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Xiaoping Jia, Arnaud Tourin, Mathias Fink, Emmanuel Fort, Antonin Eddi
To cite this version:
Vincent Bacot, Sander Wildeman, Surabhi Sreenivas, Maxime Harazi, Xiaoping Jia, et al.. Dissipation driven time reversal for waves. 2018. �hal-01755880�
Dissipation driven time reversal for waves
Vincent Bacot1,+, Sander Wildeman1,2, Surabhi Kottigegollahalli Sreenivas1,2, Maxime Harazi1,*, Xiaoping Jia1, Arnaud Tourin1, Mathias Fink1, Antonin Eddi2 & Emmanuel
Fort1,#
1 Institut Langevin, ESPCI Paris, CNRS, PSL Research University, 1 rue Jussieu, 75005 Paris, France
2 Laboratoire de Physique et Mécanique des Milieux Hétérogènes, CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot - Paris, France
+ Current address: LadHyX, CNRS, École polytechnique, 91128 Palaiseau, France
* Current address: Univ. Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
Abstract:
Dissipation is usually associated with irreversibility. Here we present a counter-intuitive concept to perform wave time reversal using a dissipation impulse. A sudden and strong modification of the damping, localized in time, in the propagating medium generates a counter-propagating time-reversed version of the initial wave. In the limit of a high dissipation shock, it amounts to a ‘freezing’ of the wave, where the initial wave field is retained while its time derivative is set to zero at the time of the impulse. The initial wave then splits into two waves with identical profiles, but with opposite time evolution. In contrast with other time reversal methods, the present technique produces an exact time reversal of the initial wave field, compatible with broad-band time reversal. Experiments performed with interacting magnets placed on a tunable air cushion give a proof of concept. Simulations show the ability to perform time reversal in 2D complex media.
Dissipation in physical systems introduces a time-irreversibility by breaking the time symmetry of their dynamics [1] [2] [3] [4] [5]. Similarly damping in wave propagation deteriorates time-reversal (TR) operations. However, for short times compared to the characteristic dissipation time, the dissipation-free wave-propagation model remains a good approximation of the wave dynamics [6]. Thus, increasing dissipation can generally be seen as effectively reducing the reversibility of the system. On the other hand, dissipation can be seen as the time reverse of gain. This is exemplified in the concept of coherent perfect absorbers (CPA), combining an interplay between interferences and absorption to produce the equivalent of a time-reversed laser [7] [8]. CPA addresses the TR of sources into sinks [9].
In this paper, we present a concept in which dissipation produces a time reversed wave. The generation process is similar to that employed in the Instantaneous Time Mirror (ITM) approach [10] in which a sudden change in the wave propagation speed in the entire medium produces a time reversed wave. Here, we consider instead a dynamical change of the dissipation coefficient. We will show that in this case also the production mechanism can be interpreted as a modification of the initial conditions characterizing the system at a given instant by virtue of Cauchy’s theorem [11]. In the following we will refer to this new, dissipation based, TR concept as DTR, for Dissipation Time Reversal. DTR can, in principle, be implemented for any type of wave propagating in a (dissipative) medium. For example, for acoustic waves, through a modulation of the viscosity, and for EM waves, through a sudden change in conductivity in the medium. Because the source term involved is a first order time derivative, this new TR technique is able to create an exact TR wave, i.e. precisely proportional to the initial wave field. This results in a higher fidelity and enhanced broadband capabilities compared to other methods [10] [12]. In the following, we demonstrate the validity of DTR concept performing a proof-of-concept 1D
experiment using a chain of coupled magnets. In addition, we show the performance of DTR in a complex 2D highly scattering media using computer simulations.
Theory of Dissipation driven Time Reversal (DTR)
Waves in homogeneous non-dissipative medium are usually governed by d’Alembert wave equation [13]. More generally, the wave fields can be described by an equation of the same structure, but in which the Laplacian operator is replaced by a more complex spatial operator. This is the case for instance when describing acoustic waves in a non-homogeneous medium or with gravity-capillary waves at the surface of deep water. This type of equations can be written in a general manner in the spatial Fourier space for the wave vector 𝒌 [14] [15]:
!!!
!!! 𝒌, 𝑡 + 𝜔!! 𝒌 𝜙 𝒌, 𝑡 = 0, (1)
where 𝜔! 𝒌 is the dispersion relation of the waves. The time reversal symmetry is a direct consequence of the second order of the time derivative: if 𝜙 𝒌, 𝑡 is a solution of the equation, 𝜙 𝒌, −𝑡 obviously satisfies the same equation. The damping effect induced by dissipation is usually described by an additional first order derivative term in this equation:
!!!
!!! 𝒌, 𝑡 + 𝜁 𝒌, 𝑡
!!
!" 𝒌, 𝑡 + 𝜔!
! 𝒌 𝜙 𝒌, 𝑡 = 0, (2)
where 𝜁 𝒌, 𝑡 is time-dependent damping coefficient. This additional dissipation term breaks the time symmetry of the equation which is precisely why dissipation is generally associated with irreversibility. For simplicity, we remove the dissipation k-dependence in the following notations. If 𝜁 𝑡 remains small compared to 𝜔!, an approximate reversibility is retained for times smaller
than 1/𝜁 𝑡 [10]. In the following, we consider a medium where the damping coefficient is initially small or negligible, i.e. 𝜁~0. At a time 𝑡!, the dissipation coefficient is set to a very high
value over the entire medium 𝜁 ≫ 𝜔! and stays at this value until a later time 𝑡! = 𝑡!+ 𝛥𝑡, where it is set back to its original value. 𝜁 𝑡 can thus be written as 𝜁 𝑡 = 𝜁. Π(𝑡), where Π(𝑡) is a unit rectangle function spanning from 𝑡! to 𝑡!. An initial wave 𝜙! with a Fourier transform 𝜙! is
originally present in the medium. During the dissipation impulse, the last term of equation (2) is negligible and one may write the approximate expressions for the wave field and its time derivative at times t after the damping impulse at time 𝑡! as:
𝜙 𝑡 = 𝜙! 𝑡! +!!!!! !" 𝑡! 1 − 𝑒!! !!!! !! !" 𝑡 = !!! !" 𝑡! 𝑒 !! !!!! . (3)
Taking 𝜁 towards infinity, we obtain that the field remains approximately constant 𝜙 𝑡 ≈ 𝜙! 𝑡! during this dissipation impulse. The system is described as an overdamped harmonic oscillator that returns very slowly to a steady equilibrium state without oscillating. In this regime, the characteristic oscillations corresponding to wave motion are stopped and the time it takes for the system to relax increases with dissipation so that in the high dissipation limit we are considering, amplitude damping does not have time to occur. A more detailed calculation shows that in the
long run the amplitude decreases as exp −!!!
!! 𝑡 − 𝑡! (see supplemental material). For the
strong dissipation limit to hold and for the wave amplitude to be retained, the duration 𝛥𝑡 of the dissipation pulse should thus satisfy 1/𝜁 < 𝛥𝑡 < 𝜁/𝜔!!. If the damping is strong enough, the
duration of the dissipation phase may be large compared to the period of the original wave. At time 𝑡!, when the dissipation ends, the wave field starts evolving again according to equation (2), with the initial conditions:
The DTR process can be interpreted as a change of the initial Cauchy conditions which characterizes the future evolution of the wave field from this initial time 𝑡!. As in the case of ITM [10], this state can be decomposed into two counter-propagative wave components using the superposition principle:
𝜙 𝑡! ,!!!" 𝑡! = !! 𝜙! 𝑡! ,!!! !" 𝑡! + ! ! 𝜙! 𝑡! , − !!! !" 𝑡! . (5)
The first term is associated (up to a factor one half) to the exact state of the incident wave field before the DTR, it corresponds to the same wave shifted in time: 𝜙! 𝑡 = !
!𝜙! 𝑡 − 𝑡!+ 𝑡! . The
second term is associated to a wave whose derivative has a minus sign. It corresponds to the time reversed wave: 𝜙! 𝑡 = !!𝜙! 𝑡!+ 𝑡! − 𝑡 .
Figure 1 shows the principle of DTR with a chirped pulse containing several frequencies. The pulse created at time 𝑡 = 0 propagates in a dispersive medium undergoing spreading (see figure 1a). At time 𝑡!"#, a damping impulse is applied and the pulse is frozen (see figure 1b). The damping is then removed resulting in the creation of two counter-propagating pulses with half the amplitude of the initial wave field. The forward propagating pulse is identical to the initial propagating pulse as if no DTR was applied apart from the amplitude factor. The backward propagating pulse is the TR version of the initial pulse. It thus narrows as it propagates, reversing the dispersion, until at time 2𝑡!"# it returns to the initial profile (with a factor of one half in the amplitude).
1D proof-of-concept experiment using a chain of magnets
We have performed a 1D experiment using a chain of magnets as a proof-of-concept for DTR. Figure 2a shows the experimental set-up composed of a circular chain of 41 plastic disks. A small
magnet oriented vertically with their North pole upward is glued on each 1 cm disk. The disks orientation being constrained, it induces a repulsive interaction between them. The disks are confined horizontally on a circle of 30 cm diameter by an underlying circle of fixed magnets with their North pole oriented upward. In addition, the friction of the disks on the table is drastically reduced by the use of an air cushion system. This allows waves to propagate in this coupled oscillator chain. A computer-controlled electromagnet is used to trigger the longitudinal waves in the chain. The change in dissipation is obtained by a sudden stop the air flow using the controlled valve which results in “freezing” the disk chain. Thus, in that case the damping coefficient 𝜁 can be considered infinite.
For small amplitude oscillations, the chain can be modeled has a system of coupled harmonic oscillators with a rigidity constant κ which can be fitted from the dispersion curve measurements. When the air pressure is slightly increased, the disks are set in the random motion as shown in Figure 2b, which presents the displacements of the disks as a function of time in gray scale. It is possible to retrieve from this noise motion the dispersion curve of the chain by performing a space-time Fourier transform. The color scale is given in degree along the circle. One degree is approximately equivalent to 2.6 mm in length. Figure 2c shows the resulting experimental dispersion curve together with the fit of the harmonic model (dashed line). The good agreement of the fit confirms the validity of and harmonic interaction and permits to give a value for the rigidity constant of κ=0.13 Ν.m-1.
Figure 3a shows the propagation of an initially localized perturbation of the magnetic chain (the magnet #37 is acting as a source). A wave packet is launched in the magnetic chain. At time t0 the
chain is suddenly damped by turning off the air flow resulting in a freezing of the disks motion. At time t1 the air flow is restored resulting in the creation of two counter propagating wave
packets: i) one which resembles the initial wave packet as if no DTR had been applied apart from a decrease of the global amplitude by a factor of approximately 2; ii) a time reversed wave packet refocusing on the source. Figure 3b shows the disk displacements simulated using the harmonic interaction model with the fitted coupling constant. The initial magnet displacement is also taken from the experiment at time tinit. This signal processing enables one to remove the forward
propagating wave packet after the DTR. The resulting displacement pattern clearly shows the two counter propagating waves after the DTR impulse. Its similarity with the experimental data also shows the validity of the model.
Simulations of DTR in 2D disordered medium
We performed computer simulations for a 2D disordered system to show the robustness of TDR technique in complex media. The simulations are based on the mass-spring model of Harazi et
al. [16] [17] introduced to simulate wave propagation in disordered stressed granular packings.
The model consists of a two-dimensional percolated network of point particle with mass m, connected by linear springs of random stiffness as shown in the schematics in Figure 4a. The masses are able to move in the plane of the network and are randomly placed on a 70x70 square lattice with a filling factor of 91%. Each mass of the network is subjected to the Newton’s equation: !!𝒓! !!! = 𝜔!"( 𝒓!" − 𝑎!) 𝒓!" 𝒓!" !∈!! , (6)
where ri is the vector position of particle i, Vi the set of neighboring particles connected to this
particle, 𝜔!" is the angular frequency and and 𝑎! the rest length of the spring, 𝒓!" and 𝒓!" are
𝜔!" are uniformly distributed between 0.5 𝜔!" and 1.5 𝜔!" , where 𝜔!! is the average angular frequency of the spring-mass systems.
Before launching the wave, the network is submitted to a static stress by pulling the four walls of the domain (strain equal to 0.2) in order to ease the propagation of transverse waves. After this phase, the boundaries of the domain are fixed (zero displacement). The network is then excited by a horizontal displacement of one of the particle during a finite time. The profile of the excitation is given by 𝑢(𝑡) = W 𝑡 cos 𝜔!𝑡 , where W 𝑡 is a temporal window restricting the oscillation to a single period and 𝜔! is the driving source pulsation chosen at 0.35 of the average angular frequency 𝜔!" of the spring-mass systems. Figure 4b and 4c show the evolution of the
horizontal displacement with time of the source and the map of the displacements of the particles at various times respectively. At t=0, the displacement is confined to the source particle. Then, the displacement of the source decreases rapidly at a noise level of approximately one tenth of its initial value. The initial perturbation propagates and is strongly scattered in the inhomogeneous mass-spring network. At time 𝑡 = 95 𝑇!, where 𝑇! = 2𝜋/𝜔! is the excitation period, the perturbation is spread over the network and the particle displacements become randomly distributed. The network acts as a complex medium due to the random spring stiffnesses and the random vacancies in the squared network.
The DTR freezing is applied at 𝑡!"# = 300 𝑇!, all the particle velocities are set to zero, as if an infinite damping was applied instantaneously, keeping only the potential energy in the system. Right after this instant, the masses are released from their frozen positions with zero velocity (see third panel Fig. 4d), and evolve according to equation (1). After a complex motion of the particles, a coherent field appears refocusing back to the initial source around time 𝑡 = 2𝑡!"#=600 𝑇! (see panel four Fig. 4d). The horizontal displacement of the source undergoes a
very sharp increase reaching approximately 80% of its initial value (see Fig. 4b). Right after this time, the converging wave diverges again, yielding to a new complex displacement field (see the movie of the propagation in supplemental material). In addition to the spatial focusing on the initial source, a time refocusing is also observed showing a temporal shortening of the initial impulsion at time 𝑡 = 2𝑡!"# (see Fig. 4c). The time width of the refocusing is approximately 6𝑇! (FWHM) i.e. twice the one of the source signal.
Discussion
The DTR process is associated with a decrease of the amplitude of the initial wave field. The kinetic energy associated to the time derivative of the field 𝜕𝜙! 𝜕𝑡 vanishes during the dissipation impulse leaving the potential energy, associated to the wave field 𝜙! unaffected. In
the case of an initially propagating wave, the energy of the wave is equally partitioned between potential and kinetic energy. Thus, half of the initial energy is lost in the DTR process resulting in a quarter of the initial energy being TR while an other quarter is retained in the initial wave. It is interesting to note that, for standing waves, the effect of the DTR depends on its relative time phase relative to the impulse since wave energy alternates between kinetic and potential.
The Cauchy analysis in terms of initial conditions to determine the wave field evolution enable one to make a link with Loschmidt’s Gedankenexperiment for particle evolution. Loschmidt imagined a deamon capable of instantaneously reversing the velocity of the particles of a gas while keeping their position unaffected and thus time reversing the gas evolution [18] [19]. Although this scheme is impossible in the case of particles due to the extreme sensitivity to initial conditions, it is more amenable for waves because they can often be described with a linear operator and any error in initial conditions will not suffer from chaotic behavior. The wave analogue of this Loschmidt daemon in terms of Cauchy’s initial condition 𝜙!, 𝜕𝜙! 𝜕𝑡 is to
change the sign of the wave field time derivative 𝜙!, − 𝜕𝜙! 𝜕𝑡 . Because of the superposition principle, the DTR is thus acting as a Loschmidt daemon by decoupling the wave field from its derivative.
The DTR concept is generic and applies in the case of complex inhomogeneous materials as shown by the 2D simulations (see Fig. 3). This can be shown directly from the Cauchy theorem. After the freezing the wave field initial condition are reset with a time derivative of the wave the wave field equal to zero. The superposition principle given in Eq. 5 holds.
In contrast with the ITM approach based on wave velocity changes [10] and standard time reversal cavities [12], the backward propagating wave is directly proportional to the TR of the original wave and not to the time reversal of its time derivative or antiderivative. From that perspective, DTR has thus no spectral limitations and can be applied for the TR of broadband wave packets removing one of the limitation of the existing TR techniques. The limitation in the TR spectral range comes from the ability to freeze the field sufficiently rapidly compared with the phase change in the wave packet, resulting in the maximum value for the time pulsation 𝜁 ≫ 𝜔!.
Conclusion
This paper presents a new way to perform an instantaneous time mirror using a dissipation modulation impulse. This concept is generic and could be applied to other type of waves. In optics, DTR could be induced by changing abruptly the conductivity of the medium as in graphene [20], in acoustics it could be obtained by changing electrorheological medium [21].
We are very grateful to Y. Couder for fruitful and stimulating discussions. We thank A. Fourgeaud for their help in building the experimental set-up. S. K. S. acknowledge the French Ambassy in India for a Charpak scholarship. The authors acknowledge the support of the AXA research fund and LABEX WIFI (Laboratory of Excellence ANR-10-LABX-24) within the French Program ‘Investments for the Future’ under reference ANR-10-IDEX-0001-02 PSL*
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Figure captions:
Figure 1: Principle of DTR mirror: a) Propagation of a chirped pulse in a dispersive medium at 3
different times. b) At time tDTR a strong dissipation impulse is applied, freezing the propagating
pulse (red curve). The pulse splits into two counter propagating pulses when the damping is stopped. At time 2tDTR, the counter-propagating pulse (blue on the left) is identical to the initial
one (at t=0, dotted line) but propagating in the other direction and with half the initial amplitude. It is thus time reversed (compensating the dispersion). The profile of the forward propagating pulse (blue on the right) is identical to that of the initial pulse propagating during 2tDTR (dotted
line) in a medium without DTR impulse, with half the amplitude.
Figure 2: a) Schematics of the experimental set-up composed of a circular chain of 41 plastic
disks with glued magnets oriented vertically. The disks are confined horizontally on a circle of 30 cm diameter by an underlying circle of fixed magnets and separated by approximately 8.8° (i.e. 2.3 cm). In addition, the friction of the disks on the table is drastically reduced by the use of an air cushion system. The change in dissipation is obtained by a sudden stop the air flow; b) displacement amplitudes of the magnets as a function of time represented excited by the air flow (noise). The displacement color map is in degrees along the circle taken from the rest position of the magnets; c) experimental dispersion curve obtained by Fourier transform of the noise induced magnet displacements (b). Dashed lines: fit of the harmonic model.
Figure 3: a) Displacement amplitudes of the magnets as a function of time represented. The
displacement color map is in degrees along the circle taken from the rest position of the magnets. The excitation is initially a localized perturbation of the magnetic chain, the magnet #37 is acting as a source. At time t0 the chain is suddenly damped by turning off the air flow resulting in a
counter propagating wave packets; b) Simulations of the magnet displacements using the harmonic interaction model with the fitted coupling constant. The initial magnet displacement is also taken from the experiment at time tinit.
Figure 4: a) Schematic view of the 2D spring model. The model consists of a 2D percolated
network of punctual masses placed in a squared lattice and connected by linear springs of random stiffnesses. The masses are free to move in the plane of the network. The network used in the simulations has dimensions 70x70 and the masses are placed at each site with a filling probability of 0.91; b) Evolution in time of the horizontal amplitude displacement of the source particle. The source is excited horizontally at time 𝑡 = 0 by a short pulse at 0.35 the average angular frequency of the spring-mass systems. The vertical dashed line represents the time of the DTR impulse 𝑡!"# = 300 𝑇! when the particle velocities are set to zero; c) Close up of the evolution of the source horizontal displacement (b) at the refocusing time 𝑡!"# = 600 𝑇! showing the time focusing; d) Panel representing the field displacement of the particles at various times 𝑡 = 0 (initial time), 95𝑇!, 𝑡!"# = 300 𝑇! (during the freezing impulse) and 𝑡 = 600 𝑇! (refocusing time);
Supplemental Material For 𝑡 ∈ 𝑡!, 𝑡! : !!! !!! 𝒌, 𝑡 + 𝜁 𝒌 !! !" 𝒌, 𝑡 + 𝜔! ! 𝒌 𝜙 𝒌, 𝑡 = 0, (S1)
We consider the regime of high dissipation, so that we assume ζ 𝒌 > 2𝜔! 𝒌 . (S1) is thus the equation of a damped harmonic oscillator in the overdamped regime, whose solutions are given by: 𝜙 𝒌, 𝑡 = 𝐴𝑒 !! 𝒌! !! !!!!!! 𝒌! 𝒌 (!!!!) + 𝐵𝑒 !! 𝒌! !! !!!!!! 𝒌! 𝒌 (!!!!) , (S2)
with 𝐴 and 𝐵 two constants. Given the initial conditions of continuity of the field and its time derivative, we obtain: 𝜙 𝒌, 𝑡 = !!!!!! 𝒌! 𝒌!! !! 𝒌,!! !! 𝒌! !!!!" 𝒌,!! ! !!!!!! 𝒌! 𝒌 𝑒 !! 𝒌! !! !!!!! 𝒌 !! 𝒌 (!!!!) + !!!!!! 𝒌! 𝒌!! !! 𝒌,!! !! 𝒌! !!!!" 𝒌,!! ! !!!!!! 𝒌! 𝒌 𝑒 !! 𝒌! !! !!!!!! 𝒌! 𝒌 (!!!!) . (S3)
Developing at first order in !! 𝒌
! 𝒌 → 0 in front of and (at order 2) inside the exponential terms:
𝜙 𝒌, 𝑡 = −! 𝒌! !!!
!" 𝒌, 𝑡! 𝑒
!! 𝒌! !!!!!!!! 𝒌! 𝒌!! !!!! 𝒌! 𝒌 !!!!
𝜙! 𝒌, 𝑡! +! 𝒌! !!! !" 𝒌, 𝑡! 𝑒 !! 𝒌! !!!!!! 𝒌! 𝒌!! !!!! 𝒌! 𝒌 !!!! , (S4) where we used the fact that ! 𝒌! !!!
!" 𝒌, 𝑡! is of order one in 𝜔! 𝒌 /ζ 𝒌 . Taking the zero th
order inside the exponential yields equation (4) of the main text. Equation (S4) also reveals that the wave amplitude decreases like 𝑒!!!!! 𝒌! 𝒌 !!!! in the long run.
