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Time Reversal and Holography with Spacetime Transformations

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Time Reversal and Holography with Spacetime Transformations ARTICLES PUBLISHED ONLINE: 11 JULY 2016 | DOI: 10.1038/NPHYS3810 Time reversal and holography with spacetime transformations Vincent Bacot1, Matthieu Labousse1,2†, Antonin Eddi3, Mathias Fink1*‡ and Emmanuel Fort1*‡ Wave control is usually performed by spatially engineering the properties of a medium. Because time and space play similar roles in wave propagation, manipulating time boundaries provides a complementary approach. Here, we experimentally demonstrate the relevance of this concept by introducing instantaneous time mirrors. We show with water waves that a sudden change of the eective gravity generates time-reversed waves that refocus at the source. We generalize this concept for all kinds of waves, introducing a universal framework which explains the eect of any time disruption on wave propagation. We show that sudden changes of the medium properties generate instant wave sources that emerge instantaneously from the entire space at the time disruption. The time-reversed waves originate from these ‘Cauchy sources’, which are the counterpart of Huygens virtual sources on a time boundary. It allows us to revisit the holographic method and introduce a new approach for wave control. olographic methods are based on the time-reversal Our approach is related to the Cauchy theorem, which states that invariance of wave equations. They rely on the fact that any the wave field evolution can be deduced from the knowledge of this wave field can be completely determined within a volume wave field (and its time derivative) at one single time (the so-called H 29 by knowing the field (and its normal derivative) on any enclosing initial conditions) . It is the dual time equivalent of standard time surface1,2. Hence, information reaching the two-dimensional (2D) reversal based on spatial boundaries. We use a sudden modification surface is sufficient to recover all information inside the whole of the wave propagation properties of the medium to create a time- volume. Based on these properties, Denis Gabor introduced the reversed wave. This time disruption realizes an instantaneous time holographic method, which provides an elegant way to back- mirror (ITM) in the entire space without the use of any antenna propagate a monochromatic wave field and obtain 3D images. or memory. The information stored in the whole medium at one More recently, time-reversal mirrors exploited the same principles instant plays the role of a bank of memories. extended to a broadband spectrum to create time-reversed waves. We will subsequently introduce the concept of the ITM and show This latter approach has been implemented with acoustic3,4, its first experimental demonstration. The experiment is spectacular elastic5, electromagnetic6 and water waves7,8. It requires the use of because it is conducted with water waves and can therefore be emitter–receptor antennas positioned on an arbitrary enclosing observed with the naked eye. We first interpret the backward wave surface. The wave is recorded, digitized, stored, time-reversed propagation in the ITM as an emission by isotropic sources created and rebroadcast by the antenna array. If the array intercepts the during the time disruption. These `Cauchy sources' define a new entire forward-propagating wave with a good spatial sampling, set of initial conditions for the wave field propagation after the it generates a perfect backward-propagating copy. Note that this ITM, allowing us to revisit the Huygens–Fresnel principle. We process is difficult to implement in optics9,10, and the standard then discuss this experiment in terms of time discontinuities and solution is to work with monochromatic light and use nonlinear conservation laws. Finally, we analyse the spacetime symmetries of regimes such as three-wave or four-wave mixing11,12. ITMs compared to standard mirrors. Here, within the general concept of spacetime In the nineteenth century, Loschmidt challenged Boltzmann's transformations13–16, we completely revisit the holographic attempt to describe irreversible macroscopic processes with method and introduce a new way to create wideband time-reversed reversible microscopic equations30,31. He imagined a daemon wave fields in 2D or 3D by manipulating time boundaries. Time capable of instantaneously reversing all velocities of all particles boundaries have recently received much attention because they in a gas. Such an operation can be ascribed to a change in initial have been shown to play a major role in several phenomena, such conditions resulting in a time-reversed motion of all particles that as time refraction, the dynamic Casimir effect, Hawking radiation, would return to their initial positions. The extreme sensitivity photon acceleration and self-phase modulation17–26. In addition, to initial conditions that lies at the heart of chaotic phenomena different suggestions to process wideband time reversal have been in nonlinear dynamics renders any such particulate scheme proposed in optics to associate both time and spatial modulation impossible. Waves are more amenable, because they can be of the medium refractive index. These suggestions, mainly for 1D described in many situations by a linear operator, and any error propagation, rely on a dynamic tuning of photonic crystals and in initial conditions will not suffer from chaotic behaviour. The metamaterials27,28. wave analogue of this Loschmidt daemon is related to the Cauchy 1Institut Langevin, ESPCI, CNRS, PSL Research University, 1 rue Jussieu, 75005 Paris, France. 2Matière et Systèmes Complexes, Université Paris Diderot, CNRS 7057, Sorbonne Paris Cité, 10 Rue A. Domon et L. Duquet, 75013 Paris, France. 3Laboratoire de Physique et Mécanique des Milieux Hétérogènes, ESPCI, CNRS, PSL Research University, 10 rue Vauquelin, 75005 Paris, France. †Present address: Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot, Sorbonne Paris Cité, 10 rue A. Domon et L. Duquet, 75013 Paris, France. ‡These authors contributed equally to this work. *e-mail: mathias.fi[email protected]; [email protected] 972 NATURE PHYSICS | VOL 12 | OCTOBER 2016 | www.nature.com/naturephysics © ƐƎƏƖɥMacmillan Publishers LimitedƦɥ/13ɥ.$ɥ/1(-%#1ɥ341#. All rights reservedƥ NATURE PHYSICS DOI: 10.1038/NPHYS3810 ARTICLES ITM ab4 Δt Δt Δ t0 tITM t0 + 2 t Time 2 Laser 0 Source −2 Refocusing Emission Δ −4 t Vertical position (mm) Shaker 0 g / −10 γ γ m −20 Shaker 0.0 0.1 0.2 0.3 0.4 t t Figure 1 | Schematic of the instantaneous time mirror. A wave source 0 ITM Time (s) c emits at time t0 a wavepacket which propagates in a given medium. A sudden spatially homogeneous disruption of the wave propagation properties occurs in the entire medium at time tITM Dt0 C1t. It results in the production of a counter-propagating time-reversed wave in addition to the initial forward-propagating wave. The counter-propagating wave refocuses at the source position at time t0 C21t. Δ t0 tITM t0 + 2 t Time Δt = 60 ms Δt = 60 ms theorem. The latter states that the future evolution of any wave field Emission ITM φ.r,t/ at position r and time t can be inferred from the knowledge Figure 2 | ITM experimental implementation. a, Experimental set-up. of the set of initial conditions (φ,@φ=@t/tm , with the field amplitude φ.r, tm/ and time derivative @φ[email protected], tm/ at a given time tm, in the A bath of water is placed on a shaker to apply a vertical jolt. Another shaker whole space. The analogue of the particle velocity reversal is to is used to hit the water surface with a tip to generate surface waves. The − deflection of a laser beam is used to measure the local surface slope. The take the new set of initial conditions (φ, .@φ=@t//tm that causes a time-reversed wave whose time dependence is inverted. However, laser is placed on a computer-controlled translation device to scan the because of the wave superposition principle, the emergence of surface. b, Typical time variations of the vertical position of the emitter this time-reversed wave is not limited to this choice of initial (a tip) in blue and of the bath (in red) together with the bath acceleration γ in an ITM experiment. γm is the maximum downward acceleration. 1t is the conditions. For instance, the new initial condition (φ, 0/tm can = (φ @φ=@ / time delay between the wave emission and the jolt. c, Image sequence of an be split into 1 2 , t tm associated with a forward wave and = (φ −.@φ=@ // ITM experiment (top view) with a point source, showing the divergent wave 1 2 , t tm associated with a backward time-reversed wave. This particular choice erases the arrow of time by starting and the time-reversed wave which diverges again after focusing back at the γ D− 1 D from a `frozen' picture of the wave field at time tm with no favoured source position. m 21g and t 60 ms. Scale bar, 1 cm (see direction of propagation. Similarly, a new set of initial conditions Supplementary Video 1). .0, @φ=@t/tm in which the wave field is null would also comprise a backward-propagating wave with a negative sign. More generally, induced by dispersion is clearly visible. The ITM generates a time- the superposition of backward- and forward-propagating waves reversed wavepacket propagating in the opposite direction. The results from the decoupling of the wave field from its time derivative resulting surface profile can be decomposed into the propagating at a given time (see Fig.1 and the Supplementary Information). and counter-propagating wavepackets using Fourier analysis. We Because both are bound together by the wave celerity, its disruption observe that the shape of the backward wavepacket is very similar to can lead to such decoupling. This offers a straightforward way to that of the initial wavepacket. Both profiles almost superimpose in experimentally implement an ITM. shape and position when measured at symmetrical times 1t from In this study, we use gravity–capillary waves to implement the the ITM.
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