time reversed acoustics - mathias fink
Post on 10-May-2015
31.096 Views
Preview:
TRANSCRIPT
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
top related