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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
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