Time Reversed Time reversed signals x dB cm x-10 -5 0 5 10-30-25-20-15-10-5 0 Distance from the...

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Transcript of Time Reversed Time reversed signals x dB cm x-10 -5 0 5 10-30-25-20-15-10-5 0 Distance from the...

  • 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

    1 0

    grad p p r div

    r c r t ρ

    ρ   ∂− =  ∂ 

    r r 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−r is 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 tv ,

    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 λ Acoustic 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)

    A m

    p li

    tu d

    e A

    m p

    li tu

    d e

    A m

    p li

    tu d

    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

    c md B

    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 Phase Im

    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 50 100 150 200 250 Time (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 ~ λ 5 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

    elementary transducers

    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 ocean B. 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

    3 0 m

    1 0 0

    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

    accelerometer 100Hz

  • a m

    p li

    tu d

    e

    Green’s function:

    GA(t)

    Time Reversal:

    GA(-t)

    1 m

    1 m

    A

  • MEMORY

    10msa m

    p . GA(t)

    A

    B

    C

    a m

    p . GB(t)

    10ms

    a m

    p . GC(t)

    10ms

    Training step: library of Green functions

  • MEMORY

    a m

    p . GA(-t)

    a m

    p . GB(-t)

    a m

    p . GC(-t)

    a m

    p . GB

    ’(t)

    B

    am p

    . am

    p .

    am p

    .

    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 converging and diverging

    waves interfere Diverging only

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

    Monochromatic

    exp {j(kr+ωωωωt)} / r with singularity

    exp {j(-kr+ωωωωt)} / r with 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

    Propagating term

    Point-like source

    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.