Abstract

Speckle-correlation imaging is a family of methods that makes use of the “memory effect” to image objects hidden behind visually opaque layers. Here, we show that a correlation analysis can be applied to quantitative imaging of an ensemble of dynamic fluorescent beads diffusing on a 2D surface. We use an epi-fluorescence microscope where both the illumination and detection light patterns are speckled, due to light scattering by a thin disordered layer. The spatio-temporal cross-correlation of the detection speckle pattern is calculated as a function of lag time and spatial shift and is used to determine the diffusion constant and number of fluorescent particles in the sample without requiring any phase retrieval procedure. It is worth to note that the “memory effect” range is not required to extend beyond a distance of few speckle grains, thus making our method potentially useful for nearly arbitrary values of the thickness of the scattering layer.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

From the very beginning of its invention, the concept of “looking through walls and around corners,” as originally phrased by Freund [1], not only raised the issue of image formation in the presence of multiple scattering optical barriers, but also that of assessing the dynamics of hidden objects [2]. As the interference speckle pattern generated by a wave propagating through a scattering layer is determined by the positions of all the scatterers in the system, it should be, in principle, possible to retrieve information about a scatterer that moves with respect to the others. Of course, this idea is not limited to optics and can be extended to other kinds of waves, such as radio waves or acoustic waves. As demonstrated with ultrasounds inside a reverberating cavity, the temporal behavior of the autocorrelation of an acoustic wave field depends on the dynamics of the scatterers moving inside the cavity [3].

To image objects hidden behind optically scattering layers, methods relying on the so-called angular memory effect has recently received particular interest. Contrary to those based on wavefront shaping [4], they do not require an access to the far side of the scattering medium for calibration and, hence, are promising for in-depth imaging in tissues. The angular memory effect refers to the property of the speckle pattern, formed in a given observation plane after the scattering layer, to shift while remaining otherwise unchanged, when the corresponding incident wave is tilted [5]. When the tilt angle is beyond a range that scales like $\lambda /e$ (where $\lambda$ is the wavelength and $e$, the thickness of the scattering medium), the memory effect does not hold anymore, i.e. the speckle pattern decorrelates. This sets a practical limit to the accessible Field of View (FoV). The image of an incoherently emitting object hidden by the scattering layer can be reconstructed from the autocorrelation of the detected scrambled pattern, using a phase retrieval algorithm [6]. This approach is based on the fact that the autocorrelation of the speckled Point Spread Function (PSF) is a diffraction-limited peak, according to the theory of speckle pattern formation [7]. Although memory-effect based image reconstruction can successfully retrieve the object’s shape, it usually does not indicate its spatial location, since only angular (k-space) information is exploited [8]. Hence, it cannot be directly applied to assess the motion of objects.

Although moving objects has been used to provide additional information for speckle correlation imaging [912], few studies have been dedicated to measuring their motion in itself. A method, using digital optical phase conjugation, has been proposed to track and count the number of coherently scattering particles [13]. Recently, Guo et al. have presented a method to measure the lateral, axial and rotational movement of a single known object using the temporal correlation of the observed dynamical speckle pattern [14], but this approach is limited to a single object.

In this work, we present a method to estimate the number and diffusion of fluorescent particles totally hidden by a scattering layer. This method is not restricted to single objects and is carried out with a standard epi-fluorescence microscope, and hence compatible with most biological and medical applications. Contrary to a recent study dedicated to imaging behind a scattering layer on a epi-fluorescence microscope [15], we do not assume that the sample is evenly illuminated and examine the general case where both the illumination and detected light are speckled. Our approach is inspired by correlation methods, hereafter referred as Fluorescence Fluctuation Microscopy (FFM), that derive from classical Fluorescence Correlation Spectroscopy [16]. This family of techniques is mostly used in cell biology and biochemistry to assess molecular diffusion, interaction, aggregation, etc. It exploits the spatial and/or temporal intensity fluctuations generated by fluorescent molecules or particles that cross the observation volume of a microscope. These fluctuations scale inversely with the concentration of particles: the smaller the average number of particles within the observation volume, the larger the relative intensity fluctuations. Additionally, the characteristic correlation time of these fluctuations provides an estimation of the diffusion constant. To our knowledge, in all previous implementations of FFM, the Point Spread Function (PSF) of imaging is limited by diffraction (typically that of a confocal microscope), which defines the size of the observation volume. In the case where the wave-front is strongly deformed by a scattering medium, the PSF becomes a speckle pattern that can extend across the whole FoV. However, its autocorrelation is still diffraction-limited and this property can be exploited, as in the case of imaging, to measure the number and dynamics of the diffusing particles.

Here, we derive a mathematical expression of the spatio-temporal cross-correlation of the observed speckle patterns as a function of the autocorrelation of the PSF and the diffusion propagator, which is analogous to that of standard FFM. Using this formalism, we experimentally show, in the case of an ensemble of beads floating at a water/air interface, that a simple correlation analysis of the detected speckle temporal dynamics (i.e. without phase retrieval) enables to determine the particle number and 2D diffusion constant. Our method can be extended beyond optical microscopy to all cases where one is interested in obtaining statistical information about dynamic parameters of moving objects behind a wall. Its main requirement is that the acquisition frame rate should be faster than the typical time scale of the involved dynamics (including the possible dynamics of the wall itself), and that the objects are bright enough to generate a superposition of moving speckle patterns with sufficient signal-to-noise ratio.

Finally, it is worth noting that, although numerous laser speckle imaging methods have been developed to measure blood flow [17], they are in fact very different from the present study. Indeed, they rely on the spatio-temporal analysis of light that is elastically scattered by moving particles. The process is thus fully coherent and there is no wall hiding the objects of interest. We thus believe that no fruitful analogy can be made with our approach.

2. Material and methods

2.1 Microscopy set-up

The experimental set up was built around an epi-fluorescence configuration, as shown in Fig. 1(a). Output of a diode pumped solid state laser at $\lambda = 561$ nm was used to excite the sample at a typical incoming power of $10$ mW. The laser beam was collimated before entering the microscope (Olympus IX71) so that the tube lens could focus it at the back focal plane of the objective (Olympus, air, $20\times$, NA=0.4). Then, the excitation laser beam illuminates the scattering medium (a TiO$_{2}$ layer) as a collimated beam with diameter $D\sim 500$ $\mu$m, subsequently producing an illumination speckle pattern. Fluorescent beads confined at the water/air interface were excited by this illumination intensity distribution and the fluorescence emission from the excited beads traveled back through the same diffuser, producing a detection speckle pattern that was collected by the same objective. This speckle pattern can be observed at various distances from the bead plane (above or below). However, when focusing too close to the bead plane, the speckle pattern is squeezed and bright, containing a very small number of speckle grains. Conversely, when moving away, the FoV gets progressively filled by the speckle pattern while its intensity decreases. The speckle pattern extension results from a combination of the diffusion cone at the exit of the scatterer plane and of the geometry of the collection optics. We thus had to find a compromise between good enough speckle grain statistics and sufficiently bright fluorescence intensity, which we did by focusing between 150 and 300 $\mu$m below the bead plane. In addition, prior to any signal acquisition, the actual number of fluorescent beads within the region of interest (ROI) was estimated by focusing at the bead plane where the individual speckle patterns shrink to spatially separated small and bright patches. Light collected by the objective was filtered by a dichroic mirror and a notch filter to block any excitation light. The fluorescent detection speckle pattern was filtered further by using a band pass filter ($600/25$ nm) to reduce spectral decorrelation and was finally imaged with a high-speed sCMOS camera (Hamamatsu ORCA-Flash4.0, $2048\times 2048$ pixels and $6.5$ $\mu$m pixel size) running at 25 fps. Besides the measurement of the detection speckle pattern, a corresponding background signal was also recorded from the scattering sample by collecting signal from a location without any beads. The average intensity of the background signal was subtracted from its corresponding detection speckle pattern for all correlation analysis. Additionally to performing speckle correlation measurements and analysis, the properties of the illumination speckle pattern produced by the TiO$_{2}$ diffuser were investigated. The diffuser was illuminated by the same excitation laser ($\lambda = 561$ nm) and objective ($20\times$, NA=0.4), while the generated illumination speckle patterns were recorded at different distances from the scatterer plane with a CCD camera (BASLER ACE, $1280\times 1024$ pixels and $4.8$ $\mu$m pixel size) and an effective $37.5\times$ magnification (obtained by combining an air, $50\times$, NA=0.6 Nikon objective with a 150 mm lens). A schematic of this illumination speckle study set-up is presented in Fig. 1(c). Finally we stress the fact that, taking into account the magnifications and camera pixel sizes, all the speckle patterns were always recorded with a minimum of $4$ pixels per speckle grain.

 

Fig. 1. (a) Schematic of the experimental set up for studying fluorescence correlations through a strong scattering medium. (b) Inset shows the magnified version of the sample region where the focal plane can be moved through the sample. (c) Schematic of the set up for characterizing the illumination speckle at varying distance above the sample plane.

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2.2 Diffusing sample

Inset (b) of Fig. 1 depicts a magnified version of the sample region of the experiment, with the 2D fluorescent model system used for our current study. This system was realized by confining hydrophobic fluorescent beads at a water/air interface, as recently used for studying Brownian diffusion [18]. We used hydrophobic fluorescent polystyrene beads of 1 $\mu$m diameter, with a sulphate modified surface, purchased from Sigma Aldrich. Indeed, as can be observed in Visualization 1, these beads undergo an unrestricted 2D Brownian diffusive motion, provided the trajectories last not too long, so that the beads’ displacements are not biased by convection or mechanical vibrations. By controlling this condition we could estimate Mean Square Displacements (MSD) as shown in Appendix A, where the slope of the MSD corresponds to a diffusion constant of $0.11 \pm 0.07$ $\mu$m$^{2}$/s.

2.3 Scattering layer

For the experiments, we used a disordered TiO$_{2}$ film as the diffuser, deposited on the substrate of an eight-well labtek (Thermo Fisher Scientific). First, a 0.1 M solution of TiO$_{2}$ nano-powder (Sigma Aldrich) with 21 nm primary particle size was prepared in deionized water (DIW). 100 $\mu$l of this solution was dispersed in a labtek-well prefilled with 300 $\mu$l of DIW, and the final solution was kept at room temperature to let the particles sediment and to obtain a dry film. The film thickness was measured by an optical microscope using a $50\times$ objective and observed to be $\sim 6$ $\mu$m in thickness. This kind of highly scattering TiO$_{2}$ diffuser usually has its transport mean-free-path within the range of 0.55 $\mu$m - 0.95 $\mu$m [19,20], which results in multiple scattering of light when passing the layer. As so, the scattering layer does not let through any ballistic light, which is consistent with the fact that, when focusing on the bead plane, we could never observe any single bright spot at the center of the speckle pattern. To perform speckle correlation analysis through a scattering layer, a thin layer of DIW ($\sim 1$ mm thick) film was deposited on the TiO$_{2}$ coated substrate of the labtek. Absolute methanol was used as the dispersion medium to transfer the fluorescent beads on the DIW surface by utilizing an air spray.

3. Principle of speckle correlation analysis

As already mentioned, two kinds of speckle patterns have to be considered in our study. The illumination speckle pattern is static and modulates the excitation intensity of the moving fluorescent beads, while each of the beads generates its own emission speckle pattern that is detected on the camera sensor, as shown in Fig. 2. These speckle patterns play the roles of the usual illumination and detection Point Spread Functions (PSFs), apart from the fact that these patterns are random and widely spread. Moreover, as they do not cover the whole FoV (especially the detection one), we crop the images such that the mean intensity is constant over the ROI. In addition, we hypothesize that, within the analyzed ROI, the speckle patterns are spatially stationary, thus making the beads invisible. By reasoning along this direction, one implicitly assumes that the detection speckle pattern is translationally invariant, although we will see later that, to some extent, this assumption can be relaxed. Of course, because the illumination intensity is a speckle pattern, the fluorescence intensity of the speckle pattern from each single randomly moving bead fluctuates, as can be observed in Visualization 2. More details about the properties of these two kinds of speckle patterns, excitation and detection, are provided in the Results section.

 

Fig. 2. Schematics of the illumination and detection speckle patterns. Within the angular memory effect range, the moving beads emit identical patterns (in red, below) but, due to the random distribution of excitation intensity (in green, above), their overall time-varying intensities are uncorrelated. In practice, the detection patterns of all the beads overlap each other in the analyzed ROI.

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When several beads diffuse together with overlapping detection speckle patterns, the resulting pattern, as detected in epi-fluorescence, is an incoherent superposition of single, independently moving and temporally fluctuating speckle patterns (see an example in Visualization 3). As expected from such a superposition, we immediately see that the contrast of a multiple-bead speckle pattern is lower than that of a single-bead speckle pattern.

Our spatio-temporal analysis consists in cross-correlating pairs of images separated by a given lag time $\tau _j = t_{i+j} - t_i$ as shown in Fig. 3(a,b), and then averaging the corresponding cross-correlation over the times $t_i$. More precisely, we aim at estimating the normalized spatio-temporal cross-correlation of the fluorescence intensity at lag time $\tau$ and spatial shift $\boldsymbol {\sigma }$, classically defined following [16]:

$$G(\boldsymbol{\sigma},\tau) = \frac{\left \langle I\left (\boldsymbol{\rho},t \right )I\left (\boldsymbol{\rho}+\boldsymbol{\sigma}, t+\tau \right ) \right \rangle_{\boldsymbol{\rho},t}}{\left \langle I\left ( \boldsymbol{\rho},t \right ) \right \rangle_{\boldsymbol{\rho},t}\left \langle I\left (\boldsymbol{\rho}+\boldsymbol{\sigma}, t+\tau \right ) \right \rangle_{\boldsymbol{\rho},t}}$$
where the brackets stand for spatial and temporal averaging, $\rho$ being the transverse coordinate in the object plane. At very short lag time, the beads have not enough time to move, so that the images are the same, apart from noise. Therefore, although two speckle patterns are involved (illumination and detection), the 2D shape of the spatio-temporal cross-correlation is basically that of a speckle autocorrelation with a central peak above a baseline, the width of which is related to the speckle grain size. At longer lag times, the beads are randomly displaced due to diffusion and the peak widens, while its amplitude decreases. To emphasize this phenomenon, we plot the mean radial profile (also averaged over the times $t_i$) of the 2D spatio-temporal cross-correlation versus lag time, as shown in Fig. 3(c). To summarize: i) the overall amplitude of the correlation at zero lag time is a measure of the contrast of the detected speckle pattern, which decreases with the numbers of beads simultaneously present; ii) with increasing time, the amplitude decreases due to the decorrelation of bead positions. We next consider the theoretical derivation of the spatio-temporal cross-correlation as a function of time.

 

Fig. 3. Analysis of the detected speckle pattern and spatio-temporal cross-correlation. (a) Recording scheme of the detection speckle patterns versus time. (b) Series of single 2D spatio-temporal cross-correlations between speckle patterns separated by a lag time $\tau$ of 2 frame times. (c) Series of radial line profiles of the averaged 2D spatio-temporal cross-correlations versus the lag time $\tau$.

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4. Derivation of the spatio-temporal cross-correlation

To calculate the position- and time-dependent fluorescence intensity $I(\boldsymbol {\rho },t)$ that appears in Eq. (1) all the particles have to be taken into account, that is:

$$I\left ( \boldsymbol{\rho},t \right ) =\sum_{i=1}^{N} \mathrm{PSF_{exc}}\left [ \boldsymbol{\rho}_i(t) \right ] \mathrm{PSF_{det}}\left [\boldsymbol{\rho}_i(t) ;\boldsymbol{\rho}-\boldsymbol{\rho}_i(t) \right ]$$
where $\mathrm {PSF_{exc}}\left [ \boldsymbol {\rho }_i(t) \right ]$ is the speckled excitation PSF, and $\mathrm {PSF_{det}}\left [\boldsymbol {\rho }_i(t) ;\boldsymbol {\rho }-\boldsymbol {\rho }_i(t) \right ]$ is the speckled detection PSF, which play here a similar role as usual Point Spread Functions. However, here we do not assume that the detection PSF is translationally invariant, and we explicitly write its dependence upon position $\boldsymbol {\rho }_i(t)$ of the emitting particle. We will discuss below the corresponding approximations, which are related to the memory effect. Note that the incoming laser intensity, absorption cross section, fluorescence quantum yield, and detection efficiency do not appear in Eq. (2), because the spatio-temporal cross-correlation as defined by Eq. (1) is a ratio where all multiplicative factors cancel out. The sum runs over the total number $N$ of particles located at positions $\boldsymbol {\rho }_i(t)$ that contribute to the intensity at point $\boldsymbol {\rho }$ within the ROI. Under the assumption that the ROI covers the full sample area, the number of particles $N$ is constant.

In what follows, we derive the correlation and normalization terms (numerator and denominator, respectively) of Eq. (1), assuming that the ROI is much larger than the correlation length of the PSFs so that boundary effects can be neglected.

4.1 Normalization term

First, we express the temporal averaging of the discrete sum in Eq. (2) by an averaging over all possible particle positions $\boldsymbol {\rho }'$ (thus assuming ergodicity):

$$\left \langle I\left ( \boldsymbol{\rho},t \right ) \right \rangle_{t} = C \int \mathrm{PSF_{exc}}\left ( \boldsymbol{\rho}' \right ) \mathrm{PSF_{det}}\left (\boldsymbol{\rho}' ; \boldsymbol{\rho}-\boldsymbol{\rho}' \right) d\boldsymbol{\rho}'$$
where $C = N/S$ is the surface density of particles over the ROI of area $S =\int d\boldsymbol {\rho }'$. Next, we introduce the normalized pseudo-convolution $\mathrm {CV_{exc,det}}$ of the excitation and detection PSFs by:
$$\mathrm{CV_{exc,det}}(\boldsymbol{\rho}) = \frac{1}{S} \int \mathrm{PSF_{exc}}\left ( \boldsymbol{\rho}' \right ) \mathrm{PSF_{det}}\left (\boldsymbol{\rho}' ; \boldsymbol{\rho}-\boldsymbol{\rho}' \right) d\boldsymbol{\rho}'$$
so that we can rewrite the two terms of the denominator in contracted form:
$$\left\langle I(\boldsymbol{\rho},t) \right\rangle_{\boldsymbol{\rho},t} = N \left\langle \mathrm{CV_{exc,det}}(\boldsymbol{\rho}) \right\rangle_{\boldsymbol{\rho}}$$
Here, “pseudo” signifies that the $\mathrm {PSF_{det}}$ depends also on $\boldsymbol {\rho }'$ in addition to $\boldsymbol {\rho }-\boldsymbol {\rho }'$. Finally, assuming stationarity of the signal, the overall denominator becomes:
$$\left\langle I ( \boldsymbol{\rho},t) \right\rangle_{\boldsymbol{\rho},t} \left\langle I(\boldsymbol{\rho}+\boldsymbol{\sigma}, t+\tau ) \right \rangle_{\boldsymbol{\rho},t} = N^2 \left \langle \mathrm{CV_{exc,det}} (\boldsymbol{\rho}) \right\rangle_{\boldsymbol{\rho}} \left\langle \mathrm{CV_{exc,det}}(\boldsymbol{\rho} + \boldsymbol{\sigma} )\right\rangle_{\boldsymbol{\rho}}$$

4.2 Numerator

From the fluorescence intensity expression given by Eq. (2), it follows that the numerator of Eq. (1) contains a double sum over the particles, that is:

$$ \begin{aligned} \left\langle I\left( \boldsymbol{\rho},t \right)I\left (\boldsymbol{\rho}+\boldsymbol{\sigma}, t+\tau \right ) \right\rangle_{\boldsymbol{\rho},t} &= \biggl\langle \sum_{i,j}^N \mathrm{PSF_{exc}} \left[ \boldsymbol{\rho}_i(t) \right] \mathrm{PSF_{det}}\left[\boldsymbol{\rho}_i(t) ;\boldsymbol{\rho}-\boldsymbol{\rho}_i(t) \right] \\ & \mathrm{PSF_{exc}}\left[ \boldsymbol{\rho}_j(t+\tau) \right] \mathrm{PSF_{det}}\left[ \boldsymbol{\rho}_j(t+\tau); \boldsymbol{\rho} + \boldsymbol{\sigma}-\boldsymbol{\rho}_j(t+\tau) \right] \biggr\rangle_{\boldsymbol{\rho},t} \end{aligned}$$

This sum can be separated into a diagonal contribution, with $N$ terms with $i=j$, and a non-diagonal cross-contribution, with $N(N-1)$ terms with $i\neq j$. Since different particles move independently, the latter can be written as a product of averages over particle positions:

$$ \begin{aligned} C_{cross}(\boldsymbol{\sigma},\tau) &= \frac{N(N-1)}{S^2}{\left\langle{\int \mathrm{PSF_{exc}}(\boldsymbol{\rho}') \mathrm{PSF_{det}}(\boldsymbol{\rho}'; \boldsymbol{\rho}-\boldsymbol{\rho}') d\boldsymbol{\rho}'} \right\rangle}_{\boldsymbol{\rho}}\\ & \left\langle \int \mathrm{PSF_{exc}}(\boldsymbol{\rho}') \mathrm{PSF_{det}}(\boldsymbol{\rho}'; \boldsymbol{\rho}+ \boldsymbol{\sigma}-\boldsymbol{\rho}') d\boldsymbol{\rho}'\right\rangle_{\boldsymbol{\rho}}\end{aligned}$$

The integrals over the $\boldsymbol {\rho }'$ are nothing but the previously introduced pseudo-convolution products of the normalization term. Therefore, the cross term can be rewritten as:

$$C_{cross}(\boldsymbol{\sigma},\tau) = N(N-1)\left\langle \mathrm{CV_{exc,det}}(\boldsymbol{\rho})\right\rangle_{\boldsymbol{\rho}} \left\langle \mathrm{CV_{exc,det}}(\boldsymbol{\rho}+\boldsymbol{\sigma}) \right\rangle_{\boldsymbol{\rho}}$$
We now consider the diagonal contribution that contains the correlation information related to the particles displacements:
$$ \begin{aligned} C_{diag}(\boldsymbol{\sigma},\tau) &= \frac{C}{S} \iiint \mathrm{PSF_{exc}}( \boldsymbol{\rho}_1 ) \mathrm{PSF_{det}}(\boldsymbol{\rho}_1;\boldsymbol{\rho}-\boldsymbol{\rho}_1) \mathrm{PSF_{exc}}(\boldsymbol{\rho}_2)\\ & \mathrm{PSF_{det}}(\boldsymbol{\rho}_2; \boldsymbol{\rho} - \boldsymbol{\rho}_2 + \boldsymbol{\sigma} ) P(\boldsymbol{\rho}_2-\boldsymbol{\rho}_1,\tau) d\boldsymbol{\rho}_1 d\boldsymbol{\rho}_2 d\boldsymbol{\rho} \end{aligned}$$
where $P(\boldsymbol {\rho }_2-\boldsymbol {\rho }_1,\tau )$ is the probability density distribution that a particle has moved from $\boldsymbol {\rho }_1$ to $\boldsymbol {\rho }_2$ during the lag time $\tau$. Obviously, this distribution is normalized as
$$\int d\boldsymbol{\rho} P(\boldsymbol{\rho}-\boldsymbol{\rho}',\tau) = \int d\boldsymbol{\rho}' P(\boldsymbol{\rho}-\boldsymbol{\rho}',\tau)= 1$$
Next, we introduce the normalized pseudo-autocorrelation of the detection PSF:
$$\mathrm{AC_{det}}(\boldsymbol{\rho}_1,\boldsymbol{\rho}_2;\boldsymbol{\sigma}-\boldsymbol{\rho}_2 +\boldsymbol{\rho}_1)= \frac{1}{S}\int \mathrm{PSF_{det}}(\boldsymbol{\rho}_1 ; \boldsymbol{\rho}-\boldsymbol{\rho}_1) \mathrm{PSF_{det}}(\boldsymbol{\rho}_2 ;\boldsymbol{\rho}-\boldsymbol{\rho}_2+\boldsymbol{\sigma} )d\boldsymbol{\rho}$$
Here “pseudo” signifies again that the detection PSF is not assumed to be translationally invariant. This leads to the general expression of the diagonal contribution:
$$ \begin{aligned} C_{diag}(\boldsymbol{\sigma},\tau) & = C \iint \mathrm{PSF_{exc}}( \boldsymbol{\rho}_1 ) \mathrm{PSF_{exc}}( \boldsymbol{\rho}_2) \mathrm{AC_{det}}(\boldsymbol{\rho}_1,\boldsymbol{\rho}_2; \boldsymbol{\sigma}-\boldsymbol{\rho}_2 + \boldsymbol{\rho}_1)\\ & P(\boldsymbol{\rho}_2-\boldsymbol{\rho}_1,\tau) d\boldsymbol{\rho}_1 d\boldsymbol{\rho}_2 d\boldsymbol{\rho} \end{aligned}$$

We now consider a first approximation, where the detection pseudo-autocorrelation depends only on the difference between the particle positions, $\boldsymbol {\rho } =\boldsymbol {\rho }_2 -\boldsymbol {\rho }_1$ and not on their absolute values in the ROI, so that the autocorrelation reads $\mathrm {AC_{det}}\left (\boldsymbol {\rho };\boldsymbol {\sigma }-\boldsymbol {\rho } \right )$. Said another way, we discard situations where the detection PSF of two particles would differ independently of their relative positions. Even if isoplanatism does not hold, it is nevertheless possible to make use of the normalized standard autocorrelation of the excitation PSF, given by:

$$\mathrm{AC_{exc}}(\boldsymbol{\rho})= \frac{1}{S}\int \mathrm{PSF_{exc}}(\boldsymbol{\rho}' )\mathrm{PSF_{exc}} (\boldsymbol{\rho}' +\boldsymbol{\rho})d\boldsymbol{\rho'}$$
to obtain the following expression of the diagonal contribution to the numerator of Eq. (1):
$$C_{diag}(\boldsymbol{\sigma},\tau) = N\int \mathrm{AC_{exc}}\left ( \boldsymbol{\rho} \right )\mathrm{AC_{det}}\left (\boldsymbol{\rho};\boldsymbol{\sigma}-\boldsymbol{\rho} \right)P(\boldsymbol{\rho},\tau)d\boldsymbol{\rho}$$
Finally, the normalized spatio-temporal cross-correlation is given by:
$$G(\boldsymbol{\sigma},\tau) = \frac{1}{N} \frac{\int \mathrm{AC_{exc}}\left ( \boldsymbol{\rho} \right ) \mathrm{AC_{det}}\left( \boldsymbol{\rho};\boldsymbol{\sigma}-\boldsymbol{\rho} \right)P(\boldsymbol{\rho},\tau)d\boldsymbol{\rho}}{\left\langle \mathrm{CV_{exc,det}}\left ( \boldsymbol{\rho} \right) \right\rangle_{\boldsymbol{\rho}} \left\langle \mathrm{CV_{exc,det}} \left( \boldsymbol{\rho} + \boldsymbol{\sigma} \right) \right\rangle_{\boldsymbol{\rho}}} +\frac{N-1}{N}$$
The next approximation consists in writing both speckled PSFs (excitation and detection) as random, space invariant and independent functions, with mean values arbitrarily set to 1 (which is of no importance because of the overall normalization). Consequently, the denominator in the first term of Eq. (16), which involves a convolution of the excitation and detection PSFs, is equal to 1. More precisely, the detection PSF now depends only on the distance to the particle center and thus, reads (and analogously for the excitation PSF):
$$\mathrm{PSF_{det}}(\boldsymbol{\rho})=1+\sqrt{a_{\mathrm{det}}}\mathrm{\delta f_{det}}(\boldsymbol{\rho})$$
with an autocorrelation given by:
$$\begin{aligned}\mathrm{AC_{det}}(\boldsymbol{\rho}) & =\left \langle \mathrm{PSF_{det}}(\boldsymbol{\rho}')\mathrm{PSF_{det}}(\boldsymbol{\rho}'+\boldsymbol{\rho}) \right \rangle_{\boldsymbol{\rho}'} \\ & =1+a_{\mathrm{det}}\mathrm{C_{det}}(\boldsymbol{\rho}) \end{aligned}$$
where:
$$\mathrm{C_{det}}(\boldsymbol{\rho}) =\frac{1}{S}\int{\mathrm{\delta f_{det}}(\boldsymbol\rho')\mathrm{\delta f_{det}}(\boldsymbol{\rho}'+\boldsymbol{\rho})d\boldsymbol{\rho}'}$$
In this simplified description, the detection autocorrelation, $\mathrm {AC_{det}}$, that appears in Eq. (16), only depends on $\boldsymbol {\sigma }-\boldsymbol {\rho }$ and not on $\boldsymbol {\rho }$, which corresponds to an infinite range of the memory effect. $\mathrm {\delta f_{det}}(\boldsymbol {\rho })$ is a random function with zero mean value and an autocorrelation that obeys $\mathrm {C_{det}}(0) = 1$ and $\mathrm {C_{det}}(\pm \infty ) = 0$ (and analogously for excitation). The parameter $a_{\mathrm {det}}$ is the speckle autocorrelation contrast and is equal to 1 in case of a fully developed speckle [7] while, according to the general theory of speckle formation, $\mathrm {C_{det}}(\boldsymbol {\rho })$ is related to the Fourier transform of the incident intensity distribution [21]. Finally, the numerator of the first term of Eq. (16) splits into several contributions, leading to the following expression of the normalized spatio-temporal cross-correlation:
$$\begin{aligned} G(\boldsymbol{\sigma},\tau) = 1 + \frac{1}{N}\int\left[\right. & a_{\mathrm{exc}}\mathrm{C_{exc}}( \boldsymbol{\rho}) + a_{\mathrm{det}}\mathrm{C_{det}}(\boldsymbol{\sigma}-\boldsymbol{\rho} ) \\ & + a_{\mathrm{exc}}a_{\mathrm{det}}\mathrm{C_{exc}}(\boldsymbol{\rho})\mathrm{C_{det}}(\boldsymbol{\sigma}-\boldsymbol{\rho} )\left. \right]P(\boldsymbol{\rho},\tau )d\boldsymbol{\rho} \end{aligned}$$
where we have used the fact that $\int P(\boldsymbol {\rho },\tau )d\boldsymbol {\rho }=1$. This expression highlights three kinds of contributions to the spatio-temporal cross-correlation: the first one depends only on the excitation speckle autocorrelation and is independent of the spatial shift, $\boldsymbol {\sigma }$; the second one depends on the detection speckle autocorrelation and on the spatial shift; the last one involves both excitation and detection speckle autocorrelations and depends on the spatial shift. We stress the fact that if our above-made assumption concerning the validity of the memory effect would break down, it would affect the last two contributions but not the first one, which takes into account only the excitation speckle autocorrelation.

In order to yield a tractable analytical expression of the spatio-temporal cross-correlation, we now describe the speckle autocorrelation shapes with Gaussian distributions, e.g. for the detection:

$$\mathrm{C_{det}}(\boldsymbol{\rho}) =\exp\left(-\frac{\rho^2}{\omega_{\mathrm{det}}^2}\right)$$
where $\omega _{\mathrm {det}}$ is the detection speckle grain size. The expression for excitation is obviously analogous, but with the parameter $\omega _{\mathrm {exc}}$. As we consider the case of 2D free diffusion, the propagator term has the well known expression:
$$P(\boldsymbol{\rho},\tau)=\frac{1}{4\pi D\tau}\exp\left(-\frac{\rho^2}{4D\tau}\right)$$
One can now insert Eq. (21) and 22 in Eq. (20) to obtain the final expression of the normalized spatio-temporal cross-correlation:
$$G(\boldsymbol{\sigma},\tau) =1+\frac{g(\boldsymbol{\sigma},\tau)}{N}$$
where:
$$\begin{aligned}g(\boldsymbol{\sigma},\tau)&=\frac{a_{exc}}{1+\tau/\tau_{\mathrm{exc}}}+\frac{a_{\mathrm{det}}}{1+\tau/\tau_{\mathrm{det}}}\exp\left({-\frac{\sigma^2/\omega_{\mathrm{det}}^2}{1+\tau/\tau_{\mathrm{det}}}}\right) \\ & +\frac{a_{\mathrm{exc}}a_{\mathrm{det}}}{1+\tau/\tau_{M}}\exp\left({-\frac{\sigma^2}{\omega_{\mathrm{det}}^2}}\frac{1+\tau/\tau_{\mathrm{exc}}}{1+\tau/\tau_{M}}\right) \end{aligned}$$
with:
$$\begin{array}{ccc} \tau_{\mathrm{exc}}=\frac{\omega_{\mathrm{exc}}^2}{4D} &\tau_{\mathrm{det}}=\frac{\omega_{\mathrm{det}}^2}{4D} &\frac{1}{\tau_M}=\frac{1}{\tau_{\mathrm{det}}} + \frac{1}{\tau_{\mathrm{exc}}} \end{array}$$
To conclude, let’s discuss two special cases of the spatio-temporal cross-correlation. First, we consider a nil spatial shift ($\sigma = 0$), leading to:
$$G(0,\tau) =1+\frac{1}{N}\left(\frac{a_{\mathrm{exc}}}{1+\tau/\tau_{\mathrm{exc}}}+\frac{a_{\mathrm{det}}}{1+\tau/\tau_{\mathrm{det}}}\ +\frac{a_{\mathrm{exc}}a_{\mathrm{det}}}{1+\tau/\tau_{M}}\right)$$
This expression looks very similar to the standard autocorrelation encountered for 2D FCS, except that three characteristic times come into play, which are related to the excitation and detection speckled $PSF$, plus a combination of both. Secondly, in the case of zero lag time, the cross-correlation becomes $G(\boldsymbol {\sigma },0) =1+\frac {1}{N}\left [(1+a_{\mathrm {exc}})a_{\mathrm {det}}\exp \left ({-\sigma ^2/\omega _{\mathrm {det}}^2}\right ) + a_{\mathrm {exc}}\right ]$. Especially interesting is the case that corresponds to the speckle produced by a unique fluorescent particle ($N=1$), which simplifies as:
$${G(\boldsymbol{\sigma},0)_{N=1}}=(1+a_{\mathrm{exc}})\left[1+a_{\mathrm{det}}\exp\left({-\frac{\sigma^2}{\omega_{\mathrm{det}}^2}}\right)\right]$$
At a first glance, this cross-correlation expression seems surprising since it does not solely depend on the parameters, $a_{\mathrm {det}}$ and $\omega _{\mathrm {det}}$, of the detected speckle but also on the contrast, $a_{\mathrm {exc}}$, of the excitation speckle. This is due to single particle diffusion through the speckled illumination pattern, which induces fluctuations of the particle excitation intensity, that is of the overall intensity of the detected speckle. Therefore, due to the implicit temporal averaging that was performed to evaluate the spatial cross-correlation, the baseline and the absolute amplitude of the latter are increased, as shown in Fig. 7.

5. Results

5.1 Illumination speckle

The horizontal and vertical profiles of the normalized spatial autocorrelation of the illumination speckle shown in Fig. 4(a) can be seen in Fig. 4(b), together with the profiles of its 2D Lorentzian fit performed to evaluate the amplitude and Full Width at Half Maximum (FWHM) values. Clearly, the autocorrelation amplitude of the illumination speckle is very close to that of a fully developed speckle pattern (i.e. a maximum that is about twice the baseline), with a FWHM of 2.5 $\mu$m (that is, an evaluation of the speckle grain size). Moreover, the intensity histogram clearly obeys the negative exponential distribution of a fully developed speckle, as reported in [21] (data not shown). The illumination speckle pattern of Fig. 4 has been recorded at a distance of $Z = 950$ $\mu$m from the scatterer surface, a distance close to the water thickness used to study the fluorescent beads dynamics. However, the FWHM of the illumination speckle autocorrelation depends upon its distance to the scatterer plane, as presented in Fig. 5.

 

Fig. 4. (a) Illumination speckle pattern at a distance of Z = 950 $\mu$m from the scatterer surface. The analysed ROI ($1024\times 1024$) covers most of the FoV ($1280\times 1024$). (b) $\sigma _x$ and $\sigma _y$ profiles of the normalized autocorrelation of the speckle along with their Lorentzian fit.

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Fig. 5. Evolution of the FWHM of the illumination speckle autocorrelation with the distance from the scatterer.

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The speckle grain size remains almost constant at distances very close to the scatterer surface and then increases linearly with a slope of $0.001$, which is in agreement with the far field speckle grain size relation $\lambda Z/D$ of a fully developed speckle [22], using our measured $D$ value ($\sim 500$ $\mu$m).

5.2 Detection speckle due to a single diffusing bead

Rather than showing the noisy snapshot of the detection speckle pattern of a moving bead, we display in Fig. 6(a) the detection speckle pattern (recorded at $250$ $\mu$m below the bead plane) of an immobile bead, recorded with a relatively long exposure time. Note that we analysed a $256\times 256$ ROI out of the FoV ($2048\times 2048$), such that the mean intensity is constant over the ROI. The corresponding normalized autocorrelation can be seen in Fig. 6(b). It is worth to remark that its maximum is significantly less than twice its baseline ($\sim 1.6$), contrarily to the autocorrelation of the illumination speckle, while both are generated by the same scattering layer. Several reasons can explain this fact: Firstly, the fluorescent beads are not very small (1 $\mu$m) compared to the wavelength. Secondly, although we use a band pass filter, the emission residual bandwidth decreases the speckle contrast [21,23]. Interestingly, the intensity histogram is not inconsistent with the approximate form given for polychromatic speckles, as reported in [21] (data not shown). Finally, the fluorescence emission is partially depolarized, which may also decrease the speckle contrast [21]. We have also observed that the detection speckle grain size slightly increases with the defocus from the sample plane. However, we do not provide any further analysis about this point, as it depends upon the detailed characteristics of the radiation pattern of a scatterer which goes much beyond the core subject of the present study.

 

Fig. 6. (a) Detection speckle pattern observed at 150 $\mu$m below the bead plane and produced by a single bead, immobilized on a 1 mm microscope slide put above the scatterer (integration time 600 ms). By doing it in this manner, one obtains an image equivalent, but with a better S/N, to the snapshot of a moving bead detection speckle pattern. (b) $\sigma _x$ and $\sigma _y$ profiles of the normalized autocorrelation of the speckle along with their Lorentzian fit.

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When a single bead moves, the overall intensity of the emission speckle pattern would blink due to the illumination speckle (see Visualization 2). Let us now point to the importance of the correlation normalization method: if each raw autocorrelation at time $t_i$ is normalized by the mean of the corresponding instantaneous intensity $I(\boldsymbol {\rho },t_i)$ and then averaged, the results is similar to the autocorrelation produced by an immobile fluorescent bead as shown in Fig. 6(b) (see Appendix B). However, if the raw autocorrelations are first averaged and then normalized with the average intensity of all images (we typically recorded 4000 detection speckle patterns lasting 160 s), which corresponds to the expression of the spatio-temporal correlation defined in Eq. (1), the result is quite different, as shown on Fig. 7. The baseline is now higher than 1 and the maximum is over 2, in agreement with the theoretical considerations presented above (cf. Equation (27)). The reason is that the autocorrelation now contains information about the illumination speckle pattern, due to fluorescent bead diffusing through it. If the illumination intensity pattern was homogeneous, both averaging methods would give the same result. In all the following, the spatio-temporal cross-correlation of the detected images will be normalized by the mean intensity of all the images, whatever the number of beads, in agreement with the definition of Eq. (1). It is interesting to note that this kind of situation, where different sources of fluctuations are compounded, has been termed "speckled speckle" by Goodman [21].

 

Fig. 7. $\sigma _x$ and $\sigma _y$ line profiles of the detection speckle autocorrelation due to a single diffusing bead. Temporal averaging has been performed prior to normalization with square of the mean intensity of all the speckle patterns in the image stack. Central noise peak, due to the limited S/N ratio of the detected speckle pattern, has been removed and a cubic spline interpolation put instead of it.

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5.3 Speckle pattern analysis with a single diffusing bead

When a single bead diffuses in the ROI, the spatio-temporal cross-correlation of the dynamic speckle pattern weakens and widens with lag time, as can be seen in Fig. 8(a). Although the temporal decay (Fig. 8(b)) at the center of the cross-correlation depends on well-defined parameters, as given by Eqs. (25) and (26), the limited signal-to-noise ratio prevents to reliably fit all of them. Therefore, with the goal in mind to estimate the diffusion constant $D$, the parameters $a_{\mathrm {exc}}$, $\omega _{\mathrm {exc}}$, $a_{\mathrm {det}}$ and $\omega _{\mathrm {det}}$ were estimated independently by fitting the excitation and detection speckle autocorrelations. However, as exemplified by Fig. 4(b)), we observed that these autocorrelations were better described by Lorentzian than Gaussian functions (although the latter leads to simple analytical formulas of the spatio-temporal cross-correlation). We thus fit the excitation and detection speckle autocorrelations with Lorentzians, which yields the amplitude parameters, $a_{\mathrm {exc}}$ and $a_{\mathrm {det}}$, and the HWHM (Half Width at Half Maximum) that were eventually converted into parameter values for $\omega _{\mathrm {exc}}$ and $\omega _{\mathrm {det}}$, using $\omega = \mathrm {HWHM}/\ln (2)$. We believe this approximation of the autocorrelation profile does not impede a quantitative analysis of the temporal decay of the cross-correlation at zero spatial shift, $G(0,\tau )$. It is also worth to stress that we fit the autocorrelation of the detection speckle pattern obtained by averaging the normalized instantaneous autocorrelations, as described in section 5.2 and Appendix B (for obtaining a detection autocorrelation that is independent of the excitation speckle contrast). A typical temporal decay is shown in Fig. 8(b), together with its fit where the only free parameters are the effective number $N$ of particles, and the diffusion constant $D$ (see Eqs. (25) and (26)). Note that the recovered number of particles slightly differs from 1, which may be due to the limited accuracy of the fixed input parameters $a_{\mathrm {exc}}$ and $a_{\mathrm {det}}$. Next, we consider the widening of the spatio-temporal cross-correlation as a function of time by displaying the $\mathrm {FWHM}^2$ of $G(\boldsymbol {\sigma },\tau )$ in Fig. 8(c). We stress the fact that at sufficiently short time values the evolution of the $\mathrm {FWHM}^2$ versus time is in very good agreement with a linear increase of the MSD (the $8\ln (2)$ factor stands for the correspondence between the $\mathrm {FWHM}^2$ and the variance of a Gaussian distribution). However, in practice, a quantitative exploit of the broadening of the spatio-temporal cross-correlation is delicate because it relies on the proper description of the speckle autocorrelation profiles.

 

Fig. 8. Spatio-temporal image cross-correlation of single bead speckle patterns. (a) Evolution of the spatio-temporal cross-correlation at different lag times. (b) Decay curve of the spatio-temporal cross-correlation amplitude at zero spatial shift, obtained from a single particle dynamic speckle. The gray solid line was obtained by fitting the data with the theoretical model of Eqs. (25) and (26), using fixed parameter values for $a_{\mathrm {exc}}$, $\omega _{\mathrm {exc}}$, $a_{\mathrm {det}}$ and $\omega _{\mathrm {det}}$ as given in the inset. (c) Evolution of the square of the FWHM of the cross-correlations depicted in (a) and comparison with a linear evolution corresponding to a MSD given by $4D\tau$.

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Finally, collecting the data recorded by focusing at different distances from the bead plane (from 150 to 250 $\mu$m below), we did not observe any bias of the diffusion constant measurements. As a whole, we found $D = 0.12 \pm 0.02$ $\mu$m$^2$/s, which has to be compared with the value estimated from the MSD measured in absence of any speckle patterns ($0.11 \pm 0.07$ $\mu$m$^2$/s, see Appendix A). It is interesting to note that, although the measured detection speckle grain size can significantly vary from one experiment to another (e.g. we measured $\omega _{\mathrm {det}} = 1.9 \pm 0.6$ $\mu$m), the diffusion constant varies much less, because its estimation takes into account the input parameters ($a_{\mathrm {exc}}$, $\omega _{\mathrm {exc}}$, $a_{\mathrm {det}}$ and $\omega _{\mathrm {det}}$).

5.4 Speckle pattern analysis with multiple diffusing beads

We then studied the evolution of the spatio-temporal cross-correlation decay at zero spatial shift versus the number of beads in the ROI. Obviously, the larger the number of diffusing particles, the smaller the amplitude, while the estimated diffusion constant does not show any systematic trend (see Fig. 9).

We nevertheless observe that the estimated number of beads deviates from the number of those that are detected by focusing at the bead plane (see section 2.1). Two main reasons can explain this fact: first, the smaller the correlation amplitude, the more one is sensitive to any parasitic signal, instability of bead motion, etc. Also, it might happen that beads diffusing close to the FoV are not detected when focusing in the bead plane, but do induce speckle patterns that reach into the ROI. In a way, speckle spatio-temporal cross-correlation has the same limits than standard FFM techniques: although the number of particles or molecules is the least model-dependent parameter to estimate, it is more likely to be biased by many difficult to control parameters, especially at high numbers of particles.

 

Fig. 9. Spatio-temporal image cross-correlation of multiple bead speckles. The evolution of the temporal decay of the spatio-temporal cross-correlation at zero spatial shift as obtained from two, three, five, seven, and twelve beads dynamic speckle patterns along with their theoretical fits.

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6. Discussion

It is worth noting that our method can be generalized in a straightforward way to analyze the motion of particles or molecules without speckle pattern (i.e. without scattering layer), in which case our approach is formally the same as that of FFM [16]. In practice, the difference with the speckle case is that the relevant area containing the diffusing particles that contribute to the amplitude of the spatio-temporal cross-correlation is no longer the chosen ROI, but the much smaller, diffraction limited, PSF. One can find more details about the connection between both situations (with and without speckle) in Appendix C.

Throughout our data analysis, we assumed that the memory effect always applies. From the estimated thickness of the scattering layer ($\sim 6$ $\mu$m), we deduce an angular range of the memory effect of about 0.9$^\circ$ (half-width) [5]. With a typical distance of 1 mm between the plane of the diffusing beads and the scatterer, this would correspond to a width of 30 $\mu$m in the sample plane. As this range is smaller than our ROI (about 80 $\mu$m in width), we may wonder why our analysis is apparently not impaired by the limited spatial extension of memory effect. In fact, we believe our method is only marginally dependent on the memory effect range. The reason lies in the fact that the characteristic decay time of the cross-correlation without spatial shift, $G(0,\tau )$, depends only on the diffusion time of a particle through a speckle grain. In case that the memory effect does not apply, the autocorrelation of the detection PSF is a function of $\boldsymbol {\sigma }-\boldsymbol {\rho }$ and $\boldsymbol {\rho }$, as written in Eq. (16). To further scrutinize this idea, let us assume that the additional dependence on $\boldsymbol {\rho }$ of the detection PSF can be taken into account by replacing the constant $a_{\mathrm {det}}$ in Eq. (20) by a function $a_{\mathrm {det}}\exp \left (-\rho ^2/\omega _{\mathrm {me}}^2\right )$, where $\omega _{\mathrm {me}}$ quantifies the spatial range of the memory effect. As a consequence, this leads to an additional decay time, $\tau _{\mathrm {me}} = \omega _{\mathrm {me}}^2/4D$, in the spatio-temporal cross-correlation. However, as long as this characteristic time is longer than $\tau _{\mathrm {det}}$ and $\tau _{\mathrm {exc}}$ (equivalently, as long as $\omega _{\mathrm {me}}$ is larger than $\omega _{\mathrm {det}}$ and $\omega _{\mathrm {exc}}$), the consequences are negligible for the decay evolution of the spatio-temporal cross-correlation. Further analysis not reported here shows that the broadening of the spatio-temporal cross-correlation saturates at very long lag times, i.e. when its spatial extent reaches the memory effect range, $\omega _{\mathrm {me}}$. The observed long-time values of the curvature of the $\mathrm {FWHM}^2$ plot, see Fig. 8(c), might originate from the memory effect limit, but this analysis is beyond the scope of the present paper. However, our memory effect considerations hold whatever the type and speed of transport (diffusion, convection, etc.). Note that the situation is totally different for image retrieval, which relies on the stability and convergence of the iterative procedure applied over the spatial extent of the object to be reconstructed.

7. Conclusion

Assessing the motions of an ensemble of single point objects hidden behind a multiple scattering layer or curtain is of general interest for biomedical applications, or in aerial, terrestrial and naval transport control and security. Although we have considered here only the case of fluorescent particles, our approach can be extended to other kinds of emitters (e.g. in the infrared), provided their emission is temporally incoherent with respect to each other. Basically, we showed that the diffusion constant and the number of invisible point-wise fluorescent particles can be assessed from the spatio-temporal correlation properties of their detected dynamic speckle pattern, whatever is the illumination (uniform or speckled). Interestingly, our model is applicable for speckle patterns with a contrast lower than that of the fully developed speckle limit. Of course, our method can be extended to other kinds of particle motions (drift, anomalous diffusion, compartmentalization, etc.), provided a model depending on a few parameters is available. We stress the fact that our approach does not require the memory effect to extend beyond a few speckle grains, the size of which being the characteristic length over which the correlation time has to be estimated.

Since a very common situation is that of a dynamically scattering medium, one may wonder how to adapt the proposed method to such a situation. We think that the correlation time of the speckle pattern due the scattering medium dynamics must be longer than that due to the moving particles of interest.

Another important question is that of the generalization of our method to three-dimensional fluorescent samples hidden behind a scattering layer. Equation (20) gives us an idea about such a situation: among the three terms contributing to the spatio-temporal cross-correlation, the single excitation and detection speckle autocorrelations correspond to kinds of backgrounds, while the product of the excitation speckle autocorrelation by the detection one introduces confocality $z$-sectioning. However, the experimental feasibility will depend upon the magnitude and noise of the non-sectioned and $z$-sectioned contributions.

Appendix A. Single particle tracking

In the absence of speckled illumination, two dimensional (2D) diffusion constant of single fluorescent particle has been evaluated by particle tracking method. For this experiment, the optical set up was similar to that used in the study of diffusion with speckle. A thin layer ($\sim 1$mm) of water film was deposited in a labtek-well without any scattering layer and the hydrophobic beads were then air sprayed on the water surface. Plane-wave geometry was employed to illuminate a single fluorescent bead within the ROI and its image was recorded by the $20\times$, NA=$0.4$ objective and the CMOS camera. Movement of the bead was tracked by capturing $4000$ images at $25$ fps and subsequently analysing its position and the mean square displacement (MSD) by utilising Fiji image analysis software [24] at different lag times. Average MSD obtained from four different experiments has been plotted in Fig. 10 and the 2D diffusion constant has been evaluated to be $D = 0.11 \pm 0.07$ $\mu$m$^{2}$/s from its slope.

 

Fig. 10. Average mean square displacement (MSD) of single particle diffusion on water/air surface with lag time. Linear fit of the MSD provides the diffusion constant $D = 0.11 \pm 0.07$ $\mu$m$^{2}$/s. Shaded region corresponds to the standard deviation.

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Appendix B. Detection speckle pattern normalization and averaging

When the excitation PSF is speckled, each of the diffusing beads emits a speckle pattern whose overall intensity fluctuates in time. Consequently, if one averages a stack of single bead detected speckle autocorrelations and then, normalizes it, one obtains an autocorrelation that embeds information about the excitation speckle (see Fig. 7). However, it is possible to get rid of this effect by normalizing each instantaneous detected speckle autocorrelation before averaging the stack. By doing so, one obtains an autocorrelation with a baseline of 1 and an amplitude that solely depends upon the detection path, as shown in Fig. 11.

 

Fig. 11. Exemple of mean autocorrelation obtained from a stack of normalized instantaneous autocorrelations, produced by a single diffusing bead. Conceptually speaking, this is equivalent to the speckle autocorrelation due to a single immobile bead, as the one shown in Fig. 6.

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Appendix C. Connection between speckle correlation analysis and fluorescence fluctuation microscopy

Here we give some details about the connection between our approach and Fluorescence Fluctuation Microscopy (FFM) [16]. Let us consider again Eq. (16), with an even illumination (i.e. $a_{exc} = 0$) and a detection PSF that is no more speckled, but given by $\exp \left ({-2\rho ^2/\omega _{det}^2}\right )$. Doing so, we obtain (assuming $N$ » 1):

$$G(\boldsymbol{\sigma},\tau)=1+\frac{1}{N_\mathrm{{PSF}}}\frac{\exp\left({\frac{-\sigma^2/\omega_{det}^2}{1+\tau/\tau_{det}}}\right)}{1+\tau/\tau_{det}}$$
where $N_\mathrm {{PSF}} = \frac {\pi \omega _{det}^2}{S}N$ is the mean number of particles within the detection PSF, much smaller than the total number of particles, $N$, within the ROI of area $S$. The corresponding much larger amplitude of the spatio-temporal cross-correlation without speckle can be seen in Fig. 12.

 

Fig. 12. Spatio-temporal cross-correlation obtained without scatterer (i.e. without speckle patterns). (a) shows the temporal decay curve at zero spatial shift, obtained from multiply particle images. The grey solid line has been obtained by fitting the data with the theoretical model, using the input value of the $\omega _{det}$ parameter given in the inset. (b) shows the evolution of the square of the FWHM of the spatio-temporal cross-correlations.

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It corresponds to a mean number of beads in the PSF of about 0.04. To derive the theoretically corresponding number of diffusing beads in the ROI, we must take into account the fact that the latter is not evenly covered by the beads. Doing so, using an estimated effective ROI size of $54.8 \mu$m and a detection PSF with $\omega _{det} =2.18 \mu$m, we derive about 8 beads in the ROI, while we observed 6 diffusing beads. In fact, the relevant area that contains the number of particles that correspond to the fluctuation amplitude can be defined in a unique way for FFM and speckle correlations: this is the area of the PSF, whether it is a standard one, limited by diffraction, or a speckled PSF. However, the experiment must last long enough for the sample particles to evenly explore the space. This time being too long for small concentrations of 1 $\mu$m beads, we rather chose ROIs containing a constant number of beads.

In the frame of FFM, the method is called space-time image correlation spectroscopy (STICS) [25]. It has been mostly developed for biological applications to assess diffusion coefficients of molecules, but also velocity vectors (magnitude and direction) in living cells, in which case Eq. (28) has to be modified to incorporate a velocity.

Funding

Deutsche Forschungsgemeinschaft (EXC 2067/1 - 390729940, Microscattab); Agence Nationale de la Recherche (ANR-16-CE92-0003-01 (Microscattab)).

Acknowledgments

AD thanks A. Sentenac for a stimulating discussion.

Disclosures

The authors declare no conflicts of interest.

References

1. I. Freund, “Looking through walls and around corners,” Phys. A 168(1), 49–65 (1990). [CrossRef]  

2. R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65(25), 3120–3123 (1990). [CrossRef]  

3. J. de Rosny, P. Roux, M. Fink, and J. H. Page, “Field fluctuation spectroscopy in a reverberant cavity with moving scatterers,” Phys. Rev. Lett. 90(9), 094302 (2003). [CrossRef]  

4. I. M. Vellekoop, “Feedback-based wavefront shaping,” Opt. Express 23(9), 12189 (2015). [CrossRef]  

5. I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988). [CrossRef]  

6. J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012). [CrossRef]  

7. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer Berlin Heidelberg, 1975).

8. K. T. Takasaki and J. W. Fleischer, “Phase-space measurement for depth-resolved memory-effect imaging,” Opt. Express 22(25), 31426–31433 (2014). [CrossRef]  

9. J. A. Newman and K. J. Webb, “Imaging optical fields through heavily scattering media,” Phys. Rev. Lett. 113(26), 263903 (2014). [CrossRef]  

10. J. A. Newman, Q. Luo, and K. J. Webb, “Imaging hidden objects with spatial speckle intensity correlations over object position,” Phys. Rev. Lett. 116(7), 073902 (2016). [CrossRef]  

11. M. Cua, E. H. Zhou, and C. Yang, “Imaging moving targets through scattering media,” Opt. Express 25(4), 3935 (2017). [CrossRef]  

12. K. J. Webb and Q. Luo, “Theory of speckle intensity correlations over object position in a heavily scattering random medium,” Phys. Rev. A 101(6), 063827 (2020). [CrossRef]  

13. E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1(4), 227 (2014). [CrossRef]  

14. C. Guo, J. Liu, T. Wu, L. Zhu, and X. Shao, “Tracking moving targets behind a scattering medium via speckle correlation,” Appl. Opt. 57(4), 905 (2018). [CrossRef]  

15. M. Hofer, C. Soeller, S. Brasselet, and J. Bertolotti, “Wide field fluorescence epi-microscopy behind a scattering medium enabled by speckle correlations,” Opt. Express 26(8), 9866 (2018). [CrossRef]  

16. E. L. Elson, “Fluorescence correlation spectroscopy: Past, present, future,” Biophys. J. 101(12), 2855–2870 (2011). [CrossRef]  

17. D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15(1), 011109 (2010). [CrossRef]  

18. G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015). [CrossRef]  

19. I. M. Vellekoop and A. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007). [CrossRef]  

20. R. H. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79(22), 4369–4372 (1997). [CrossRef]  

21. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, Second Edition (SPIE, 2020).

22. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation functiong(r),” Phys. Rev. Lett. 85(7), 1416–1419 (2000). [CrossRef]  

23. O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014). [CrossRef]  

24. J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017). [CrossRef]  

25. B. Hebert, S. Costantino, and P. W. Wiseman, “Spatiotemporal imag correlation spectroscopy (STICS) theory, verification, and application to protein velocity mapping in living CHO cells,” Biophys. J. 88(5), 3601–3614 (2005). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. I. Freund, “Looking through walls and around corners,” Phys. A 168(1), 49–65 (1990).
    [Crossref]
  2. R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65(25), 3120–3123 (1990).
    [Crossref]
  3. J. de Rosny, P. Roux, M. Fink, and J. H. Page, “Field fluctuation spectroscopy in a reverberant cavity with moving scatterers,” Phys. Rev. Lett. 90(9), 094302 (2003).
    [Crossref]
  4. I. M. Vellekoop, “Feedback-based wavefront shaping,” Opt. Express 23(9), 12189 (2015).
    [Crossref]
  5. I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
    [Crossref]
  6. J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
    [Crossref]
  7. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer Berlin Heidelberg, 1975).
  8. K. T. Takasaki and J. W. Fleischer, “Phase-space measurement for depth-resolved memory-effect imaging,” Opt. Express 22(25), 31426–31433 (2014).
    [Crossref]
  9. J. A. Newman and K. J. Webb, “Imaging optical fields through heavily scattering media,” Phys. Rev. Lett. 113(26), 263903 (2014).
    [Crossref]
  10. J. A. Newman, Q. Luo, and K. J. Webb, “Imaging hidden objects with spatial speckle intensity correlations over object position,” Phys. Rev. Lett. 116(7), 073902 (2016).
    [Crossref]
  11. M. Cua, E. H. Zhou, and C. Yang, “Imaging moving targets through scattering media,” Opt. Express 25(4), 3935 (2017).
    [Crossref]
  12. K. J. Webb and Q. Luo, “Theory of speckle intensity correlations over object position in a heavily scattering random medium,” Phys. Rev. A 101(6), 063827 (2020).
    [Crossref]
  13. E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1(4), 227 (2014).
    [Crossref]
  14. C. Guo, J. Liu, T. Wu, L. Zhu, and X. Shao, “Tracking moving targets behind a scattering medium via speckle correlation,” Appl. Opt. 57(4), 905 (2018).
    [Crossref]
  15. M. Hofer, C. Soeller, S. Brasselet, and J. Bertolotti, “Wide field fluorescence epi-microscopy behind a scattering medium enabled by speckle correlations,” Opt. Express 26(8), 9866 (2018).
    [Crossref]
  16. E. L. Elson, “Fluorescence correlation spectroscopy: Past, present, future,” Biophys. J. 101(12), 2855–2870 (2011).
    [Crossref]
  17. D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15(1), 011109 (2010).
    [Crossref]
  18. G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
    [Crossref]
  19. I. M. Vellekoop and A. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
    [Crossref]
  20. R. H. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79(22), 4369–4372 (1997).
    [Crossref]
  21. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, Second Edition (SPIE, 2020).
  22. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation functiong(r),” Phys. Rev. Lett. 85(7), 1416–1419 (2000).
    [Crossref]
  23. O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
    [Crossref]
  24. J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
    [Crossref]
  25. B. Hebert, S. Costantino, and P. W. Wiseman, “Spatiotemporal imag correlation spectroscopy (STICS) theory, verification, and application to protein velocity mapping in living CHO cells,” Biophys. J. 88(5), 3601–3614 (2005).
    [Crossref]

2020 (1)

K. J. Webb and Q. Luo, “Theory of speckle intensity correlations over object position in a heavily scattering random medium,” Phys. Rev. A 101(6), 063827 (2020).
[Crossref]

2018 (2)

2017 (2)

M. Cua, E. H. Zhou, and C. Yang, “Imaging moving targets through scattering media,” Opt. Express 25(4), 3935 (2017).
[Crossref]

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

2016 (1)

J. A. Newman, Q. Luo, and K. J. Webb, “Imaging hidden objects with spatial speckle intensity correlations over object position,” Phys. Rev. Lett. 116(7), 073902 (2016).
[Crossref]

2015 (2)

I. M. Vellekoop, “Feedback-based wavefront shaping,” Opt. Express 23(9), 12189 (2015).
[Crossref]

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

2014 (4)

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

K. T. Takasaki and J. W. Fleischer, “Phase-space measurement for depth-resolved memory-effect imaging,” Opt. Express 22(25), 31426–31433 (2014).
[Crossref]

J. A. Newman and K. J. Webb, “Imaging optical fields through heavily scattering media,” Phys. Rev. Lett. 113(26), 263903 (2014).
[Crossref]

E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1(4), 227 (2014).
[Crossref]

2012 (1)

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref]

2011 (1)

E. L. Elson, “Fluorescence correlation spectroscopy: Past, present, future,” Biophys. J. 101(12), 2855–2870 (2011).
[Crossref]

2010 (1)

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15(1), 011109 (2010).
[Crossref]

2007 (1)

2005 (1)

B. Hebert, S. Costantino, and P. W. Wiseman, “Spatiotemporal imag correlation spectroscopy (STICS) theory, verification, and application to protein velocity mapping in living CHO cells,” Biophys. J. 88(5), 3601–3614 (2005).
[Crossref]

2003 (1)

J. de Rosny, P. Roux, M. Fink, and J. H. Page, “Field fluctuation spectroscopy in a reverberant cavity with moving scatterers,” Phys. Rev. Lett. 90(9), 094302 (2003).
[Crossref]

2000 (1)

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation functiong(r),” Phys. Rev. Lett. 85(7), 1416–1419 (2000).
[Crossref]

1997 (1)

R. H. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79(22), 4369–4372 (1997).
[Crossref]

1990 (2)

I. Freund, “Looking through walls and around corners,” Phys. A 168(1), 49–65 (1990).
[Crossref]

R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65(25), 3120–3123 (1990).
[Crossref]

1988 (1)

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref]

Bednarek, S. Y.

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

Berkovits, R.

R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65(25), 3120–3123 (1990).
[Crossref]

Bertolotti, J.

M. Hofer, C. Soeller, S. Brasselet, and J. Bertolotti, “Wide field fluorescence epi-microscopy behind a scattering medium enabled by speckle correlations,” Opt. Express 26(8), 9866 (2018).
[Crossref]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref]

Blanc, C.

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref]

Boas, D. A.

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15(1), 011109 (2010).
[Crossref]

Boniello, G.

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Brasselet, S.

Carpineti, M.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation functiong(r),” Phys. Rev. Lett. 85(7), 1416–1419 (2000).
[Crossref]

Costantino, S.

B. Hebert, S. Costantino, and P. W. Wiseman, “Spatiotemporal imag correlation spectroscopy (STICS) theory, verification, and application to protein velocity mapping in living CHO cells,” Biophys. J. 88(5), 3601–3614 (2005).
[Crossref]

Cua, M.

de Rosny, J.

J. de Rosny, P. Roux, M. Fink, and J. H. Page, “Field fluctuation spectroscopy in a reverberant cavity with moving scatterers,” Phys. Rev. Lett. 90(9), 094302 (2003).
[Crossref]

de Vries, P.

R. H. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79(22), 4369–4372 (1997).
[Crossref]

Dunn, A. K.

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15(1), 011109 (2010).
[Crossref]

Eliceiri, K. W.

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

Elson, E. L.

E. L. Elson, “Fluorescence correlation spectroscopy: Past, present, future,” Biophys. J. 101(12), 2855–2870 (2011).
[Crossref]

Fedorenko, D.

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Feng, S.

R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65(25), 3120–3123 (1990).
[Crossref]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref]

Fink, M.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

J. de Rosny, P. Roux, M. Fink, and J. H. Page, “Field fluctuation spectroscopy in a reverberant cavity with moving scatterers,” Phys. Rev. Lett. 90(9), 094302 (2003).
[Crossref]

Fleischer, J. W.

Freund, I.

I. Freund, “Looking through walls and around corners,” Phys. A 168(1), 49–65 (1990).
[Crossref]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref]

Gigan, S.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

Giglio, M.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation functiong(r),” Phys. Rev. Lett. 85(7), 1416–1419 (2000).
[Crossref]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, Second Edition (SPIE, 2020).

Gross, M.

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Guo, C.

Hebert, B.

B. Hebert, S. Costantino, and P. W. Wiseman, “Spatiotemporal imag correlation spectroscopy (STICS) theory, verification, and application to protein velocity mapping in living CHO cells,” Biophys. J. 88(5), 3601–3614 (2005).
[Crossref]

Heidmann, P.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

Hofer, M.

Hoopes, G. M.

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

In, M.

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Judkewitz, B.

Katz, O.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

Kop, R. H.

R. H. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79(22), 4369–4372 (1997).
[Crossref]

Lagendijk, A.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref]

R. H. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79(22), 4369–4372 (1997).
[Crossref]

Laplantine, E.

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

Liu, J.

Luo, Q.

K. J. Webb and Q. Luo, “Theory of speckle intensity correlations over object position in a heavily scattering random medium,” Phys. Rev. A 101(6), 063827 (2020).
[Crossref]

J. A. Newman, Q. Luo, and K. J. Webb, “Imaging hidden objects with spatial speckle intensity correlations over object position,” Phys. Rev. Lett. 116(7), 073902 (2016).
[Crossref]

Mbarek, N. B.

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Medfai, M.

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Mosk, A.

Mosk, A. P.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref]

Newman, J. A.

J. A. Newman, Q. Luo, and K. J. Webb, “Imaging hidden objects with spatial speckle intensity correlations over object position,” Phys. Rev. Lett. 116(7), 073902 (2016).
[Crossref]

J. A. Newman and K. J. Webb, “Imaging optical fields through heavily scattering media,” Phys. Rev. Lett. 113(26), 263903 (2014).
[Crossref]

Nobili, M.

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Page, J. H.

J. de Rosny, P. Roux, M. Fink, and J. H. Page, “Field fluctuation spectroscopy in a reverberant cavity with moving scatterers,” Phys. Rev. Lett. 90(9), 094302 (2003).
[Crossref]

Perry, N.

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

Reynolds, G. D.

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

Rosenbluh, M.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref]

Roux, P.

J. de Rosny, P. Roux, M. Fink, and J. H. Page, “Field fluctuation spectroscopy in a reverberant cavity with moving scatterers,” Phys. Rev. Lett. 90(9), 094302 (2003).
[Crossref]

Ruan, H.

Schindelin, J.

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

Shao, X.

Shorte, S. L.

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

Soeller, C.

Sprik, R.

R. H. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79(22), 4369–4372 (1997).
[Crossref]

Stocco, A.

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Takasaki, K. T.

Tinevez, J.-Y.

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

Vailati, A.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation functiong(r),” Phys. Rev. Lett. 85(7), 1416–1419 (2000).
[Crossref]

van Putten, E. G.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref]

Vellekoop, I. M.

Vos, W. L.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref]

Webb, K. J.

K. J. Webb and Q. Luo, “Theory of speckle intensity correlations over object position in a heavily scattering random medium,” Phys. Rev. A 101(6), 063827 (2020).
[Crossref]

J. A. Newman, Q. Luo, and K. J. Webb, “Imaging hidden objects with spatial speckle intensity correlations over object position,” Phys. Rev. Lett. 116(7), 073902 (2016).
[Crossref]

J. A. Newman and K. J. Webb, “Imaging optical fields through heavily scattering media,” Phys. Rev. Lett. 113(26), 263903 (2014).
[Crossref]

Wiseman, P. W.

B. Hebert, S. Costantino, and P. W. Wiseman, “Spatiotemporal imag correlation spectroscopy (STICS) theory, verification, and application to protein velocity mapping in living CHO cells,” Biophys. J. 88(5), 3601–3614 (2005).
[Crossref]

Wu, T.

Yang, C.

Zhou, E. H.

Zhu, L.

Appl. Opt. (1)

Biophys. J. (2)

E. L. Elson, “Fluorescence correlation spectroscopy: Past, present, future,” Biophys. J. 101(12), 2855–2870 (2011).
[Crossref]

B. Hebert, S. Costantino, and P. W. Wiseman, “Spatiotemporal imag correlation spectroscopy (STICS) theory, verification, and application to protein velocity mapping in living CHO cells,” Biophys. J. 88(5), 3601–3614 (2005).
[Crossref]

J. Biomed. Opt. (1)

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15(1), 011109 (2010).
[Crossref]

Methods (1)

J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri, “Trackmate: An open and extensible platform for single-particle tracking,” Methods 115, 80–90 (2017).
[Crossref]

Nat. Mater. (1)

G. Boniello, C. Blanc, D. Fedorenko, M. Medfai, N. B. Mbarek, M. In, M. Gross, A. Stocco, and M. Nobili, “Brownian diffusion of a partially wetted colloid,” Nat. Mater. 14(9), 908–911 (2015).
[Crossref]

Nat. Photonics (1)

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

Nature (1)

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Optica (1)

Phys. A (1)

I. Freund, “Looking through walls and around corners,” Phys. A 168(1), 49–65 (1990).
[Crossref]

Phys. Rev. A (1)

K. J. Webb and Q. Luo, “Theory of speckle intensity correlations over object position in a heavily scattering random medium,” Phys. Rev. A 101(6), 063827 (2020).
[Crossref]

Phys. Rev. Lett. (7)

R. H. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79(22), 4369–4372 (1997).
[Crossref]

R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65(25), 3120–3123 (1990).
[Crossref]

J. de Rosny, P. Roux, M. Fink, and J. H. Page, “Field fluctuation spectroscopy in a reverberant cavity with moving scatterers,” Phys. Rev. Lett. 90(9), 094302 (2003).
[Crossref]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref]

J. A. Newman and K. J. Webb, “Imaging optical fields through heavily scattering media,” Phys. Rev. Lett. 113(26), 263903 (2014).
[Crossref]

J. A. Newman, Q. Luo, and K. J. Webb, “Imaging hidden objects with spatial speckle intensity correlations over object position,” Phys. Rev. Lett. 116(7), 073902 (2016).
[Crossref]

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation functiong(r),” Phys. Rev. Lett. 85(7), 1416–1419 (2000).
[Crossref]

Other (2)

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer Berlin Heidelberg, 1975).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, Second Edition (SPIE, 2020).

Supplementary Material (3)

NameDescription
» Visualization 1       Movie showing a single fluorescent micro-sized bead undergoing an unrestricted 2D Brownian diffusive motion, as seen in a epi-fluorescence configuration, without any scattering layer on the optical paths.
» Visualization 2       Movie showing the dynamic speckle observed when a single fluorescent micro-sized bead, undergoing an unrestricted 2D Brownian diffusive motion, is illuminated through a fully scattering layer and observed through the very same layer, using a epi-fluo
» Visualization 3       Movie showing the dynamic speckle observed when multiply fluorescent micro-sized beads, undergoing an unrestricted 2D Brownian diffusive motion, are illuminated through a fully scattering layer and observed through the very same layer, using a epi-fl

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Figures (12)

Fig. 1.
Fig. 1. (a) Schematic of the experimental set up for studying fluorescence correlations through a strong scattering medium. (b) Inset shows the magnified version of the sample region where the focal plane can be moved through the sample. (c) Schematic of the set up for characterizing the illumination speckle at varying distance above the sample plane.
Fig. 2.
Fig. 2. Schematics of the illumination and detection speckle patterns. Within the angular memory effect range, the moving beads emit identical patterns (in red, below) but, due to the random distribution of excitation intensity (in green, above), their overall time-varying intensities are uncorrelated. In practice, the detection patterns of all the beads overlap each other in the analyzed ROI.
Fig. 3.
Fig. 3. Analysis of the detected speckle pattern and spatio-temporal cross-correlation. (a) Recording scheme of the detection speckle patterns versus time. (b) Series of single 2D spatio-temporal cross-correlations between speckle patterns separated by a lag time $\tau$ of 2 frame times. (c) Series of radial line profiles of the averaged 2D spatio-temporal cross-correlations versus the lag time $\tau$.
Fig. 4.
Fig. 4. (a) Illumination speckle pattern at a distance of Z = 950 $\mu$m from the scatterer surface. The analysed ROI ($1024\times 1024$) covers most of the FoV ($1280\times 1024$). (b) $\sigma _x$ and $\sigma _y$ profiles of the normalized autocorrelation of the speckle along with their Lorentzian fit.
Fig. 5.
Fig. 5. Evolution of the FWHM of the illumination speckle autocorrelation with the distance from the scatterer.
Fig. 6.
Fig. 6. (a) Detection speckle pattern observed at 150 $\mu$m below the bead plane and produced by a single bead, immobilized on a 1 mm microscope slide put above the scatterer (integration time 600 ms). By doing it in this manner, one obtains an image equivalent, but with a better S/N, to the snapshot of a moving bead detection speckle pattern. (b) $\sigma _x$ and $\sigma _y$ profiles of the normalized autocorrelation of the speckle along with their Lorentzian fit.
Fig. 7.
Fig. 7. $\sigma _x$ and $\sigma _y$ line profiles of the detection speckle autocorrelation due to a single diffusing bead. Temporal averaging has been performed prior to normalization with square of the mean intensity of all the speckle patterns in the image stack. Central noise peak, due to the limited S/N ratio of the detected speckle pattern, has been removed and a cubic spline interpolation put instead of it.
Fig. 8.
Fig. 8. Spatio-temporal image cross-correlation of single bead speckle patterns. (a) Evolution of the spatio-temporal cross-correlation at different lag times. (b) Decay curve of the spatio-temporal cross-correlation amplitude at zero spatial shift, obtained from a single particle dynamic speckle. The gray solid line was obtained by fitting the data with the theoretical model of Eqs. (25) and (26), using fixed parameter values for $a_{\mathrm {exc}}$, $\omega _{\mathrm {exc}}$, $a_{\mathrm {det}}$ and $\omega _{\mathrm {det}}$ as given in the inset. (c) Evolution of the square of the FWHM of the cross-correlations depicted in (a) and comparison with a linear evolution corresponding to a MSD given by $4D\tau$.
Fig. 9.
Fig. 9. Spatio-temporal image cross-correlation of multiple bead speckles. The evolution of the temporal decay of the spatio-temporal cross-correlation at zero spatial shift as obtained from two, three, five, seven, and twelve beads dynamic speckle patterns along with their theoretical fits.
Fig. 10.
Fig. 10. Average mean square displacement (MSD) of single particle diffusion on water/air surface with lag time. Linear fit of the MSD provides the diffusion constant $D = 0.11 \pm 0.07$ $\mu$m$^{2}$/s. Shaded region corresponds to the standard deviation.
Fig. 11.
Fig. 11. Exemple of mean autocorrelation obtained from a stack of normalized instantaneous autocorrelations, produced by a single diffusing bead. Conceptually speaking, this is equivalent to the speckle autocorrelation due to a single immobile bead, as the one shown in Fig. 6.
Fig. 12.
Fig. 12. Spatio-temporal cross-correlation obtained without scatterer (i.e. without speckle patterns). (a) shows the temporal decay curve at zero spatial shift, obtained from multiply particle images. The grey solid line has been obtained by fitting the data with the theoretical model, using the input value of the $\omega _{det}$ parameter given in the inset. (b) shows the evolution of the square of the FWHM of the spatio-temporal cross-correlations.

Equations (28)

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G ( σ , τ ) = I ( ρ , t ) I ( ρ + σ , t + τ ) ρ , t I ( ρ , t ) ρ , t I ( ρ + σ , t + τ ) ρ , t
I ( ρ , t ) = i = 1 N P S F e x c [ ρ i ( t ) ] P S F d e t [ ρ i ( t ) ; ρ ρ i ( t ) ]
I ( ρ , t ) t = C P S F e x c ( ρ ) P S F d e t ( ρ ; ρ ρ ) d ρ
C V e x c , d e t ( ρ ) = 1 S P S F e x c ( ρ ) P S F d e t ( ρ ; ρ ρ ) d ρ
I ( ρ , t ) ρ , t = N C V e x c , d e t ( ρ ) ρ
I ( ρ , t ) ρ , t I ( ρ + σ , t + τ ) ρ , t = N 2 C V e x c , d e t ( ρ ) ρ C V e x c , d e t ( ρ + σ ) ρ
I ( ρ , t ) I ( ρ + σ , t + τ ) ρ , t = i , j N P S F e x c [ ρ i ( t ) ] P S F d e t [ ρ i ( t ) ; ρ ρ i ( t ) ] P S F e x c [ ρ j ( t + τ ) ] P S F d e t [ ρ j ( t + τ ) ; ρ + σ ρ j ( t + τ ) ] ρ , t
C c r o s s ( σ , τ ) = N ( N 1 ) S 2 P S F e x c ( ρ ) P S F d e t ( ρ ; ρ ρ ) d ρ ρ P S F e x c ( ρ ) P S F d e t ( ρ ; ρ + σ ρ ) d ρ ρ
C c r o s s ( σ , τ ) = N ( N 1 ) C V e x c , d e t ( ρ ) ρ C V e x c , d e t ( ρ + σ ) ρ
C d i a g ( σ , τ ) = C S P S F e x c ( ρ 1 ) P S F d e t ( ρ 1 ; ρ ρ 1 ) P S F e x c ( ρ 2 ) P S F d e t ( ρ 2 ; ρ ρ 2 + σ ) P ( ρ 2 ρ 1 , τ ) d ρ 1 d ρ 2 d ρ
d ρ P ( ρ ρ , τ ) = d ρ P ( ρ ρ , τ ) = 1
A C d e t ( ρ 1 , ρ 2 ; σ ρ 2 + ρ 1 ) = 1 S P S F d e t ( ρ 1 ; ρ ρ 1 ) P S F d e t ( ρ 2 ; ρ ρ 2 + σ ) d ρ
C d i a g ( σ , τ ) = C P S F e x c ( ρ 1 ) P S F e x c ( ρ 2 ) A C d e t ( ρ 1 , ρ 2 ; σ ρ 2 + ρ 1 ) P ( ρ 2 ρ 1 , τ ) d ρ 1 d ρ 2 d ρ
A C e x c ( ρ ) = 1 S P S F e x c ( ρ ) P S F e x c ( ρ + ρ ) d ρ
C d i a g ( σ , τ ) = N A C e x c ( ρ ) A C d e t ( ρ ; σ ρ ) P ( ρ , τ ) d ρ
G ( σ , τ ) = 1 N A C e x c ( ρ ) A C d e t ( ρ ; σ ρ ) P ( ρ , τ ) d ρ C V e x c , d e t ( ρ ) ρ C V e x c , d e t ( ρ + σ ) ρ + N 1 N
P S F d e t ( ρ ) = 1 + a d e t δ f d e t ( ρ )
A C d e t ( ρ ) = P S F d e t ( ρ ) P S F d e t ( ρ + ρ ) ρ = 1 + a d e t C d e t ( ρ )
C d e t ( ρ ) = 1 S δ f d e t ( ρ ) δ f d e t ( ρ + ρ ) d ρ
G ( σ , τ ) = 1 + 1 N [ a e x c C e x c ( ρ ) + a d e t C d e t ( σ ρ ) + a e x c a d e t C e x c ( ρ ) C d e t ( σ ρ ) ] P ( ρ , τ ) d ρ
C d e t ( ρ ) = exp ( ρ 2 ω d e t 2 )
P ( ρ , τ ) = 1 4 π D τ exp ( ρ 2 4 D τ )
G ( σ , τ ) = 1 + g ( σ , τ ) N
g ( σ , τ ) = a e x c 1 + τ / τ e x c + a d e t 1 + τ / τ d e t exp ( σ 2 / ω d e t 2 1 + τ / τ d e t ) + a e x c a d e t 1 + τ / τ M exp ( σ 2 ω d e t 2 1 + τ / τ e x c 1 + τ / τ M )
τ e x c = ω e x c 2 4 D τ d e t = ω d e t 2 4 D 1 τ M = 1 τ d e t + 1 τ e x c
G ( 0 , τ ) = 1 + 1 N ( a e x c 1 + τ / τ e x c + a d e t 1 + τ / τ d e t   + a e x c a d e t 1 + τ / τ M )
G ( σ , 0 ) N = 1 = ( 1 + a e x c ) [ 1 + a d e t exp ( σ 2 ω d e t 2 ) ]
G ( σ , τ ) = 1 + 1 N P S F exp ( σ 2 / ω d e t 2 1 + τ / τ d e t ) 1 + τ / τ d e t

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