Nonlinear Time Reversal in a Wave Chaotic System

Abstract

Exploiting the time-reversal invariance and reciprocal properties of the lossless wave equation enables elegantly simple solutions to complex wave-scattering problems, and is embodied in the time-reversal mirror. Here we demonstrate the implementation of an electromagnetic time-reversal mirror in a wave chaotic system containing a discrete nonlinearity. We demonstrate that the timereversed nonlinear excitations reconstruct exclusively upon the source of the nonlinearity. As an example of its utility, we demonstrate a new form of secure communication, and point out other applications.PACS numbers: 05.45. Mt, 05.45.Vx, 41.20.Jb, 42.25.Dd Wave chaos concerns the study of solutions to linear wave equations that display classical chaos in their short-wavelength limit. Such systems are endowed with many universal wave properties, such as eigenvalue and scattering-matrix statistics, by virtue of their classically chaotic counterparts.[1] Although wave chaotic systems are strongly scattering and have complex behavior, they can be elegantly studied by exploiting the time-reversal invariance and reciprocal properties of the linear wave equation. [2-9] Adding objects with complex nonlinear dynamics to linear wave chaotic systems has only recently been considered, [10] and represents an exciting new direction of research. Here we examine a wave chaotic system with a single discrete nonlinear element, and create a new nonlinear electromagnetic time-reversal mirror that shows promise for both fundamental studies and novel applications.A time-reversal mirror works by taking advantage of the invariance of the lossless wave equation under timereversal; for a time-forward solution of the wave equation representing a wave travelling in a given direction, there is a corresponding time-reversed solution representing a wave travelling in the same direction backwards in time, or in the opposite direction forward in time. This can be realized by transmitting a waveform at a particular source location and recording the reverberating waveforms (sona) with an array of receivers; the recorded waveforms are reversed in time and retransmitted back from the receivers, propagating to and reconstructing a time-reversed version of the original waveform at the source [3]. Time-reversal mirrors have been demonstrated for both acoustic [2-9, 11, 12] and electromagnetic waves [6,8,13], and exploited for applications such as lithotripsy [2,4], underwater communication [2, 14, 15], sensing perturbations [11,12], and achieving sub-wavelength imaging [6][7][8]16].An ideal time-reversal mirror in an open environment would collect the forward-propagating wave at every point on a closed surface enclosing the transmitter, requiring a very large number of receivers. The receiving array can be simplified, without significant loss of fidelity of the reconstruction, if there is a closed, ray-chaotic environment where a propagating wave (with wavelength much smaller than the size of the enclosure) will eventually reach every point in the environment, allowin...

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“…In superposition of time reversed electromagnetic wave in chaotic system [33], one can send linear (non-linear) port and receive linear (on-linear) port. In 1971, L. Chua cathegorized electronic elementsas resistor (R), capacitor (C), inductor (L) and memristor (M) [37,38].…”

“…A superposition of a nonlinear sound wave and time reversed nonlinear sound wave can reduce improper superposition of lobes around theproper peak produced by a superposition of reflected waves [24][25][26][27][28][29]33].…”

Nonlinear Electromagnetic Time Reversal in an Open Semireverberant System

“…However, in order to employ a TRM in a broader range of applications, it is desirable to obtain the channel response without physically sending a short pulse from the source location. To this end, a recent demonstration with a single-antenna electromagnetic TRM in a closed wave-chaotic system has shown that the reconstruction of a short pulse can be achieved when a localized nonlinearity is enclosed in the system [30]. That is, when a short pulse enters the system, its interaction with the nonlinear element generates harmonics, and the element becomes a new source originating from the location of the nonlinear element.…”

E. Cartan’s and A. Connnes’ Supersymmetry and Differences of Understanding Physical Phenomena

“…A superposition of a non linear sound wave and time reversed non-linear sound wave can reduce improper superposition of lobes around the proper peak produced by a superposition of reflected waves [19][20][21][22][23][24]38]. …”

“…We find interesting superposition properties of memristor wave, which was studied inRefs. [37][38][39][40][41]47].…”