## Abstract

Time-reversal invariance can be exploited in wave physics to control wave propagation in complex media. Because time and space play a similar role in wave propagation, time-reversed waves can be obtained by manipulating spatial boundaries or by manipulating time boundaries. The two dual approaches will be discussed in this paper. The first approach uses ‘time-reversal mirrors’ with a wave manipulation along a spatial boundary sampled by a finite number of antennas. Related to this method, the role of the spatio-temporal degrees of freedom of the wavefield will be emphasized. In a second approach, waves are manipulated from a time boundary and we show that ‘instantaneous time mirrors’, mimicking the Loschmidt point of view, simultaneously acting in the entire space at once can also radiate time-reversed waves.

## 1. Introduction

It is well known that, although classical mechanics equations are reversible on the microscopic scale, it is difficult to test the time-reversal invariance of a complex system of particles. It requires introducing a ‘mirror’ of the time variable. In an ‘image world’, such mirror exists by taking a motion picture of the system and then running the film backward. In the ‘real world’, the reversal of the motion can be ascribed to the sudden change in initial conditions of the complex system. It was the Austrian physicist J. Loschmidt who first challenged Boltzmann’s attempt to describe irreversible macroscopic processes with reversible microscopic equations that imagined daemons capable of inversing instantaneously all velocities of all particles in a gas. The daemons have to prepare the system in a new initial condition where all the velocities are reversed. However, such a proposal remains mainly a Gedanken experiment; the extreme sensitivity to initial conditions that lies at the heart of chaotic phenomena in nonlinear dynamics renders any such particulate scheme impossible to realize.

Time-reversal invariance not only occurs in the physics of massive particles, but also occurs in wave physics to the extent that waves propagate without any dissipative process. Waves are more amenable to time-reversal schemes since they can be described in many situations by a linear operator and any error in the initial conditions will not suffer from chaotic behaviour. It is interesting to note that both the holographic principle and the time-reversal mirror (TRM) approach are mainly based on the time-reversal invariance of wave equations. They also rely on the fact that any wavefield can be completely determined within a volume knowing the field (and its normal derivative) on any enclosing surface [1,2]. Hence, information reaching the two-dimensional surface is sufficient to recover all the fields inside the whole volume. Based on these properties, Gabor [1] introduced the holographic method providing an elegant way to back-propagate a monochromatic wavefield towards its initial source and to obtain a three-dimensional image of any radiating object. TRMs exploit also the same principles for broadband wavefield to physically create a time-reversed wave that exactly refocuses back, in space and time, to the original source regardless of the complexity of the medium as if time were going backwards. This latter approach has been implemented with acoustic [3,4], electromagnetic [5] and water waves [6,7]. It requires the use of emitter–receptor antennas positioned on an arbitrary enclosing surface. The wave is recorded, digitized, stored, time-reversed and rebroadcasted by the same antenna array. If the array intercepts the entire forward wave with a good spatial sampling, it generates a perfect backward-propagating copy. Note that for optical waves, this processing is difficult to implement [8,9] and the standard solution is to work with monochromatic light and to use nonlinear regimes such as three-wave of four-wave mixing [10,11].

We have seen that the Loschmidt approach to design a ‘time machine’ for particles is based on the sudden change in initial conditions of a complex system. The motion or ‘state’ of an *N* particle system is determined at one time by the *N* positions and *N* velocities of the particles as a point in a 6*N* phase space described by {**r**_{i},**v**_{i}}. The Loschmidt daemons prepare a new initial condition for the *N* particles as {**r**_{i},−**v**_{i}} by reversing the *N* velocities.

In order to understand the way to design such a ‘time machine’ for waves, we have to take into account the exact nature of the wave equation. It is a partial differential equation of order 2 and the wavefield *φ*(**r**,*t*) is a continuous function of four variables (three dimensions in space and one in time). Therefore, the solution to the wave equation has to be a function described over a ‘hypervolume’ with four variables and its boundary is a ‘hypersurface’ with three variables [12].

In fact, there are two possibilities for the three variables of the hypersurface. It can be two spatial dimensions and one time dimension. This is the so-called Cauchy boundary conditions (BCs) that prescribe both the field *φ*(**r**,*t*) and its normal derivative ∂*φ*(**r**,*t*)/∂*n* for **r** along a two-dimensional surface **S** for all time *t*. Once we know {*ϕ*(**r**∈**S**,*t*),∂*ϕ*(**r**∈**S**,*t*)/∂*n*}, we can use the Huygens–Fresnel–Helmholtz theorem to predict the field in all the hypervolume. The other possibility is a hypersurface described by three spatial dimensions where both the field *φ*(**r**,*t*_{m}) and its time derivative ∂*φ*(**r**,*t*_{m})/∂*t* for all **r** are prescribed at given time *t*_{m}. This is the Cauchy initial conditions (ICs) that are described by the wavefield state {*ϕ*,∂*φ*(**r**,*t*)/∂*t*}_{tm}. This is the initial value theorem that is classically called the Cauchy problem. These two possibilities of Cauchy conditions (BCs or ICs) give rise to two dual approaches to create a ‘time machine’ for waves.

The first approach uses the TRM concept that was initially developed and experimented with acoustic waves and later with microwaves. It refers to the manipulation of boundary conditions on a two-dimensional surface sampled by a finite number of antennas. In §2, it is described both theoretically and experimentally with an emphasis on the use of TRM in complex media to beat diffraction limits, thanks to medium complexity. The importance of the spatio-temporal degrees of freedom of a wavefield will be discussed.

Section 3 will be devoted to a second approach ‘à la Loschmidt’ with wave control by initial conditions manipulation. It refers to the concept of ‘instantaneous time mirrors’ that mimics the role of Loschmidt daemons, simultaneously acting in the entire space at once to radiate time-reversed waves and we will show experimental evidence of this process with water waves.

## 2. The time-reversal mirror approach ‘à la Huygens’

We have seen that there are two dual approaches to predict a wavefield inside a hypervolume. We will now focus on the first choice where we used BCs instead of ICs. We know that if we record an incoming wavefield along a two-dimensional surface during a sufficient time, the Huygens–Fresnel–Helmholtz–Kirchoff integral theorem allows recovery of the wavefield at any time (past and future) in the whole volume.

Therefore, a time-reversal experiment ‘à la Huygens’ can be conceived in the following way. During a first step an initial source radiates a transient wavefield inside a volume surrounded by a two-dimensional surface **S** along which the field and its normal derivative are recorded and stored as {*φ*(**r**∈**S**,*t*),∂*φ*(**r**∈**S**,*t*)/∂*n*} during a sufficient amount of time to be sure that the incoming field as completely disappear along **S**.

Employing a source term *s*(**r**,*t*), the radiated wavefield verifies the wave equation
2.1As we are in a causal universe *φ*(**r**,*t*) is of course the causal solution of (2.1). Besides, due to the time-reversal symmetry of the wave operator, there is also an anti-causal solution (the advanced one) that is never observed.

In the second step of the time-reversal experiment, our goal is to radiate from the boundary such an advanced solution. New boundary conditions have to be created along surface *S* with a wavefield oscillating in a time-reversed chronology compared with the chronology of the causal field. These time-reversed boundary conditions along *S* can be written as {*φ*(**r**∈**S**,*T*−*t*),∂*φ*(**r**∈**S**,*T*−*t*)/∂*n*}, where *T* is a causal delay needed to record and re-emit the signals. Experimentally, it requires the use of emitter–receptor antennas positioned on surface *S* that recorded, digitized, stored, time-reversed and rebroadcasted by the same antenna array both the field and its time derivative (figure 1).

These new boundary conditions radiated a wave that is going backward to the source: the so-called time-reversed field *φ*_{tr}(**r**,*t*). It can be computed using the Helmholtz–Kirchhoff integral that is valid inside a zone without source. Note that the refocusing time is expected at 2*T*, which is redefined in the following text as the new time origin.

We have to note that that this new wavefield verifies a homogeneous wave equation with the time-reversed BCs (the original source is no longer present):
2.2Therefore, it cannot be equal to *φ*(**r**,−*t*). In fact, it is not enough to time-reverse the wavefield on the boundary *S*. You also have to time-reverse the source term on equation (2.1) and *φ*(**r**,−*t*) indeed verifies the following equation deduced from equation (2.1) where *t* is replaced by −*t*:
2.3This means that in order to achieve a perfect time-reversal, both *the source has to be transformed into a sink* (the time-reversal of a source *s*(**r**,−*t*)) and also the *field and its normal derivative on surface S have to be time-reversed.* Note that if the source is impulsive one writes *s*(**r**,*t*)=*S*(**r**)*δ*(*t*) and as *δ*(*t*)=*δ*(−*t*) we easily find that [13]
2.4This important result is, in some way, disappointing because it means that reversing a wavefield using a closed TRM is not enough to radiate only the advanced wavefield. Complete time-reversal requires not only to time-reverse the field but also the original source. Equation (2.4) can be interpreted as the difference of advanced and retarded waves centred on the initial source position. The converging wave (advanced) collapses at the origin and is followed by a diverging (retarded) wave. Thus, the time-reversed field observed as a function of time shows two wavefronts of opposite sign. The wave re-emitted by the TRM looks like a convergent wavefield during a given period, but a wavefield *doesn’t know how to stop.* When the converging wavefield reaches the location of the initial source location, it collapses and then continues its propagation as a diverging wavefield.

In the case of punctual source term located at **r**_{0}, we obtain for the time-reversed field
2.5where *G*(**r**,**r**_{0};*t*) is the spatio-temporal Green’s function solution of equation (2.1) with an impulsive point-like source located in **r**_{0}. If the point-like source is monochromatic with pulsation *ω*, we obtain through a Fourier transform a field pattern
2.6with being the monochromatic Green’s function. The time-reversed field is then proportional to the imaginary part of the Green’s function and it is a universal result valid for any complex media provided that there is no dissipation. In free space, as the Green’s function is a diverging spherical wave, the focal spot is limited to one-half wavelength as is well known. Note that in complex media the field amplitude at the focal point is directly proportional to the LDOS, the so-called *local density of states* that depends on the medium complexity. Thus, the time-reversed field at the source point and at the focal time is directly proportional to the number of modes excited by the source.

To achieve a perfect time-reversal both the field on the surface of the cavity has to be time-reversed, and the source has to be transformed into a sink [14]. In this manner, one may achieve time-reversed focusing below the diffraction limit. The role of the new source term in equation (2.3) is to transmit a diverging wave that exactly cancels the outgoing spherical wave.

In a monochromatic approach, taking into account the evanescent waves concept, the necessity of replacing a source by a sink in the complete time-reversed operation can be interpreted as follows. In the first step, a point-like source of size much smaller than a wavelength radiates a field that can be described as a superposition of homogeneous plane waves propagating in the various directions and of decaying, nonpropagating, evanescent plane waves. The evanescent waves contain information on fine scale features of the source; they decay exponentially with distance and do not contribute in the far field. If the TRM is located in the far field of the source, the time-reversed field retransmitted by the mirror does not contain these evanescent components. The role of the sink is to radiate exactly, with good timing, the evanescent waves that have been lost during the first step. The resulting field contains the evanescent part that is needed to focus below diffraction limits. Time-reversal below the diffraction limit has been experimentally demonstrated in acoustics, using an acoustic sink placed at the focal point. Focal spots of size λ/14 have been observed by de Rosny & Fink [14]. One drawback is the need to use an active source at the focusing point to exactly cancel the usual diverging wave created during the focusing process.

Another solution is to create a passive sink that behaves as a perfect absorber of subwavelength size and it is a concept that is currently studied by different teams in optics and in acoustics [15–17].

### (a) Time-reversal mirrors in complex media: from theory to experiment

This theoretical approach described an ideal experiment where one covers the whole surface *S* with a collection of transmit/receive antenna measuring and transmitting the field and its normal derivative respecting the Nyquist criterion. It requires a considerable amount of electronics and it is then obviously not suitable for a practical implementations.

A first important simplification is to work with a TRM located in the far field of the source and of the medium heterogeneities. In this case, as the normal derivative of the field is proportional to the field, one only needs to measure the field on the boundary and time-reverse it.

A second important simplification occurs when the time-reversal operation is performed only on a limited angular area, thus apparently limiting the focusing quality. A TRM consists typically of a small number of elements, or time-reversal channels. However, as we will show, the major interest of TRM, compared with classical focusing devices (lenses and beam forming) is certainly the relation between the medium complexity and the size of the focal spot.

A TRM acts as an antenna that uses complex environments to appear wider than it is, resulting in a refocusing quality that does not depend on the TRM aperture. The reference experiment conducted with ultrasound using a small aperture TRM is the following [18–21]. A multiple scattering medium made of a forest of steel rods is placed between a point-like transducer and a TRM which are separated by a distance much larger than a wavelength (figure 2*a*). A short pulse of 1 μs is emitted by the point source, and the set of impulse responses from this point to the array of sensors (piezoelectric transducers) constituting the TRM is acquired. Note that those responses are spread over more than 200 μs, i.e. 200 times the initial pulse duration. This is due to the diffusive property of the multi-scattering medium. In such sample, the typical spreading time (the so-called Thouless time) is equal to *τ*_{thouless}=*L*^{2}/*D*, where *D* is a diffusion coefficient related to the mean free path and *L* is the thickness of the sample. Those responses are then flipped in time and sent back by the TRM acting now as a source. It is then possible to scan the field produced by this time-reversal focusing by translating the point source sensor parallel to the exit of the multiple scattering media. An impressive time compression is observed, since the time-reversed wave field at the focal point lasts about 1 μs compared with 200 μs. Besides, the typical spatial dimension of the focus obtained through the scattering sample is an order of magnitude lower than that measured in water (free space) as can be seen in figure 2*b*. This super-focusing result can be explained by the fact that the diffusive medium acts as a lens. When the scattering medium is inserted between the source and the focus, the waves emitted from the source diffuse. The diffusive halo acts as an array of Huygens secondary sources at the exit of the complex medium, along the interface that faces the focus. Now, since all those secondary sources have been phase-matched by time-reversal (or phase conjugation) to focus at the desired location, this secondary array of sources has the size of the whole medium, and a correspondingly smaller diffraction limit, 10 times smaller in free space. In this way, TRMs use complex environments to appear wider than they are, resulting in a refocusing quality that no longer depends on their own angular aperture but rather on the angular aperture of the scattering medium, which can be 4*π* steradians. The temporal and spatial signal to noise ratios of such focusing experiments have been shown to depend linearly on the total number of spatio-temporal degrees of freedom of the medium, i.e. *the number of uncorrelated sensors in the array times the number of uncorrelated frequencies in the bandwidth* [19]. In the bandwidth Δ*ω* of the transducers, the number of uncorrelated frequencies depends on the spectral correlation length *δω* of the scattered waves. Frequency components separated by *δω* are different degrees of freedom of the incident field. With an average spectral correlation function given by the Fourier transform of the ‘time of flight’ distribution, we get *δω*=*D*/*L*^{2} so the number of uncorrelated frequencies grows with *L*^{2}.

Very interestingly, the optics community has realized only recently that focusing through diffusive media can be performed either with monochromatic light or with broadband light, by means of spatial light modulators or with time-shaping light control. Focusing well below the Rayleigh limit of conventional lenses has been demonstrated using this concept [22–25].

Note that in all focusing experiments through diffusive media, we assume that the propagation medium does not change between the recording and the re-emitting steps. The applicability of the time-reversal process to focus in such medium is strongly relevant to the concept of fidelity introduced by Peres and to the study of Loschmidt echo [26–30].

What happens when the number of sensors of the TRM is reduced down to a single one? Regarding the mentioned optical experiments, the answer is quite obvious: a single laser source that is shone through a multiple scattering medium gives rise to a so-called speckle pattern, i.e. a random interference pattern. It is clear that no focusing can occur whatsoever in this case since changing the phase of this single source only changes the phase of the output speckle, hence precluding any focusing effect. The solution to this problem lies in the use of interference between statistically independent speckles, or equivalently between speckle patterns at frequencies which are uncorrelated. This is where time-reversal strongly differs from its monochromatic ancestor, namely, phase conjugation [25].

This very simple yet fascinating experiment is easier to understand considering another experiment conducted with elastic waves propagating inside a two-dimensional solid cavity as described in the set-up of figure 3 taken from [31]. An aluminium cone coupled with a transducer generates surface waves (in fact flexural waves) at one point **r**_{0} of a non-symmetrical and ergodic silicon cavity. A second transducer is used as a receiver. The central frequency *f*_{0} of the transducers is set to 1 MHz and their relative bandwidth to 100% (Δ*f*=1 MHz). The source is considered point-like and isotropic because the cone tip is much smaller than the central wavelength. A 1 μs pulse is emitted by the source and the impulse response is acquired at **r**, digitized. Then a 1000 μs long portion of the signal (the Heisenberg time of the cavity) is time-reversed and sent from point **r**. The time-reversed field generated is then scanned using an optical interferometer around point **r**_{0}. The spatio-temporal behaviour of time-reversed field is represented in figure 3*b*. At the collapse time, it is focused onto a spot whose width is about half a wavelength, limited by the diffraction limit. In other words, in this cavity, it is possible to reach the diffraction limit using a single sensor, thanks to time-reversal. This can actually be explained through a simple physical picture. Such a cavity can indeed be described in terms of modes, which are eigenmodes of the propagation and whose eigenvalues are their resonant frequencies. When a point source emits a short pulse in this medium (at point **r**_{0}), it excites all the modes that are non-zero at this location, each one at its resonant frequency, just as a guitar cord being slapped. Measuring the field at point **r** then simply consists of acquiring the phases and amplitudes of the relative eigenmodes of the cavity at this specific position, at every resonance frequency, and with the condition that all those modes are in phase at point **r**_{0} because of the pulsed excitation. The time-reversal focusing naturally comes out considering that time reversing an impulse response is equivalent to phase conjugating the signal at every frequency of the bandwidth: *f*(*t*) to *f*(−*t*) or *F*(*ω*) to *F**(*ω*). Indeed, sending the time-reversed impulse response from point **r**, we compensate the phase of each eigenmode at point **r**_{0}. This, in turn, ensures that all eigenmodes within the bandwidth interfere constructively at this point and at a specific time, the collapse time. From this very simple approach, one can also infer the signal-to-noise ratio of this focusing operation. This ratio is defined as the time-reversal peak energy divided by the standard deviation of the field measured away from **r**_{0} and at a time different from the collapse one; at this specific time and at point **r**_{0}, all the modes interfere coherently while everywhere else and at any other time they add up incoherently: the signal-to-noise ratio roughly equals the number of eigenmodes of the cavity within the bandwidth. Of course the focal spot width is given by the average correlation length of the modes, around half of the central wavelength. At this point, it is worth making the link between this modal approach and the results of the time-reversal cavity [21,32] linking the time-reversal produced field to the imaginary part of the Green’s functions of the medium. This result, independent of the complexity of the medium, was a monochromatic one which can easily be integrated over the bandwidth in order to obtain the spatially varying field after time-reversal and at the collapse time:
2.7Within a cavity or a complex medium where a modal analysis can be used, and if there is no degenerated modes (which is the case for a high enough quality factor chaotic cavity), Green’s functions are proportional to the eigenmodes. Hence integrating equation (2.7) is formally equivalent to adding coherently at point **r**_{0} the eigenmodes of the cavity that resonate in the bandwidth. Here also the cavity transforms a single point-like transducer in a ‘virtual lens’ of 4*π* steradians resulting in a spot as small as the diffraction limit. All these principles can be extended with other kind of waves such as microwaves and optical waves.

### (b) Focusing from the far field below the diffraction limit with resonant metalenses

We have seen that super-resolution can be achieved with an acoustic sink, but it has a severe drawback. It needs to use an active source at the focusing point to exactly cancel the usual diverging wave created during the focusing process or to locate a perfect absorber at the focusing point which limits strongly the interest of these techniques to applications.

We know that the time-reversal focusing spot at each frequency depends on the imaginary part of the Green’s function for any heterogeneous medium; another approach consists of putting in the near field of the source a medium of finite size made of a distribution of subwavelength resonators. By diffracting off the subwavelength resonators, evanescent waves can convert into propagative waves, which can then be detected in the far field. Therefore, the subwavelength coupling between a set of subwavelength resonators modifies the spatial dependence of the imaginary part of the Green’s function that now oscillates on scales much smaller than the wavelength. For a broadband pulse with enough frequency diversity, a time-reversal operation will generate at the focal time the coherent superposition of the imaginary parts of the Green’s functions at each frequency. At the source point, the time-reversed field is directly proportional to the number of modes excited inside the microstructure from the source. While at the other points, the oscillations of the imaginary parts at different frequencies cancel their effects. To predict the behaviour of time-reversal in such medium, we have to know the field correlations. A wave propagating in any medium can be characterized by a spatial correlation length that represents, at a given frequency, the smallest distance between two points which exhibits statistically different wavefields. If the correlation length of the medium is much smaller than the wavelength, one can achieve a focusing on a scale of the order of the correlation length of the medium. This is the concept of a resonant metalens, a lens capable of achieving subwavelength resolution from the far field for imaging and focusing applications.

We exploited this idea first in the field of microwaves [33–36] to create subwavelength focal spots. We consider eight possible focusing points placed in a strong reverberating chamber (figure 4*a*). Eight electromagnetic sources are placed at these eight locations to be used in the learning step of the time-reversal process. These sources consist of wire antennas used at a central frequency of 2.45 GHz (i.e. λ=12 cm), with a bandwidth of 100 MHz. The pitch between them is λ/30. These eight antennas form an array which will be referred to as the receiving array. Each antenna in this array is surrounded by a microstructure consisting here of a *random distribution of thin copper wires* (figure 4*b*). The mean distance between the thin copper wires was of the order of 1 mm (correlation length λ/100), while the frequency correlation *δω* was of the order of 30 MHz, resulting in three independent speckle patterns in the whole bandwidth. A TRM made of eight commercial dipolar antennas is placed in the far field, 10 wavelengths apart from the receiving array. The set ‘reverberant chamber/TRM’ acts as a *virtual far field time-reversal cavity*. When antenna marked #3 in figure 4 sends a short electromagnetic pulse (10 ns), the eight signals received at the TRM are much longer than the initial pulse due to strong reverberation in the chamber (typically 500 ns). As an example the signal received at one of the antennas of the TRM is shown in figure 5*a*. When antenna marked #4 is in its turn used as a source, it is remarkable to point out that now the signal received at the same antenna in the TRM (shown in figure 5*b*) looks significantly different although sources #3 and #4 were λ/30 apart from each other. When these signals are time-reversed and transmitted back, the resulting waves converge, respectively, to antennas #3 and #4 where they recreate pulses as short as the initial ones (figure 5*c*,*d*). Measuring the signal received at the other antennas of the receiving array gives access to the spatial focusing around antennas #3 and #4 (figure 5*e*). The remarkable result is that the two antennas can be addressed independently as the focusing spots created around them have a size much less than the wavelength (here typically λ/30): the diffraction limit is exceeded although the focusing points are in the far field of the TRM.

This concept, based on coupled subwavelength resonators, needs polychromatic sources to take advantage of the spatio-temporal degrees of freedom offered by arrays of subwavelength resonant scatterers. Using this approach, it was also shown that sound can be controlled from the far field in order to focus onto spots as thin as λ/8 [37], in an array of soda cans, which are Helmholtz resonators (figures 6 and 7). A monochromatic approach for light was also demonstrated using numerical simulation based on this idea [38], to focus optical waves onto subwavelength spots from the far field using wavefront shaping.

## 3. The instantaneous time mirror mimicking Loschmidt for waves

To mimic the Loschmidt approach in wave physics, one has first to measure an incoming wavefield at one specific time *t*_{m} in the whole volume {*φ*,∂*φ*/∂*t*}_{tm}. Then the analogue for wave of the particle velocity reversal is to prepare a new set of initial conditions {*φ*,−∂*φ*/∂*t*}_{tm}, where the sign of the time derivative has been reversed. Such initial conditions will give rise to a time-reversed wave. Even if this solution is appealing, it is not clear how to prepare such a new wavefield pattern. It is in this context that an interesting solution was recently proposed in the field of water waves [37]. Because of the wave superposition principle, the emergence of this time-reversed wave is, however, not limited to this choice of new initial conditions. For instance, a new initial condition {*φ*,0}_{tm} can be split into 1/2{*φ*,∂*φ*(**r**,*t*)/∂*t*}_{tm} associated with a forward wave and 1/2{*φ*,−∂*φ*(**r**,*t*)/∂*t*}_{tm} associated with a backward time-reversed wave. This particular disruption erases the arrow of time starting from a ‘frozen’ picture of the wavefield at time *t*_{m} with no favoured direction of propagation. Similarly, a new set of initial condition {0,∂*φ*(**r**,*t*)/∂*t*}_{tm} in which the wavefield is null would also comprise a backward-propagating wave with negative sign. More generally, any superposition of the old initial conditions {*φ*,∂*φ*/∂*t*}_{tm} with a new set of initial conditions {0, *f*(∂*φ*/∂*t*)}_{tm} with *f* being any function of ∂*φ*/∂*t* results in the superposition of a forward and backward-propagating wave.

To prepare such a new wave pattern, we use the fact that a sudden modification of the wave celerity in the whole medium at time *t*_{m} results in a new source term that depends on the incoming wavefield observed at time *t*_{m}. This offers a straightforward way to experimentally implement an instantaneous TRM. To understand the origin of this source term in the d’Alembert wave equation, let us introduce a time-dependent phase velocity *c*(*t*)=*c*_{0}/*n*(*t*), where *n*(*t*) is a time-dependent index and *c*_{0} is the initial phase velocity. The disruption undergone by the medium at time *t*_{m} can be modelled by a *δ*-Dirac function such that . The wave equation can be written as a nonhomogeneous equation in which the equivalent source term *s*(** r**,

*t*) is induced by the velocity disruption: 3.1with The source term is localized in time but delocalized in space. It corresponds to an instantaneous source that is proportional to the second time derivative of the wavefield at the instant

*t*

_{m}of the disruption. This source term suddenly creates a set of

*real monopolar sources*

*s*(

**,**

*r**t*), instantaneously in the whole space which isotropically radiate, generating in all directions an additional wavefield. This instantaneous source term in the wave equation is equivalent to a change in the initial conditions that become now the superposition of the original state of the unperturbed wavefield {

*φ*,∂

*φ*/∂

*t*}

_{tm}plus an added term [12,39]. This last term generates both a forward-propagating wavefield and a time-reversed backward-propagating wavefield that are proportional to the time derivative of the original incident wavefield.

It is in the field of gravity-capillary waves that such an experiment was recently described [39]. Since the surface wave celerity depends on the effective gravity *g*, the disruption of the celerity is simply achieved by applying a vertical impulsive acceleration to the whole liquid bath that change *g* in *g*+*γ*_{m}. A bath of water is placed on a shaker to control its vertical motion. A tip is used to hit the liquid surface and generate a point-like source of water waves. Figure 8*a* shows a sequence of images of the wave propagation on the bath taken from above. A circular wave packet centred on the impact point is emitted as the tip hits the surface. Owing to the dispersive nature of the gravity-capillary wave, the wave profile is modified with time. The average wave propagation velocity is of the order of magnitude of 10 cm s^{−1}. After a time *t*_{m}=60 ms, a vertical downwards acceleration is applied to the bath. It reaches *γ*_{m}=−18 *g* in approximately 2 ms. Such an impulsive change of wave celerity can be described by a delta function in time. The propagation of the initial outwards propagating wave is not significantly affected by this disruption. However, at the time of disruption, we observe the radiation of a backwards converging circular wave packet which focuses back to the initial source and that diverges again after. Such behaviour is analogous to classical TRMs except that it is obtained without any antenna array or memory. The information stored in the whole medium at one instant plays the role of a bank of memories. We observed the emergence of the advanced Green’s function from the bath and it is followed after the collapse by a diverging wave. These results can be extended to any type of source and figure 8*b* shows an impressive example with a metallic ‘smiley’ hitting the liquid surface and radiating a complex field pattern. The instantaneous time mirror is activated at a time *t*_{m}, where the field structure has apparently completely lost the shape of the smiley and magically, by virtue of the time-reversal symmetry, one observes a backward wave that recreates a real image of the smiley in the bath.

A careful analysis of this process can be conducted in terms of time reflection and time refraction [39,40] and it shows that the time discontinuity created by the vertical acceleration of the bath created a time-reversed wave that is proportional to the time derivative of the initial wavefield. Irrespective of source of complex shape this backwards wave recreated a perfect image of the initial source as in a holographic experiment, except that here the hologram is spread in all space and it is read by the action of the temporal discontinuity.

## 4. Conclusion

In this paper, we presented the two main approaches that can be used to generate time-reversed waves. The first approach uses TRMs with wave manipulation along a spatial boundary sampled by a finite number of antennas. Related to this method, the role of the spatio-temporal degrees of freedom of the wavefield has been emphasized and we show that a one channel TRM can be extremely efficient in dispersive media provided that the used bandwidth is large enough. There are plenty of applications of this approach for telecommunication, imaging and defence. In the second approach, we showed that waves can be manipulated on time boundaries by sudden modifications of the wave celerity in the whole medium. Such an approach of ‘instantaneous time mirrors’ mimics the Loschmidt point of view and is very efficient for radiating time-reversed waves without the use of any antenna.

## Competing interests

The author declares that there are no competing interests.

## Funding

I received no funding for this study.

## Acknowledgements

The author thanks Emmanuel Fort, Vincent Bacot, Geoffroy Lerosey, Matthieu Labousse and Antonin Eddi for stimulating discussions.

## Footnotes

One contribution of 12 to a theme issue ‘Loschmidt echo and time reversal in complex systems’.

- Accepted February 8, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.