Time Reversed Acoustics

( ),p r tr

acoustic pressure field (scalar)

is the density and is the sound velocity( )rρ r ( )c rr

Spatial reciprocity Time reversal invariance

( ) ( )( ) ( )2

2

2

10

grad p pr div

r c r tρ

ρ ∂− = ∂

rr rIn linear acoustics

This equation contains only ( )2

2

,p r t

t

∂∂

r

Then if is a solution( ),p r tr

( ),p r t−ris also a solution

because( ) ( )2 2

2 2

, ,p r t p r t

t t

∂ ∂ −=

∂ ∂

r r

t0

t1 t0

t1

( )p r tv,−( )p r t

v,

Acoustic propagation in a non dissipative fluid

Elementary transducers

RAMsACOUSTIC SOURCE

Heterogeneous Medium

ACOUSTIC SINK ??

( )p r tiv ,

( )p r T tiv

, −TRANSMIT MODE

RECEIVE MODE

Time Reversal Cavity

Elementary transducers

RAMsACOUSTIC SOURCE

Heterogeneous Medium ( )p r tiv,

( )p r T tiv , −

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 scattering

medium

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

The experimental setup

Linear array, 128 transducers

Element size ¾ λAcoustic source

ν=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

pli

tud

eA

mp

litu

de

Am

pli

tud

e

Spatial Focusing

Focal spot : beamwidth at -6 dB : 35 mm / 1 mm

Spatial resolution does not depend of the array aperture

MRT

Random medium

Time reversed signals

x

cmdB

x

-10 -5 0 5 10-30

-25

-20

-15

-10

-5

0

Distance from the source (mm)

dB

Directivity patterns of the time-reversed waves

around 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 m onochrom atic

( , ) R e ( ) ( ) ( )

w ith ( ) com plex function

( ) = ( )

T hus the .

( , ) ( ) ( )

or

( ) ( )

j t j t j t

j x

j t j t

T R operation p x t p x t

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

P x

P x P x e

T R operation

p x t P x e P x e

P x P x

ω ω ω

φ

ω ω

−

−

→ ⇔

= ∝ +

→− ∝ +

⇔ ( ) ( ), o r, x xφ φ⇔ −

x

Max p(x,t)

.

Source location 1 channel TRM

( )P x Pointlike

Phase Conjugated Mirror

Time Reversal versus Phase Conjugation

Field modulus

TR

PC

Im

Re

Complex Representation

Field Modulus and PhaseIm

Source location

Off axis

Field Modulus

Re

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

t

-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

∆ω/δω =150δτ =L2/D ~ 150 µs

Focusing in monochromatic mode : the lens

D

F

λF/D

Spatial Diversity

Communications in diffusive media with TRM

20-element Array

pitch ~ λ5 receivers

4 λ apart

Central frequency 3.2 MHz (λ=0.46 mm)

Distance 27 cm (~ 600 λ)

L=40 mm, 4.8mm=*l

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

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

Homogeneous

medium 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

1

2

3

4

5

10 µs

16

mm

Diffusive medium water

N independant channels, higher in a diffusive medium,

Shannon Capacity in diffusive media

The number of informations that one can send per unit of time

from an array to a volume depends on the number of independant

focal spots that one can create inside the volume of interest.

Multiple scattering and reverberation allow to obtain smaller

focal spots

Homogeneous

medium

Diffusive medium

( )p r , tiv

acoustic source

elementarytransducers

reflecting boundaries

( )p r ,T tiv −

Receive mode

Transmit mode

The effect of boundaries on Time Reversal Mirror

TRM Experiment in the oceanTRM Experiment in the oceanB. Kuperman group, SCRIPPSB. Kuperman group, SCRIPPS

Experiment Area Source-Receive Array

SRA: 29 transducers, 78 m, 3-4 kHz, 174 dB/1uPa

3.5 kHz tranceiver

3.5 kHz SRA (’99 and ’00)

L = 78 m

N = 29

Up-slope Experiment: Elba

1 m

Diffraction limit

30 m

100

m

10 km

Time-Reversal in chaotic billiards

Silicon wafer – chaotic geometry

Transducers

Coupling tips

Carsten Draeger, J de Rosny, M. Fink

Ergodicity

The Carsten Draeger Experiment

2 ms : Heisenberg time of the cavity : time for any ray to reach the

vicinity of any point inside the cavity (in a wavelength)

Time-reversed field observed with an optical probe

1 m

1 m

accelerometer100Hz <∆Ω < 10kHz

timea

mp

litu

de

Green’s function:

GA(t)

A

A nice application : Interactive Objects

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

How to transform any object in a tactile screen ?

am

pli

tud

e

Green’s function:

GA(t)

Time Reversal:

GA(-t)

1 m

1 m

A

MEMORY

10msam

p. GA(t)

A

B

C

am

p. GB(t)

10ms

am

p. GC(t)

10ms

Training step: library of Green functions

MEMORY

am

p. GA(-t)

am

p. GB(-t)

am

p. GC(-t)

am

p. GB

’(t)

B

amp

.am

p.

amp

.

0.21

0.98

0.33

maxima:

POINT B

Source Localisation by cross correlation

mimicking a time reversal experiment

Tactile Objects

Origin of the diffraction limit

Wave focusing : 3 steps

Converging only

Both convergingand diverging

waves interfereDiverging only

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

Monochromatic

exp j(kr+ωωωωt) / rwith singularity

exp j(-kr+ωωωωt) / rwith singularity

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

Goal

converging

No interference

and singularity

« Perfect » TR - the acoustic sink

No diffraction limit

exp j(kr+ωωωωt) / r

with singularity

Principle of the acoustic sink

Out of phase

The Acoustic Sink Formalism

Propagatingterm

Point-likesource

Source at r0 excited by f(-t)

(TR source)

Converging

wave

)()(),(1

022 rrtftr p

t

c∆

rrr −−=−

∂∂− δ

)()(),(1

022 rrtftr p

t

c∆

rrr −=

∂∂− δ

Field Time Reversal Field Time Reversal and Source

Time Reversal : the Sink

Experimental results

Focal spots with and without an acoustic sink

λ/14 tip

Some applications of ultrasonic time reversal

with leaky cavities and waveguides

• Smart transducer design

Time reversal compression in a solid waveguide

G. Montaldo, P. Roux, A. Derode, M. Fink

0 5 10 15 20 25 30-20

-10

0

10

20

30

40

50

Shock wave and lithotripsy

Time (µs)

Pre

ssure

(B

ar)

1 cm

150 shots 300 shots 600 shots

+/- 40 Volts, F = 2 cm Pmax ~ 600 bars

Scatterer

Aberrating mediumEMISSION

RECEPTION

diverging wave

EMISSION

converging wave

window selection

Transducer array

Time Reversal in Pulse Echo mode : 1 target

Multi target mediumTransmission 1

A

B

Reception 1

a

b

Transmission 2

a

b

Reception 2

a2

b2

Transmission 3

a2

b2

Reception 3

a3

Time reversal

Time reversal

Iterative Time Reversal on multi target medium

J.L. Thomas, F. Wu, M. Fink

Application of TRM to Lithotripsy

E m iss ion 1

defect

E m iss ion 2 : after tim e reversal

transducers a rray so lid sam ple

tim e

R éception 1

tim e

tim e

R éception 2

tim e

Applications to defect detectionin titanium alloy (SNECMA)

Time Reversal Mirror in non-destructive testing

F. Wu, D. Cassereau, N. Chakroun, V. Miette, M. Fink

86 mm

103 mm

Axe y

Axe x

iteration 0

iteration 1

iteration 2

Zone witha flat bottom holeat 140mm depth

Zone withoutdefect(speckle)

ch

an

nels

Iterative time reversal in titanium alloy

1

128

time

High Power Time Reversal Mirror for Therapy

Electronic channels (18 W per channel)

Initial Prototype (200 elements)

Single element (8 mm diameter, 1 MHz)

Electrical matching 50 Ohms, 50 % efficiency

(Collaboration IMASONIC, France)

200 Emission boards for THERAPY

100 Emission/Reception boards for THERAPY+IMAGING

Aperture 180 mm

Focal dist. 140 mm

Correction of skull aberrations using an implanted hydrophone

Experimental scan

without correctionExperimental scan with correction

(TR + Amplitude compensation)

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

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

High Power Time Reversal Mirror for Therapy

Résultats

• Examen IRM

n Examen histologique

Sonoluminescence

Validity of the model:

L < Lshock Time reversal invariance

L > Lshock Discontinuity formation and

dissipative phenomena

021

2

22

4

02

2

2

0

=∂

∂+

∂

∂−∆

t

p

ct

trp

c

trp

ρ

β),(),(

r

r

Non-linear acoustics and time reversal invariance

),(),(00

0 txpc

ctxc aρβ+=

Z= z -c0 t

P(Z)

0

02shock

cL

v

λπ β

=

M. Tanter, J.L. Thomas, F. Coulouvrat, M. Fink

Westervelt, homogeneous medium1963

1D experimental results : Reversibility before shockformation

6.5 7 7.5 8 8.5 9 9.5 10-2

0

2

6.5 7 7.5 8 8.5 9 9.5 10-2

0

2

6.5 7 7.5 8 8.5 9 9.5 10-2

0

2

Z = -0,75 m

6.5 7 7.5 8 8.5 9 9.5 10-2

0

2

6.5 7 7.5 8 8.5 9 9.5 10-2

0

2

6.5 7 7.5 8 8.5 9 9.5 10-2

0

2

Z = -0,4 m

Z = -0,02 m

Z = 0,75 m

Z = 0,4 m

Z = 0,02 m

Fo

rward

Pro

paga

tion

Ba

cw

ard

Pro

paga

tion

L/Lshock = 0,75

p = 1,5 Atm

Experimental results : Irreversibility after shockformation

6.5 7 7.5 8 8.5 9 9.5 10

-5

0

5

6.5 7 7.5 8 8.5 9 9.5 10

-5

0

5

6.5 7 7.5 8 8.5 9 9.5 10

-5

0

5

6.5 7 7.5 8 8.5 9 9.5 10

-5

0

5

6.5 7 7.5 8 8.5 9 9.5 10

-5

0

5

6.5 7 7.5 8 8.5 9 9.5 10

-5

0

5

Z = -0,75 m

Z = -0,4 m

Z = -0,02 m

Z = 0,75 m

Z = 0,4 m

Z = 0,02 m

Fo

rward

Pro

paga

tion

Ba

cw

ard

Pro

paga

tion

L/Lshock = 2,3

p = 5 Atm

Application to medical imaging : tissue harmonics cancellation ?

Transmit Focus M.I. = 1 Record backscattered echoes Time reversal + emission

f0f0 + 2f0f0 + 2f0

Frequency (MHz)4.33.2 6.4

70 % bandwidth

First Problem : signals suffer two times the transducer bandwidth

Correct the effects of the transducer bandwidth

Fully programmable

E/R electronics !!

Harmonic cancellation by Time Reversal

128 elts., 4.3 MHz, pitch 0.33 mm

F = 40 mm

Thin copper

filament

Water

Initial

Emission(f = 3.2 MHz)

Backscattered

signals

Backscaterred

signals

Classical medical ultrasound probe

Time (µs)

T.R. +

Bw correction

+ Re-emission

Amax

A1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

A1/Amax

Harmonic ampl.

Fundamental ampl.

Electromagnetic TRM

In dissipative medium

Breaking time reversal symmetry : Dissipation

• Time reversal remains a matched filter :

For a given emission energy, it maximizes the

acoustic pressure received at focus

• BUT, it is no more an inverse filter of the

propagation

G. Montaldo; M. Tanter, M. Fink

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 the

objective

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

plitu

de 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

Absorbing

And aberrating

Ureol sample

128 elts.

1.5 MHz

Pitch 0.5 mm

Distance in mm

Tim

e in

s

10 20 30 40 50

1

2

3

4

5

6

7 -35

-30

-25

-20

-15

-10

-5

0

Distance in mm

10 20 30 40 50

1

2

3

4

5

6

7

Time Reversal

Focusing

Focusing after 30

iterations

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 Skull

Optimal signal to transmit Classical Cylindrical law

Transducernumber j

1 12

8Transducernumber 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)

Pre

ssu

re (

dB

)

Spatial focusing

Limits of classical Iterative Time Reversal

Plane wave

illumination

Backscattered

echoes

Strongest scatterer

is finally selected

Transmited time

reversed echoes

frequency

PROBLEM

1) Temporal spreading of the signals due to the transducers

bandwidth (signals become step by step monochromatic)

2) How to focus on the others scatterers

1

2

3

1 23

DistanceT

ime

The Modified Iterative Method

Plane wave illumination

of 3 targetsEchoes of the

3 targets

G. Montaldo, M. Tanter, M. Fink

Tim

e

Distance

Time Rev.

Focusing

1st diffuser

1 wavefront

selection

Filtering

1st diffuser

Time Rev.

Focusing

2nd diffuser

1 wavefront

selection

1 wavefront

selection

Time Rev.

Focusing

3rd diffuser

Filtering

1st and 2nd diffuser

The Modified Iterative Method

Tim

e in

s0

20

0 50Distance in mm

3 scatterers

of 0.5

Aberrating

mask

Application : Focusing through aberrating media

Echoes of a

plane wave

illumination

10 20 30 40 50

2

4

6

Tim

e (

µs)

(a)

00

Axial position (mm)

10 20 30 40

2

4

6

8

10

12

Axial position (mm)

Dep

th(m

m)

(c)

0

Multiple targets identification in speckle noise

Echographic image of the

Phantom with 8 wires

Pulse echo signals

Identification of the

8 waveforms

Calculated positions of

the targets after identification

of the waveforms.

Experiments at 4 MHz on a medical test phantom

Could be achieved in a few ms !!!!

Influence of the trabecular bone on the acousticpropagation

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

Experimental results

Time reversal through the skull

-20 -10 0 10 20-35

-30

-25

-20

-15

-10

-5

0

Distance from the initial point source (mm)

Pre

ssu

re (

dB

)

in waterthrough the skull: cylindrical lawthrough the skull: time reversal

Theory

Time reversal in a dissipative medium

Wave equation in fluids :

0),(

)(

1

)(

),()()(1

2

2

2=

∂∂−

∂∂+

t

trp

rcr

trpgraddivr

tr

ρρτ

1Tra

nsu

cer

nu

mb

er

i

0

Received wave front Corrected wave front127

Time (µs) Time (µs)

Tra

nsu

cer

nu

mb

er

i

0

127

Hydrophone

Thin Aberrating

and Absorbing Layer

Array of transducers

t

Loss

Amplitude

Ai

Gain

Amplitude

1/Ai

Amplitude compensation

Breaking the time reversal invariance

Experimental results

Time reversal in a dissipative medium

-20 -10 0 10 20-35

-30

-25

-20

-15

-10

-5

0

Distance from the initial point source (mm)

Pre

ssu

re (

dB

)

in waterthrough the skull: time reversalthrough the skull: time reversal + amplitude compensation

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 the

objective

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

plitu

de 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

Absorbing

And aberrating

Ureol sample

128 elts.

1.5 MHz

Pitch 0.5 mm

Distance in mm

Tim

e in

s

10 20 30 40 50

1

2

3

4

5

6

7 -35

-30

-25

-20

-15

-10

-5

0

Distance in mm

10 20 30 40 50

1

2

3

4

5

6

7

Time Reversal

Focusing

Focusing after 30

iterations

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 Skull

Optimal signal to transmit Classical Cylindrical law

Transducernumber j

1 12

8Transducernumber 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)

Pre

ssu

re (

dB

)

Spatial focusing

Time Reversal in Optics

There is no linear and instanteneous detector in optics :

1. the detectors are slow compared to the period of the wave and there are only sensitive to the average energy (quadratic)

2. We have to use monochromatic wave that will interfere with a reference plane

wave to give a phase information on a non linear material.

Principle of Monochromatic Holography

How to measure the phase of

any incident wave and to phase conjugated ? :

A two step process :

1. recording with the nonlinearproperty of the film the interference

between the incident wave and a

reference plane wave

2. Illuminating the film with a contrapropagating plane wave and

The interationwith the Hologram

creates the phase-conjugated wave

PCM

PCM

1

1

The Magic Mirror

Stationnary Regime

C1 C2

1

1

Scatterer

Aberrating mediumEMISSION

RECEPTION

diverging wave

EMISSION

converging wave

window selection

Transducer array

Time Reversal in Pulse Echo mode : 1 target

Multi target mediumTransmission 1

A

B

Reception 1

a

b

Transmission 2

a

b

Reception 2

a2

b2

Transmission 3

a2

b2

Reception 3

a3

Time reversal

Time reversal

Iterative Time Reversal on multi target medium

J.L. Thomas, F. Wu, M. Fink

Application of TRM to Lithotripsy

E m iss ion 1

defect

E m iss ion 2 : after tim e reversal

transducers a rray so lid sam ple

tim e

R éception 1

tim e

tim e

R éception 2

tim e

Applications to defect detectionin titanium alloy (SNECMA)

Time Reversal Mirror in non-destructive testing

F. Wu, D. Cassereau, N. Chakroun, V. Miette, M. Fink

86 mm

103 mm

Axe y

Axe x

iteration 0

iteration 1

iteration 2

Zone witha flat bottom holeat 140mm depth

Zone withoutdefect(speckle)

ch

an

nels

Iterative time reversal in titanium alloy

1

128

time

Limits of classical Iterative Time Reversal

Plane wave

illumination

Backscattered

echoes

Strongest scatterer

is finally selected

Transmited time

reversed echoes

frequency

PROBLEM

1) Temporal spreading of the signals due to the transducers

bandwidth (signals become step by step monochromatic)

2) How to focus on the others scatterers

1

2

3

1 23

DistanceT

ime

The Modified Iterative Method

Plane wave illumination

of 3 targetsEchoes of the

3 targets

G. Montaldo, M. Tanter, M. Fink

Tim

e

Distance

Time Rev.

Focusing

1st diffuser

1 wavefront

selection

Filtering

1st diffuser

Time Rev.

Focusing

2nd diffuser

1 wavefront

selection

1 wavefront

selection

Time Rev.

Focusing

3rd diffuser

Filtering

1st and 2nd diffuser

The Modified Iterative Method

SD>

-20 -15 -10 -5 0 5(dB ref. max level)

10

20

30

40

50

60

70

80

90

1000 10 20 30 40 50

Time (ms)D

ep

th (m

)

Range= 7.195 km21-JUL-1999 17:14:14.00

10

20

30

40

50

60

70

80

90

1000 10 20 30 40 50

Time (ms)

Dep

th (m

)

Range= 7.97 km21-JUL-1999 17:13:41.00

f=3500 Hz

SRA VRA

TIME REVERSAL OF A DISPERSED PULSE

Time

Sp

ace

Sp

ace

Sp

ace

How to build the cancellation operator ?

e(x,t)

Original signal with some targets

)()]([),( 111 xAxttxw τδ +=A ‘single waveform’ is selected

xdtdttxwtxetP ′′−′′′′= ∫∫ ),(),()( 1

Weight of the waveform at each time t

∫ ′′−′= tdttxwtPtxD ),()(),( 11

Building the echoes of the first target

),(),(),( 1

1 txDtxetxe −=F

Substraction from the original signal

Tim

e in

s0

20

0 50Distance in mm

3 scatterers

of 0.5

Aberrating

mask

Application : Focusing through aberrating media

Echoes of a

plane wave

illumination

A general approach : Backscattering Operator

array of N*Ntransmittersreceivers)(tδ : transmitted on

channel m

: received onchannel l.

R( )=K( )E( )

E( ) and R( ) vector signals,

K( ) is the N × transfer matrix .

Transmitted signals: em(t)

Received signals:

rl(t) = ∑=

N

m 1

klm(t) ⊗ em(t) , Ll ≤≤1

NxN inter element impulse responses : k lm(t)

N

C. Prada, M. Fink

klm(t)

Spatial Reciprocity => K( ) is symmetrical

In the frequency domain

The Backscattering Time Reversal Operator

Transmission

Input E

Reception

Output K E

Transmission

Input K* E*

time reversal

Reception

Output K K* E*

K*

K:

Time Reversal Operator

Iterations of the Time Reversal Operation

EMISSION 0

RECEPTION 0

Eo

Ro=KEo

Iteration 0

EMISSION 1

RECEPTION 1

E1=K*Eo*

R1=KE1

Iteration 1

...

⇒E2n=[K*K]nEoIteration 2n:...

Eigenvalues : depend on target reflectivities

Eigenvectors : waveforms transmitted by the array to focus on each target

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

0

0.5

1

-50 -25 0 25 50

time (µs)

-30

-25

-20

-15

-10

-5

0

-12 -6 0 6 12 (mm)

dB 8 bit

One bit

Inverse filter through Skull

NmJ

jjmjm tethtf ≤≤

=∑ ⊗= 1

1

)()()( ωω ωdetej

tj∫= )(Ej)(

Fourier Transform Inverse Fourier Transform

)()(H)( ωωω EF = )()()( ωωω FE1-

H=Inversion

at each frequency

H(ω)

1

m

j

N

1

N

hmj(t)

Array of transmittersSet of receivers in the focal plane

E(ω) F(ω)

LD

F

One channel time reversal mirror as an estimateof a spatial correlator

R0

R1

O Time reversal mirrorSource

Observation point

ΦΦΦΦTR (R1,R0,t) = g(O,R0,-t) * g(R1,O, t)

Time reversal field observed at point R1 coming from a source at R0

t

R0

R1

O sourceObserver 0

Observer 1

C (R1,R0,t) = g(R0,O,-t) * g(R1,O, t)

)sin(

)()(),,( 00 ∑=n n

nnn

tROtROg

ωωψψ

)()()()()()( 210

210 ORRORR

nnnnnnψψψψψψ =

∫ +=2

1

),,(),,(),( 1010

t

t

TR

R ORgtORgdtR τττφ

∑==n

nnn

n

TRORRtR )()()(

1)0,( 2

102

1 ψψψω

φ

)/2( 010 n

RRJ λπ −

R0O Time reversal mirror

Source

Observation point

Self averaging in a Chaotic Cavity

R1

nψ eigenmodesGreen function

A one channel TR experiment gives a TR field

Average over realizations of chaotic cavities ?)0,( 1RTR

φ

If chaotic rays support irregular modes, Berry Conjecture

Experimentally one realization is enough to observe the spatial correlation

TIME REVERSAL IS SELF –AVERAGING many uncorellated eigenmodes =400

C. Draeger, J.de Rosny, M. Fink

The Cavity Formula

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

AB

In terms of the cavity modesA and B cannot exchange

all informations, because

A and B are always at theAntinodes of some modes

nψ eigenmodes

)sin(

)()(),,( ∑=n n

nnn

tBAtABg

ωωψψ

Carsten Draeger