Abstract

The ultimate objective of laser speckle flowmetry (and a host of specific implementations such as laser speckle contrast analysis, LASCA or LSCA; laser speckle spatial contrast analysis, LSSCA; laser speckle temporal contrast analysis, LSTCA; etc.) is to infer flow velocity from the observed speckle contrast. Despite numerous demonstrations over the past 25 years of such a qualitative relationship, no convincing quantitative relationship has been proven. One reason is a persistent mathematical error that has been propagated by a host of workers; another is a misconception about the proper autocorrelation function for ordered flow. Still another hindrance has been uncertainty in the specific relationship between decorrelation time and local flow velocity. Herein we attempt to dispel some of these errors and misconceptions with the intent of turning laser speckle flowmetry into a quantitative tool. Specifically we review the underlying theory, explore the impact of various analytic models for relating measured intensity fluctuations to scatterer motion, and address some of the practical issues associated with the measurement and subsequent data processing.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2008 (3)

2007 (5)

H. Cheng and T. Q. Duong, “Simplified laser-speckle-imaging analysis method and its application to retinal blood flow imaging,” Opt. Lett. 32, 2188-2190 (2007).
[CrossRef] [PubMed]

F. Scheffold and R. Cerbino, “New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12, 50-57 (2007).
[CrossRef]

S. J. Kirkpatrick, D. D. Duncan, R. K. Wang, and M. K. Hinds, “Quantitative temporal contrast imaging for tissue mechanics,” J. Opt. Soc. Am. A 24, 3728-3734 (2007).
[CrossRef]

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

T. M. Le, J. S. Paul, H. Al-Nashash, A. Tan, A. R. Luft, F. S. Sheu, and S. H. Ong, “New insights into image processing of cortical blood flow monitors using laser speckle imaging,” IEEE Trans. Med. Imaging 26, 833-842 (2007).
[CrossRef]

2006 (3)

M. S. D. Smith, E. F. Packulak, and M. G. Sowa, “Development of a laser speckle imaging system for measuring relative blood flow velocity,” Proc. SPIE 6343, 634304 (2006).
[CrossRef]

E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor & Francis, 2006).
[CrossRef]

P. Zakharov, A. Völker, A. Buck, B. Weber, and F. Scheffold, “Quantitative modeling of laser speckle imaging,” Opt. Lett. 31, 3465-3467 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (1)

T. Durduran, M. G. Burnett, G. Yu, C. Zhou, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenburg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

2003 (2)

H. Isono, S. Kishi, Y. Kimura, N. Hagiwara, N. Konishi, and H. Fujii, “Observation of choroidal circulation using index of erythrocytic velocity,” Arch. Ophthalmol. (Chicago) 121, 225-231 (2003).

H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, “Modified laser speckle imaging method with improved spatial resolution,” J. Biomed. Opt. 8, 559-564 (2003).
[CrossRef] [PubMed]

2002 (1)

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

1999 (3)

Y. Aizu and T. Asakura, “Coherent optical techniques for diagnostics of retinal blood flow,” J. Biomed. Opt. 4, 61-75 (1999).
[CrossRef]

P. A. Lemieux and D. J. Durian, “Investigation of non-Gaussian scattering processes by using nth-order intensity correlation functions,” J. Opt. Soc. Am. A 16, 1651-1664 (1999).
[CrossRef]

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

1996 (1)

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): A nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1, 174-170 (1996).
[CrossRef]

1995 (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

D. J. Durian, “Accuracy of diffusing-wave spectroscopy theories,” Phys. Rev. E 51, 3350-3358 (1995).
[CrossRef]

1993 (1)

M. E. Thomas and D. D. Duncan, “Atmospheric transmission,” in Atmospheric Propagation of Radiation, Vol. 2 of the Infrared & Electro-Optical Systems Handbook, F.G.Smith, ed., (ERIM Infrared Information Analysis Center and SPIE Optical Engineering Press, 1993).

1991 (1)

B. Chu, Laser Light Scattering: Basic Principles and Practice, 2nd ed. (Academic, 1991).

1990 (1)

1986 (1)

1985 (1)

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

1984 (1)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics, 2nd ed., J.C.Dainty, ed. (Springer-Verlag, 1984).

1981 (1)

A. R. Fercher and J. D. Briers, “Flow visualization by means of single-exposure speckle photography,” Opt. Commun. 37, 326-330 (1981).
[CrossRef]

1970 (1)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).

Aizu, Y.

Y. Aizu and T. Asakura, “Coherent optical techniques for diagnostics of retinal blood flow,” J. Biomed. Opt. 4, 61-75 (1999).
[CrossRef]

Al-Nashash, H.

T. M. Le, J. S. Paul, H. Al-Nashash, A. Tan, A. R. Luft, F. S. Sheu, and S. H. Ong, “New insights into image processing of cortical blood flow monitors using laser speckle imaging,” IEEE Trans. Med. Imaging 26, 833-842 (2007).
[CrossRef]

Asakura, T.

Y. Aizu and T. Asakura, “Coherent optical techniques for diagnostics of retinal blood flow,” J. Biomed. Opt. 4, 61-75 (1999).
[CrossRef]

Bandyopadhyay, R.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Boas, D. A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).

Briers, J. D.

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): A nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1, 174-170 (1996).
[CrossRef]

A. R. Fercher and J. D. Briers, “Flow visualization by means of single-exposure speckle photography,” Opt. Commun. 37, 326-330 (1981).
[CrossRef]

Buck, A.

Buck, F.

Burnett, M. G.

T. Durduran, M. G. Burnett, G. Yu, C. Zhou, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenburg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

Cen, J.

H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, “Modified laser speckle imaging method with improved spatial resolution,” J. Biomed. Opt. 8, 559-564 (2003).
[CrossRef] [PubMed]

Cerbino, R.

F. Scheffold and R. Cerbino, “New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12, 50-57 (2007).
[CrossRef]

Chaikin, P. M.

Chen, S.

H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, “Modified laser speckle imaging method with improved spatial resolution,” J. Biomed. Opt. 8, 559-564 (2003).
[CrossRef] [PubMed]

Cheng, H.

H. Cheng and T. Q. Duong, “Simplified laser-speckle-imaging analysis method and its application to retinal blood flow imaging,” Opt. Lett. 32, 2188-2190 (2007).
[CrossRef] [PubMed]

H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, “Modified laser speckle imaging method with improved spatial resolution,” J. Biomed. Opt. 8, 559-564 (2003).
[CrossRef] [PubMed]

Choi, B.

Chu, B.

B. Chu, Laser Light Scattering: Basic Principles and Practice, 2nd ed. (Academic, 1991).

Detre, J. A.

T. Durduran, M. G. Burnett, G. Yu, C. Zhou, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenburg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

Devor, A.

Dixon, P. K.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Dörschel, K.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Duncan, D. D.

D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, “Statistics of local speckle contrast,” J. Opt. Soc. Am. A 25, 9-15 (2008).
[CrossRef]

S. J. Kirkpatrick, D. D. Duncan, R. K. Wang, and M. K. Hinds, “Quantitative temporal contrast imaging for tissue mechanics,” J. Opt. Soc. Am. A 24, 3728-3734 (2007).
[CrossRef]

M. E. Thomas and D. D. Duncan, “Atmospheric transmission,” in Atmospheric Propagation of Radiation, Vol. 2 of the Infrared & Electro-Optical Systems Handbook, F.G.Smith, ed., (ERIM Infrared Information Analysis Center and SPIE Optical Engineering Press, 1993).

Dunn, A. K.

Duong, T. Q.

Durduran, T.

T. Durduran, M. G. Burnett, G. Yu, C. Zhou, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenburg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

Durian, D. J.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

P. A. Lemieux and D. J. Durian, “Investigation of non-Gaussian scattering processes by using nth-order intensity correlation functions,” J. Opt. Soc. Am. A 16, 1651-1664 (1999).
[CrossRef]

D. J. Durian, “Accuracy of diffusing-wave spectroscopy theories,” Phys. Rev. E 51, 3350-3358 (1995).
[CrossRef]

Fercher, A. R.

A. R. Fercher and J. D. Briers, “Flow visualization by means of single-exposure speckle photography,” Opt. Commun. 37, 326-330 (1981).
[CrossRef]

Friebel, M.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Fujii, H.

H. Isono, S. Kishi, Y. Kimura, N. Hagiwara, N. Konishi, and H. Fujii, “Observation of choroidal circulation using index of erythrocytic velocity,” Arch. Ophthalmol. (Chicago) 121, 225-231 (2003).

Furuya, D.

T. Durduran, M. G. Burnett, G. Yu, C. Zhou, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenburg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

Gittings, A. S.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Gong, H.

H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, “Modified laser speckle imaging method with improved spatial resolution,” J. Biomed. Opt. 8, 559-564 (2003).
[CrossRef] [PubMed]

Goodman, J. W.

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

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

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics, 2nd ed., J.C.Dainty, ed. (Springer-Verlag, 1984).

Gopal, A.

Greenburg, J. H.

T. Durduran, M. G. Burnett, G. Yu, C. Zhou, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenburg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

Guizar-Iturbide, I.

Hagiwara, N.

H. Isono, S. Kishi, Y. Kimura, N. Hagiwara, N. Konishi, and H. Fujii, “Observation of choroidal circulation using index of erythrocytic velocity,” Arch. Ophthalmol. (Chicago) 121, 225-231 (2003).

Hahn, A.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Hinds, M. K.

Huang, J. S.

Isono, H.

H. Isono, S. Kishi, Y. Kimura, N. Hagiwara, N. Konishi, and H. Fujii, “Observation of choroidal circulation using index of erythrocytic velocity,” Arch. Ophthalmol. (Chicago) 121, 225-231 (2003).

Jakeman, E.

E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor & Francis, 2006).
[CrossRef]

Kimura, Y.

H. Isono, S. Kishi, Y. Kimura, N. Hagiwara, N. Konishi, and H. Fujii, “Observation of choroidal circulation using index of erythrocytic velocity,” Arch. Ophthalmol. (Chicago) 121, 225-231 (2003).

Kirkpatrick, S. J.

Kishi, S.

H. Isono, S. Kishi, Y. Kimura, N. Hagiwara, N. Konishi, and H. Fujii, “Observation of choroidal circulation using index of erythrocytic velocity,” Arch. Ophthalmol. (Chicago) 121, 225-231 (2003).

Konishi, N.

H. Isono, S. Kishi, Y. Kimura, N. Hagiwara, N. Konishi, and H. Fujii, “Observation of choroidal circulation using index of erythrocytic velocity,” Arch. Ophthalmol. (Chicago) 121, 225-231 (2003).

Le, T. M.

T. M. Le, J. S. Paul, H. Al-Nashash, A. Tan, A. R. Luft, F. S. Sheu, and S. H. Ong, “New insights into image processing of cortical blood flow monitors using laser speckle imaging,” IEEE Trans. Med. Imaging 26, 833-842 (2007).
[CrossRef]

Lemieux, P. A.

Luft, A. R.

T. M. Le, J. S. Paul, H. Al-Nashash, A. Tan, A. R. Luft, F. S. Sheu, and S. H. Ong, “New insights into image processing of cortical blood flow monitors using laser speckle imaging,” IEEE Trans. Med. Imaging 26, 833-842 (2007).
[CrossRef]

Luo, Q.

H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, “Modified laser speckle imaging method with improved spatial resolution,” J. Biomed. Opt. 8, 559-564 (2003).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Martinez-Niconoff, G.

Müller, G.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Ong, S. H.

T. M. Le, J. S. Paul, H. Al-Nashash, A. Tan, A. R. Luft, F. S. Sheu, and S. H. Ong, “New insights into image processing of cortical blood flow monitors using laser speckle imaging,” IEEE Trans. Med. Imaging 26, 833-842 (2007).
[CrossRef]

Packulak, E. F.

M. S. D. Smith, E. F. Packulak, and M. G. Sowa, “Development of a laser speckle imaging system for measuring relative blood flow velocity,” Proc. SPIE 6343, 634304 (2006).
[CrossRef]

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Parthasarathy, A. B.

Paul, J. S.

T. M. Le, J. S. Paul, H. Al-Nashash, A. Tan, A. R. Luft, F. S. Sheu, and S. H. Ong, “New insights into image processing of cortical blood flow monitors using laser speckle imaging,” IEEE Trans. Med. Imaging 26, 833-842 (2007).
[CrossRef]

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Pine, D. J.

Ramirez-San-Juan, J. C.

Ramos-Garcia, R.

Ridley, K. D.

E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor & Francis, 2006).
[CrossRef]

Roggan, A.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4, 36-46 (1999).
[CrossRef]

Scheffold, F.

Sheu, F. S.

T. M. Le, J. S. Paul, H. Al-Nashash, A. Tan, A. R. Luft, F. S. Sheu, and S. H. Ong, “New insights into image processing of cortical blood flow monitors using laser speckle imaging,” IEEE Trans. Med. Imaging 26, 833-842 (2007).
[CrossRef]

Smith, M. S. D.

M. S. D. Smith, E. F. Packulak, and M. G. Sowa, “Development of a laser speckle imaging system for measuring relative blood flow velocity,” Proc. SPIE 6343, 634304 (2006).
[CrossRef]

Sowa, M. G.

M. S. D. Smith, E. F. Packulak, and M. G. Sowa, “Development of a laser speckle imaging system for measuring relative blood flow velocity,” Proc. SPIE 6343, 634304 (2006).
[CrossRef]

Suh, S. S.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Tan, A.

T. M. Le, J. S. Paul, H. Al-Nashash, A. Tan, A. R. Luft, F. S. Sheu, and S. H. Ong, “New insights into image processing of cortical blood flow monitors using laser speckle imaging,” IEEE Trans. Med. Imaging 26, 833-842 (2007).
[CrossRef]

Thomas, M. E.

M. E. Thomas and D. D. Duncan, “Atmospheric transmission,” in Atmospheric Propagation of Radiation, Vol. 2 of the Infrared & Electro-Optical Systems Handbook, F.G.Smith, ed., (ERIM Infrared Information Analysis Center and SPIE Optical Engineering Press, 1993).

Tom, W. J.

Völker, A.

Völker, A. C.

Wang, R. K.

Weber, B.

Webster, S.

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): A nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1, 174-170 (1996).
[CrossRef]

Weitz, D. A.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Zhou, C.

T. Durduran, M. G. Burnett, G. Yu, C. Zhou, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenburg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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T. Durduran, M. G. Burnett, G. Yu, C. Zhou, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenburg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518-525 (2004).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Historical result due to Fercher and Briers [1] and correction.

Fig. 2
Fig. 2

Speckle contrast as a function of relative integration time for Lorentzian and Gaussian autocorrelation functions.

Fig. 3
Fig. 3

Fractional uncertainty in decorrelation time due to uncertainty in proper correlation model.

Fig. 4
Fig. 4

Sensitivity factors for two limiting correlation behaviors.

Fig. 5
Fig. 5

Speckle contrast as a function of integration time for various intensity correlation laws.

Equations (22)

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I ( r ¯ , t ) = 1 T d t i ( r ¯ , t ) rect ( t t T ) ,
rect ( x ) { 1 x 1 2 0 else ,
K ( r ¯ ) σ I ( r ¯ ) μ I ( r ¯ ) .
E { I ( r ¯ , t ) } μ I ( r ¯ ) = 1 T d t E { i ( r ¯ , t ) } rect ( t t T ) = μ i ( r ¯ )
σ I 2 ( r ¯ ) = 1 T 0 T d τ C i ( r ¯ , τ ) [ 2 ( 1 τ T ) ] ,
C i ( r ¯ , τ ) + μ i 2 ( r ¯ ) = R i ( r ¯ , τ ) = i ( r ¯ , t ) i ( r ¯ , t + τ ) .
R i ( r ¯ , τ ) = μ i 2 ( r ¯ ) + R E ( r ¯ , τ ) 2 .
R E ( r ¯ , τ ) = μ i ( r ¯ ) exp { τ τ c } ,
C i ( r ¯ , τ ) = μ i 2 ( r ¯ ) exp { 2 τ τ c } ,
K ( r ¯ ) = { τ c 2 T [ 2 τ c T ( 1 e 2 T τ c ) ] } 1 2 .
K ( r ¯ ) = { τ c 2 T ( 1 e 2 T τ c ) } 1 2 .
C i ( r ¯ , τ ) = μ i 2 ( r ¯ ) exp { 2 ( τ τ c ) 2 } ,
K ( r ¯ ) = { τ c 2 T [ 2 π erf ( 2 T τ c ) τ c T ( 1 e 2 ( T τ c ) 2 ) ] } 1 2 .
S = τ c τ c K K .
τ c = C i ( r ¯ , τ ) μ i 2 ( r ¯ ) d τ ,
K 2 ( T ) = 1 T 0 T d τ β g 1 ( τ ) 2 [ 2 ( 1 τ T ) ] ,
K = S M ,
M = 1 N s i = 1 N s I i , S 2 = 1 N s 1 i = 1 N s ( I i M ) 2 ,
τ c = λ 2 π V ,
C i ( r ¯ , τ ) μ i 2 ( r ¯ ) = [ 2 J 1 ( π D V τ λ z ) ( π D V τ λ z ) ] 2 ,
τ c = w V ,
w = λ z D ,

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