Abstract

The scattering of coherent monochromatic light at an optically rough surface, such as a diffuser, produces a speckle field, which is usually described by reference to its statistical properties. For example, the real and imaginary parts of a fully developed speckle field can be modeled as a random circular Gaussian process. When such a speckle field is used to illuminate a second diffuser, the statistics of the resulting doubly scattered field are in general no longer Gaussian, but rather follow a K distribution. In this paper we determine the space–time correlation function of such a doubly scattered speckle field that has been imaged by a single lens system. A space–time correlation function is derived that contains four separate terms; similar to the Gaussian case it contains an average DC term and a fluctuating AC term. However, in addition there are two terms that are related to contributions from each of the diffusers independently. We examine how our space–time correlation function varies as the diffusers are rotated at different speeds and as the point spread function of the imaging system is changed. A series of numerical simulations are used to confirm different aspects of the theoretical analysis. We then finish with a discussion of our results and some potential applications, including controlling spatial coherence and speckle reduction.

© 2013 Optical Society of America

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References

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  1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts, 2006), pp. 47–53.
  2. S. G. Hanson, T. F. Q. Iversen, and R. S. Hansen, “Dynamic properties of speckled speckles,” Proc. SPIE 7387, 738716 (2010).
    [CrossRef]
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    [CrossRef]
  4. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7, 827–832 (1990).
    [CrossRef]
  5. T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. A 10, 324–328 (1993).
    [CrossRef]
  6. H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A 16, 1402–1412 (1999).
    [CrossRef]
  7. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields: part I. Theory and a numerical investigation,” J. Opt. Soc. Am. A 28, 1896–1903 (2011).
    [CrossRef]
  8. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields: part II. Experimental investigation,” J. Opt. Soc. Am. A 28, 1904–1908 (2011).
    [CrossRef]
  9. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
    [CrossRef]
  10. K. A. O’Donnell, “Speckle statistics of doubly scattered light,” J. Opt. Soc. Am. 72, 1459–1463 (1982).
    [CrossRef]
  11. D. Newman, “K distributions from doubly scattered light,” J. Opt. Soc. Am. A 2, 22–26 (1985).
    [CrossRef]
  12. T. Yoshimura and K. Fujiwara, “Statistical properties of doubly scattered image speckle,” J. Opt. Soc. Am. A 9, 91–95 (1992).
    [CrossRef]
  13. R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
    [CrossRef]
  14. L. G. Shirley, “Laser speckle from thin and cascaded diffusers,” Ph.D. dissertation (University of Rochester, 1988).
  15. L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6, 765–781 (1989).
    [CrossRef]
  16. T. Okamoto and T. Asakura, “Detection of the object velocity using doubly-scattered dynamic speckles under Gaussian beam illumination,” J. Mod. Opt. 38, 1821–1839 (1991).
    [CrossRef]
  17. A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
    [CrossRef]
  18. D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
    [CrossRef]
  19. Personal communication, Prof. S. G. Hanson (during a research visit at the Technische Universität Ilmenau, Germany, November2012).
  20. W. Wang, S. Hanson, and M. Takeda, “Complex amplitude correlations of dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 23, 2198–2207 (2006).
    [CrossRef]
  21. V. S. R. Gudimetla, “Moments of the intensity of a non-circular Gaussian laser speckle in the diffraction field,” Opt. Commun. 130, 348–356 (1996).
    [CrossRef]
  22. D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
    [CrossRef]
  28. D. A. Tichenor and J. W. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. 62, 293–295 (1972).
    [CrossRef]
  29. J. Ohtsubo, “Non-Gaussian speckle: a computer simulation,” Appl. Opt. 21, 4167–4175 (1982).
    [CrossRef]
  30. T. Fricke-Begemann, “Optical measurement of deformation fields and surface processes with digital speckle correlation,” Ph.D. dissertation (Carl von Ossietzky Universitat Oldenburg, 2002).
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    [CrossRef]
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    [CrossRef]
  34. Y. Kuratomi, K. Sekiya, H. Satoh, T. Tomiyama, T. Kawakami, B. Katagiri, Y. Suzuki, and T. Uchida, “Speckle reduction mechanism in laser rear projection displays using a small moving diffuser,” J. Opt. Soc. Am. A 27, 1812–1817 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013 (1)

2011 (2)

2010 (2)

2009 (2)

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

2008 (2)

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
[CrossRef]

2007 (1)

2006 (3)

1999 (1)

1996 (1)

V. S. R. Gudimetla, “Moments of the intensity of a non-circular Gaussian laser speckle in the diffraction field,” Opt. Commun. 130, 348–356 (1996).
[CrossRef]

1993 (1)

1992 (2)

1991 (1)

T. Okamoto and T. Asakura, “Detection of the object velocity using doubly-scattered dynamic speckles under Gaussian beam illumination,” J. Mod. Opt. 38, 1821–1839 (1991).
[CrossRef]

1990 (1)

1989 (1)

1986 (1)

R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
[CrossRef]

1985 (1)

1982 (3)

1981 (1)

1972 (1)

1971 (1)

1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

Asakura, T.

T. Okamoto and T. Asakura, “Detection of the object velocity using doubly-scattered dynamic speckles under Gaussian beam illumination,” J. Mod. Opt. 38, 1821–1839 (1991).
[CrossRef]

Barakat, R.

R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
[CrossRef]

Chiang, F. P.

Churnside, J. H.

Claus, D.

Ferri, F.

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

Fricke-Begemann, T.

T. Fricke-Begemann, “Optical measurement of deformation fields and surface processes with digital speckle correlation,” Ph.D. dissertation (Carl von Ossietzky Universitat Oldenburg, 2002).

Fried, D. L.

Fujiwara, K.

Gatti, A.

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

George, N.

Goodman, J. W.

D. A. Tichenor and J. W. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. 62, 293–295 (1972).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1966).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts, 2006), pp. 47–53.

Gudimetla, V. S. R.

V. S. R. Gudimetla, “Moments of the intensity of a non-circular Gaussian laser speckle in the diffraction field,” Opt. Commun. 130, 348–356 (1996).
[CrossRef]

Hansen, R. S.

S. G. Hanson, T. F. Q. Iversen, and R. S. Hansen, “Dynamic properties of speckled speckles,” Proc. SPIE 7387, 738716 (2010).
[CrossRef]

H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A 16, 1402–1412 (1999).
[CrossRef]

Hanson, S.

Hanson, S. G.

S. G. Hanson, T. F. Q. Iversen, and R. S. Hansen, “Dynamic properties of speckled speckles,” Proc. SPIE 7387, 738716 (2010).
[CrossRef]

H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A 16, 1402–1412 (1999).
[CrossRef]

Personal communication, Prof. S. G. Hanson (during a research visit at the Technische Universität Ilmenau, Germany, November2012).

Hennelly, B. M.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Iversen, T. F. Q.

S. G. Hanson, T. F. Q. Iversen, and R. S. Hansen, “Dynamic properties of speckled speckles,” Proc. SPIE 7387, 738716 (2010).
[CrossRef]

Iwamoto, S.

Joyeux, D.

Katagiri, B.

Kawakami, T.

Kelly, D. P.

D. P. Kelly and D. Claus, “Filtering role of the sensor pixel in Fourier and Fresnel digital holography,” Appl. Opt. 52, A336–A345 (2013).
[CrossRef]

D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields: part I. Theory and a numerical investigation,” J. Opt. Soc. Am. A 28, 1896–1903 (2011).
[CrossRef]

D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields: part II. Experimental investigation,” J. Opt. Soc. Am. A 28, 1904–1908 (2011).
[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Fundamental diffraction limits in a paraxial 4-f imaging system with coherent and incoherent illumination,” J. Opt. Soc. Am. A 24, 1911–1919 (2007).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Kirchner, M.

Kuratomi, Y.

Leushacke, L.

Li, D.

Li, Q. B.

Lowenthal, S.

Magatti, D.

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

Naughton, T. J.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Newman, D.

O’Donnell, K. A.

Ohtsubo, J.

Okamoto, T.

T. Okamoto and T. Asakura, “Detection of the object velocity using doubly-scattered dynamic speckles under Gaussian beam illumination,” J. Mod. Opt. 38, 1821–1839 (1991).
[CrossRef]

Pandey, N.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Fundamental diffraction limits in a paraxial 4-f imaging system with coherent and incoherent illumination,” J. Opt. Soc. Am. A 24, 1911–1919 (2007).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Rose, B.

Satoh, H.

Sekiya, K.

Sheridan, J. T.

Shirley, L. G.

L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6, 765–781 (1989).
[CrossRef]

L. G. Shirley, “Laser speckle from thin and cascaded diffusers,” Ph.D. dissertation (University of Rochester, 1988).

Suzuki, Y.

Takeda, M.

Tichenor, D. A.

Tomiyama, T.

Uchida, T.

Voelz, D.

Wang, W.

Ward, J. E.

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
[CrossRef]

Xiao, X.

Yoshimura, T.

Yura, H. T.

Appl. Opt. (3)

IRE Trans. Inf. Theory (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

J. Mod. Opt. (1)

T. Okamoto and T. Asakura, “Detection of the object velocity using doubly-scattered dynamic speckles under Gaussian beam illumination,” J. Mod. Opt. 38, 1821–1839 (1991).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (11)

Y. Kuratomi, K. Sekiya, H. Satoh, T. Tomiyama, T. Kawakami, B. Katagiri, Y. Suzuki, and T. Uchida, “Speckle reduction mechanism in laser rear projection displays using a small moving diffuser,” J. Opt. Soc. Am. A 27, 1812–1817 (2010).
[CrossRef]

L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6, 765–781 (1989).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Fundamental diffraction limits in a paraxial 4-f imaging system with coherent and incoherent illumination,” J. Opt. Soc. Am. A 24, 1911–1919 (2007).
[CrossRef]

L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7, 827–832 (1990).
[CrossRef]

T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. A 10, 324–328 (1993).
[CrossRef]

H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A 16, 1402–1412 (1999).
[CrossRef]

D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields: part I. Theory and a numerical investigation,” J. Opt. Soc. Am. A 28, 1896–1903 (2011).
[CrossRef]

D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields: part II. Experimental investigation,” J. Opt. Soc. Am. A 28, 1904–1908 (2011).
[CrossRef]

D. Newman, “K distributions from doubly scattered light,” J. Opt. Soc. Am. A 2, 22–26 (1985).
[CrossRef]

T. Yoshimura and K. Fujiwara, “Statistical properties of doubly scattered image speckle,” J. Opt. Soc. Am. A 9, 91–95 (1992).
[CrossRef]

W. Wang, S. Hanson, and M. Takeda, “Complex amplitude correlations of dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 23, 2198–2207 (2006).
[CrossRef]

Opt. Acta (1)

R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
[CrossRef]

Opt. Commun. (1)

V. S. R. Gudimetla, “Moments of the intensity of a non-circular Gaussian laser speckle in the diffraction field,” Opt. Commun. 130, 348–356 (1996).
[CrossRef]

Opt. Eng. (2)

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (2)

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

Proc. SPIE (2)

S. G. Hanson, T. F. Q. Iversen, and R. S. Hansen, “Dynamic properties of speckled speckles,” Proc. SPIE 7387, 738716 (2010).
[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
[CrossRef]

Other (6)

The Mathworks Ins., http://www.mathworks.co.uk/help/images/ref/imrotate.html (Data retrieved: November2012).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1966).

T. Fricke-Begemann, “Optical measurement of deformation fields and surface processes with digital speckle correlation,” Ph.D. dissertation (Carl von Ossietzky Universitat Oldenburg, 2002).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts, 2006), pp. 47–53.

Personal communication, Prof. S. G. Hanson (during a research visit at the Technische Universität Ilmenau, Germany, November2012).

L. G. Shirley, “Laser speckle from thin and cascaded diffusers,” Ph.D. dissertation (University of Rochester, 1988).

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Figures (8)

Fig. 1.
Fig. 1.

Optical configuration in which dynamic speckled speckles are formed. D1, the first diffuser; D2, the second diffuser. Both can be rotated around the optical axis with angular speed ω1 and ω2, respectively. L, thin lens with diameter 2a0 and focal length f.

Fig. 2.
Fig. 2.

Plot of the spatial correlation function μ(|Δp|) versus |Δp| for several values of the quantity v. Note that as v decreases, the statistics of the speckle field transforms from K distribution to Gaussian distribution.

Fig. 3.
Fig. 3.

Plot of the temporal correlation function as a function of ω2τ(ω1τ) for several values of the quantity v, under the condition that ω10 and ω20. Note that as v decreases, the statistics of the speckle field transforms from K distribution to Gaussian distribution.

Fig. 4.
Fig. 4.

Plot of the temporal correlation function μ(p,ω1,0,τ) as a function of ω1τ, for several values of the quantity v. Note that whatever the value of v is, the statistics of the speckle field is obeying Gaussian distribution.

Fig. 5.
Fig. 5.

(a) Simulating images of Gaussian speckle pattern and (b) K-distributed speckle pattern. Due to the value of v chosen, the statistics the speckle field is transforming from Gaussian distribution to K distribution. The contrast of the speckle pattern in (a) is 1.06 and in (b) is 1.6.

Fig. 6.
Fig. 6.

Simulation results of the spatial correlation function μ(|Δp|) versus |Δp| for several values of the quantity v. Solid curve: theoretical perditions from Eq. (18).

Fig. 7.
Fig. 7.

Simulation results of the temporal correlation function as a function of ω1τ for several values of the quantity v, under the condition that ω10 and ω20. Dots: mean correlation values of running the simulation 10 times. Error bar: the standard deviation of the correlation values. Solid curve: theoretical perditions from Eq. (19).

Fig. 8.
Fig. 8.

Simulation results of the temporal correlation function as a function of ω1τ for several values of the quantity v, under the condition that ω10 and ω2=0. Dots: mean correlation values of running the simulation 10 times. Error bar: the standard deviation of the correlation values. Solid curve: theoretical perditions from Eq. (20).

Equations (26)

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

I(r1,t1)I(r2,t2)=I(r1,t1)I(r2,t2)+|E*(r1,t1)E(r2,t2)|2,
E*(r2,t1+τ)E(r1,t1)=exp[ik2d(r22r12)]exp{w02k28d2[r1(r2cosω1τ+Rr2sinω1τ)]2},
E*(r2,t1)E(r1,t1)=exp[ik2d(r22r12)]exp[(r1r2)22ζ2],
ζ=2dw0k.
I(p1,t)I(p2,t+τ)=A*(p1,t)A(p1,t)A*(p2,t+τ)A(p2,t+τ),
A(p,t)=E(r,t)exp[iψ(r,t)]T2(p,r)d2r.
T2(p,r)=exp[iπλ(r2di+p2do)]exp[(rρ1+pρ2)2],
ρ1=λdiπa0
ρ2=λdoπa0,
I(p1,t)I(p2,t+τ)=E*(r1,t)E(r2,t)E*(r3,t+τ)E(r4,t+τ)×exp[iψ(r1,t)+iψ(r2,t)iψ(r3,t+τ)+iψ(r4,t+τ)]×T2*(p1,r1)T2(p1,r2)T2*(p2,r3)T2(p2,r4)d2r1d2r2d2r3d2r4.
E*(r1,t)E(r2,t)E*(r3,t+τ)E(r4,t+τ)=E*(r1,t)E(r2,t)E*(r3,t+τ)E(r4,t+τ)+E*(r1,t)E(r4,t+τ)E*(r3,t+τ)E(r2,t).
exp[iψ(r1,t)+iψ(r2,t)iψ(r3,t+τ)+iψ(r4,t+τ)]=δ(r1r2)δ(r3r4)+δ(r4r1)δ(r3r2),
r1=r1cosω2τRr1sinω2τ
r2=r2cosω2τRr2sinω2τ,
I(p1,t)I(p2,t+τ)=Γ0+Γ1(p1,p2,ω1,τ)+Γ2(p1,p2,ω2,τ)+Γ12(p1,p2,ω1,ω2,τ),
Γ0=π2ρ144,
Γ1(p1,p2,ω1,τ)=π2ρ144vexp[(1v)(p2p1cosω1τ+Rp1sinω1τ)2ρ22],
Γ2(p1,p2,ω2,τ)=π2ρ144vexp[(p2p1cosω2τ+Rp1sinω1τ)2ρ22]
Γ12(p1,p2,ω1,ω2,τ)=π2ρ14v2[1+v(1v)cos(ω1τω2τ)]2exp{2v(p2p1cosω2τ+Rp1sinω2τ)2ρ22[1+v(1v)cos(ω1τω2τ)]}×exp{2(1v)[1cos(ω1τω2τ)]+(p12+p22)ρ22[1+v(1v)cos(ω1τω2τ)]}.
v=ζ2(ρ12+ζ2),
N=1v1.
μG(p1,p2,ω1,ω2,τ)=|A*(p1,t)A(p2,t+τ)|G2I(p1,t)I(p2,t+τ)G=Γ12(p1,p2,ω1,ω2,τ)Γ0,
μ(|Δp|)=I(p1)I(p2)Γ0Γ0=vexp[(1v)Δp2ρ22]+vexp(Δp2ρ22)+exp(Δp2ρ22),
μ(|p|,ω1,ω2,τ)=I(p)I(p)Γ0Γ0=vexp[2(1v)p2(1cosω1τ)ρ22]+vexp[2p2(1cosω2τ)ρ22]+4v2[1+v(1v)cos(ω1τω2τ)]2exp{4vp2(1cosω2τ)ρ22[1+v(1v)cos(ω1τω2τ)]}×exp{4(1v)p2[1cos(ω1τω2τ)]ρ22[1+v(1v)cos(ω1τω2τ)]}.
μ(|p|,ω1,0,τ)=I(p)I(p)Γ0Γ2Γ0+Γ2=v1+vexp[2(1v)p2(1cosω1τ)ρ22]+4v2exp[4(1v)p21cosω1τρ22(1+v(1v)cosω1τ](1+v)[1+v(1v)cosω1τ]2.
μ=|I1{I{I1}×{I{I2}}*}|I¯1×I¯2I¯1×I¯2,

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