Recurrent scattering and memory effect at the Anderson transition

Transcription

1 Recurrent scattering and memory effect at the Anderson transition Alexandre Aubry Institut Langevin CNRS UMR 7587, ESPCI ParisTech Paris, France Collaborators: John Page, Laura Cobus (University of Manitoba, Winnipeg, Canada) Sergey Skipetrov, Bart van Tiggelen (LPMMC, Grenoble, France) Arnaud Derode (Institut Langevin, Paris, France)

2 Introduction Experimental flexibility of ultrasound Multi-element array Time-resolved measurement of the amplitude and phase of the wave Numerous applications Ultrasound Imaging: Medical diagnosis Non destructive evaluation Focusing: Medical therapy Telecommunications Time reversal: Imaging and focusing through complex media Interest of a matrix approach Acquisition of the inter-element matrix K. Source i Array of transducers This matrix contains all the information available on the medium under investigation Receiver j k ij

3 Introduction Matrix K in «simple» media C. Prada and M. Fink, Wave Motion, 20: , 1994 Singular value decomposition of K : Diagonal matrix containing the N singular values ( 1 > 2 > > N ) U and V : Unitary matrices whose columns are the singular vectors Simple media: one eigenstate one scatterer 1 =reflectivity E=V 1 E=V 2 2 =reflectivity Each eigenvector V i back-propagates respectively towards each scatterer of the medium

4 Introduction Matrix K in complex media? Practical interest: Interest for detection and imaging (separation single / multiple scattering) Characterization of scattering media Fundamental interest: Link with random matrix theory Study of Anderson localization (recurrent scattering)

5 Introduction One relevant parameter: the scattering mean free path l e Single scattering (t <<l e /c) Classical imaging techniques (ultrasound imaging, D.O.R.T method, Kirchhoff migration) Multiple scattering (t >>l e /c) Nightmare for imaging Statistical approach, diffusion equation Measurements of transport parameters (l e, D ) Strong interference effects halt the wave within the scattering medium Return probability, recurrent scattering

6 Introduction Which frequency? Ultrasound 1 5 MHz Which medium, which scattering regime? Ballistic regime Diffusive regime Anderson localization Disorder Soft tissues Bone Coarse grain steels Mesoglasses 1

7 Outline Propagation operator in weakly scattering media Link with random matrix theory Single/multiple scattering Memory effect Propagation operator in strongly disordered media Memory effect Recurrent scattering Coherent backscattering Return probability

8 Propagation operator in weakly scattering media A. Aubry, A. Derode, Phys. Rev. Lett., 2009 J. Appl. Phys., 2009 WRCM, 2010 J. Acoust. Soc. Am., 2011

9 Experimental procedure 1/ Acquisition of the inter-element matrix 2/ Time-frequency analysis Keep the temporal resolution provided by ultrasonic measurements while working in the Fourier domain time frequency

10 Statistical properties of K / multiple scattering Experiment in a multiple scattering medium Real part of K Matrix K (32 32) Random distribution of scatterers K(t,f) = random matrix SVD No equivalence between eigenstates and scatterers of the medium Quarter circle law Numerical simulations Experiment Statistical approach Distribution of singular values = Quarter circle law V. Marcenko and L. Pastur, Math. USSR-Sbornik 1, 457, 1967

11 Statistical properties of K / single scattering Experiment in a single scattering medium Soft tissues In the single scattering regime, we are far from the expected quarter circle law Single scattering Multiple scattering Deterministic coherence along the antidiagonals of K Random feature in the multiple scattering regime

12 Optical memory effect in backscattering Speckle is not random at it might seems s a s a' s a W W s b Scattering medium s b' s b Scattering medium Small rotation of the incident field Small rotation of the speckle Despite disorder, it remains an information on the nature of the incident beam. Single scattering: the memory effect persists over the whole angular domain Multiple scattering: The memory effect is restricted to a small angular domain Freund et al., Phys. Rev. Lett., 1988 Feng et al., Phys. Rev. Lett., 1988

13 Memory effect in backscattering Far-field s a' s b' s b s a Memory effect condition: W Scattering medium Intermediate field i i' j' j Array of transducers a Scattering medium Δs a = Δs b i+j=i'+j' (Antidiagonals of K) W Single scattering: Multiple scattering: Memory effect Deterministic coherence whatever the distance between i and j Short-range correlations governed by the size of the diffusive halo Coherence length Spatial coherence along the antidiagonals of K

14 Propagation operator in strongly scattering media A. Aubry, L. A. Cobus, S. E. Skipetrov, B. A. van Tiggelen, A. Derode, and J. H. Page, arxiv: , 2013

15 Anderson localization of elastic waves H. Hu, A. Strybulevych, J.H. Page, S.E. Skipetrov, and B. Van Tiggelen, Nature Physics 4, 945, 2008 Mesoglasses fabricated by brazing aluminum beads together to form a solid porous 3D elastic network. Pulsed transmission measurements: Localization between 1.2 and 1.25 MHz P.W. Anderson, Phys. Rev., 1959 scattering loops constructive interference Figure courtesy of John Page

16 Statistical properties of K / strong disorder Source i Real part of K t=185 µs, f=1.25 MHz Receiver j Long-range correlations even at long times of flight in the strongly scattering regime Recurrent scattering

17 Spatial intensity profile Mean backscattered intensity Typical spatial intensity profiles in the single/multiple scattering regime Single scattering Multiple scattering distance source receiver S R S=R p + p - Incoherent contribution Coherent backscattering peak Wolf and Maret, Phys. Rev. Lett., 1985 van Albada and Lagendijk, Phys. Rev. Lett., 1985

18 Normalized intensity Spatial intensity profile Normalized intensity Spatial intensity profile t=185 µs, f=1.25 MHz distance source - receiver Coherent backscattering enhancement is below 2 even at long time of flight distance source - receiver Recurrent scattering

19 Source i Source i Separation recurrent scattering / conventional multiple scattering Normalized intensity Source i Raw matrix K 1 Conventional multiple scattering Receiver j Recurrent scattering Receiver j Receiver j total intensity Recurrent scattering Conventional multiple scattering distance source - receiver

20 time [µs] Conventional multiple scattering / Coherent backscattering Diffusive regime Space-time evolution of the mean backscattered intensity at f=1.8 MHz The CB peak width scales as f=1.8 MHz diffusive behavior x~λa/w x 2 W~Dt distance source receiver [mm] time [µs]

21 time [µs] Conventional multiple scattering / Coherent backscattering (CB) Manifestation of Localization Saturation of the growth of the diffusive halo x~λa/w Space-time evolution of the mean backscattered intensity at f=1.215 MHz The CB peak narrowing saturates MHz localized regime distance source receiver [mm] time [µs]

22 Recurrent scattering ratio Recurrent scattering ratio Recurrent scattering intensity ratio = recurrent scattering intensity total backscattered intensity Recurrent scattering intensity ratio VS time time [µs] The recurrent scattering contribution is substantial even at long times of flight in the localization band (>70% at t=200 µs)

23 Time decay of the return probability Return probability Recurrent scattering intensity Return probability = Probability for a wave to come back close to its starting spot Key quantity in self-consistent theory of Anderson localization (renormalization of the diffusion constant) time [µs] MHz localized regime Diffusive regime: Localized regime: S.E. Skipetrov and B.A. van Tiggelen, Phys. Rev. Lett., 2006 The measured return probability displays a very slow decay around the mobility edge (unpredicted by SC theory, multifractality?)

24 Mean of the first singular values Intense recurrent scattering paths in the localized band DORT analysis of the array response matrix K (t=150 µs) Discrepancy between the two first (=largest) singular values and RMT predictions around 1.2 MHz Singular value decomposition of K K = UΛV i λ 1 λ 2 dots: experiment dashed line: random matrix theory (Hankel) Frequency [MHz] The largest singular values may be associated to intense recurrent scattering paths λ 3

25 Intense recurrent scattering paths in the localized band Weakly scattering media: one eigenstate one scatterer Strongly scattering media: one eigenstate one recurrent scattering path Numerical back-propagation of the first singular vector at the surface of the scattering sample (f=1.2 MHz) Hot spot = entry-exit point of a recurrent scattering path Hot spots switch on at regular intervals of time Hypothesis: The same RS path travelled several times

26 Conclusion & Perspectives

27 Conclusion & Perspectives Propagation operator in scattering media Statistical behaviour of the array response matrix (Random Matrix Theory) Deterministic coherence of the single scattering contribution Memory effect Perspectives: Separation single/multiple scattering for non-destructive testing applications (Coll. EDF) Extension to other fields of wave physics: optics (Amaury Badon, Dayan Li), seismology (?) etc. Scattering matrix - transmission/reflection open/closed channels (Benoît Gérardin) Recurrent scattering in strongly scattering media Recurrent scattering and memory effect Manifestation of Anderson localization: Coherent backscattering, Return probability Correspondence between eigenstates of K and scattering loops Perspectives: Theoretical understanding, random lasers (?)

28 Posters Benoît Gérardin Matrix approach of wave propagation in disordered elastic wave guides Scattering matrix, transmission/reflection matrix Open/closed scattering channels Bimodal law Amaury Badon Passive measurements of Green s function in optics Correlations of incoherent scattered wave-fields

29 Thanks for your attention! Alexandre Aubry Institut Langevin CNRS UMR 7587, ESPCI ParisTech Paris, France Collaborators: John Page, Laura Cobus (University of Manitoba, Winnipeg, Canada) Sergey Skipetrov, Bart van Tiggelen (LPMMC, Grenoble, France) Arnaud Derode (Institut Langevin, Paris, France)