Abstract

Sensing and imaging methods based on the dynamic scattering of coherent light (including laser speckle, laser Doppler, diffuse correlation spectroscopy, dynamic light scattering, and diffusing wave spectroscopy) quantify scatterer motion using light intensity fluctuations. The underlying optical field autocorrelation, rather than being measured directly, is typically inferred from the intensity autocorrelation through the Siegert relationship, assuming that the scattered field obeys Gaussian statistics. Here, we demonstrate interferometric near-infrared spectroscopy for measuring the time-of-flight (TOF) resolved field and intensity autocorrelations in turbid media. We find that the Siegert relationship breaks down for short TOFs due to static paths whose optical field does not decorrelate over experimental time scales. We also show that eliminating such paths by polarization gating restores the validity of the Siegert relationship. The unique capability of measuring optical field autocorrelations, as demonstrated here, enables the study of non-Gaussian and non-ergodic light scattering processes. Moreover, direct measurements of field autocorrelations are more efficient than indirect measurements based on intensity autocorrelations. Thus, optical field measurements may improve the quantification of scatterer dynamics with coherent light.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (1)

2014 (1)

T. Durduran and A. G. Yodh, “Diffuse correlation spectroscopy for non-invasive, micro-vascular cerebral blood flow measurement,” NeuroImage 85, 51–63 (2014).
[Crossref]

2010 (1)

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15, 011109 (2010).
[Crossref]

2008 (2)

2004 (1)

P. Jacquod and E. V. Sukhorukov, “Breakdown of universality in quantum chaotic transport: The two-phase dynamical fluid model,” Phys. Rev. Lett. 92, 116801 (2004).
[Crossref]

2002 (1)

V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: A tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
[Crossref]

2001 (1)

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[Crossref]

2000 (1)

P.-A. Lemieux and D. J. Durian, “From avalanches to fluid flow: A continuous picture of grain dynamics down a heap,” Phys. Rev. Lett. 85, 4273–4276 (2000).
[Crossref]

1999 (1)

1998 (2)

I. Flammer, G. Bucher, and J. Rička, “Diffusive wave illumination: light-scattering study of colloidal dynamics in opaque media,” J. Opt. Soc. Am. A 15, 2066–2077 (1998).
[Crossref]

V. A. Gopar and P. A. Mello, “The problem of quantum chaotic scattering with direct processes reduced to the one without,” Europhys. Lett. 42, 131–136 (1998).
[Crossref]

1997 (1)

1996 (1)

H. U. Baranger and P. A. Mello, “Short paths and information theory in quantum chaotic scattering: transport through quantum dots,” Europhys. Lett. 33, 465–470 (1996).
[Crossref]

1995 (1)

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref]

1993 (1)

1992 (2)

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[Crossref]

B. Ackerson, R. Dougherty, N. Reguigui, and U. Nobbman, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[Crossref]

1990 (2)

A. G. Yodh, P. D. Kaplan, and D. J. Pine, “Pulsed diffusing-wave spectroscopy: High resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
[Crossref]

J. G. H. Joosten, E. T. F. Geladé, and P. N. Pusey, “Dynamic light scattering by nonergodic media: Brownian particles trapped in polyacrylamide gels,” Phys. Rev. A 42, 2161–2175 (1990).
[Crossref]

1989 (2)

F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[Crossref]

P. Pusey and W. V. Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
[Crossref]

1988 (2)

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref]

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[Crossref]

1987 (1)

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[Crossref]

1981 (1)

1970 (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[Crossref]

1962 (1)

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

Ackerson, B.

B. Ackerson, R. Dougherty, N. Reguigui, and U. Nobbman, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[Crossref]

Baranger, H. U.

H. U. Baranger and P. A. Mello, “Short paths and information theory in quantum chaotic scattering: transport through quantum dots,” Europhys. Lett. 33, 465–470 (1996).
[Crossref]

Berne, B. J.

B. J. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, 2000).

Boas, D. A.

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15, 011109 (2010).
[Crossref]

D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192–215 (1997).
[Crossref]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref]

Bonner, R.

Borycki, D.

Bucher, G.

Campbell, L. E.

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref]

Chaikin, P. M.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[Crossref]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref]

Chong, S. P.

Cowan, M.

J. Page, M. Cowan, D. Weitz, and B. van Tiggelen, “Diffusing acoustic wave spectroscopy: Field fluctuation spectroscopy with multiply scattered ultrasonic waves,” in Wave Scattering in Complex Media: From Theory to Applications, B. A. van Tiggelen and S. E. Skipetrov, eds., Vol. 107 of NATO Science Series (Springer, 2003), pp. 151–174.

Cummins, H.

H. Cummins and H. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1970), Vol. 8, pp. 133–200.

Dainty, J. C.

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[Crossref]

Dougherty, R.

B. Ackerson, R. Dougherty, N. Reguigui, and U. Nobbman, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[Crossref]

Dunn, A. K.

Durduran, T.

T. Durduran and A. G. Yodh, “Diffuse correlation spectroscopy for non-invasive, micro-vascular cerebral blood flow measurement,” NeuroImage 85, 51–63 (2014).
[Crossref]

Durian, D. J.

P.-A. Lemieux and D. J. Durian, “From avalanches to fluid flow: A continuous picture of grain dynamics down a heap,” Phys. Rev. Lett. 85, 4273–4276 (2000).
[Crossref]

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

Flammer, I.

Foschum, F.

Geladé, E. T. F.

J. G. H. Joosten, E. T. F. Geladé, and P. N. Pusey, “Dynamic light scattering by nonergodic media: Brownian particles trapped in polyacrylamide gels,” Phys. Rev. A 42, 2161–2175 (1990).
[Crossref]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications (Roberts and Company, 2006).

Gopal, A.

Gopar, V. A.

V. A. Gopar and P. A. Mello, “The problem of quantum chaotic scattering with direct processes reduced to the one without,” Europhys. Lett. 42, 131–136 (1998).
[Crossref]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref]

Jacquod, P.

P. Jacquod and E. V. Sukhorukov, “Breakdown of universality in quantum chaotic transport: The two-phase dynamical fluid model,” Phys. Rev. Lett. 92, 116801 (2004).
[Crossref]

John, S.

F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[Crossref]

Joosten, J. G. H.

J. G. H. Joosten, E. T. F. Geladé, and P. N. Pusey, “Dynamic light scattering by nonergodic media: Brownian particles trapped in polyacrylamide gels,” Phys. Rev. A 42, 2161–2175 (1990).
[Crossref]

Kaplan, P. D.

A. G. Yodh, P. D. Kaplan, and D. J. Pine, “Pulsed diffusing-wave spectroscopy: High resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
[Crossref]

Kholiqov, O.

Kienle, A.

Lemieux, P.-A.

P.-A. Lemieux and D. J. Durian, “From avalanches to fluid flow: A continuous picture of grain dynamics down a heap,” Phys. Rev. Lett. 85, 4273–4276 (2000).
[Crossref]

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

Lequeux, F.

V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: A tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
[Crossref]

MacKintosh, F. C.

F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[Crossref]

Mandel, L.

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

Maret, G.

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[Crossref]

Megen, W. V.

P. Pusey and W. V. Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
[Crossref]

Mello, P. A.

V. A. Gopar and P. A. Mello, “The problem of quantum chaotic scattering with direct processes reduced to the one without,” Europhys. Lett. 42, 131–136 (1998).
[Crossref]

H. U. Baranger and P. A. Mello, “Short paths and information theory in quantum chaotic scattering: transport through quantum dots,” Europhys. Lett. 33, 465–470 (1996).
[Crossref]

Michels, R.

Milner, S. T.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[Crossref]

Nobbman, U.

B. Ackerson, R. Dougherty, N. Reguigui, and U. Nobbman, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[Crossref]

Nossal, R.

Page, J.

J. Page, M. Cowan, D. Weitz, and B. van Tiggelen, “Diffusing acoustic wave spectroscopy: Field fluctuation spectroscopy with multiply scattered ultrasonic waves,” in Wave Scattering in Complex Media: From Theory to Applications, B. A. van Tiggelen and S. E. Skipetrov, eds., Vol. 107 of NATO Science Series (Springer, 2003), pp. 151–174.

Parthasarathy, A. B.

Pecora, R.

B. J. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, 2000).

Pine, D. J.

V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: A tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
[Crossref]

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[Crossref]

A. G. Yodh, P. D. Kaplan, and D. J. Pine, “Pulsed diffusing-wave spectroscopy: High resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
[Crossref]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref]

Pusey, N.

N. Pusey, “Dynamic light scattering,” in Neutron, X-rays and Light. Scattering Methods Applied to Soft Condensed Matter, T. Zemb and P. Lindner, eds. (2002), Chap. 9.

Pusey, P.

P. Pusey and W. V. Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
[Crossref]

Pusey, P. N.

J. G. H. Joosten, E. T. F. Geladé, and P. N. Pusey, “Dynamic light scattering by nonergodic media: Brownian particles trapped in polyacrylamide gels,” Phys. Rev. A 42, 2161–2175 (1990).
[Crossref]

Reed, I. S.

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

Reguigui, N.

B. Ackerson, R. Dougherty, N. Reguigui, and U. Nobbman, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[Crossref]

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” in Noise and Stochastic Processes, N. Wax, ed. (Dover, 1954), p. 133.

Ricka, J.

Romer, S.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[Crossref]

Schätzel, K.

Scheffold, F.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[Crossref]

Schurtenberger, P.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[Crossref]

Siegert, A. J. F.

A. J. F. Siegert, MIT Rad. Lab. Rep. (1943).

Skipetrov, S. E.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[Crossref]

Srinivasan, V. J.

Stephen, M. J.

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[Crossref]

Sukhorukov, E. V.

P. Jacquod and E. V. Sukhorukov, “Breakdown of universality in quantum chaotic transport: The two-phase dynamical fluid model,” Phys. Rev. Lett. 92, 116801 (2004).
[Crossref]

Swinney, H.

H. Cummins and H. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1970), Vol. 8, pp. 133–200.

Tom, W. J.

van Tiggelen, B.

J. Page, M. Cowan, D. Weitz, and B. van Tiggelen, “Diffusing acoustic wave spectroscopy: Field fluctuation spectroscopy with multiply scattered ultrasonic waves,” in Wave Scattering in Complex Media: From Theory to Applications, B. A. van Tiggelen and S. E. Skipetrov, eds., Vol. 107 of NATO Science Series (Springer, 2003), pp. 151–174.

Viasnoff, V.

V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: A tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
[Crossref]

Weitz, D.

J. Page, M. Cowan, D. Weitz, and B. van Tiggelen, “Diffusing acoustic wave spectroscopy: Field fluctuation spectroscopy with multiply scattered ultrasonic waves,” in Wave Scattering in Complex Media: From Theory to Applications, B. A. van Tiggelen and S. E. Skipetrov, eds., Vol. 107 of NATO Science Series (Springer, 2003), pp. 151–174.

Weitz, D. A.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref]

Wolf, E.

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

Wolf, P. E.

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[Crossref]

Wu, X.-L.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[Crossref]

Xue, J.-Z.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-L. Wu, and P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[Crossref]

Yodh, A. G.

T. Durduran and A. G. Yodh, “Diffuse correlation spectroscopy for non-invasive, micro-vascular cerebral blood flow measurement,” NeuroImage 85, 51–63 (2014).
[Crossref]

D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192–215 (1997).
[Crossref]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref]

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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. (a) A Mach–Zehnder interferometer with a tunable, temporally coherent light source (DFB, distributed feedback laser centered at 855 nm; APP—anamorphic prism pair; OI—optical isolator; FOPC, fiber optical polarization controller; DBD—dual balanced detector; L1, L2, L3, L4—lenses; PC—personal computer). (b) Idealized visualization of the real part of the cross-spectral density for different photon paths. (c) Coherent light scattering from moving particles causes fluctuations in the mutual coherence function, Γrs. Fluctuation dynamics increase with photon TOF due to increasing momentum transfer. (d) TPSF is obtained by temporally averaging consecutive measurements of |Γrs|2 [see Eq. (13)]. (e) Field and intensity autocorrelations are determined from Γrs and |Γrs|2, respectively.
Fig. 2.
Fig. 2. Transmitted field and intensity dynamics as a function of TOF and scatterer concentration for parallel polarization: (a) cp=5.2%, (b) cp=5.5%, and (c) cp=5.8%. The first column shows the TPSFs, with constituent static and dynamic components, and the second column shows complex field time courses for three τs values, marked on the TPSF plots. The third column shows that the directly estimated field autocorrelations, g1(iNIRS), calculated using Eq. (10), disagree with g1,I(iNIRS), the field autocorrelations inferred from the intensity autocorrelations using the Siegert relation [Eq. (1)] with variable β. Discrepancies are most notable for short paths and low scattering. However, as shown in the fourth column, both the field-based and intensity-based dynamic-component autocorrelation estimates (γ1(iNIRS) and γ1,I(iNIRS), respectively) agree better if the non-ergodic component is accounted for [cf. Eqs. (5) and (7)]. In this figure, the superscript (iNIRS) was omitted to improve readability.
Fig. 3.
Fig. 3. Transmitted intensity and phase distributions for a cp=5.2% phantom and parallel polarizations [Fig. 2(a)]. Intensity histograms for: (a) τs=50  ps, (b) τs=80  ps, and (c) τs=110  ps are fitted with modified Rician [Eq. (16)] and negative exponential distributions. Both fits overlap for long paths. I¯s denotes the mean value of Is. (d) Optical field phases are uniform for long paths.
Fig. 4.
Fig. 4. Transmitted field and intensity dynamics as a function of TOF for perpendicular polarization and cp=5.2% [same concentration as in Fig. 2(a)]. (a) TPSF overlaps with If(τs) due to the absence of the static component. Note that normalization was performed to the TPSF maximum for parallel polarization [cf. Fig. 2(a)]. (a) Complex field time courses. (c, d) Directly estimated field autocorrelations agree with those inferred from intensity autocorrelations.
Fig. 5.
Fig. 5. Transmitted intensity and phase distributions for cp=5.2% phantom and perpendicular polarizations [Fig. 4]. (a) Intensity histogram for a short path (τs=50  ps) is fitted with negative exponential and modified Rician distributions, which overlap. (b) Optical field phases are uniform for even the shortest paths [cf. Fig. 3].

Equations (20)

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g2(τd)=1+β|g1(τd)|2,
g1(τd)=Us*(td)Us(td+τd)Is(td),
g2(τd)=Is(td)Is(td+τd)Is(td)2,
Us(td)=Uf(td)+Uc,
g1(τd)=η+(1η)γ1(τd),
g2(τd)=1+2η(1η)Re[γ1(τd)]+(1η)2|γ1(τd)|2.
γ1(τd)=η2+g2(τd)1η1η,
S(ν,td)=SDC(ν,td)+2Re[Wrs(ν,td)],
F1{2Re[Wrs(ν,td)]}=F1{Wrs(ν,td)}+F1{Wrs*(ν,td)}=Γrs(τs,td)+Γrs*(τs,td),
Γrs(τs,td)=Ur*(ts,td)Us(ts+τs,td)ts=limT12TTTUr*(ts,td)Us(ts+τs,td)dts,
g1(iNIRS)(τs,τd)=G1(iNIRS)(τs,τd)G1(iNIRS)(τs,0),
G1(τs,τd)=Us*(τs,td)Us(τs,td+τd)td,
G1(iNIRS)(τs,τd)=G1(τs,τd)I0(τsτs)dτs,
I¯s(iNIRS)(τs)=G1(iNIRS)(τs,0)=Is(τs,td)td=Is(τs,td)td*I0(τs),
G1(τs,τd)=Ic(τs)+G1,f(τs,τd).
G1(iNIRS)(τs,τd)=[Ic(τs)+G1,f(τs,τd)]*I0(τs)=Ic(iNIRS)(τs)+G1,f(iNIRS)(τs,τd),
I¯s(iNIRS)(τs)=Ic(iNIRS)(τs)+I¯f(iNIRS)(τs)
g2(iNIRS)(τs,τd)=Is(τs,td)Is(τs,td+τd)tdIs(τs,td)td2.
pI(Is)=ξexp[ξ(Is+Ic)]I0(α),Is0,
pϕ(ϕs)=ξ2πeξIc0exp[αcos(ϕs)ξIs]dIs,πϕs<π,

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