Time Reversed Acoustics

( ),p r t acoustic pressure field (scalar)is the density and is the sound velocity in an heterogeneous medium( )rρ ( )c r

This equation contains only ( )2

2

,p r tt

∂∂

Then if is a solution( ),p r t

( ),p r t− is also a solution

because ( ) ( )2 2

2 2

, ,p r t p r tt t

∂ ∂ −=

∂ ∂

t0

t1 t0

t1

( )p r t,−( )p r t,

Acoustic propagation in a non dissipative fluid

In Linear Acoustics 0),()()(

1)(

),((2

2

2 =∂

∂−

ttrp

rcrrtrpgraddiv

ρρIn Non Linear Acoustics 0),(

),(),(1

),(),((

2

2

2 =∂

∂−

ttrp

prcprprtrpgraddiv

ρρ

Spatial Reciprocity Time Reversal Invariance

Elementary transducers

RAMsACOUSTIC SOURCE

Heterogeneous Medium

ACOUSTIC SINK ??

( )p r ti,

( )p r T ti, −TRANSMIT MODE

RECEIVE MODE

Time Reversal Cavity

Elementary transducers

RAMsACOUSTIC SOURCE

Heterogeneous Medium ( )pr ti,

( )p r T ti, −TRANSMIT MODE

RECEIVE MODE

DIFFRACTION LIMITED FOCAL SPOT

DEPENDING ON THE MIRROR ANGULAR APERTURE

INFORMATION LOST

Theory by D. Cassereau, M. Fink, D. Jackson, D.R. Dowling

Time Reversal Mirror

Source

Time reversed signals

Time Reversal in a multiple scattering medium

?

TRM array

Multiple scatteringmedium

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

The experimental setup

Linear array, 128 transducersElement size ¾ λ

Acoustic sourceν=3 MHz, λ=0.5 mm Steel rods forest

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

Transmitted signal through the rods recorded on transducer 64

Time reversed wave recorded at the source location

Transmitted signal through water recorded on transducer 64

Time (µs)

Time (µs)

Am

plitu

deA

mpl

itude

Am

plitu

de

Spatial focusing of the time reversed wave

Mobile hydrophone

-10 - 5 0 5 10-30-25-20-15-10-50

Distance (mm)

Am

plitu

de

One-bit versus 8-bit time reversal

One-bit time reversal, L=40 mm

-3

-1.5

0

1.5

3

-50 -25 0 25 50

time (µs)

8 bit time-reversal, L=40 mm

-1-0.5

00.5

1

-50 -25 0 25 50time (µs)

-30

-25

-20

-15

-10

-5

0-12 -6 0 6 12 (mm)

dB 8 bitOne bit

-10 -5 0 5 10-30

-25

-20

-15

-10

-5

0

Distance from the source (mm)

dB

Directivity patterns of the time-reversed wavesaround the source position with 128 transducers(blue line) and 1 transducer (red line).

One channel time reversal mirror

Time reversed signal

S

Time Reversal versus Phase Conjugation

( )

*

*

*

. ( , ) ( ,- )

If the source is monochromatic( , ) Re ( ) ( ) ( )

with ( ) complex function

( ) = ( )Thus the .

( , ) ( ) ( )or

( ) ( )

j t j t j t

j x

j t j t

TR operation p x t p x t

p x t P x e P x e P x eP x

P x P x eTR operation

p x t P x e P x e

P x P x

ω ω ω

φ

ω ω

−

−

→ ⇔

= ∝ +

→− ∝ +

⇔ ( ) ( ),or, x xφ φ⇔ −

xMax p(x,t)

.Source location 1 channel TRM

( )P x PointlikePhase Conjugated Mirror

Time Reversal versus Phase Conjugation

Field modulus

TR

PC

Im

Re

A Complex Representation of the Field

Im

Source location

Off axis

Field Modulus

Focusing quality depends on the field to field correlation ()( * δω) ωω +ΨΨ

t

Polychromatic Focusing

-Field-field correlation )()(ω * δωω+Ψ Ψ = fourier transform of the travel time distribution )(tI

0 50 100 150 200 250Time (µs)

)(tI

δω = 8 kΗz

2 2.5 3 3.5 4 4.5 5

MHz

ω∆

?δω

Thoules time, δτ =D2/L ~ 150 µs ∆ω/δω =150

FT

2 2.5 3 3.5 4 4.5 5

MHz

δω

FT

How many uncorrelated speckles ?

Focusing in monochromatic mode : the lens

D

F

λF/D

Spatial Diversity

Spatial and frequency diversity

One element time reversal mirror

-10 -5 0 5 10-25

-20

-15

-10

-5

0

-10 -5 0 5 10-25

-20

-15

-10

-5

0

dB

Phase conjugation Time-reversal

128 elements time reversal mirror

-10 -5 0 5 10-35

-30

-25

-20

-15

-10

-5

0

-10 -5 0 5 10-35

-30

-25

-20

-15

-10

-5

0

dB

TR1

PC1

PC128 TR128

Spatial and Frequency Diversity

Communications in diffusive media with TRM

20-element Arraypitch ~ λ

5 receivers4 λ apart

Central frequency 3.2 MHz (λ=0.46 mm)

Distance 27 cm (~ 600 λ)

L=40 mm, 4.8mm=*

A.Derode, A. Tourin, J. de Rosny, M. Tanter, M. Fink, G. Papanicolaou

T0 = 3.5 µs

-1+1

0.7µs

Transmission of 5 random sequences of 2000 bits to the receivers

#1 #2 #3 #4 #5 Error rate

Diffusive medium 0 0 0 1 0 10-4

Homogeneousmedium 489 640 643 602 503 28.77 %

Modulation BPSK

Spatial focusing

- 1 5 - 1 0 - 5 0 5 1 0 1 5

- 2 5

- 2 0

- 1 5

- 1 0

- 5

0

1 2 3 4 5

12345

10 µs

16mm

Diffusive medium water

Shannon Capacity (MIMO)

C = Log2 det (I+SNR × tH* H) bits/s/Hz

(Cover and Thomas 1991, Foschini 1998)

R = H E

Propagation OperatorFT

thij H(ω)

The Time Reversal Operator tH* H

T R

tHH* E*

array

EH

HE

H*E*tHTR

array

tH* H E

M. Tanter

C = Log2 det (I+SNR × tH* H )

∗= VDUH T

C = Log2 det (I+SNR × tU* D2 U )

U tU*= I

C = Log2 det (I+SNR × D2 )

C = Log2 1 + SNR × λi2∑

= Ni ..1

N independant channels, N degrees of freedom

Shannon Capacity in Diffusive Media

Experimental results : singular values distribution

Homogeneous medium Diffusive medium40×40 inter-element impulse responses

→ At 3.2 MHz : 34 / 6 singular values (–32 dB)

The number of singular values is equal to the number ofindependant focal spots that one can create on the receivingarray

( )p r , tiacoustic source

elementarytransducers

reflecting boundaries

( )p r ,T ti −

Receive mode

Transmit mode

The effect of boundaries on Time Reversal Mirror

-1

-0,5

0

0,5

1

-40 0 40

Time (µs)

Ampl

itude

-50

-40

-30

-20

-10 0 -20-10

010

20m

m

dB

80

0

40

Time (µs)

Dep

th(m

m)

0

-40

1 -Time Reversal in an Ultrasonic Waveguide

Hau

teur

du

guid

e (m

m)

80µs0 40

40

0

0

-40

S

O y

x

L

H

vertical transducer

array

water

reflecting boundaries128 elements

P. Roux, M. Fink

The Kaleidoscopic Effect : Virtual Transducers

pointsource S real

TRM

TRMimage

apertureof the TRM

in free water

apertureof the TRM

in thewaveguide

-50

-40

-30

-20

-10

0-20 -10 0 10 20

mmA

mpl

itude

(dB

)

guide d'ondeeau libre

A comparison between the focal spot with and without the waveguide

Open space

Waveguide effect

If the pitch is to large : grating lobes

3.5 kHz tranceiver

3.5 kHz SRA (’99 and ’00)

L = 78 mN = 29

B. Kuperman, SCRIPPSTime Reversal in Ocean Acoustics

Up-slope Experiment: Elba

1 m

Diffraction limit

30 m

100 m

10 km

2 - Time-Reversal in a Chaotic Billiard

Silicon wafer – chaotic geometry

Transducers

Coupling tips

Carsten Draeger, J de Rosny, M. Fink

Ergodicity

2 ms : Heisenberg time of the cavity : time for any ray to reachthe vicinity of any point inside the cavity (in a wavelength)

Time-reversed field observed with an optical probe

15 mm15 mm

waferscannedregion

(a) (b)

(c) (d)

(f)(e)

R R

R R

R R

How many uncorrelated speckle in the frequency bandwidth of thetransducers ?

For an ergodic cavity it is equalto the number of modes in thebandwidth :

In our case 400 modes : thusthe SNR is the square of the modenumber = 20

Why Ergodicity does not garantee aperfect time reversal ?

Waves are not particles and even notrays : Modal theory only

With a one channel TRM, what is the SNR ?

The Cavity Formula

),,(),,(),,( ),,( tBBgtAAgtABgtABg ⊗−=⊗−

AB

In terms of the cavity modesA and B cannot exchangeall informations, becauseA and B are always at theantinodes of some modes

nψ eigenmodes

)sin()()(),,( ∑=n n

nnn

tBAtABgωωψψ

Carsten Draeger

Origin of the diffraction limit

Wave focusing : 3 steps

Converging only

Both convergingand diverging

waves interfereDiverging only

Diffraction limit(λ/2)J. de Rosny, M. Fink

Monochromatic

exp j(kr+ωt) / rwith singularity

exp j(-kr+ωt) / rwith singularity

Sin (kr)/r . exp(jωt)without singularity

Goal

convergingNo interferenceand singularity

« Perfect » TR - the acoustic sink

No diffraction limit

exp j(kr+ωt) / r

with singularity

Principle of the acoustic sinkOut of phase

The Acoustic Sink Formalism

Propagatingterm

Point-likesource

Source at r0 excited by f(-t)

(TR source)Converging

wave

)()(),(1022 rrtftr p

t

c∆ −−=−

∂∂

− δ

)()(),(1022 rrtftr p

t

c∆ −=

∂∂

− δ

Focal spots with and without an acoustic sink

λ/14 tip

1 m

1 m

accelerometer100Hz <∆Ω < 10kHz

timeam

plitu

de

Green’s function:GA(t)

A

A nice application of Chaos : Interactive Objects

R. Ing, N. Quieffin, S. Catheline, M. Fink

How to transform any object in a tactile screen ?

ampl

itude

Green’s function:GA(t)

Time Reversal:GA(-t)

1 m

1 m

A

MEMORY

10msamp. GA(t)

A

B

C

amp. GB(t)

10ms

amp. GC(t)

10ms

Training step: library of Green functions

MEMORY

amp. GA(-t)

amp. GB(-t)

amp. GC(-t)

amp. GB

’(t)

B

amp.

amp.

amp.

0.21

0.98

0.33

maxima:

POINT B

Localisation step by cross correlation

Tactile Objects

Some other examples

• A new concept of smart transducer design with reverberation and programmable transmitters

• What happens if the source is outside the waveguide ?

Time Reversal in Leaky Cavities andWaveguides

A first example : the D shape billiard

half-cylinder hydrophone needlecontact transducer

h( , t)

Principle of time reversal focusing

Hydrophone needle

Time reversal process: )t,r(h)t,r(h)t,r(u 0t

−⊗=

u( , t)

h( , -t)x

y

z

0r

0r

0r

r

Time Reversal Focusing with steering

Abs

ciss

ax

(mm

)

-25

0

25

75 100 125Time of arrival

xy

z

contact transducer

100mm

130mmmoving pulsed source

A second example : the SINAI BILLIARD

30 emissiontransducers(1.5 MHz , 5mm x 8 mm pitch 1 mm)

Motors

z

xy

hydrophone electronics

Electronics :

Fully programmable multi-channelsystem.

Large transducer

element, not optimized

2 µs

400 µs400 µs

Hydrophone

Emission transducers

Principle of TR Focusing

2 µs

-45

-5

-15

-25

-35

0dB

FUN

DA

ME

NT

AL

0 20 40 60 80 100-0.4

-0.2

0

0.2

0.4

0.6

0.8A

mpl

itude

Time (µs) Distance(mm)

Dis

tanc

e(m

m)

TR Kaleidoscope (Fundamental and Harmonics)

-45

-5

-15

-25

-35

0dB

Dis

tanc

e(m

m)

Distance(mm)

HA

RM

ON

IC

0 20 40 60 80 100

-0.4

-0.2

0

0.2

0.4

0.6

Am

plitu

de

Time (µs)

Spatial lobes : - 30 dB

Spatial Lobes ~ - 50 dB

Temporal lobes : - 38 dB

Temporal lobes : -60 dB

Building a 3D Image

Emissiontransducers

Receptiontransducer(harmonic)

Object0

20

40

0

20

40-80

-60

-40

-20

0

Distance (mm)Distance (mm)

Dis

tanc

e (m

m)

In a dissipative medium

The effect of dissipation on Time Reversal :an example : the skull and brain therapy

G. Montaldo; M. Tanter, M. Fink

Influence of the trabecular bone on the acoustic propagation

Diploë :Porous zone(c = 2700 m.s- 1)

External wall(c = 3000 m.s- 1)

Internal wall(c = 3000 m.s- 1) 0

),(

)(

1

)(

),()()(1

2

2

2=

∂

∂−

∂

∂+

t

trp

rcr

trpgraddivr

tr

ρρτ

Breaking the time reversal invariance

o(-t) e(t)T

h e

i t

e r

a t i

v e

m

e t

h o

d

d(t)

First transmit step :the objective

c(t)Emission of thelobes

e(-t)o(t)+d(t) T.R and reemission :reconstruction of theobjective

c(-t)-d(t)+d(t)

Lobes reconstruction

e(-t)-c(-t)o(t)-d(t)-The differenceeliminates the lobes

0 20 30 40 50 60-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Distance in mm

Am

plit

ud

e in

Db

10

10

1

20

30

Experiments : improvement of the focal spot

128 elts.1.5 MHz

Pitch 0.5 mm

D = 60 mm

F = 60 mm

Water

AbsorbingAnd aberratingUreol sample

128 elts.1.5 MHz

Pitch 0.5 mm

Distance in mm

Tim

e in

µs

10 20 30 40 50

1

2

3

4

5

6

7 -35

-30

-25

-20

-15

-10

-5

0

Distance in mm10 20 30 40 50

1

2

3

4

5

6

7

Time ReversalFocusing

Focusing after 30iterations

Experiments : Spatial and temporal focusing

• Very simple operations : time reversal + signal substraction• Inversion just limited by the propagation time• Here, optimal focusing can be achieved in a few ms !!!

Focusing through the SkullOptimal signal to transmit Classical Cylindrical law

Transducernumber j

1 128

Transducernumber j

1

0

25

0

25

-20 -10 0 10 20-35

-30

-25

-20

-15

-10

-5

0

Distance from the initial point source (mm)

Pres

sure

(dB

)

Spatial focusing

300 elements Time Reversal Mirror (Therapy/Imaging)

Front view(300 elements and C 4-2 echographic probe)

Spherical active surface:Aperture 180 mm

Focal dist. 140 mm

Global view(300 elements and C 4-2 echographic probe)

128 Channels of a HDI 1000 scanner

200 Emission boards for THERAPY

100 Emission/Reception boards for THERAPY+IMAGING

Coupling + cooling system

Correction of skull aberrations using an implanted hydrophone

Experimental scanwithout correction

Experimental scanwith correction

(Time reversal + AmplitudeCompensation)

Acoustic Pressure measured at focus : - 70 Bars, 1600 W.cm-2 (with correction)

- 15 Bars, 80 W.cm-2 (without correction)

Transkull in vivo experiments

MRI Histology

Transkull in vivo experiments

Transkull in vivo thermally induced necrosis