Abstract

Imaging the propagation of light in time and space is crucial in optics, notably for the study of complex media. We demonstrate the passive measurement of time-dependent Green’s functions between every point at the surface of a strongly scattering medium by means of low coherence interferometry. 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. In particular, the spatio-temporal spreading of the diffusive halo and the coherent backscattering effect can be locally investigated in the vicinity of each point acting as a virtual source. On the one hand, this approach allows quantitative imaging of the diffusion constant in 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 study opens important perspectives both in optical diffuse tomography with potential applications to biomedical imaging and in fundamental physics for the experimental investigation of Anderson localization.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]

2016 (3)

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]

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).

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]

2015 (3)

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

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]

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]

2014 (3)

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]

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]

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]

2013 (1)

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]

2012 (2)

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]

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

2011 (1)

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]

2010 (2)

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

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]

2009 (3)

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

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]

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

2008 (4)

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

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]

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

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]

2007 (1)

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]

2006 (3)

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

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]

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]

2004 (3)

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]

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

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

2003 (4)

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

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]

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

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]

2002 (1)

2001 (1)

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

2000 (2)

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]

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]

1999 (2)

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]

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

1997 (1)

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

1996 (1)

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]

1991 (1)

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]

1989 (1)

1985 (2)

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]

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

1984 (1)

1958 (1)

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

Ackermann, M.

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]

Aegerter, C. M.

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]

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]

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]

Akkermans, E.

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

Albada, M. P. V.

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]

Alerstam, E.

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

Anderson, P. W.

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

Arridge, S. R.

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

Aubry, A.

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]

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]

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]

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]

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]

Aubry, G. J.

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]

Azizi, L.

Badon, A.

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]

Baker, W.

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

Barthelemy, P.

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]

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

Bartolini, P.

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

Beaurepaire, E.

Bertolotti, J.

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]

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

Boccara, A. C.

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]

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]

Boso, G.

Bührer, W.

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]

Burresi, M.

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).

Campillo, M.

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]

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

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Supplementary Material (1)

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Figures (4)

Fig. 1.
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 Sα(r,r+Δr,t) to extract the full-field interferogram C(r,r+Δr,t). (c) Examples of full-field interferograms acquired at different OPDs in the TiO2 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, gAB(t)gAB(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.
Fig. 2.
Fig. 2. Isotropic diffusion from a TiO2 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 Δx=0. (f) Measured spatio-temporal diffuse intensity for Δx=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 W2 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 m2·s1.
Fig. 3.
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 m2·s1.
Fig. 4.
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.

Equations (8)

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Sα(r,r+Δr,t)=0T|eiαψ(r,t+τ)+ψ(r+Δr,τ)|2dτ,
C(r,r+Δr,t)=1T0Tψ(r,t+τ)ψ*(r+Δr,τ)dτ.
tC(r,r+Δr,t)Tg(r,r+Δr,t)g(r,r+Δr,t),
I(Δr,t)={|tC(r,r+Δr,t)|2+|tC(r,r+Δr,t)|2}={|g(r,r+Δr,t)|2+|g(r,r+Δr,t)|2},
Iinc(Δr,t)=Iz(t).exp(Δr24Dt),
Icoh(Δr,t)=Iz(t)·exp(Δr2lc2).
IMS(Δr,t)IMS(0,t)=12[exp(Δr24Dt)+exp(Δr2lc2)].
δrDtmax,

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