Nonlinear Time Reversal in a Wave Chaotic System

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 time-reversed 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.

Figure 1

(a) Schematic of the experimental setup. A Gaussian-shaped pulse ($f_{\mathrm{pulse}}=3.8\mathrm{GHz}$, $t=50\mathrm{ns}$) is transmitted into the ray-chaotic enclosure through the linear port and reverberates through the scattering environment, interacting with the nonlinear element. (b) An example of a full sona signal measured by the oscilloscope at the transceiver port, including the 3.8 GHz carrier tone and modulation envelope. Lower inset: short segment (150 ns) of the sona in detail. Upper inset: magnitude of the fast Fourier transform of the sona shown in (b) as a function of frequency. The inset shows a closeup around the center frequency of the pulse, indicating the frequency components arising from the pulse ($f_{\mathrm{pulse}}$) and the nonlinear element ($f_{\mathrm{pulse}}+f_{\mathrm{diode}}$).

Figure 2

Reconstructions collected at the linear port and at the nonlinear port after broadcast of time-reversed sonas from the transceiver port. (a) Reconstruction of the linear sona at the linear port (only the positive part of the waveform is shown); inset shows the reconstructed pulse in detail. (b) The (lack of) reconstruction of the nonlinear signal at the linear port; (c) similar (lack of) reconstruction of the linear sona at the nonlinear port. Inset of (c) shows the pulse initially incident on the linear port in the time-forward step. (d) Reconstruction of the nonlinear signal at the nonlinear port; inset shows the reconstructed pulse in detail.

Figure 3

(a) Transmission of a four-color (two bits per pixel) image using the time-reversed linear sona, which is reconstructed only at the linear port. The lack of reconstructed pulses at the nonlinear port corresponds to the transmission of “00” that decodes as a black pixel. Inset: an example constructed sona, displaying the combination of “1” and “0” component sonas, to generate a reconstruction representing “1101.” (b) Transmission of a different four-color image using the time-reversed nonlinear sona, which is reconstructed only at the nonlinear port.