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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 14 — Jul. 5, 2010
  • pp: 14519–14534
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Noise in laser speckle correlation and imaging techniques

S. E. Skipetrov, J. Peuser, R. Cerbino, P. Zakharov, B. Weber, and F. Scheffold  »View Author Affiliations


Optics Express, Vol. 18, Issue 14, pp. 14519-14534 (2010)
http://dx.doi.org/10.1364/OE.18.014519


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Abstract

We study the noise of the intensity variance and of the intensity correlation and structure functions measured in light scattering from a random medium in the case when these quantities are obtained by averaging over a finite number N of pixels of a digital camera. We show that the noise scales as 1/N in all cases and that it is sensitive to correlations of signals corresponding to adjacent pixels as well as to the effective time averaging (due to the finite integration time) and spatial averaging (due to the finite pixel size). Our results provide a guide to estimation of noise levels in such applications as multi-speckle dynamic light scattering, time-resolved correlation spectroscopy, speckle visibility spectroscopy, laser speckle imaging etc.

© 2010 Optical Society of America

1. Introduction

The statistical properties of optical speckle patterns resulting from scattering of light from a random medium are largely independent of the nature of the latter [1

1. J.W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).

]. Rearrangements of scatterers due to Brownian motion or flow lead to characteristic fluctuations of the speckle intensity with time. Probing these intensity fluctuations is a very common and highly sensitive non-invasive method to study the dynamics of complex fluids and biological systems. One of the earliest applications is dynamic light scattering (DLS), also known as photon correlation spectroscopy (PCS), where the fluctuations of the far-field speckle are analyzed [2

2. B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

]. The method is widely used to study polymers in solution or for the sizing of sub-micron particles. The same method applied to turbid media is known as diffusing-wave spectroscopy (DWS). In both DLS and DWS, one usually makes use of the ergodicity of intensity fluctuations and replace ensemble averaging by time averaging. Both are typically applied to study fluctuations in the (sub-) millisecond range. Thus, for a total measurement time of a few minutes, millions of fluctuations are sampled which provides an excellent signal to noise ratio. Using a single photon counter and a digital correlator, nanosecond time resolution is commonly achieved. Even though at the shortest times the signal to noise ratio is limited by photon shot noise [3

3. K. Schätzel, “Noise in photon correlation and photon structure functions,” J. Mod. Opt. 30, 155–166 (1983).

, 4

4. C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006). [CrossRef] [PubMed]

], in most practical cases this fundamental limitation is not the main concern and other sources of noise (such as sample purity, stability of the experimental setup, etc.) play the main role. Over the last decade, a wealth of new laser-speckle based experimental techniques have been introduced [5

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

]. Most of them are made possible by recent advances in optical sensor technology. A modern digital CCD (charge-coupled device) or CMOS (complementary metal oxide semi-conductor) sensor contains millions of detectors which allow massive parallel processing of a large number of signals corresponding to intensities of distinct speckle spots. The availability of an area detector has led to numerous new applications in traditional far-field speckle detection such as multispeckle dynamic light scattering [6–10

6. S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996). [CrossRef]

], time-resolved correlation spectroscopy (TRC) [11

11. L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (2003). [CrossRef]

], speckle visibility spectroscopy (SVS) [12

12. P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90, 184302 (2003). [CrossRef] [PubMed]

, 13

13. 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]

] and has also enabled new speckle imaging techniques such as the near-field scattering (NFS) [14

14. R. Cerbino and A. Vailati, “Near-field scattering techniques: novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Curr. Opin. Colloid Interface Sci. 14, 416–425 (2009). [CrossRef]

], laser speckle imaging (LSI) [15–18

15. 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, 174–179 (1996). [CrossRef]

] or echo speckle imaging (ESI) [14

14. R. Cerbino and A. Vailati, “Near-field scattering techniques: novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Curr. Opin. Colloid Interface Sci. 14, 416–425 (2009). [CrossRef]

, 19

19. P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in: A. A. Kokhanovsky, ed. Light Scattering Reviews 4 (Springer, Heidelberg, 2009).

, 20

20. P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Materials, to appear (2010).

].

The obvious advantages of massive parallel detection provided by an image sensor come at a price of specific and often critical limitations. The temporal resolution of digital cameras (typically, in the millisecond range) is inferior to the resolution of a traditional single photon counter such as a photo-multiplier (nanoseconds). In addition, to maximize the signal to noise ratio the experimental setup is usually designed in such a way that the size of an individual speckle spot is comparable to the size of the individual active area (pixel) of the camera. Hence, in contrast to the traditional PCS experiment, a camera pixel is not an ideal point-like detector and, furthermore, fluctuations detected by neighboring pixels are correlated. It would be, of course, desirable to increase the speckle size with respect to the size of the camera pixel (to mimic a point-like detector) and to reduce correlation of signals detected by neighboring pixels. However, these two objectives are mutually exclusive, at least if one wishes to exploit signals from all available pixels of the image sensor.

The goal of this article is to provide guidelines for a better understanding of the measured quantities and their fluctuations in speckle-based optical techniques for essentially all cases of practical interest. Whenever possible we provide analytical expressions that can be directly applied to the analysis of experimental data. In particular, we analyze the influence of a limited time resolution and provide an approximate treatment of spatial correlations between signals corresponding to neighboring pixels. We start by considering the variance of intensities in a stationary speckle patterns and then extend our analysis to the time correlation function and the intensity structure function of dynamic speckle patterns.

2. Properties of stationary speckle patterns

Consider coherent laser light scattered by a random sample in a typical light scattering experiment in reflection (see Fig. 1). Usually the cross polarized channel is analyzed in order to suppress specular and low order scattering reflection. A digital camera provides us with N values of integrated (over the area of a pixel) intensity Iα, α = 1, . . .N, corresponding to N distinct pixels. N could be a (typically, large) total number of available pixels, but it could also be just a small subset of pixels carrying information about only a part of the optical field, scattered by a small part of the sample (see Fig. 2). Our purpose is to characterize the statistical properties of the optical field based on this information. Note that quite generally the statistical properties of the fully developed speckle patterns considered here are the same for measurements in the image plane or in the far-field [1

1. J.W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).

]. Therefore our results should find applications in many research areas where speckle based techniques are employed.

Fig. 1. Experimental setup. A flat sample is illuminated with a polarized expanded laser beam. The diffuse reflected light is detected in the cross polarization channel by imaging the surface of the sample with a digital camera. The size of individual speckle spots in the image plane can be adjusted by changing the aperture of the camera objective.

2.1. Negative exponential distribution of integrated intensities

We start by considering a speckle pattern resulting from scattering of light in a solid sample, where the positions of scattering centers are fixed. It is known that under quite general conditions, the intensity I of light scattered from such a random medium follows the negative exponential distribution: P(I) = (1/〈I〉)exp(−I/〈I〉) [1

1. J.W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).

]. In the experiment we only have access to

Iα=1a2pixelαd2rI(r),
(1)

which is the intensity integrated over an area a 2 of a pixel of the camera (d 2 r denotes twodimensional integration from here on). Here α = 1, . . .,N indexes different pixels of the camera. We added 1/a 2 in front of the integral in Eq. (1) to ensure that Iα and I(r) have the same average value. As the first and the simplest example, we assume that Iα follows the same negative exponential distribution as I and, in addition, that integrated intensities corresponding to different pixels are uncorrelated. The average value of intensity I and the variance of its fluctuations can be estimated from the data {Iα} as

i=1NΣα=1NIα,
(2)
c=1N1Σα=1N(Iαi)2.
(3)
Fig. 2. Schematic representation of the matrix of pixels of a digital camera. The full matrix is divided in a set of meta-pixels Nx × Ny = N pixels each. Spatially-varying statistical properties of the speckle pattern imaged by the camera (i.e., the variance of intensities c) are estimated by averaging over all pixels within the same meta-pixel. In our calculation (Sec. 2.3), we are taking into account correlations between intensities at neighboring pixels in all directions. A pixel which is not at the boundary of the square matrix (pixel 1 in the figure) has 8 neighbors: 4 neighbors of type 2 and 4 neighbors of type 3.

These statistical estimators are unbiased, i.e. the mean values of i and c found by averaging over an ensemble of realizations are equal to the actual values of the average intensity and of its variance, respectively:

i=I,
(4)
c=(II)2=I2.
(5)

Here the angular brackets 〈. . .〉 denote ensemble averaging. The normalized variance 〈c〉/〈I2 is equal to one in this case. The speckle contrast K can be defined as

K=cI.
(6)

The values of i and c obtained in a series of measurements fluctuate around their means Eqs. (4) and (5). The variances of these fluctuations are

σi2i2=i2i2i2=1N,
(7)
σc2c2=c2c2c2=8N×N34N1.
(8)

Note that the only assumption we have made is that the integrated intensities Iα are uncorrelated random variables obeying the negative-exponential distribution. In the limit of a very large number of pixels N, Eq. (8) simplifies:

σc2c2N=8N.
(9)

To which extent are Eqs. (4)–(9) relevant to realistic experiments? For a standard square matrix of adjacent pixels (see Fig. 2), two situations are possible depending on the relation between the speckle size (i.e. the correlation range b of the scattered intensity) and the pixel size a. For small speckles (ba), different pixels are uncorrelated, but the statistical distribution of integrated intensity Iα differs from the negative exponential one. In the opposite case of large speckles (ba) the statistical distribution of Iα approaches the negative exponential one, but intensities corresponding to neighboring pixels become correlated. The situation described in the present section — uncorrelated pixels with negative exponential distribution of Iα — can be reached by working with large speckles (ba) but using only a subset of pixels (i.e., only pixels distanced by 1, 2 or even more pixels) in the analysis. However, using only a subset of all available pixels means that the total number of pixels is reduced. It is therefore desirable to extend Eqs. (4)–(9) to a more realistic model for the distribution function of Iα.

2.2. Gamma distribution of integrated intensities

For small speckles (ba) the integrated intensities Iα corresponding to different pixels can be considered independent, whereas the distribution of each Iα, according to Goodman [1

1. J.W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).

], can be modeled with the so-called gamma distribution:

P(Iα)=1Γ(μ)(μI)μIαμ1exp(μIαI).
(10)

Here the parameter μ ≥ 1 depends on the pixel shape and size, as well as on the spatial correlation function g 2r) of intensity I(r):

1μ=1a4pixeld2rpixeld2r'[g2(rr')1],
(11)

where both integrations run over the area of the same pixel. The parameter μ provides a measure of how the ratio of speckle size b to the pixel size a influences the statistical properties of the measured intensity. It is worthwhile to note that in the photon correlation spectroscopy, 1/μ is equal to the intercept (usually denoted by β) of the autocorrelation function: g 2(0) = 1+1/μ (see also Section 3.1).

If the sample is illuminated by a Gaussian beam, the intensity correlation function of the far-field speckle pattern is

g2(Δr)=I(r)I(r+Δr)I2=1+exp(Δr2b2),
(12)

whereas for a plane wave passed through a circular aperture,

g2(Δr)=1+[2J1(Δrb)(Δrb)]2.
(13)

In our experiments, the correlation function of intensity in the speckle pattern is controlled by the (circular) aperture of the camera objective, so that Eq. (13) is used throughout this work. In both cases b quantifies the extent of spatial correlations of I(r) (i.e., the size of a single speckle spot). An analytic expression for μ can only be obtained for very small speckles ba: μ ⋍ (a/b)2/(4π). Figure 3 (left) shows a speckle image of light scattered from solid Teflon taken for the case of large speckles that can be resolved in space (f/# = 32). Be Ĩ(q) the Fourier transform of the image I(r). From the inverse Fourier transform of the power spectrum ∣I(q)∣2 we directly obtain the normalized intensity correlation function g 2r) [1

1. J.W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).

] [see Fig. 3 (right)]. In our example, for speckles of finite size b/a = 1.24, the intercept of the correlation function is given by 1+1/μ = 1.92. In Fig. 4 we compare Eq. (11) with experimental results and find an excellent agreement.

Fig. 3. Left: False-color image of light intensities in a speckle pattern (60 × 60 pixels) obtained by using the smallest available aperture setting (f/# = 32). Intensity scale from 0 to 104 in arbitrary units. Right: Intensity correlation function obtained by the inverse Fourier transform of the speckle power spectrum (full frame 640×480 pixels). The data is quantitatively described by Eq. (13) with b = 1.24a and μ = 1.09.
Fig. 4. Speckle parameter μ as a function of the speckle size b divided by the size of the camera pixel a. Symbols: experimental results for the case of scattering from a solid sample (Teflon). Solid line: Eq. (11). Dashed line: the approximate result in the small-speckle limit ba.

The expectations and the variances of i and c defined by Eqs. (2) and (3) can be calculated for the gamma distribution using the known expression for its statistical moments:

Iαn=Γ(μ+n)Γ(μ)(Iμ)n.
(14)

We obtain

i=I,σi2i2=1μN,
(15)
c=I2μ,
(16)
σc2c2=2N1[1+3μ(11N)].
(17)

These equations are exact provided that the integrated intensities Iα are independent random variables obeying the gamma distribution [Eq. (10)]. As expected, they coincide with Eqs. (4), (5), (7), and (8) for μ = 1, when Eq. (10) becomes identical to the negative exponential distribution. When μ > 1, the variance of intensities c of the speckle pattern is reduced due to the effective spatial averaging over the area of a pixel that becomes comparable to the typical size of the speckle spot b [see Eq. (16)]. At the same time, fluctuations of c from one meta-pixel to another are suppressed [see Eq. (17)]. In the limit of large N we have

σc2c2N=2N(1+3μ).
(18)

2.3. Role of correlations between neighboring pixels

1μ2=1a4pixel1d2rpixel2d2r[g2(rr)1],
(19)
1μ3=1a4pixel1d2rpixel3d2r[g2(rr)1],
(20)

With these definitions in hand, we are ready to compute the average value and the variance of i. The average 〈i〉=〈I〉 is the same as in the absence of correlations between pixels. To compute the variance of i, σ 2 i = 〈i 2〉−〈i2, we need to know 〈i 2〉 which is the ensemble average of the square of Eq. (2):

i2=1N2Σα=1NΣα=1NIαIα.
(21)

Using the definition [Eq. (1)] of Iα and the parameters μ, μ 2 and μ 3 that we introduced in Eqs. (11), (19), and (20), the correlation function 〈Iα I α〉 that appears in Eq. (21) is equal to

I2=1a4pixeld2rpixeld2r'I(r)I(r')
=I2(1+1μ),
(22)

when α = α′,

I1I2=1a4pixel1d2rpixel2d2r'I(r)I(r')
=I2(1+1μ2),
(23)

when α and α′ correspond to two neighboring pixels situated side by side as the pixels 1 and 2 in Fig. 2,

I1I3=1a4pixel1d2rpixel3d2r'I(r)I(r')
=I2(1+1μ3),
(24)

i2=1N2{NI2+4(NN)I1I2
+4(N1)2I1I3+[N2(3N2)2]I2}.
(25)

Using Eqs. (22)–(25) and the definition σ 2 i = 〈i 2〉−〈i2 we obtain the final result:

σi2i2=1N[1μ+4μ2(11N)+4μ3(12N+1N)].
(26)

The average value of c can be found by noting that the definition of c, Eq. (3), can be rewritten as

c=1N1Σα=1NIα2i211N.
(27)

The ensemble average of this equation is readily found by using Eq. (22) for 〈I 2 α〉 ≡ 〈I 2〉 and Eq. (25) for 〈i 2〉:

c=I2[1μ4μ2N(N+1)4(N1)μ3N(N+1)].
(28)

For N → ∞, Eq. (28) reduces to Eq. (16).

The variance of c, σ 2 c, can be calculated along the same lines as 〈c〉. The calculation, however, is much more involved and rapidly leads to cumbersome equations. Instead of dealing with this lengthy and, anyway, approximate analytic calculation, we estimate σ 2 c by a numerical simulation. We use a computer to generate a set of 4096×4096 speckle patterns with correlation lengths b = 2, 4, and 8, respectively. These speckle patterns are then used to generate integrated speckle patterns with pixel sizes a varying from 2 to 64. The integrated speckle patterns model outputs of a typical CCD or CMOS camera. Finally, we numerically compute the dependencies of 〈c〉 and σ 2 c on the number N of pixels, for fixed ratios b/a. For b/a ≪ 1 the parameter μ characterizing integrated speckle patterns [Eq. (11)] is a function of b/a only. Hence, by inverting this function we can express b/a and hence σ 2 c / 〈c2 as a function of μ instead of b/a. We know that Eq. (17) is valid in the limit of μ → ∞ (or, equivalently, b/a→0) which suggests that σ 2 c/〈c2 can be expressed as a series in 1/μ of which the terms of order (1/μ)0 and (1/μ)1 are already known from Eq. (17). In the limit of large N, the results of our simulations can be fit by adding a term quadratic in 1/μ:

σc2c22N[1+3μ+12μ2]=H(μ)N,
(29)

where we introduced H(μ)≃2(1+3/μ+12/μ 2). This equation appears to describe our results quite well for 0 < 1/μ < 0.8 (see Sec. 2.5). For uncorrelated pixels H(μ) ≃ 2(1+3/μ) from Eq. (18).

2.4. Properties of time-integrated speckle patterns

If the scattering medium is not stationary like, e.g., a liquid suspension of dielectric particles, the intensity I of scattered light fluctuates in time. In such an experiment the intensity I of scattered light changes not only as a function of position r, but also as a function of time t: I = I(r, t). Fluctuations of I in time can be characterized by its autocorrelation function,

g2(τ)=I(t)I(t+τ)I2,
(30)

Under the assumption of small pixel size ab and statistically independent Iα following the negative exponential distribution, the effect of finite integration time can be described by a parameter ν [13

13. 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]

, 27

27. K. Schätzel, “Noise on photon correlation data. i. autocorrelation functions,” Quantum Opt.: J. Eur. Opt. Soc. B 2, 287–305 (1990). [CrossRef]

]:

1v=1T20Tdt0Tdt'[g2(tt')1].
(31)

This definition is analogous to that of μ in Eq. (11), with the integrations over space (i.e. over a pixel) being replaced by integrations over time. Both integrations represent (partial) averaging over position or time due to either the finite area of the pixel [in Eq. (11)] or non-zero integration time T [in Eq. (31)]. Because the integrand of Eq. (31) is a function only of the difference tt′, we can change the variables of integration to t¯=12(t+t) and τ = tt′ and perform integration over . The expression for ν then becomes

Fig. 5. Average variance of the intensity fluctuations 〈c〉 and its noise σ 2 c as functions of the number of pixels N. Speckle pattern is recorded in the image plane for light reflected from a solid piece of Teflon. Exposure time is 1 ms, transport mean free path in Teflon l* ≃ 0.25 mm, camera pixel size a = 9.9 µm, magnification one. Left panel: analysis using a subset of pixels (all neighboring pixels omitted, full symbols) leads to a constant value of 〈c〉 [Eq. (16), dotted lines]. Analysis using all pixels (open symbols) is compared to the prediction of Eq. (28) (solid lines). Right panel: thick black and thin blue lines show predictions of Eqs. (18) and (29), respectively. The inset shows a comparison of the experimental values (symbols) and theoretical predictions for H(μ) = 2 c/〈c2: Eq. (18) (blue line) and Eq. (29) (red line).
1v=2T0T[g2(τ)t](1τ/T)dτ.
(32)

The integrated intensity Iα follows the gamma distribution [Eq. (10)] with ν substituted for μ [1

1. J.W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).

]. The average value and the normalized variance of c are given by Eqs. (16) and (17), respectively.

For pixel size a that becomes comparable to the correlation length b of the speckle pattern, we can take into account correlations between integrated intensities corresponding to neighboring pixels following the approach of Sec. 2.3. The outcome of our analysis is quite simple: all results of Sec. 2.3—and, in particular, Eqs. (26), (28), and (29)—still apply but with μ, μ 2 and μ 3 replaced by μ × ν, μ 2 × ν and μ 3 × ν, respectively. Here μ, μ 2 and μ 3 describe the impact of spatial correlations, whereas ν accounts for time integration. The effects of spatial and time correlations thus fully decouple. This is due to the fact that the spatio-temporal correlation function g 2r,τ)−1 = 〈I(r, t)I(rr, t + τ)〉/〈I2 − 1 decouples, in its turn, into a product of position- and time-dependent parts.

2.5. Comparison with experiment

In our experiment, linearly polarized light from a solid state laser (Verdi V5 from Coherent, wavelength λ = 532 nm) is expanded and collimated to a beam of several centimeters in waist to create an approximately homogeneous illumination spot on the sample surface (see Fig. 1). Liquid samples are contained in large glass containers whereas solid samples are measured in air. The diffuse reflected light is monitored in the image plane in the crossed polarization channel with a CCD camera PCO Pixelfly (640×480 pixels of 9.9×9.9 µm2 size, 12 bit) at magnification one. The estimated depth-of-focus of the imaging system is close to 0.1 mm. The actual size of speckle spots on the CCD chip can be adjusted by varying the aperture of the camera objective. For the simple configuration of a single lens placed at a distance l from the screen and a circular aperture with radius q the speckle correlation length is [26

26. T. Yoshimara, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am A 3, 1032–1054 (1986). [CrossRef]

] b = 2l/q = [4l/kf] × f/#.

We first analyze the static speckle pattern of light reflected from a solid piece of Teflon (thickness ≈1 cm) with a typical photon transport mean free path l* ≃ 0.25 mm. In Fig. 5, we compare our experimental results for 〈c〉 (left panel) and its normalized variance σ 2 c/〈c2 (right panel) with the predictions of the theoretical model developed above. The parameter μ is determined from the best fit to the data via the relation μ = 〈I2/〈c〉 for N → ∞. The values obtained for μ are found in excellent agreement with the theoretical predictions as shown in Fig. 4. Here the parameter b for all settings has been obtained from the relation bf/# using the known value of b/a = 1.24 for f/# = 32 (Fig. 3). For consistency, in the following discussion, for dynamic media, we use the fitted values of μ for each setting.

For the static samples, good agreement with theory is found for the N-dependence of the mean and the variance of 〈c〉. In particular, we clearly see that 〈c〉 is indeed independent of N for a subset of uncorrelated pixels, whereas it acquires an N-dependence for correlated pixels. The expected 1/N dependence of the variance of c is observed for both uncorrelated and correlated pixels, though with different prefactors. As follows from Eq. (29), the prefactor H(μ) depends on the degree of correlation between neighboring pixels that we quantify by the parameter μ. It is shown in the inset of Fig. 5. The values of H following from the experiment are close to those expected theoretically. Small but visible discrepancies between data and theory in Fig. 5 are most probably due to the approximate nature of our theoretical model: even for uncorrelated pixels the statistical distribution of intensity does not follow the gamma distribution exactly [1

1. J.W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).

] and weak correlations certainly exist not only between neighboring but between distant pixels as well.

3. Noise in photon correlation spectroscopy

3.1. Fluctuations of the intensity correlation function

We now turn our attention to imaging and spectroscopy with speckle correlations. Detailed information on temporal fluctuations of intensity can be obtained by choosing Tτc and correlating speckle images separated by a time lag τ larger than T. When intensities Iα(t) are measured for N pixels, the degree of intensity correlation can be estimated as

c(τ)=1N1Σα=1N[Iα(t)i(t)][Iα(t+τ)i(t+τ)],
(33)

where

i(t)=1NΣα=1NIα(t)
(34)

is an unbiased estimator of the average intensity 〈I〉; it is equivalent to i of Sec. 2. However, it will be important for us to keep track of the difference between i(t) and i(t+τ) in the definition [Eq. (33)] of c(τ).

The average and the variance of c(τ) for independent pixels with Iα(t) distributed according to the negative exponential distribution are:

c(τ)=I2[g2(τ)1],
(35)
σc(τ)2c(τ)2=g2(τ)2(32N)2[g2(τ)1N](N1)[g2(τ)1]2.
(36)

In the limit of large N we have

σc(τ)2c(τ)2N=1N×g2(τ)[3g2(τ)2][g2(τ)1]2.
(37)

c(τ)=I2[1μ4μ2N(N+1)4(N1)μ3N(N+1)][g2(τ)1]
(38)

for the average of c(τ). Here μ, μ 2 and μ 3 account for spatial correlations of the instantaneous speckle pattern and are given by Eqs. (11), (19) and (20), respectively.

The calculation of the variance of fluctuations of c(τ) with account for correlations between neighboring pixels appears to be much more involved. However, an approximate result can still be obtained by an ad hoc combination of Eqs. (29) and (36):

σc(τ)2c(τ)2H(μ)8N×g2(τ)[3g2(τ)2][g2(τ)1]2.
(39)

To compare Eqs. (35)–(39) with measurements we have carried out a time correlation experiment on a slowly relaxing sample. We dispersed about 3.3% of polystyrene microspheres (diameter 710 nm) in an aqueous solution of cetylpyridinium chloride/sodium salicylate (100 mM CPyCl/60 mM NaSal). The resulting surfactant solution strongly scatters light with the transport mean free path l* ≃ 73 µm at 532 nm. A detailed characterization of the system is given in Ref. [29

29. N. Willenbacher, C. Oelschlaeger, M. Schšpferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007). [CrossRef] [PubMed]

]. It displays strongly viscoelastic properties with a slow terminal relaxation. For our measurement we keep the sample at room temperature T ≃ 22° C which sets the relaxation time to several tens of seconds. We first determine the full autocorrelation function from a combined photon PCS and CCD-camera based experiment [9

9. 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]

]. This yields the temporal autocorrelation function g 2(τ) of scattered light (see the inset of Fig. 6). We then perform measurements of the noise of the speckle correlation coefficient c(τ) for different sizes of the camera objective aperture corresponding to the values of μ ranging from 1.29 to 5.15, and as a function of the number of pixels N. In all cases we observe that the normalized variance of c(τ) scales with 1/N as expected from the theory. In Fig. 6 we show 2 c/〈c2 as a function of the time lag τ. The general rule is that the normalized variance of the correlation coefficient c(τ) increases with τ. Experiment and theory are in fairly good agreement if the analysis is performed on a subset of pixels (right panel of Fig. 6), especially at small μ, when the statistics of intensity fluctuations is close to negative exponential. The agreement is worse when all pixels are analyzed (left panel of Fig. 6), even though the theory describes the general trend of our data quite well. This suggests that either we were unable to eliminate residual experimental errors or that additional factors, not taken into account in our theoretical model, enter into play when important correlations between signals corresponding to different pixels are present.

Fig. 6. Noise of the speckle correlation coefficient as a function of the time lag τ (symbols). Lines show predictions by Eqs. (39) (left) and (36) (right). Inset of the left panel shows the intensity correlation function g 2(τ).

3.2. Fluctuations of the intensity structure function

Instead of the intensity correlation function g 2(τ), fluctuations of intensity can be characterized by a structure function

D(τ)=[I(t)I(t+τ)]2=I2d(τ),
(40)

where for the case of Gaussian statistics

d(τ)=[I(t)I(t+τ)]2I2=2[g2(0)g2(τ)].
(41)

s(τ)=1NΣα=1N[Iα(t)Iα(t+τ)]2.
(42)
Fig. 7. Noise of the speckle structure coefficient as a function of time lag τ (symbols). As predicted by the theory, the noise is independent of τ. Lines show predictions of Eqs. (46) (left) and (44) (right). The inset of the right panel shows the intensity structure function d(τ).

For the negative exponential distribution of integrated intensities Iα(t), assuming no correlation between intensities at different pixels, we obtain

s(τ)=D(τ),
(43)
σs(τ)2s(τ)2=5N.
(44)

Accounting for finite pixel size a that can become comparable to the correlations length b of the speckle pattern, we obtain

s(τ)=D(τ)μ.
(45)

Once again, the calculation of the variance of fluctuations of s(τ) taking into account correlations between neighboring pixels appears to be much more involved. However, an approximate result can still be obtained by a (though ad-hoc) combination of Eqs. (29) and (44):

σs(τ)2s(τ)258NH(μ).
(46)

A comparison between Eqs. (44), (46) and experimental data is presented in Fig. 7. The theory correctly predicts that the noise of s(τ) scales with 1/N and that it is independent of τ. However, theory and experiment do not show quantitative agreement: theoretical lines lie either higher (blue and red lines in the left panel of Fig. 7) or lower (all other lines in Fig. 7) than the data. The reasons behind this disagreement are the same as in the case of intensity correlation coefficient c(τ) in Sec. 3.1.

4. Conclusions

Acknowledgements

This work has been supported by the Swiss-French Germaine de Staël collaboration project No. 19126RL and the Swiss National Science Foundation projects 200020–126772 and 117762. RC acknowledges financial support from the European Union (Marie Curie Intra-European Fellowship, Contract No. EIF–038772)

References and links

1.

J.W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).

2.

B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

3.

K. Schätzel, “Noise in photon correlation and photon structure functions,” J. Mod. Opt. 30, 155–166 (1983).

4.

C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006). [CrossRef] [PubMed]

5.

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

6.

S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996). [CrossRef]

7.

L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, “Universal aging features in the restructuring of fractal colloidal gels,” Phys. Rev. Lett. 84, 2275–2278 (2000). [CrossRef] [PubMed]

8.

A. Knaebel, M. Bellour, J.P. Munch, V. Viasnoff, F. Lequeux, and J.L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000). [CrossRef]

9.

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]

10.

P. Zakharov, S. Bhat, P. Schurtenberger, and F. Scheffold, “Multiple scattering suppression in dynamic light scattering based on a digital camera detection scheme,” Appl. Opt. 45(8), 1756–1764 (2006) [CrossRef] [PubMed]

11.

L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (2003). [CrossRef]

12.

P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90, 184302 (2003). [CrossRef] [PubMed]

13.

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]

14.

R. Cerbino and A. Vailati, “Near-field scattering techniques: novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Curr. Opin. Colloid Interface Sci. 14, 416–425 (2009). [CrossRef]

15.

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, 174–179 (1996). [CrossRef]

16.

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009). [CrossRef]

17.

C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E 71, 066603 (2005). [CrossRef]

18.

D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, “Statistics of local speckle contrast,” J. Opt. Soc. Am. A 25(1), 9–15 (2008). [CrossRef]

19.

P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in: A. A. Kokhanovsky, ed. Light Scattering Reviews 4 (Springer, Heidelberg, 2009).

20.

P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Materials, to appear (2010).

21.

J. D. Briers, “Laser doppler, speckle and related techniques for blood perfusion mapping and imaging,” Physiol. Meas.22, R35–R66 (2001). [CrossRef]

22.

A.K. Dunn, H. Bolay, M.A. Moskowitz, and D.A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21, 195–201 (2001). [CrossRef] [PubMed]

23.

B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004). [CrossRef] [PubMed]

24.

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “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]

25.

A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage 27(15), 279–290 (2005). [CrossRef] [PubMed]

26.

T. Yoshimara, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am A 3, 1032–1054 (1986). [CrossRef]

27.

K. Schätzel, “Noise on photon correlation data. i. autocorrelation functions,” Quantum Opt.: J. Eur. Opt. Soc. B 2, 287–305 (1990). [CrossRef]

28.

M. Erpelding, A. Amon, and J. E. Crassous, “Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid,” Phys. Rev. E 78, 046104 (2008). [CrossRef]

29.

N. Willenbacher, C. Oelschlaeger, M. Schšpferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007). [CrossRef] [PubMed]

OCIS Codes
(030.6140) Coherence and statistical optics : Speckle
(100.4550) Image processing : Correlators
(110.6150) Imaging systems : Speckle imaging
(170.3880) Medical optics and biotechnology : Medical and biological imaging

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: May 17, 2010
Revised Manuscript: June 15, 2010
Manuscript Accepted: June 20, 2010
Published: June 22, 2010

Virtual Issues
Vol. 5, Iss. 11 Virtual Journal for Biomedical Optics

Citation
S. E. Skipetrov, J. Peuser, R. Cerbino, P. Zakharov, B. Weber, and F. Scheffold, "Noise in laser speckle correlation and imaging techniques," Opt. Express 18, 14519-14534 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14519


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References

  1. J. W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).
  2. .B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  3. K. Schatzel, “Noise in photon correlation and photon structure functions,” J. Mod. Opt. 30, 155–166 (1983).
  4. C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, ”Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006). [CrossRef] [PubMed]
  5. F. Scheffold and R. Cerbino, ”New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12, 50–57 (2007). [CrossRef]
  6. S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996). [CrossRef]
  7. L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, “Universal aging features in the restructuring of fractal colloidal gels,” Phys. Rev. Lett. 84, 2275–2278 (2000). [CrossRef] [PubMed]
  8. A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000). [CrossRef]
  9. 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]
  10. P. Zakharov, S. Bhat, P. Schurtenberger, F. Scheffold, “Multiple scattering suppression in dynamic light scattering based on a digital camera detection scheme,” Appl. Opt. 45(8), 1756–1764 (2006) [CrossRef] [PubMed]
  11. L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (2003). [CrossRef]
  12. P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90, 184302 (2003). [CrossRef] [PubMed]
  13. 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]
  14. R. Cerbino and A. Vailati, “Near-field scattering techniques: novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Curr. Opin. Colloid Interface Sci. 14, 416–425 (2009). [CrossRef]
  15. 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, 174–179 (1996). [CrossRef]
  16. P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009). [CrossRef]
  17. . C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E 71, 066603 (2005). [CrossRef]
  18. D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, ”Statistics of local speckle contrast,” J. Opt. Soc. Am. A 25(1), 9–15 (2008). [CrossRef]
  19. P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in: A. A. Kokhanovsky, ed,. Light Scattering Reviews 4 (Springer, Heidelberg, 2009).
  20. P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Materials, to appear (2010).
  21. . J. D. Briers, “Laser doppler, speckle and related techniques for blood perfusion mapping and imaging,” Physiol. Meas. 22, R35–R66 (2001). [CrossRef]
  22. A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21, 195–201 (2001). [CrossRef] [PubMed]
  23. B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004). [CrossRef] [PubMed]
  24. T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “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]
  25. . A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage 27(15), 279–290 (2005). [CrossRef] [PubMed]
  26. T. Yoshimara, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am A 3, 1032–1054 (1986). [CrossRef]
  27. K. Sch¨atzel, “Noise on photon correlation data. i. autocorrelation functions,” Quantum Opt.: J. Eur. Opt. Soc. B 2, 287–305 (1990). [CrossRef]
  28. . M. Erpelding, A. Amon, and J. E. Crassous, “Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid,” Phys. Rev. E 78,046104 (2008). [CrossRef]
  29. . N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007). [CrossRef] [PubMed]

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