Playing with ultrasound in complex media

Arnaud Tourin

Mesoscopic Physics in Complex Media, 01002 (2010)

© Owned by the authors, published by EDP Sciences, 2010

This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.

Institut Lang in - ESPCI aris, Franceve P

DOI:10.1051/iesc/2010mpcm01002

Article published online by EDP Sciences and available at http://www.iesc-proceedings.org or http://dx.doi.org/10.1051/iesc/2010mpcm01002

Playing with ultrasound in complex media

Arnaud Tourin

Cargèse – July, 13, 2010

The Langevin Institute

Founded in january 2009 to gather people from

LOA (M. FINK) : ultrasound, sound, seismic waves, microwaves

LOP (C. BOCCARA) : optics, ultrasound

New devices for manipulating waves in various kinds of media

Time Reversal Mirror

Multiwave Imaging

IR SNOM, thermal radiation STM

Medical imaging & therapy,

non destructive testing,

underwater acoustics,

seismology,

telecommunications,

tactile objects,…

Fundamental physics/applications/companies

Echosens, Sensitive Object, SSI, TRCOM, LLTech (~200)

92 people

34 permanent researchers

14 for the technical & administrative support

29 Phd students

15 Post-Doc

One Director (M. Fink) and two vice directors (R. Carminati and AT)

5 groups

Acoustic waves & microwaves in complex media

Nanophotonics & Optical waves in diffusive media

Wave Physics for medicine & Biology

Detection, Imaging, and characterization

Optical physics and wave theory

The Langevin Institute

Acoustic waves & microwaves in complex media

Acoustic waves & microwaves in random media

Julien de Rosny

(Research scientist)

Geoffroy Lerosey

(Research scientist)

Arnaud Tourin

(Professor)

Mathias Fink

(Professor)

Agnès Maurel

(Research scientist)

Arnaud Derode

(Professor)

Alice Bretagne

(PhD student)

Marie Müller

(Associate professor)

Abdel Ourir

(Engineer)

OUTLINE

Wave propagation in random media

The ballistic and coherent waves

The incoherent wave and the diffusion approximation

Beyond the DA : the backscattering cone

Using disorder for controlling wave propagation

Disorder for guiding ultrasound

Disorder for focusing ultrasound

A random (or ordered) bubbly medium for filtering US

σ

<<

L

Mesoscopic physics with ultrasound

Experimental approach : measuring the S matrix

i

j

++

Transducer arrays

f ~ 500 kHz à 5 MHz

λ ∼λ ∼λ ∼λ ∼ mm

=

−+++

−−+−

S ?

+ -

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

Experimental approach : measuring the S matrix

(average) time-resolved transmitted amplitude / intensity

(average) backscattered amplitude or intensity (BS Cone)

Perform a SVD and study the statistics of the singular values

Focusing using time reversal, wavefront shaping, inverse filter

19 rods / cm²Density :

Diameter : 0.8 mm

5mm < L < 80mm

Experimental approach :

what can be measured with ultrasound ?

128-transducer arrayPitch : 0.42 mm

Single transducerνννν=3.2 MHz, λλλλ=0.48 mm

2D random sample

Experimental approach

0 10Time (µs)

Water

Time (µs)0 10

127

1

# t

ransducer

Am

plit

ude

Spatio-temporal distribution of the amplitude

Time (µs)0 10

127

1

0 10Time (µs)

L=7 mm

Am

plit

ude

# t

ransducer

L=15 mm L=30 mm

0 10Time (µs)0 10Time (µs)

127

1

127

10 10Time (µs)0 10Time (µs)

Am

plit

ude

Am

plit

ude

Spatio-temporal distribution of the amplitude# t

ransducer

# t

ransducer

0 225Time (µs)

0 225Time (µs)

L=70 mm

Am

plit

ude

127

1

Spatio-temporal distribution of the amplitude# t

ransducer

ωωσ −=

Quantum waveClassical (acoustic) wave

ω

=

ωσωω Ψ=Ψ+∆Ψ

∂

∂

Ψ=∆Ψ

The wave equation in a heterogeneous medium

Ψ−+Ψ=Ψ σωωω

=σ

∂

∂ Ψ=Ψ

+∆−

ω =

+

Ψσ−σ−ω+

Ψσ−ω+

Ψ=

Ψ

single scattered wave

unscattered wave

The Born expansion

double scattered wave

The wave equation in a heterogeneous medium

−+=+∆ δωσωω

ωσωωω −+−=

The Dyson equation

−−Σ−+−=− ωωωωω

σ =

Beer-Lambert

−−

−

−∝=

πω

Scattering mean free path

Source Receiving

array

How to build an estimator of the ensemble average ?

average on 80

Configurations

L=30 mm

Time (µs)

127

1

Time (µs)

127

1

Time (µs)

Time (µs)

The coherent wave# t

ransducer

# t

ransducer

L(mm)

=

Ln

[T

A(L

)]

Ballistic amplitude averaged over 80 random sample positions

Ballistic amplitude

Ballistic amplitude averaged over the 127 transducers

+=

The coherent wave

127

1Time (µs)

Time (µs)

Am

plit

ude

FT

# t

ransducer

The coherent wave

0

1

2

3

4

0 5 10 15 20 25 30 35

Thickness (mm)

2,7 MHz, l = 8,3mm

3,2 MHz, l = 4mm

Ln (

TA

(L))

Frequency (MHz)

Mean

free p

ath

( mm

)

The coherent wave

Scattering from a single inclusion

θ

+→Ψ∞→

==π

θθθπσ

π

σ =

Frequency (MHz)

σ (

mm

)

θ

( )θ θθσ =

θθσσπ

=

Scattering from a single inclusion

+=

+σ+σ=σ→

MHz

Scattering from a single inclusion

σ (

mm

)

The incoherent wave

0 225Time (µs)

0 225Time (µs)

L=70 mm

Am

plit

ude

127

1

# t

ransducer

The Bethe Salpeter equation

+Ω+Ω−=Ω+Ω− ωωωω

=

Ω+Ω− ωω

Ω+Ω− ωω

• Boltzmann approximation Transfer radiative equation for the specific intensity

• Diffusion approximation Diffusion equation

∆=

∂

∂

=

The incoherent intensity

=

=

=

=τ

Time-of-flight distribution

Total transmission varies as ~ /L (ohm’s law)

The incoherent wave

=

=

Time(µs)

0 100 200 300 400 500 600

=

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

=∆=In

tensity

θ -10 -5 0 5 10

Speckle

One realisation

θ -10 -5 0 5 10

0

1

2

Averaging over

80 realisations

Inte

nsity

The backscattering cone

Speckle Coherent contribution

==

Incoherent contribution

The backscattering cone

φ °

∝

0

1

2

0

1

2

0

1

2

-10 -5 0 5 10

The backscattering cone

D=3,2 mm²/µs

-2,5

-2

-1,5

-1

-0,5

0

0 1 2 3 4

ln(t)

∆θ

)

Dynamic CBE

Norm

alis

ed

inte

nsity

θ (degree)

=∆

0

1

2

-15 -10 -5 0 5 10 15

=

Stationary CBE

The backscattering cone

L

cf. John Page’s lecture

Weak localisation

Using disorder

for controlling wave propagation

1. A random fiber

H. De Raedt, Ad Lagendijk, Pedro de Vries

« Transverse localization of light »

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

ψ

ψ∂

=∂

( )

( )

∂ ∂≡ + +

∂ ∂

≡ −

π

ξ

=

ω−≡ ∝

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

1. A random fiber

Schwartz et al., Nature 446, 52-55 (2007)

Intensity distribution after L=10 mm propagation

hexagonal lattice 15% positional disorder(average over 100

realizations)

Steel rods (0.8 mm)

embedded in a PVA matrix

Transducer array

fc=7.5 MHz Point-like

receiving

transducer

1. A random fiber

Ordered sample

(triangular lattice, a=2.4mm)

Ordered (blue)

Disordered (red)Disordered sample

Energ

y(A

U)

Lateral distance

(cm)

Tim

e (

µs)

Tim

e (

µs)

Lateral distance

(cm)

Alice Bretagne, AT

2. A random lens

<

Transducer arrays

f ~ 500 kHz à 5 MHz

λ ∼λ ∼λ ∼λ ∼ mm

2. A random lens

Source

Time reversed waveforms

TRM

Multiple scatteringmedium

A.Derode, A. Tourin, P. Roux, M. Fink

>>

2. A random lens

20 40 60 80 100 120 140 160

20 40 60 80 100 120 140 160

Time (µs)20 40 60 80 100 120 140 160

Waveform transmitted through the rods and received at transducer #64

Time reversed wave recorded at the source location

Waveform transmitted in water and received at transducer # 64

Time (µs)

Time (µs)

Am

plit

ud

eA

mp

litu

de

Am

plit

ud

e

2. A random lens

Beamwidth at -12 dB : 1 mm (scattering medium) / 35 mm (free space)

Spatial resolution does not depend on the array aperture !

TRM

Time reversed waveforms

x

Spatial Focusing : the hyperfocusing effect

F

D

dB

λF/D

One-channel time reversal mirror

Distance from the source (mm)

dB

Directivity patterns of the time-reversed waves

128 transducers

1 transducer

Time reversed signal

2. A random lens

2. A random lens

• I.M. Vellekoop and A.P. Mosk

Optics Lett. 32, 2309 (2007)

• E. Herbert, M. Pernot, M. Tanter,G. Montaldo, M. Fink IEEE transactions on ultrasonics,

ferroelectrics, and frequencycontrol 56 (11), 2388 (2009)

α=∝∆

πϕ <<

a lot of iterations are neededTwo problematic issues

low sensitivity (one by one)

An alternative approach : wave front shaping

2. A random lens

In the case of two transducers, the intensity at the intended focus is :

Φ−Φ++=

ρ

Φ+=

+Φ+=

When a phase x is added to the 2nd transducer, the intensity at focus is :

At least 3 measurements must be performed

i.e.,

Finally,Φ−

and are transmitted from the two transducers

Extension to a N-transducer array using spatial coded excitations

Transmission ofV0(x,ωωωω) + ejx.Vi(x,ωωωω)

Recording of I = I(x,ωωωω)at focus

Repeat the transmissionfor four different values

(x=0, π, π, π, π, ππππ/2, -ππππ/2)

−−

−−

−−

=

Hadamard matrix

<<

2. A random lens

=

Ψ

Ψ

Ψ

Ψ

Φ

Φ

Φ

Φ

=−=

−=−==Ψ

π

ππ

πππ −+−+=

2. A random lens

10 20 30 40 50 60 70 80

20

40

60

80

100

120

140

160

10 20 30 40 50 60 70 80

20

40

60

80

100

120

140

160

10 20 30 40 50 60 70 80

50

60

70

80

90

100

110

120

130

140

150

V1(t) V31(t) V41(t)

10 20 30 40 50 60 70 80

10

20

30

40

50

60

70

80

Definition of spatio-temporal

vectors

2. A random lens

Multiple scattering

medium

Transducer array

f=3.2 MHz

0 100 200 300 400 500 600 700 800 900-0.5

0

0.5

Time (ms)

Am

plit

ude (

AU

)

Typical waveform received at the focal point

after a Hadamard emission

A. Bretagne, J. Aulbach, M. Tanter, M. Fink, A. Tourin

2. A random lens

-60 -40 -20 0 20 40 600

0.2

0.4

0.6

0.8

1

Distance from the focal point (l)

Monochromatic case

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80.5

0.6

0.7

0.8

0.9

Norm

aliz

ed

am

plit

ud

e

TRWFS WFS (phase only)

2. A random lens

Distance from focal point (λ)

Tim

e (

µs)

-60 -40 -20 0 20 40 60

880

885

890

895

900

905

910

915

920

-80

-70

-60

-50

-40

-30

-20

-10

0

Time Reversal Wavefront shaping

Broadband case

dB

-60 -40 -20 0 20 40 60

400

500

600

700

800

900

1000

1100

1200

Distance from focal point (λ)

2. A random lens

-60 -40 -20 0 20 40 600

0.2

0.4

0.6

0.8

1

Distance from the focal point (λ)

Norm

aliz

ed

am

plit

ude

-1.5 -1 -0.5 0 0.5 1 1.5

0.5

0.6

0.7

0.8

0.9

TRWFS WFS (phase only)

Broadband case

2. A random lens

3. A random (or periodic) bubbly medium

M. Kafesaki et al.

Air Bubbles in water : a strongly Multiple

Scattering Medium for Acoustic Waves

Phys. Rev. Lett. 84, 6050 (2000)

σ σ>>

ρω

βair

==

= =

Minnaert Resonance

10 % volume fraction

V. Leroy, A. Bretagne, M. Fink, H. Willaime, P. Tabeling, A. Tourin

« Design and characterization of bubble phononic crystals »

App. Phys. Lett. 95, 171904 (2009)

h=78 µm, ax=200 µm, az=360 µm

3. A random (or periodic) bubbly medium

Thank you for your attention !