2005).
[Crossref]

Li, X.G.

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, 021803:1–4 (2007).
[Crossref]

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, 021803:1–4 (2007).
[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, 093602:1–4 (2004).
[Crossref]

Lugiato, L.A.

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

M., Bache

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

Magatti, D.

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

F. Ferri, D. Magatti, Gatti A., M. Bache, E. Brambilla, and Lugiato L. A., “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602:1–4 (2005).
[Crossref]

Meyers, R.

R. Meyers, K.S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A,  77, 041801:1–4 (2008).
[Crossref]

Ottonello, P.

L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. 89, 091109:1–3 (2006).
[Crossref]

Pittman, T.B.

T.B. Pittman, Y.H. Shih, D.V. Strekalov, and A.V. Sergienko, “Optical imaging by meansof two-photon quantum entanglement,” Phys. Rev. A 52, 3429–3432 (1995).
[Crossref]

Ricklin, J.C.

Salem, M.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).
[Crossref]

Scarcelli, G.

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106:1–3 (2006).
[Crossref]

G. Scarcelli, V. Berardi, and Shih Y., “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations,” Phys. Rev. Lett. 96, 063602:1–4 (2006).
[Crossref]

Sergienko, A.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] [PubMed]

T.B. Pittman, Y.H. Shih, D.V. Strekalov, and A.V. Sergienko, “Optical imaging by meansof two-photon quantum entanglement,” Phys. Rev. A 52, 3429–3432 (1995).
[Crossref]

Sermutlu, E.

H.T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profilesafter propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[Crossref]

Shapiro, J.H.

B.I. Erkmen and J.H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A,  77, 043809:1–13 (2008).
[Crossref]

Shen, X.

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, 021803:1–4 (2007).
[Crossref]

Shih, Y.

R. Meyers, K.S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A,  77, 041801:1–4 (2008).
[Crossref]

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106:1–3 (2006).
[Crossref]

A. Valencia, Scarcelli G., M. D’Angelo, and Y. Shih, “Two-Photon Imaging with Thermal Light,” Phys. Rev. Lett. 94, 063601:1–4 (2005).
[Crossref]

Shih, Y.H.

T.B. Pittman, Y.H. Shih, D.V. Strekalov, and A.V. Sergienko, “Optical imaging by meansof two-photon quantum entanglement,” Phys. Rev. A 52, 3429–3432 (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] [PubMed]

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] [PubMed]

T.B. Pittman, Y.H. Shih, D.V. Strekalov, and A.V. Sergienko, “Optical imaging by meansof two-photon quantum entanglement,” Phys. Rev. A 52, 3429–3432 (1995).
[Crossref]

Tang, H.Q.

Tyler, G.A.

Valencia, A.

A. Valencia, Scarcelli G., M. D’Angelo, and Y. Shih, “Two-Photon Imaging with Thermal Light,” Phys. Rev. Lett. 94, 063601:1–4 (2005).
[Crossref]

Wang, K.G.

D.Z. Cao, J. Xiong, and K.G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801:1–5 (2005).
[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, 021803:1–4 (2007).
[Crossref]

Wolf, E.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).
[Crossref]

Wu, L.A.

Xiong, J.

D.Z. Cao, J. Xiong, and K.G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801:1–5 (2005).
[Crossref]

Y., Shih

G. Scarcelli, V. Berardi, and Shih Y., “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations,” Phys. Rev. Lett. 96, 063602:1–4 (2006).
[Crossref]

Yura, H.T.

Zhai, Y.H.

Zhang, D.

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, 021803:1–4 (2007).
[Crossref]

Zhao, D.M.

Zheng, X.J.

Zhou, G.Q.

Zhu, K.C.

Zhu, Y.B.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106:1–3 (2006).
[Crossref]

L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. 89, 091109:1–3 (2006).
[Crossref]

J. Mod. Opt. (1)

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

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

Opt. Commun. (1)

H.T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profilesafter propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (7)

D.Z. Cao, J. Xiong, and K.G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801:1–5 (2005).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1.

Geometry of a LGI system through turbulent atmosphere. The source field f(u) is split into two beams by the beam splitter (BS). z 0, z 1 and z 2 are the distances from the source to the unknown object, from the source to the reference detector Dr , and from the object to the test detector Dt , respectively. u, y, x 1, x 2 are the coordinators at the source plane, object plane, reference detector plane and test detector plane.

Fig. 2.
Fig. 2.

Ghost images of a double-slit. λ = 0.785μm, ρs = 2.5cm, C 2 n = 10-14m-2/3. From the left to the right, the red dashed lines correspond to z = 500, 1000, 2000 m. For comparison, the blue solid lines are ghost images in free space.

Fig. 3.
Fig. 3.

Rz as the function of z, λ = 0.785μm. (a)ρs = 2.5cm, from the top to the bottom, C 2 n = 10-13m-2/3, 10-14m-2/3, 10-15m-2/3, 10-16m-2/3. (b)C 2 n = 10-14m-2/3, from the top to the bottom, ρs = 2.5, 5, 10, 15, 20cm, the lowest 3 lines are too close to be distinguished.

Equations (24)

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E1(x1)=(jλz1)1/2 duf (u) eλz1(x1u)2 eϕ1 (u,x1) ,
E2(x2)=(jλz0)1/2 (jλz2)1/2 dudy f (u)eλz0(yu)2eϕ0(u,y)t(y)eλz2(yx2)2eϕ2(x2,y),
I1(x1)I2(x2)
=1λ3z0z1z2du1du1du2du2dydyf(u1)f*(u1)f(u2)f(u2)f*(u2)t(y)t*(y)
eλz1[(x1u1)2(x1u1)2] eλz0[(yu2)2(yu2)2]eλz2[(x2y)2(x2y)2]
eϕ1(u1,x1)+ϕ1*(u1,x1) eϕ0(u2,y)+ϕ0*(u2,y) eϕ2(x2,y)+ϕ2*(x2,y)
=1λ3z0z1z2 d u1 d u1 d u2 d u2 dydy f(u1)f*(u1)f(u2)f(u2) eϕ1(u1x1)+ϕ1*(u1,x1)
eϕ0(u2,y)+ϕ0*(u2,y) eϕ2(x2,y)+ϕ2*(x2,y)
t (y) t* eλz1[(x1u1)2(x1u1)2] eλz0[(yu2)2(yu2)2] eλz2[(x2y)2(x2y)2]
eϕi(x,y)+ϕi*(x,y)=exp{(xx)2+(xx)(yy)+(yy)2ρi2},
f(u1)f*(u1)f(u2)f*(u2)=f(u1)f*(u1)f(u2)f*(u2)+f(u1)f*(u2)f(u2)f*(u1) .
G(x1,x2)=I1(x1)I2(x2)I1(x1)I2(x2)
=γ0γ1γ2π3du1du2dydyt(y)t*(y)I(u1)I(u2)
ejγ1[(x1u1)2(x1u2)2]+jγ0[(yu2)2(yu1)2]+jγ2[(x2y)2(x2y)2]
eβ1(u1u2)2β0[(u2u1)2+(u2u1)(yy)+(yy)2]β2(yy)2 ,
G(x1,x2)=γ0γ1γ2π3 dydy t (y) t* (y) ejγ0(y2y2)+jγ2(x2y)2jγ2(x2y)2(β0+β2)(yy)2 h (y,y,x1) ,
h(y,y,x1)= d u1 d u2 eAu12Bu1u2Cu22Du1Eu2 =2π4AcB2eBDECD2AE24ACB2 ,
h(y,y,x1)=πexp{[4γ2(β0+β1)2αβ02](yy)2+4αγ(γjβ0)(x1y)2+4αγ(γ+jβ0)(x1y)24α(α+β0+β1)}α(α+β0+β1),
M(x1)=dx2 G (x1,x2)
=γ2π2 dydy δ (yy) t (y) t* (y) h (y,y,x1) ej(γ0+γ2)(y2y2)(β0+β2)(yy)2
=γ2π2 dy t(y)2 h (y,y,x1)
M(x1)=1λz t(z)2 hz(y),
hz(y)=1πRzey2/Rz2 ,
Rz=(α+β0+β1)/2γ=λz2πρs1+ρs2(0.55k2z)6/5[(Cn2(0))6/5+(Cn2(1))6/5] .

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