Abstract

Reflective ghost diffraction (GD) for objects with rough surfaces is investigated theoretically and numerically. Using a speckle model to describe the reflectivity of the object with variable rough surfaces, the incident and reflective angle-dependent diffraction condition is obtained, and the analytical expression of the reflective GD is derived. Numerical simulations are given to show how the height variance and correlation length of the object and the reflective angle in the experimental scheme can affect the quality of the reflective GD. Specifically, we find that the changes of diffraction patterns of both the simple objects and the complicated objects are sensitive to parameters such as the reflective angle and the surface fluctuation. Additionally, comparative studies on both reflective GD and ghost imaging (GI) have also been performed. We find that reflective GI is more robust against system parameters than reflective GD. These results may be useful for future experimental works.

© 2013 Optical Society of America

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References

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  1. M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
    [CrossRef]
  2. J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
    [CrossRef]
  3. M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging schemes: fast and broadband,” Opt. Express 12, 6067–6081 (2004).
    [CrossRef]
  4. A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
    [CrossRef]
  5. J. Cheng and S. Han, “Classical correlated imaging from the perspective of coherent-mode representation,” Phys. Rev. A 76, 023824 (2007).
    [CrossRef]
  6. B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010).
    [CrossRef]
  7. J. Cheng, “Transfer functions in lensless ghost-imaging systems,” Phys. Rev. A 78, 043823 (2008).
    [CrossRef]
  8. B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
    [CrossRef]
  9. T. Shirai, H. Kellock, T. Setälä, and A. T. Friberg, “Imaging through an aberrating medium with classical ghost diffraction,” J. Opt. Soc. Am. A 29, 1288–1292 (2012).
    [CrossRef]
  10. W. Gong and S. Han, “Multiple-input ghost imaging via sparsity constraints,” J. Opt. Soc. Am. A 29, 1571–1579 (2012).
    [CrossRef]
  11. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916–7921 (2009).
    [CrossRef]
  12. J. Lin and J. Cheng, “Lensless ghost diffraction with partially coherent sources: effects of the source size, transverse coherence, detector size and defocusing length,” Chin. Phys. Lett. 28, 094203 (2011).
    [CrossRef]
  13. R. Meyers, K. S. Deacon, and Y. shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008).
    [CrossRef]
  14. L. Basano and P. Ottonello, “Diffuse-reflection ghost imaging from a double-strip illuminated by pseudo-thermal light,” Opt. Commun. 283, 2657–2661 (2010).
    [CrossRef]
  15. N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “Experimental observation of lensless ghost imaging by measuring reflected photons,” Opt. Lasers Eng. 48, 671–675 (2010).
    [CrossRef]
  16. N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
    [CrossRef]
  17. C. F. Wang, D. W. Zhang, Y. F. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
    [CrossRef]
  18. N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
    [CrossRef]
  19. N. D. Hardy, “Analyzing and improving image quality in reflective ghost imaging,” M. S. thesis (Massachusetts Institute of Technology, 2011).
  20. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2007).
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  22. J. W. Goodman, Statistical Optics (Wiley, 1985).

2012

2011

J. Lin and J. Cheng, “Lensless ghost diffraction with partially coherent sources: effects of the source size, transverse coherence, detector size and defocusing length,” Chin. Phys. Lett. 28, 094203 (2011).
[CrossRef]

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

2010

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

L. Basano and P. Ottonello, “Diffuse-reflection ghost imaging from a double-strip illuminated by pseudo-thermal light,” Opt. Commun. 283, 2657–2661 (2010).
[CrossRef]

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “Experimental observation of lensless ghost imaging by measuring reflected photons,” Opt. Lasers Eng. 48, 671–675 (2010).
[CrossRef]

N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
[CrossRef]

C. F. Wang, D. W. Zhang, Y. F. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[CrossRef]

2009

2008

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

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

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

2007

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

2006

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

2004

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[CrossRef]

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

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging schemes: fast and broadband,” Opt. Express 12, 6067–6081 (2004).
[CrossRef]

Bache, M.

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

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[CrossRef]

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging schemes: fast and broadband,” Opt. Express 12, 6067–6081 (2004).
[CrossRef]

Bai, Y. F.

C. F. Wang, D. W. Zhang, Y. F. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[CrossRef]

Basano, L.

L. Basano and P. Ottonello, “Diffuse-reflection ghost imaging from a double-strip illuminated by pseudo-thermal light,” Opt. Commun. 283, 2657–2661 (2010).
[CrossRef]

Bisht, N. S.

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “Experimental observation of lensless ghost imaging by measuring reflected photons,” Opt. Lasers Eng. 48, 671–675 (2010).
[CrossRef]

Brambilla, E.

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

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging schemes: fast and broadband,” Opt. Express 12, 6067–6081 (2004).
[CrossRef]

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[CrossRef]

Chen, B.

C. F. Wang, D. W. Zhang, Y. F. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[CrossRef]

Cheng, J.

J. Lin and J. Cheng, “Lensless ghost diffraction with partially coherent sources: effects of the source size, transverse coherence, detector size and defocusing length,” Chin. Phys. Lett. 28, 094203 (2011).
[CrossRef]

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

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

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

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

Deacon, K. S.

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

Erkmen, B. I.

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

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

Ferri, F.

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

Friberg, A. T.

Gatti, A.

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

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[CrossRef]

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging schemes: fast and broadband,” Opt. Express 12, 6067–6081 (2004).
[CrossRef]

Gong, W.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2007).

J. W. Goodman, Statistical Optics (Wiley, 1985).

Han, S.

W. Gong and S. Han, “Multiple-input ghost imaging via sparsity constraints,” J. Opt. Soc. Am. A 29, 1571–1579 (2012).
[CrossRef]

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

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

Hardy, N. D.

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

N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
[CrossRef]

N. D. Hardy, “Analyzing and improving image quality in reflective ghost imaging,” M. S. thesis (Massachusetts Institute of Technology, 2011).

Kandpal, H. C.

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “Experimental observation of lensless ghost imaging by measuring reflected photons,” Opt. Lasers Eng. 48, 671–675 (2010).
[CrossRef]

Kellock, H.

Lin, J.

J. Lin and J. Cheng, “Lensless ghost diffraction with partially coherent sources: effects of the source size, transverse coherence, detector size and defocusing length,” Chin. Phys. Lett. 28, 094203 (2011).
[CrossRef]

Lugiato, L. A.

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

Luguato, L. A.

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging schemes: fast and broadband,” Opt. Express 12, 6067–6081 (2004).
[CrossRef]

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[CrossRef]

Magatti, D.

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

Meyers, R.

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

Ottonello, P.

L. Basano and P. Ottonello, “Diffuse-reflection ghost imaging from a double-strip illuminated by pseudo-thermal light,” Opt. Commun. 283, 2657–2661 (2010).
[CrossRef]

Setälä, T.

Shapiro, J. H.

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

N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
[CrossRef]

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

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

Sharma, E. K.

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “Experimental observation of lensless ghost imaging by measuring reflected photons,” Opt. Lasers Eng. 48, 671–675 (2010).
[CrossRef]

shih, Y.

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

Shirai, T.

Wang, C. F.

C. F. Wang, D. W. Zhang, Y. F. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[CrossRef]

Zhang, D. W.

C. F. Wang, D. W. Zhang, Y. F. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[CrossRef]

Adv. Opt. Photon.

Chin. Phys. Lett.

J. Lin and J. Cheng, “Lensless ghost diffraction with partially coherent sources: effects of the source size, transverse coherence, detector size and defocusing length,” Chin. Phys. Lett. 28, 094203 (2011).
[CrossRef]

J. Mod. Opt.

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

J. Opt. Soc. Am. A

Opt. Commun.

L. Basano and P. Ottonello, “Diffuse-reflection ghost imaging from a double-strip illuminated by pseudo-thermal light,” Opt. Commun. 283, 2657–2661 (2010).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “Experimental observation of lensless ghost imaging by measuring reflected photons,” Opt. Lasers Eng. 48, 671–675 (2010).
[CrossRef]

Phys. Rev. A

C. F. Wang, D. W. Zhang, Y. F. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[CrossRef]

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

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

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

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

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

M. Bache, E. Brambilla, A. Gatti, and L. A. Luguato, “Ghost imaging using homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[CrossRef]

Phys. Rev. Lett.

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

Proc. SPIE

N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE 7815, 78150L (2010).
[CrossRef]

Other

N. D. Hardy, “Analyzing and improving image quality in reflective ghost imaging,” M. S. thesis (Massachusetts Institute of Technology, 2011).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2007).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (5)

Fig. 1.
Fig. 1.

Scheme of the reflective GD or GI. BS denotes a nonpolarizing beam splitter. x, α, xr, xt are the coordinates at the source plane, object plane, reference detector plane, and test detector plane. θi, θo are the incident angle and reflective angle, respectively.

Fig. 2.
Fig. 2.

Reflective GD of a double-strip with different parameters, such as the reflective angle θo, surface height variance σh2 and correlation length lc, λ=0.658μm, θi=π/15, L0=60cm, L1=8cm, L2=52cm. Each curve is normalized according to its maximum value. To distinguish these curves clearly, they are shifted from the bottom to the top with bias 0, 1, 2, 3. (a) Effects of σh2 and lc θo=π/15. From the bottom to the top, (σh,lc)=(0.1λ,λ),(0.1λ,100λ),(0.2λ,λ),(0.2λ,100λ). (b–d) The different reflective angles θo are π/8, π/4, 3π/8, and other conditions are the same as (a).

Fig. 3.
Fig. 3.

Same as Fig. 2, but the reflective object is an eight-strip.

Fig. 4.
Fig. 4.

Reflective GI of a double-strip with different parameters, such as the reflective angle θo, surface height variance σh2 and correlation length lc. λ=0.658μm, re=8mm, D=10mm, θi=π/15, L0=L1=L2=60cm. Curves are shifted to be distinguished clearly, and the magnitudes of the bias for all curves from the bottom to the top are 0, 1, 2, 3 unit, respectively. (a) Effects of σh2 and lc. θo=π/15. From the bottom to the top, (σh,lc)=(0.1λ,500λ),(0.1λ,λ),(λ,500λ),(λ,λ). (b)–(d) Different reflective angles θo are π/8, π/4, 3π/8, and the other conditions are the same as (a).

Fig. 5.
Fig. 5.

Same as Fig. 4, but the reflective object is an eight-strip.

Equations (15)

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

Eo(α)=Ei(α)R(α)exp(jϕ(α)),
ϕ(α)=k(i^·n^+o^·n^)h(α),
Et(xt)=1λL1L2Es(x)exp[jπλL1(xαcosθi)2]R(α)×exp[jϕ(α)]exp[jπλL2(αcosθoxt)2]dxdα,
Er(xr)=1jλL0Es(x)exp[jπλL0(xrx)2]dx,
Es(x)Es*(x)Es(x)Es*(x)=Es(x)Es*(x)Es(x)Es*(x)+Es(x)Es*(x)Es(x)Es*(x).
G(xt,xr)=Er(xr)Er*(xr)Et(xt)Et*(xt)Er(xr)Er*(xr)Et(xt)Et*(xt)=1λ3L0L1L2dαdαt(α,α)dxdxdxdxΓ(x,x)Γ(x,x)exp{iπλL1[(xαcosθi)2(xαcosθi)2]}exp{iπλL0[(xxr)2(xxr)2]}exp{iπλL2[(αcosθoxt)2(αcosθoxt)2]},
Γ(x,x)=I0rect(xre)δ(xx),
σϕ2=[k(i^·n^+o^·n^)]2σh2,
I(xr)=G(xt=0,xr)=I02λ3L0L1L2dαdαdxdxt(α,α)exp[iπcos2θoλL2(α2α2)]rect(xre)rect(xre)exp{iπλL1[(xαcosθi)2(xαcosθi)2]}exp{iπλL0[(xxr)2(xxr)2]},
k0=πλL0,k1=πλL1,k2=πλL2cos2θocos2θi,
I(xr)=I02λ3L0L1L2cos2θidudut(u,u)re/2re/2dxdxexp[ik1((xu)2(xu)2)ik0((xxr)2(xxr)2)+ik2(u2u2)]=I02λ3L0L1L2cos2θidudut(u,u)re/2re/2dxdxexp[i((k1+k2)(u2u2)+(k1k0)x22(k1uk0xr)x(k1k0)x2+2(k1uk0xr)x)]=I02λ3L0L1L2cos2θidudut(u,u)ei(k1+k2)(u2u2)re/2re/2dxdxexp[i(k1k0)((xk1uk0xrk1k0)2(xk1uk0xrk1k0)2)i(k1uk0xr)2(k1uk0xr)2k1k0]=πI02dudut(u,u)exp[i(k1+k2)u2ik12u22k0k1uxrk1k0]exp[i(k1+k2)u2+ik12u22k0k1uxrk1k0]λ3L0L1L2cos2θi|k1k0|,
L0=L1+L2cos2θicos2θo,
I(xr)=πI02λ3L0L1L2cos2θi×dαdαt(α,α)exp[i2k0k1k1k0cosθi(α,α)xr]|k1k0|.
I(xr)=πI02λ3L0L1L2cos2θi|dαR(α)ei2πcosθiλ(L0L1)αxr|2
I(xr)=πDI02re2cosθoλ3L0L1L2cos2θit(α,α)exp[iπλ(cos2θiL1+cos2θoL2)(α2α2)]sinc(Dcosθo(αα)λL2)sinc[re(αcosθixr)λL1]sinc[re(αcosθixr)λL1]dαdα,

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