Ultrasound in random media - French National Centre for ...

Ultrasound in random media

Arnaud Tourin

Director : Arnaud Tourin Deputy director : Rémi Carminati et Mickael Tanter Honorary director : Mathias Fink

http://www.institut-langevin.espci.fr

Director of ESPCI (1925-‐1946)

Piezoelectricity Pierre and Jacques Curie 1880

Paul Langevin (1872-1946)

quartz

Interdisciplinary approach of wave physics All kinds of waves : Acoustics, Water waves, Microwaves, Optics

Temporal shaping Spatial shaping

Liquid crystals SLM

Femtosecond Coherent control

Interferometry/detection

holography

Time-‐resolved techniques

Ultrasound Microwave

Optics

Fundamental and applied Research

Anderson localiza>on,

Random lasers,

QED,

Wave chaos,

Time reversal,…

Medecine and Biology,

Non destruc>ve tes>ng,

Wireless communica>ons,

Defense industry,

Geophysics,…

•  ECHOSENS (2001) 50 employees (Mongolia Pharm) > 2000 Fibroscan sold

•  SENSITIVE OBJECT (2003) 35 employees (Tyco)

•  SUPERSONIC IMAGINE (2005) 120 employees > 800 Aixplorer sold

•  TIME-REVERSAL COM (2008) 40 employees (Bull)

•  LLTECH (2008) 8 employees

Innovation and start-up companies

€

ℓ s < L < ℓ a ,ℓ φ

L Mesoscopic physics with ultrasound

Optical speckle

Courtesy Georg Maret

John Page

University of Manitoba

Ultrasound speckle

OUTLINE

Lecture 1 : mesoscopic physics with ultrasound

❒ The ballis>c and coherent waves

❒ The diffuse (incoherent) intensity and the Diffusion Approxima>on

❒ Beyond the DA : the weak and strong localisa>on regimes

Lecture 2 : controlling ultrasound in disordered media

❒ Time Reversal focusing in mul>ple scaTering media

❒ Broadband wavefront shaping in mul>ple scaTering media

❒ A matrix method for imaging through a mul>ple scaTering media

❒ Imaging changes in scaTering media from Time Reversal of the Coda wave Difference (TRECOD)

Experimental approach : measuring the S matrix

i

j

Transducer arrays

f ~ 500 kHz - 5 MHz λ ∼ mm

S ?

+ -

In optics, S. Popoff et al., Phys. Rev. Lett. 104, 100601 (2010)

Experimental approach : measuring the S matrix

!  Ψij(t), ⟨Ψij(t)⟩, ⟨∣Ψij(t)∣2⟩, ∫Ψij2(t)dt, ∑ij∫Ψi

2(t)dt, ⟨Ψil(t) Ψ*im(t)⟩,…

!  Distribu>on of transmission eigenchannels

B. Gerardin’s talk on Saturday

Mello and Pichard, Phys. Rev. B 40, 5276 (1989)

128-‐transducer array Pitch : 0.42 mm

Single transducer ν=3.2 MHz λ=0.48 mm

2D random sample Diameter: 0.8 mm Density : 19 rods / cm

Setup

0 10 Time (µs)

Water

Time (µs) 0 10

127

1

# tra

nsdu

cer

Am

plitu

de

The ballistic wave

Time (µs) 0 10

127

1

0 10 Time (µs)

L=7 mm

Am

plitu

de

# tra

nsdu

cer

L=15 mm

0 10 Time (µs)

127

1 0 10 Time (µs)

Am

plitu

de

The ballistic wave #

trans

duce

r L=30 mm

0 10 Time (µs)

127

1 0 10 Time (µs)

Am

plitu

de

# tra

nsdu

cer

0 225 Time (µs)

0 225 Time (µs)

L=70 mm A

mpl

itude

127

1

The diffuse wave #

trans

duce

r

Quantum wave Classical (acoustic) wave

The wave equation in a heterogeneous medium

single scattered wave

unscattered wave

The Born expansion

double scattered wave

The wave equation in a heterogeneous medium

€

ΔG(ω ,r,r0)+ k02G(ω ,r,r0) = σ(r)G(ω ,r,r0)+δ(r − r0)

€

G(ω ,r,r0) = G0(ω ,r − r0)+ G0(ω ,r − r1∫ )σ(r1 )G(ω ,r1,r0) dr1

The Dyson equation

€

G(ω ,r − r0) = G0(ω ,r − r0)+ G0(ω ,r − r1∫ )Σ(ω ,r1 − r2) G(ω ,r2 − r0) dr1dr2∫

Beer-‐Lambert

€

Icoh = Gω(r,r0)2∝ exp−

r − r0ℓ S

€

G(ω ,r − r0) =exp ike r − r04π r − r0

€

ke2 ≈ k0

2 −Σ(ω)

€

G(ω ,k) =1

k 2 − k02 −Σ(ω ,k)

€

= k02 − 4πnfISA

"  Foldy (1945) first order in n

"  Lax (1952) QCA

"  Percus, Yevick (1958) pair-correlation function

"  Waterman et Truell (1961) second order in n

"  Twersky (1962) pair-correlation function

"  Frisch (1968) diagrammatic expansion

"  Keller (1964) second order

"  Gubernatis (1965) CPA

"  Lagendijk et Tiggelen (1996) ISA

V. Mamou, A. Derode, A. Tourin, Phys. Rev E. 74, 036606 (2006)

Source Receiving array

How to build an estimator of the ensemble average ?

average on 80

Configurations

L=30 mm

0 10 Time (µs)

127

1 0 10

Time (µs)

Am

plitu

de

127

1

0 10 Time (µs)

Am

plitu

de

Time (µs) 10 0

The coherent wave #

trans

duce

r

# tra

nsdu

cer

127

1 Time (µs) 10 0

0 10 Time (µs)

Am

plitu

de

2 2.5 3 3.5 4 4.5 5

MHz

FT

# tra

nsdu

cer

The coherent wave

0

1

2

3

4

0 5 10 15 20 25 30 35

2,7 MHz, l = 8,3mm

3,2 MHz, l = 4mm

Log [A

c(L)]

The coherent wave

A. Tourin, A. Derode, A. Peyre and M. Fink J. Acoust. Soc. Am. 108 (2), 503-512 (2000)

€

1ℓ ext

=1ℓ S

+1ℓ a

Thickness (mm)

Scattering from a single inclusion

θ

€

σ =PS

dPinc /dS= 2π sinθ f (k,θ)

2dθ

0

π

∫

Optical theorem

€

σ =4πk0

Im f (k0 ,0)

Scattering cross section

0

0,4

0,8

1,2

1,6

2

0 1 2 3 4 5 Frequency (MHz)

σ (m

m)

θ


€

→ σ = σrigid + σelastic+ interference

0

MHz 2 4 6 8

0

0.5

1

1.5


σ (m

m)

The dwell time

A. Derode, A. Tourin, M. Fink, Phys. Rev. E 64, 036605 (2001)

2.5 3 3.5 0.5

1

1.5

2

2.5

3

3.5

MHz

µs

Group delay

experiment

Theory

Jia, Caroli and Velicky, PRL (1999)

Another example : acousQc propagaQon in granular media

♦λ > 10d, coherent wave E ♦λ ~ d, diffuse wave S

2 µs

Glass beads d: 600-‐800 µm

0.03 – 3 Mpa

BW: 20 kHz-‐1MHZ

First loading Reloading

Time (µs) Time (µs)

The coherent wave in a dry granular medium

♦ Effec>ve medium theory

VP =p

(K + 4/3G)/⇢ VS =p

G/⇢

Keff (P ) =kn12⇡

(�Z)2/3✓6⇡P

kn

◆1/3

Geff (P ) =kn + 3/2kt

20⇡(�Z)2/3

✓6⇡P

kn

◆1/3

Goddard (1990); De Gennes (2001); Makse, Johnson, Schwartz (2000); Velicky, Caroli (2002); J.N. Roux (2000)

0 20 40 60 80 100 120 140 160 180 200450

500

550

600

650

700

750

800

850

900

950

Frequency (kHz)

Vite

sse

de P

hase

(m/s

)L1=9cm, L2=18cm

35kPa88kPa177kPa265kPa354kPa442kPa530kPa618kPa707kPa

Hertz contact

♦ Phase speed

Z : coordina>on number

P1/6

Vphase

VP,S(P ) / Z1/3P 1/6

€

P =∝ δ 3/ 2

The diffuse wave

0 225 Time (µs)

0 225 Time (µs)

L=70 mm A

mpl

itude

127

1

# tra

nsdu

cer

The Bethe Salpeter equation

€

G(r,r0 ;ω −Ω/2)G*(r,r0 ;ω +Ω/2) = G(r,r0 ;ω −Ω/2) G*(r,r0 ;ω +Ω/2) +

€

Iincoh ∝ Ψ(r; t)2

€

G(r,r1;ω −Ω/2) G*(r,r2,ω +Ω/2) U(r1,r2,r3 ,r4 )∫ G(r3,r0 ;ω −Ω/2)G*(r4 ,r0;ω +Ω/2) dr1dr2dr2dr4

€

ω , Ω

€

∂

∂t+ v.∇+ loss

⎡

⎣ ⎢

⎤

⎦ ⎥ IV(r, t) = source + scatteringThe BS equa>on is a transport equa>on

Boltzmann approximation # Transfer radiative equation for the specific intensity

€

∂Iincoh∂t

+Iincohτa

= DΔIincoh

Diffusion approximation # Diffusion equation

€

D =13VEℓ

*

The diffuse intensity

€

1Vp

∂Iu(r, t)∂t

+u∇Iu = −1lsIu(r, t)+ n dΩu'

dσdΩ

(u'∫ →u)Iu(r, t)+ source − loss

€

ℓ* =ℓ S

1−cosθ

Transmission of the diffuse wave

€

I(t) =e− t /τa

4πDte−m 2π 2Dt

(L+2z0 )2

m=1

∞

∑ sinπmz0L + 2z0

⎛

⎝ ⎜

⎞

⎠ ⎟ sin

πm(z + z0)L + 2z0

⎛

⎝ ⎜

⎞

⎠ ⎟

€

T(L) =∝ℓ*

L

Dynamic measurement

StaQonary measurement

€

G =e2

hT

Ohm’s law

Landauer formula

L

€

z0 =π

4ℓ*

Time-of-flight distribution

Transmission of the diffuse wave

Time(µs)

0 100 200 300 400 500 600

€

ls = 4.2mm

A. Tourin, A. Derode and M. Fink, « MulQple scaSering of sound » Waves in Random Media 10, R31-‐R60 (2000)

€

τD ∝L2

π 2D

@ 3.2 MHz

The backscattering cone

100 mm

128 éléments

EMISSION RECEPTION

M.P. van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985) E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985)

A. Tourin, B. A. Van Tiggelen, A. Derode, P. Roux, M. Fink, Phys. Rev. Lett 79, 3637 (1997)

θ

Beyond the diffusion approximation

Inte

nsity

θ (degree) -10 -5 0 5 10

Speckle

One realisation

θ (degree) -10 -5 0 5 10

0

1

2

Averaging over 80 realisations In

tens

ity


i

j i

j

Speckle Coherent contribution

j

i

Incoherent contribution


Ladder diagrams Most-‐crossed diagrams

φ °

t1=3 µs

0

1

2

t2=11 µs

0

1

2

t3=27 µs

0

1

2

-10 -5 0 5 10


D=3,2 mm²/µs

-2,5

-2

-1,5

-1

-0,5

0 0 1 2 3 4

ln(t) Ln

(kΔθ)

Dynamic CBE

Nor

mal

ised

inte

nsity

θ (degree)

0

1

2

-15 -10 -5 0 5 10 15

Stationary CBE


L

Weak localisation

€

D(L) L→∞⎯ → ⎯ ⎯ 0

Vanishing of the diffusion constant

SC theory

Localized wave funcQons

= dense pure point

spectrum

Vanishing of the conductance ( )

Anderson localization

€

T =ALW (ω)D(ω)

€

ℓ*

Scaling theory of localizaQon

? ?

The first experimental proof in 3D with ultrasound

See John Page’s lecture

A signature of localiza>on is usually sought in the exponen>al decrease of the average transmiTed intensity. But absorp>on has always been a major obstacle to reaching unambiguous conclusions.

An alterna>ve way is to probe the dynamic spreading of the intensity in the transverse direc>on

H. Hu, A. Strybulevych, J.H. Page, S.E. Skipetrov and B.A. van Tiggelen Nature Phys. 4, 945 (2008)

Transverse localization

H. De Raedt, Ad Lagendijk, Pedro de Vries

« Transverse localization of light »

Phys. Rev. Lett. 62, 47 (1989)

x

y z

ω0

Lee,P. A. et al., Rev. Mod. Phys. 57, 287-337 (1985)

T. Schwartz, G. Bartal, S. Fishman, M. Segev, Nature 446, 52-55 (2007)

Intensity distribuQon at the laWce output (L=10 mm propaga>on)

hexagonal lapce 15% posi>onal disorder (average over 100

realiza>ons)

Transverse localization of light

Transverse localizaQon of sound

A. Bretagne, M. Fink and A. Tourin Phys. Rev. B 88, 100302 (2013)

Low disorder Strong disorder

Strong disorder

Low disorder

Water

$ Next lecture :

•  how disorder can be turned into an ally to manipulate waves ?

•  how to image through a mul>ple scaTering media ?

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