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

1. Introduction

In recent years, ghost imaging (GI) has attracted a lot of attentions in the field of quantum optics [1]. As a kind of nonlocal imaging method, intensity correlation measurements are performed in GI experiments to get the image of an unknown object. Both quantum entangled and classically correlated incoherent light sources can be used to realize GI [2–17]. A very interesting application of the GI with classically incoherent sources is that it can be used to generate images without the use of lenses. Lensless imaging is very useful in many applications. We have studied the realization of lensless ghost diffraction and its applicability in x-ray diffraction [4, 13]. Some other groups have shown that true images also can be produced from lensless ghost imaging (LGI) systems [14, 15]. Very recently, researchers have proposed the idea of ghost camera and realized reflected ghost imaging experiment [16].

However, in all these studies cited above, the light fields are propagated in free space. To the best of our knowledge, the properties of GI through turbulent atmosphere have never been investigated so far. Beam propagation in a turbulent atmosphere is a topic of a long history and still attracts many researcher’s interests [18–24]. In this paper, we try to investigate the possibility to realize ghost imaging through turbulent atmosphere. Using the extended Huygens-Fresnel integral, we analytically derive the ghost imaging formula in atmospheric turbulence. The ghost image is a convolution of the object and a Gaussian function. Strong turbulence, large propagation distance, and small source size lead to bad imaging quality. Our results may find applications in spatial measurement and observation.

2. Theory

Let’s consider the lensless ghost imaging (LGI) scheme [14, 15] shown in Fig. 1. An incoherent thermal or pseduo-thermal source is split into two beams by the beam splitter BS and travel through two different imaging systems, one is a test imaging system which contains an unknown object and a test detector Dt without resolution, the other is a reference imaging system which is irrelevant with the object and has a high resolution reference detector Dr. The intensity distribution recorded in Dt and Dr are correlated by a correlator to obtain the correlation function of the intensity fluctuations. To get an image of the object, z 0 = z 1 is needed to be satisfied. Here we assume there exist atmospheric turbulence in all optical paths. A more realistic situation, in which only the test arm experiences atmospheric turbulence and the reference arm is local, can be viewed as a special case of the general theory by assuming the turbulence strength in the reference arm to be zero. When there is no atmosphere, the imaging properties of this LGI system are well known now.

In the reference path, based on the extended Huygens-Fresnel integral [19], the field at Dr is

E1(x1)=(jλz1)1/2duf(u)eλz1(x1u)2eϕ1(u,x1),

where f(u) is the source distribution, ϕ 1(u,x 1) represents the random part of the complex phase due to the turbulent atmosphere. In the test path, similarly at the plane of Dt, we have

E2(x2)=(jλz0)1/2(jλz2)1/2dudyf(u)eλz0(yu)2eϕ0(u,y)t(y)eλz2(yx2)2eϕ2(x2,y),

where t(y) is the object and ϕi characterize the turbulent effects.

 figure: 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.

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Then the intensity correlation function is given by

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)
=1λ3z0z1z2du1du1du2du2dydyf(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]

where we have assumed the statistics of the source, and the three propagation regimes are independent.

It is now well known that the statistical averages caused by the turbulent atmosphere can be described approximately by

eϕi(x,y)+ϕi*(x,y)=exp{(xx)2+(xx)(yy)+(yy)2ρi2},

where ρi, = (0.55C 2(i) n k 2 zi)-3/5 is the coherence length of a spherical wave propagating through a turbulent medium and C 2(i) n is the refractive-index structure parameter describing the strength of the atmospheric turbulence in the path zi. It is important to point out that a quadratic approximation of the Rytov’s phase structure function has been used in Eq. (4) in order to simplify the analysis and obtain an analytical formula [19, 20, 21, 22]. This quadratic approximation has been widely used in literatures and is usually accepted to be valid in the weak fluctuation regime. Even for strong fluctuations cases, the approximation also seems useful.

The sources used in GI generally can be described as a zero-average Gaussian random process, so

f(u1)f*(u1)f(u2)f*(u2)=f(u1)f*(u1)f(u2)f*(u2)+f(u1)f*(u2)f(u2)f*(u1).

Suppose the source is fully incoherent, we have 〈f(u)f *(u′)〉 = I(u)δ(u - u′).

Introducing several new parameters, γi = π/λzi, βi = ρ -2 i = (0.55C 2(i) n k 2 zi)6/5, then the correlation of intensity fluctuations is

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,

The image information is contained in this function.

Suppose the source intensity has a Gaussian distribution, I(u) = e -u2/ρ 2 s, defining α = ρ -2 s we can integrate Eq. (5) to get,

G(x1,x2)=γ0γ1γ2π3dydyt(y)t*(y)ejγ0(y2y2)+jγ2(x2y)2jγ2(x2y)2(β0+β2)(yy)2h(y,y,x1),

where the kernel function

h(y,y,x1)=du1du2eAu12Bu1u2Cu22Du1Eu2=2π4AcB2eBDECD2AE24ACB2,

with A = α + β 0 + β 1 + 0 - 1, B = -2β 0 - 2β 1, C = α + β 0 + β 1 - 0 + 1, D = β 0(y′ - y) + 2 1 x 1 - 2 0 γ 1, E = -β 0 (y′ - y) - 2 1 x 1 + 2 0 y.

Now, when the imaging condition of a LGI system is satisfied (z 0 = z 1 = z), γ 0 = γ 1 = γ = n/λz, the kernel function is simplified to be

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),

Combined this Eq. and Eq. (6), we obtain a general formula to describe the lensless ghost imaging through turbulent atmosphere. If the test detector Dt is a bucket detector, then the ghost image is proportional to

M(x1)=dx2G(x1,x2)
=γ2π2dydyδ(yy)t(y)t*(y)h(y,y,x1)ej(γ0+γ2)(y2y2)(β0+β2)(yy)2
=γ2π2dyt(y)2h(y,y,x1)

Eq. (9) can be represented as:

M(x1)=1λzt(z)2hz(y),

in which a z-independent factor π2α has been neglected. In Eq. (10), ⊗ means convolution hz(y) can be considered as the point-spread function (PSF) at the distance z,

hz(y)=1πRzey2/Rz2,

and the z-dependent radius

Rz=(α+β0+β1)/2γ=λz2πρs1+ρs2(0.55k2z)6/5[(Cn2(0))6/5+(Cn2(1))6/5].

Eq. (10) is the main result in this paper, it clearly determines how the ghost image is degraded in the atmospheric turbulence.

3. Numerical calculations and analysis

 figure: 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.

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For a realistic situation, in which only the test arm experiences atmospheric turbulence, by setting C 2(1) n = 0 and C 2(0) n = C 2 n, we can use numerical calculations to demonstrate atmospheric effects on ghost imaging. In the first example, we set λ = 0.785 μm, ρs = 2.5 cm, C 2 n = 10-14m-2/3. The object is a double-slit, with slit width 2 cm and separation 6 cm. From Fig. 2, we can directly see the deformation of the images when the propagation length is increased. Compared with the free space, at large distance, the atmospheric turbulence may significantly decrease the imaging quality. But when the propagation distance is small, even in turbulent atmosphere, high quality ghost images still can be formed. The test arm and reference arm have a common path before the BS, but this distance is very small compared with z 0, and can not affect our results.

Since the ghost images are determined by the PSF, changing C 2 n from the weak turbulence (10-16m-2/3) to strong turbulence (10-13m-2/3), we plot Rz as the function of z in Fig. 3(a). We can see Rz become larger and larger when C 2 n is increased, which means the imaging quality will be worse and worse. Eq. (12) also relies on the source size ρs, so we have plotted Rz(z) with 5 different ρs in Fig. 3(b). We find increasing z can decrease Rz, leading to the improvement of imaging quality. But the improvement will be saturated when ρs is enough large.

 figure: 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.

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Based on these numerical results and theoretical formulas, we find the main features of GI through turbulent atmosphere are:

  1. It is very clear, when R → 0, the PSF h 0 (y) → δ(y), so the ghost image will takes the form M(x1)1λzt(x1)2, which is exactly the image of the object, i.e. , one can realize perfectly imaging.
  2. The ghost image is the convolution of the original object and a PSF, so the imaging quality is totally determined by the PSF. Since the PSF is a Gaussian function, the image will be greatly degraded when Rz is increased to a large value.
  3. Since Rz is a monotone increasing function of z, the imaging quality will be worse when the propagating distance is longer.
  4. Strong fluctuation (large C 2 n)) will significantly degraded the imaging quality because it leads to a large Rz.
  5. Increasing the source size may improve the imaging quality, but the improvement will be saturated when ps is large enough.

4. Conclusion

In conclusion, we have investigated the ghost imaging through turbulent atmosphere. Based on the extended Huygens-Fresnel integral, using a quadratic approximation of the structure function, analytical formulas for the ghost images have been derived. We find the ghost image can be viewed as the convolution of the original object and a point-spread function. The imaging quality is determined from the size of the PSF. The images will be significantly degraded with strong turbulence and large propagation distance. Increasing the source size can improve imaging quality, but the improvement will be saturated for enough large sources.

Acknowledgments

Support from the National Natural Science Foundation of China (10774047), and South China University of Technology is acknowledged.

References and links

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]  

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]  

3. 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]  

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

5. 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]  

6. 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]  

7. 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]  

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

9. 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]  

10. 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]  

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

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

13. 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]  

14. 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]  

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

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

17. 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]  

18. H.T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972). [CrossRef]   [PubMed]  

19. J.C. Ricklin and F.M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002). [CrossRef]  

20. 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]  

21. 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]  

22. 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]  

23. 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]  

24. G.A. Tyler and R.W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009). [CrossRef]   [PubMed]  

References

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  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]
  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]
  3. 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]
  4. J. Cheng and S. Han, “Incoherent Coincidence Imaging and Its Applicability in X-ray Diffraction,” Phys. Rev. Lett. 92, 093903:1–4 (2004).
    [Crossref]
  5. 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]
  6. 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]
  7. 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]
  8. D.Z. Cao, J. Xiong, and K.G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801:1–5 (2005).
    [Crossref]
  9. 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]
  10. 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]
  11. J. Cheng and S. Han, “Classical correlated imaging from the perspective of coherent-mode representation,” Phys. Rev. A 76, 023824:1–5 (2007).
    [Crossref]
  12. J. Cheng, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823:1–5 (2008).
    [Crossref]
  13. 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]
  14. 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]
  15. L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. 89, 091109:1–3 (2006).
    [Crossref]
  16. R. Meyers, K.S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A,  77, 041801:1–4 (2008).
    [Crossref]
  17. 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]
  18. H.T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [Crossref] [PubMed]
  19. J.C. Ricklin and F.M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
    [Crossref]
  20. 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]
  21. 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]
  22. 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]
  23. 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]
  24. G.A. Tyler and R.W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).
    [Crossref] [PubMed]

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 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]

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]

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]

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, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823:1–5 (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, 021803:1–4 (2007).
[Crossref]

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]

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]

Phys. Rev. Lett. (6)

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]

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]

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]

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]

Waves Random Media (1)

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]

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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)

Equations on this page are rendered with MathJax. Learn more.

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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