Time reversal and holography with spacetime transformations

Abstract

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 effective 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 effect 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.Holographic methods are based on the time-reversal invariance of wave equations. They rely on the fact that any wave field can be completely determined within a volume by knowing the field (and its normal derivative) on any enclosing surface 1,2 . Hence, information reaching the 2D surface is sufficient to recover all information inside the whole volume. Based on these properties, Denis Gabor introduced the Holographic method, which provides an elegant way to back-propagate a monochromatic wave field and obtain 3D images. More recently, time-reversal mirrors exploited the same principles extended to a broadband spectrum to create time-reversed waves. This latter approach has been implemented with acoustic 3 , elastic 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 antenna array. If the array intercepts the entire forward wave with a good spatial sampling, it generates a perfect backwardpropagating copy. Note that this process is difficult to implement in optics 8,9 , and the standard solution is to work with monochromatic light and use nonlinear regimes such as three-wave or four-wave mixing 10,11 .Here, within the general concept of spacetime transformations [12][13][14][15][16] , we completely revisit the holographic method and introduce a new way to create wideband time-reversed wave fields in 2D or 3D by manipulating time boundaries. Time boundaries have recently received much attention because they have been shown to play a major role in several phenomena such as time refraction, dynamic Casimir effect, Hawking radiation, photon acceleration and self-phase modulation [17][18][19][20][21][22][23][24][25] . In addition, different suggestions to process wideband time-reversal have been proposed in optics to associate both time and spatial modulation of the medium refractive index. These suggestions, mainly for 1D propagation, rely on a dynamic tuning of photonic crystals ...

Quantum time mirrors for general two-band systems

“…Recently an "instantaneous time mirror", i.e. a new form of a classical time mirror circumventing the measurement process, was implemented [27]. Here the propagation of gravity-capillary water waves is inverted by a short and homogeneous perturbation, specifically a vertical acceleration of the water bath.…”

From the time-reversal mirror to the instantaneous time mirror

“…Even if this solution is appealing, it is not clear how to prepare such a new wave field pattern. It is in this context that an interesting solution was recently proposed [18] in the field of water waves. Because of the wave superposition principle, the emergence of this time reversed wave is however not limited to this choice of new initial conditions.…”

“…This last term generates both a forward-propagating wave field and a time-reversed backwardpropagating wave field that are proportional to the time derivative of the original incident wave field. It is in the field of gravity-capillary waves that such an experiment was recently described [18]. 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 .…”

“…The ITM is activated at a time t m where the field structure has apparently completely loss the shape of the smiley and by virtue of the timereversal symmetry, one observe a backward wave that recreate a real image of the smiley in the bath !!! It is also interesting to interpret this experiment in term of the two concepts of time refection and time refraction [18][19][20][21]. When we introduced an instantaneous time disruption for the wave speed or for the time-dependent index of refraction as c(t) 2 = c 2 0 (1 + αδ(t − t m )), we have to remember that a Dirac pulse is the limit of a rectangular pulse with two time discontinuities.…”

“…This instantaneous source term in the wave equation is equivalent to a change in the initial conditions (see Eq. (2)) that becomes now the superposition of the original state of the unperturbed wave field ϕ, ∂ϕ ∂t tm plus an added term 0, [12,18]. This last term generates both a forward-propagating wave field and a time-reversed backwardpropagating wave field that are proportional to the time derivative of the original incident wave field.…”