Spatio-temporal imaging of light transport in highly scattering media under white light illumination

Amaury Badon; Dayan Li; Geoffroy Lerosey; A. Claude Boccara; Mathias Fink; Alexandre Aubry

doi:10.1364/OPTICA.3.001160

1. INTRODUCTION

Light is the most common probe for investigating complex media at the mesoscopic scale as it both offers an excellent resolution and is noninvasive at moderate energies. Nonetheless, due to the inhomogeneous distribution of refractive index, light suffers multiple scattering while propagating in or through the medium. Unveiling the complexity of light scattering is then necessary to retrieve the features of an object of interest or of the surrounding environment. In an inhomogeneous medium, it is a classical approach to consider a scattering sample as one realization of a random process, and study statistical physical quantities such as the mean intensity [1–3]. Under this approach, several physical parameters are relevant to characterize wave propagation in scattering media: the scattering mean-free path $l_{s}$ , the transport mean-free path $l_{t}$ , the diffusion constant $D$ , and the absorption length $l_{a}$ . Classical backscattering imaging techniques, such as optical coherence tomography, fail when multiple scattering predominates [4]. However, one can still measure the long-scale spatial variations of the diffusive parameters. The resulting image is not an image of the refractive index $n (r)$ but, e.g., of the diffusion constant $D (r)$ with a resolution of the order of the transport mean-free path, $l_{t}$ , at best. In the literature, diffuse optical tomography is the gold standard technique to reconstruct the spatial distribution of transport parameters at each point of a volume from intensity measurements at the surface [5]. Unfortunately, this inverse problem is intrinsically nonlinear with respect to the optical properties of the medium. This method is thus computationally intensive and limited in terms of spatial resolution [6–8] (e.g., 5 mm in human soft tissues).

In this paper, we propose a simple and efficient approach that does not require any inversion procedure and provides a direct image of transport parameters with a spatial resolution of the order of a few transport mean-free paths. Experimentally, it relies on a passive measurement of time-dependent Green’s functions between every point at the surface of a scattering medium without the use of any coherent source. This method is based on the following fundamental result: the cross-correlation (or mutual coherence function) of an incoherent wave field measured at two points $A$ and $B$ can yield the time-dependent Green’s function $g_{B A} (t)$ between these two points [9–12]. Previously developed in seismology [13–15], in acoustics [16,17], and in the microwave regime [18], the Green’s function estimation from the cross-correlations of a diffuse wave field has recently been extended to optics [19]. A proof-of-concept experiment has demonstrated the passive measurement of time-dependent Green’s functions between individual scatterers using low coherence interferometry. The case of a strongly scattering medium has also been briefly discussed by measuring the autocorrelation signal at one point $A$ of the medium surface. It was shown to converge toward the self-Green’s function $g_{A A} (t)$ associated with a virtual sensor placed at point $A$ .

In this paper, this approach is generalized to the passive measurement of time-dependent Green’s functions $g_{A B} (t)$ between any point $A$ and $B$ at the surface of a scattering medium. We show in particular how a simple Michelson interferometer allows us to simultaneously acquire millions of time-dependent Green’s functions under a fully incoherent white light illumination. This nontrivial result means that any point illuminated by a broadband and incoherent light can virtually become a coherent point source as if a femtosecond micro-laser was placed at its location. Reciprocally, it can also become a micro-sensor able to measure both the amplitude and phase of light at the femtosecond time scale. Hence, one can measure the impulse response between any of these virtual micro-antennas. This set of responses, which we will refer to as the Green’s matrix, contains a wealth of information and provides a unique signature of the complicated trajectories experienced by light in the medium. Its measurement is decisive in many applications, in particular for optical imaging and the characterization of complex media.

Here, to demonstrate the potential of our approach, we take advantage of the Green’s matrix to investigate the spatio-temporal evolution of the intensity in the vicinity of each point acting as a virtual source. This experimental scheme is first applied to the study of a thick layer of ${TiO}_{2}$ nanoparticles, statistically homogenous in terms of disorder. The spatial intensity profile shows two contributions: (i) an incoherent background that results from the incoherent summation of all the multiple scattering paths that the wave can follow within the scattering medium and (ii) a coherent contribution that results from the constructive interference between reciprocal multiple scattering paths, an interference phenomenon known as coherent backscattering [20–22]. The first component directly accounts for the growth of the diffusive halo within the scattering medium, whereas the second component results in a coherent backscattering peak at the source location [23,24]. A fit of the incoherent background with a simple diffusion model allows an estimation of the diffusion constant, independently from the absorption losses. The case of a heterogeneous layer made of ZnO nanoparticles is tackled in a second part. The growth of the diffusive halo is investigated over sliding overlapping areas of the scattering medium [24,25]. It yields an image of the diffusion constant that is shown to be inversely proportional to the concentration of nanoparticles. In the last part, the spatio-temporal evolution of light transport is investigated in a Teflon tape sample. Our approach reveals the anisotropic growth of the diffusive halo induced by the orientation of fibers in the scattering medium.

2. PASSIVE MEASUREMENT OF TIME-DEPENDENT GREEN’S FUNCTIONS

The experimental setup employed for the passive measurement of the point-to-point Green’s functions at the surface of a scattering sample is displayed in Fig. 1. An incoherent broadband light source (650–850 nm) isotropically illuminates a scattering sample. This incident wave field exhibits a coherence time $τ_{c} = 10 fs$ and a coherence length $l_{c} = 3 μm$ [19]. The backscattered wave field is collected by a microscope objective and sent to a Michelson interferometer, which is used here as a spatio-temporal field correlator. The beams coming from the two interference arms are recombined and focused by a lens. A CCD camera conjugated with the sample surface records the output intensity:

S_{α} (r, r + Δ r, t) = \int_{0}^{T} {| e^{i α} ψ (r, t + τ) + ψ (r + Δ r, τ) |}^{2} d τ,

with

τ

the absolute time,

r

the position vector on the CCD screen,

ψ (r, τ)

the scattered wave field associated with the first interference arm,

T

the integration time of the CCD camera, and

α

an additional phase term controlled with a piezoelectric actuator placed on mirror

M_{1}

. The tilt of mirror

M_{2}

allows a displacement

Δ r

of the associated wave field on the CCD camera. The motorized translation of mirror

M_{1}

induces a time delay

t = δ / c

between the two interferometer arms, with

δ

the optical path difference (OPD) and

c

the speed of light. The interference term is extracted from the four intensity patterns [Eq. (1)] recorded at

α = 0

,

π / 2

,

π

, and

3 π / 2

, as depicted in Fig. 1(b) (“four phase method” [26]). It directly yields the correlation

C

of the scattered wave field

ψ

:

C (r, r + Δ r, t) = \frac{1}{T} \int_{0}^{T} ψ (r, t + τ) ψ^{*} (r + Δ r, τ) d τ .

If the incident light is spatially and temporally incoherent, the time derivative of the correlation function

C (r, r + Δ r, t)

should converge toward the difference of the causal and anticausal Green’s function [19], such that

\partial_{t} C (r, r + Δ r, t) \underset{T \to \infty}{\sim} g (r, r + Δ r, t) - g (r, r + Δ r, - t),

where

g (r, r + Δ r, t)

is the time-dependent causal Green’s function between a point source at

r

and a point detector

r + Δ r

. This fundamental result means that a passive correlation measurement mimics the following active experiment [see Fig. 1(d)]: (i) a virtual light source of size

l_{c}

located at point

r

emits a pulse of duration

τ_{c}

; (ii) a virtual point-like receiver located at point

r + Δ r

records the time-resolved scattered wave field. This property has been recently demonstrated in optics by retrieving the ballistic and multiple scattering components of the Green’s function between individual scatterers [19]. Following this experimental proof-of-concept, we investigate the case of strongly scattering media.

Fig. 1. Experimental setup and principle of measurement. (a) Experimental setup: a broadband incoherent light source isotropically illuminates a scattering sample. The spatio-temporal correlation of the scattered wave field is extracted by means of a Michelson interferometer and recorded by a CCD camera. HL, halogen lamp; MO, microscope objective; BS, beam splitter; M, mirror; PZT, piezoelectric actuator. (b) Application of the “four phase method” to the measured intensity patterns $S_{α} (r, r + Δ r, t)$ to extract the full-field interferogram $C (r, r + Δ r, t)$ . (c) Examples of full-field interferograms acquired at different OPDs in the ${TiO}_{2}$ layer. (d) Example of time derivative of the autocorrelation function $C (r, r, t)$ acquired at a point $r$ of the CCD camera [marked with the white square in (c)]. (e) Principle of our approach: the time derivative of the correlation signal between two points $A$ and $B$ directly provides the difference between the causal and anticausal Green’s functions, $g_{A B} (t) - g_{A B} (- t)$ , between these two points. This passive measurement mimics an active experiment in which a virtual point-like pulsed source is introduced at point $A$ and a virtual detector records the time-resolved scattered wave field at point $B$ .

Download Full Size | PDF

The first sample under study is an 80-μm-thick layer of ${TiO}_{2}$ nanoparticles. A set of time-dependent causal and anticausal Green’s functions, $g (r, r + Δ r, t) - g (r, r + Δ r, - t)$ , has been measured following the experimental procedure described above and depicted in Fig. 1. The time derivative of the mutual coherence function, $\partial_{t} C (r, r + Δ r, t)$ , is estimated from the finite difference of the mutual coherence function measured at time $t \pm Δ t / 2$ , with a time step $Δ t = 0.66 fs$ . As the bandwidth of the light source spans from $λ_{\min} = 400 nm$ to $λ_{\max} = 950 nm$ , the Nyquist criterion is fulfilled ( $Δ t < λ_{\min} / 2 c$ ) and the finite time difference approximation is valid. Figure 1(d) displays an example of $\partial_{t} C (r, r, t)$ in Fig. 1(d) for an integration time $T = 750 ms$ . The convergence of $\partial_{t} C (r, r, t)$ toward the difference between the causal and anticausal Green’s function [Eq. (3)] is investigated in Supplement 1. Here, we have access to a satisfying estimation of the Green’s function over 100 μm in terms of OPD, which corresponds to scattering path lengths of approximately 13 $l_{t}$ (see Section 3.A). Therefore, the recorded signal contains information on the propagation of light deep into the diffusive regime [27]. As illustrated by Fig. 1(d), it actually exhibits a long tail due to the numerous scattering events experienced by the wave during its propagation in the medium.

3. SPATIO-TEMPORAL IMAGING OF LIGHT TRANSPORT

A. Homogeneously Disordered Scattering Medium

When one tries to image an unknown medium from a set of Green’s functions, an important issue is the importance of multiple scattering relative to single scattering. Multiple scattering is actually a nightmare for classical imaging techniques, which are based on the first Born approximation. However, one can adopt a probabilistic approach and investigate statistical quantities such as the mean intensity [1]. As the causal and anticausal Green’s functions are, by definition, zero for $t < 0$ and $t > 0$ , respectively, the mean intensity can be averaged over the positive and negative times, such that

I (Δ r, t) = {⟨ {| \partial_{t} C (r, r + Δ r, t) |}^{2} ⟩ + ⟨ {| \partial_{t} C (r, r + Δ r, - t) |}^{2} ⟩} = {⟨ {| g (r, r + Δ r, t) |}^{2} ⟩ + ⟨ {| g (r, r + Δ r, - t) |}^{2} ⟩},

where the symbol

⟨ \cdot ⟩

denotes an average over the CCD pixels, i.e., over the position vector

r

of the virtual source on the sample surface. As we shall see, the evolution of this quantity as a function of the source–receiver relative position

Δ r

and the time of flight

t

allows us to quantitatively describe the transport of light through the scattering medium under study.

Figure 2 displays the spatio-temporal evolution of the mean intensity deduced from the set of point-to-point Green’s functions measured in the ${TiO}_{2}$ layer. Note that the noise background is priorly subtracted from the measured intensity (see Supplement 1). Each spatial intensity profile is then renormalized by its maximum at each time of flight $t$ . Figures 2(b)–2(d) display the result at different OPDs ( $δ = 10$ , 50, and 100 μm, respectively). Not surprisingly, we retrieve the feature of a diffusive halo whose spatial extent increases with time at the same speed in all directions. This isotropic growth is easily accounted for by the fact that the ${TiO}_{2}$ nanoparticles are small compared to the wavelength, are randomly packed, and display an isotropic differential cross-section.

Fig. 2. Isotropic diffusion from a ${TiO}_{2}$ strongly scattering layer. (a) Sketch of the spatio-temporal diffuse intensity and its 2D spatial sections for different OPDs or, equivalently, times of flight. (b)–(d) Measured spatial distribution of the mean intensity at different OPDs ( $δ = 10$ , 50, and 100 μm, respectively). Each intensity map has been renormalized with its maximum. (e) Sketch of the spatio-temporal diffuse intensity and its section for $Δ x = 0$ . (f) Measured spatio-temporal diffuse intensity for $Δ x = 0$ . The intensity is renormalized by its maximum at each time of flight. (g) Spatial intensity profile (blue disks) obtained at $δ = 40 μm$ fitted with the sum of two Gaussian curves (continuous black line) that account for the incoherent intensity (blue dashed line) and the coherent backscattering peak (red dashed line). (h) Time evolution of the mean-square $W^{2}$ of the diffusive halo (blue squares). A linear fit (green dashed line) over the OPD range (20–100 μm) (green arrows) yields an estimation of the diffusion constant: $D = 730$ $m^{2} \cdot s^{- 1}$ .

Download Full Size | PDF

Now that we have qualitatively observed the diffusive spreading of the mean intensity, we aim to precisely characterize the scattering properties of the medium with a quantitative measurement of the diffusion constant $D$ . Theoretically, the multiple scattering intensity can be split into two terms. First, the incoherent contribution, denoted as $I_{inc} (Δ r, t)$ , corresponds to the interference of the wave with itself. In the deep multiple scattering regime, this term obeys the diffusion equation (see Supplement 1). In the time domain, the corresponding reflection profile has been derived by Patterson et al. [28]:

I_{inc} (Δ r, t) = I_{z} (t) . \exp (- \frac{Δ r^{2}}{4 D t}),

with

D

the diffusion constant and

I_{z} (t)

a physical quantity that depends on the sample thickness

L

, the transport mean-free path

l_{t}

, and the absorption length

l_{a}

, but not on the source-receiver distance

Δ r

(see Supplement 1). Second, the coherent contribution, denoted

I_{coh} (Δ r, t)

, corresponds to the interference of the wave with its reciprocal counterpart [20–22] and is expressed as [25]

I_{coh} (Δ r, t) = I_{z} (t) \cdot \exp (- \frac{Δ r^{2}}{l_{c}^{2}}) .

At a given time of flight, the normalized multiple scattering intensity is thus given by

\frac{I_{MS} (Δ r, t)}{I_{MS} (0, t)} = \frac{1}{2} [\exp (- \frac{Δ r^{2}}{4 D t}) + \exp (- \frac{Δ r^{2}}{l_{c}^{2}})] .

In our configuration, the spatial distribution of the intensity at a given time has the following shape [see Fig. 2(g)]: a narrow, steep peak (the coherent contribution), on top of a wider pedestal that widens with time (the incoherent contribution). It is worth noting that due to the normalization process, the multiple scattering intensity profile does not depend on the absorption or the thickness of the sample. By fitting this intensity profile at each time of flight with two Gaussian curves, one can separate the coherent and incoherent contributions [see Fig. 2(g)]. The width

W

of the incoherent background directly accounts for the spatial extent of the diffusive halo. Figure 2(d) displays the evolution of

W^{2}

versus time. From Eq. (7), we see that the slope of

W^{2}

versus time should be equal to

2 D

. The linear fit of

W^{2}

in Fig. 2(d) allows us to measure the diffusion constant in the

{TiO}_{2}

scattering sample: we find

D = 730 m^{2} \cdot s^{- 1}

. Note that the linear fit is applied from an OPD of 20 μm, which corresponds to a typical penetration depth of more than 3

l_{t}

. Before this time, the diffusion approximation is not valid yet and the evolution of

W^{2}

is not linear.

In Supplement 1, the measured value of $D$ is compared to the result of a more conventional time-of-flight experiment, which, unlike our approach, is sensitive to absorption losses. The combination of both methods allows us to quantitatively estimate the absorption length: $l_{a} ≃ 118 μm$ . Our measurement is in qualitative agreement with transmission time-of-flight measurements performed in other ${TiO}_{2}$ samples [29]. An estimation of the transport mean-free path $l_{t}$ is also possible using the relation $D = v_{e} l_{t} / 3$ , with $v_{e}$ the energy velocity. The size of the nanoparticles being small compared to the wavelength and the medium being diluted, we can consider the energy velocity $v_{e}$ to be close to the vacuum speed of light $c$ [30]. Thus, we obtain the following qualitative estimation for the transport mean-free path: $l_{t} \sim 7.5 μm$ .

The access to transport parameters from the correlation of a diffuse wave field is a precious piece of information for the optical characterization of a scattering layer. However, this measurement is averaged over the whole field-of-view (FOV) and does not provide any local information on disorder. This issue is tackled in the next subsection.

B. Heterogeneously Disordered Scattering Medium

Until now, the disorder was statistically homogeneous in the scattering sample under study. We now investigate the case of a medium with a space-dependent concentration of scatterers. For this purpose we have synthesized a heterogeneously disordered layer of ZnO nanoparticles. Variations in the concentration of scatterers are visible at a scale smaller than 100 μm in the microscopic image [see Fig. 3(a)]. Cross-correlations of the diffuse wave field are recorded using the procedure described in Section 2. A satisfying estimation of the Green’s function is found over 130 μm in terms of OPD, and the noise background is subtracted from the measured mean intensity prior to data analysis. The spatial extent $W$ of the diffusive halo is estimated at each time of flight by fitting the normalized multiple scattering intensity profile [Eq. (7)] with two Gaussian curves. First, the mean intensity is averaged over all the active pixels of the CCD. A linear fit of $W^{2}$ versus time [Fig. 3(b)] allows an estimation of the diffusion constant averaged over a FOV of $5 {mm}^{2} : ⟨ D (r) ⟩ = 1425 m^{2} \cdot s^{- 1}$ . The FOV being at least one order of magnitude larger than the typical scale at which the concentration of scatterers fluctuates, such a measurement does not provide a satisfying characterization of the disorder in the scattering sample.

Fig. 3. Imaging of the diffusion constant in a scattering layer heterogeneous in disorder. (a) Microscopic image of the ZnO scattering layer. (b) Time evolution of the mean-squared width of the diffusive halo averaged over the whole sample surface (black squares), over areas surrounded by the blue dashed square (blue diamonds) and the red dashed square (red circles) in (a). A linear fit of each experimental data set (dashed lines) over the OPD range (40–130 μm) yields an estimation of the diffusion constant in the corresponding areas. (c) Superimposition of the measured diffusion constant map onto the sample image. The color scale is in $m^{2} \cdot s^{- 1}$ .

Download Full Size | PDF

To cope with this issue, a local approach is needed. Instead of considering the mean intensity over the whole sample surface, one can average the intensity over sub-areas of the CCD camera. For instance, one can consider sub-areas delineated with red and blue squares in Fig. 3(a) that display low and high concentrations of scatterers, respectively. The corresponding diffusive halos show a contrasted speed of expansion [see Fig. 3(b)]. Quantitatively, a different diffusion constant is measured in both areas. We find $D = 1125 m^{2} \cdot s^{- 1}$ in the blue area and $D = 2770 m^{2} \cdot s^{- 1}$ in the red area. This is in agreement with the fact that a higher concentration of scatterers implies a smaller diffusion constant. $D$ is actually proportional to the transport mean-free path $l_{t}$ , which itself scales as the inverse of the concentration of scatterers [30]. Hence, the diffusion contrast highlighted by Fig. 3(b) indicates that the blue area is twice and half more concentrated than the red one.

A further development is to build an image of the medium from local measurements of the diffusion constant. This is done by considering the mean intensity over sliding windows of $100 μm \times 100 μm$ . The spatial extent $W^{2}$ of the diffusive halo is fitted linearly over time and yields an estimation of the diffusion constant at the center of each window. The map of the diffusion constant is superimposed to the reflectivity image of the scattering medium in Fig. 3(c). Qualitative agreement is found between the two images since the diffusion constant is larger in areas where the concentration in scatterers is lower. However, both images bring different information since the microscope image is only related to the concentration of scatterers at the surface of the medium, whereas the diffusion constant also depends on the nature of disorder below the surface. A three-dimensonal (3D) image could thus be built from a passive Green’s function retrieval, but it would require inversion schemes like in optical diffuse tomography [5].

The spatial resolution $δ r$ of the image displayed in Fig. 3(c) is basically limited by the transverse spreading of the diffusive halo, such that

δ r \sim \sqrt{D t_{\max}},

where

t_{\max}

is the largest time delay investigated in the data analysis. Here

t_{\max} = 450 fs

; hence

δ r

is of the order of 25 μm. Considering a range of shorter time delays would improve the resolution but, in the meantime, would limit the precision of our measurement. Note that the resolution of the image displayed in Fig. 3(c) is also limited by the size of the spatial sliding window. A better resolution would be obtained if a smaller sliding window was considered. But, in this case, the average of the intensity would not be satisfying. Residual fluctuations of the intensity pattern would be too high because of the lack of average over disorder configurations. Consequently, a compromise has to be found between the size of the sliding window (i.e., the image resolution) and a sufficient average over disorder configurations (i.e., the signal-to-noise ratio).

C. Anisotropic Scattering Medium

Until now, we have only considered the case of isotropic scattering media. However, many complex samples such as biological tissues [31], nematic crystals [32,33], porous materials [34], or fibrous media [35] give rise to anisotropic light diffusion. In such media, the scalar diffusion constant should be replaced by a diffusion tensor (see Supplement 1). Providing that the principal directions of anisotropy are aligned with the Cartesian coordinates, this tensor is diagonal [36]. The three nonzero coefficients $D_{x x}$ , $D_{y y}$ , and $D_{z z}$ correspond to the diffusion constant along the directions $x$ , $y$ , and $z$ , respectively. To illustrate the ability of our approach in measuring different components of the diffusion tensor, we now investigate light diffusion in a piece of thread seal tape (Polytetrafluoroethylene), which is made of fibers aligned along the $x$ direction [see Fig. 4(a)]. Strong forward scattering arises along the normal to the fibers’ direction. Hence, light diffusion is supposed to be slower along the fibers: $D_{x x} < D_{y y}$ . To check that hypothesis, the experimental procedure described in Section 2 is performed. The measured Green’s matrix allows us to investigate the spatio-temporal evolution of the mean intensity [Eq. (4)] averaged over the whole FOV. Figure 4 displays the diffusive halo in the $x - y$ plane at different times of flight. The intensity distribution clearly presents an elongated shape along the $y$ direction. As expected, light diffusion is thus slower along the $x$ direction. This confirms the anisotropy of the sample and the ability of our approach to reveal it.

Fig. 4. Anisotropic diffusion in a stretched Teflon tape. (a) Microscopic image of the sample. (b)–(d) Spatial intensity distribution recorded at different OPDs ( $δ = 20$ , 30, and 50 μm, respectively). Each intensity map has been renormalized by its maximum.

Download Full Size | PDF

In Supplement 1, the growth of the diffusive halo is quantitatively investigated in both the $x$ and $y$ directions. The components of the diffusion tensor are estimated following the procedure described in Section 3.A. We find $D_{x x} = 1100 m^{2} \cdot s^{- 1}$ and $D_{y y} = 3000 m^{2} \cdot s^{- 1}$ . An anisotropy factor $g$ can be deduced from the measurement of the diffusion tensor in the $x - y$ plane [37], such that $g = 1 - (D_{x x} / D_{y y}) = 0.63$ . The anisotropy of diffusion is thus successfully quantified by our approach.

4. DISCUSSION

The first point we would like to emphasize is the potential impact of our approach in optical diffuse tomography. In contrast with most studies in the literature that only involve intensity measurements between a limited number of active sources and detectors, our method allows us to passively measure a whole set of time-dependent responses between every point at the surface of a scattering medium. Moreover, compared to similar recent experiments performed in transmission with a coherent illumination scheme [38,39], backscattering measurements open the possibility to address the very first scattering events that may reveal fine structural features hardly accessible otherwise. The access to short times of flight enables a local study of the growth of the diffusive halo and a direct two-dimensional (2D) imaging of transport parameters with a resolution not given by the thickness of the sample, as it would be in transmission, but of the order of the transport mean-free path. Finally, our approach is by no means limited to 2D imaging. One perspective of this work is actually to apply inversion schemes of the diffusion or radiative transfer equations [40] to recover a 3D image of the scattering medium.

A passive measurement of the Green’s matrix can also enable a great leap forward in fundamental physics. In this work, we have shown, for instance, how the anisotropy of light diffusion can be revealed and quantified. A further step would be to investigate exotic transport phenomena such as super- and sub-diffusion of light [41,42] or, more ambitiously, to provide a direct proof for Anderson localization [43,44]. Past experimental studies of Anderson localization in optics have been ambiguous, due to either absorption effects [45,46] or fluorescence issues [29]. In contrast, our approach would remove any possible ambiguity. Space- and time-dependent experiments can enable absorption-independent observations of localization [47]. Moreover, a white light illumination in a backscattering configuration limits the incident power required for a satisfactory signal-to-noise ratio, contrary to a transmission configuration that implies a much less energetic signal through a thick scattering layer ( $L ≫ l_{t}$ ). Our experimental scheme would thus prevent fluorescence or nonlinear phenomena [29]. It also allows us to investigate phenomena such as coherent backscattering [48,49] or recurrent scattering [50], which constitute relevant observables at the onset of Anderson localization.

5. CONCLUSION

In summary, we have proposed a novel approach to investigate light transport through complex media. Inspired by previous studies in ultrasound and seismology, we have shown how a set of time-dependent Green’s functions can be passively extracted between every point at the surface of a scattering medium placed under white light illumination. The experimental access to this Green’s matrix is essential since it contains all the information about the complex trajectories of light within the medium. Adopting a probabilistic and local approach, one can, for instance, investigate the spatio-temporal expansion of the diffusive halo in different parts of the sample. On the one hand, this allows 2D imaging of the diffusion constant at the surface of the scattering medium, with a spatial resolution of the order of a few transport mean-free paths. On the other hand, our approach is able to reveal and quantify the anisotropy of light diffusion, which can be of great interest for optical characterization purposes. This work opens important perspectives both in optical diffuse tomography with potential applications to biomedical imaging and in fundamental physics with the quest for an experimental demonstration of Anderson localization in 3D.

Funding

European Research Council (ERC) (ERC Synergy HELMHOLTZ); Agence Nationale de la Recherche (ANR) (ANR-14-CE26-0032 LOVE, ANR-10-IDEX-0001-02 PSL*, ANR-10-LABX-24, LABEX WIFI); High Council for Scientific and Technological Cooperation between France and Israel (P2R Israel 29704SC); Direction Générale de l’Armement (DGA).

Acknowledgment

We thank O. Katz for fruitful discussions.

See Supplement 1 for supporting content.

REFERENCES

1. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer, 2006).

2. M. C. W. Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves,” Rev. Mod. Phys. 71, 313–371 (1999). [CrossRef]

3. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

4. C. Dunsby and P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D 36, R207–R227 (2003). [CrossRef]

5. T. Durduran, R. Choe, W. Baker, and A. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73, 076701 (2010). [CrossRef]

6. A. Puszka, L. Di Sieno, A. Dalla Mora, A. Pifferi, D. Contini, A. Planat-Chrétien, A. Koenig, G. Boso, A. Tosi, L. Hervé, and J.-M. Dinten, “Spatial resolution in depth for time-resolved diffuse optical tomography using short source-detector separations,” Biomed. Opt. Express 6, 1–10 (2015). [CrossRef]

7. L. Azizi, K. Zarychta, D. Ettori, E. Tinet, and J.-M. Tualle, “Ultimate spatial resolution with diffuse optical tomography,” Opt. Express 17, 12132–12144 (2009). [CrossRef]

8. A. B. Konovalov and V. V. Vlasov, “Theoretical limit of spatial resolution in diffuse optical tomography using a perturbation model,” Quantum Electron. 44, 239–246 (2014). [CrossRef]

9. B. A. van Tiggelen, “Green function retrieval and time reversal in a disordered world,” Phys. Rev. Lett. 91, 243904 (2003). [CrossRef]

10. K. Wapenaar, “Retrieving the elastodynamic Green’s function of an arbitrary inhomogeneous medium by cross correlation,” Phys. Rev. Lett. 93, 254301 (2004). [CrossRef]

11. K. Wapenaar, E. Slob, and R. Snieder, “Unified Green’s function retrieval by cross correlation,” Phys. Rev. Lett. 97, 234301 (2006). [CrossRef]

12. R. Snieder, “Extracting the Green’s function from the correlation of coda waves: a derivation based on stationary phase,” Phys. Rev. E 69, 046610 (2004). [CrossRef]

13. C. Weller, “Seismic exploration method,” U.S. patent 3,812,457 (May 21, 1974).

14. M. Campillo and A. Paul, “Long-range correlations in the diffuse seismic coda,” Science 299, 547–549 (2003). [CrossRef]

15. E. Larose, L. Margerin, A. Derode, B. A. van Tiggelen, M. Campillo, N. Shapiro, A. Paul, L. Stehly, and M. Tanter, “Correlation of random wavefields: an interdisciplinary review,” Geophysics 71, SI11–SI21 (2006). [CrossRef]

16. R. L. Weaver and O. I. Lobkis, “Ultrasonics without a source: thermal fluctuation correlations at MHz frequencies,” Phys. Rev. Lett. 87, 134301 (2001). [CrossRef]

17. A. Derode, E. Larose, M. Campillo, and M. Fink, “How to estimate the Green’s function of a heterogeneous medium between two passive sensors? Application to acoustic waves,” Appl. Phys. Lett. 83, 3054–3056 (2003). [CrossRef]

18. M. Davy, M. Fink, and J. de Rosny, “Green’s function retrieval and passive imaging from correlations of wideband thermal radiations,” Phys. Rev. Lett. 110, 203901 (2013). [CrossRef]

19. A. Badon, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Retrieving time-dependent Green’s functions in optics with low-coherence interferometry,” Phys. Rev. Lett. 114, 023901 (2015). [CrossRef]

20. Y. Kuga and A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984). [CrossRef]

21. M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985). [CrossRef]

22. P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985). [CrossRef]

23. E. Larose, L. Margerin, B. A. van Tiggelen, and M. Campillo, “Weak localization of seismic waves,” Phys. Rev. Lett. 93, 048501 (2004). [CrossRef]

24. A. Aubry, A. Derode, and F. Padilla, “Local measurements of the diffusion constant in multiple scattering media: application to human trabecular bone imaging,” Appl. Phys. Lett. 92, 124101 (2008). [CrossRef]

25. A. Aubry and A. Derode, “Ultrasonic imaging of highly scattering media from local measurements of the diffusion constant: separation of coherent and incoherent intensities,” Phys. Rev. E 75, 026602 (2007). [CrossRef]

26. A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High-resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Opt. 41, 805–812 (2002). [CrossRef]

27. Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999). [CrossRef]

28. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989). [CrossRef]

29. T. Sperling, L. Schertel, M. Ackermann, G. J. Aubry, C. M. Aegerter, and G. Maret, “Can 3D light localization be reached in ‘white paint’?” New J. Phys. 18, 013039 (2016). [CrossRef]

30. M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991). [CrossRef]

31. S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Krämer, and M. S. Patterson, “Anisotropy of light propagation in human skin,” Phys. Med. Biol. 45, 2873–2886 (2000). [CrossRef]

32. D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, “Time-resolved experiments on light diffusion in anisotropic random media,” Phys. Rev. E 62, 6681–6687 (2000). [CrossRef]

33. B. A. van Tiggelen, R. Maynard, and A. Heiderich, “Anisotropic light diffusion in oriented nematic liquid crystals,” Phys. Rev. Lett. 77, 639–642 (1996). [CrossRef]

34. P. Johnson, S. Faez, and A. Lagendijk, “Full characterization of anisotropic diffuse light,” Opt. Express 16, 7435–7446 (2008). [CrossRef]

35. E. Simon, P. Krauter, and A. Kienle, “Time-resolved measurements of the optical properties of fibrous media using the anisotropic diffusion equation,” J. Biomed Opt. 19, 075006 (2014). [CrossRef]

36. S. D. Konecky, T. Rice, A. J. Durkin, and B. J. Tromberg, “Imaging scattering orientation with spatial frequency domain imaging,” J. Biomed Opt. 16, 126001 (2011). [CrossRef]

37. E. Alerstam and T. Svensson, “Observation of anisotropic diffusion of light in compacted granular porous materials,” Phys. Rev. E 85, 040301 (2012). [CrossRef]

38. L. Pattelli, R. Savo, M. Burresi, and D. S. Wiersma, “Spatio-temporal visualization of light transport in complex photonic structures,” Light Sci. Appl. 5, e16090 (2016).

39. T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Photonics 7, 48–52 (2012). [CrossRef]

40. S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009). [CrossRef]

41. J. Bertolotti, K. Vynck, L. Pattelli, P. Barthelemy, S. Lepri, and D. S. Wiersma, “Engineering disorder in superdiffusive levy glasses,” Adv. Funct. Mater. 20, 965–968 (2010). [CrossRef]

42. P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453, 495–498 (2008). [CrossRef]

43. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958). [CrossRef]

44. A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2009). [CrossRef]

45. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997). [CrossRef]

46. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006). [CrossRef]

47. H. Hu, A. Strybulevych, J. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys. 4, 945–948 (2008). [CrossRef]

48. S. Ghosh, D. Delande, C. Miniatura, and N. Cherroret, “Coherent backscattering reveals the Anderson transition,” Phys. Rev. Lett. 115, 200602 (2015). [CrossRef]

49. L. A. Cobus, S. E. Skipetrov, A. Aubry, B. A. van Tiggelen, A. Derode, and J. H. Page, “Anderson mobility gap probed by dynamic coherent backscattering,” Phys. Rev. Lett. 116, 193901 (2016). [CrossRef]

50. A. Aubry, L. A. Cobus, S. E. Skipetrov, B. A. van Tiggelen, A. Derode, and J. H. Page, “Recurrent scattering and memory effect at the Anderson localization transition,” Phys. Rev. Lett. 112, 043903 (2014). [CrossRef]

Spatio-temporal imaging of light transport in highly scattering media under white light illumination

Abstract

1. INTRODUCTION

2. PASSIVE MEASUREMENT OF TIME-DEPENDENT GREEN’S FUNCTIONS

3. SPATIO-TEMPORAL IMAGING OF LIGHT TRANSPORT

A. Homogeneously Disordered Scattering Medium

B. Heterogeneously Disordered Scattering Medium

C. Anisotropic Scattering Medium

4. DISCUSSION

5. CONCLUSION

Funding

Acknowledgment

REFERENCES

Supplementary Material (1)

Cited By

Figures (4)

Equations (8)

Optica