Nonlinear time reversal of classical waves: Experiment and model

We consider time reversal of electromagnetic waves in a closed, wave-chaotic system containing a discrete, passive, harmonic-generating nonlinearity. An experimental system is constructed as a time-reversal mirror, in which excitations generated by the nonlinearity are gathered, time-reversed, transmitted, and directed exclusively to the location of the nonlinearity. Here we show that such nonlinear objects can be purely passive (as opposed to the active nonlinearities used in previous work), and we develop a higher data rate exclusive communication system based on nonlinear time reversal. A model of the experimental system is developed, using a star-graph network of transmission lines, with one of the lines terminated by a model diode. The model simulates time reversal of linear and nonlinear signals, demonstrates features seen in the experimental system, and supports our interpretation of the experimental results.

Figure 1

(a) The experimental setup in the time-forward step, containing a passive nonlinear circuit, showing the Gaussian pulse injected into the linear port and the resulting sona measured at the receiving port. (b) The experimental setup in the time-reversed step, showing the time-reversed nonlinear sona injected into the receiving port and reconstructing as a time-reversed excitation at the nonlinear port.

Figure 2

(a) Fourier transform of the sona measured at the receiving port, indicating the linear sona at $f_0$ and nonlinear sona at $2f_0$. (b) Reconstruction of the linear sona at the linear port. (c) There is no appreciable reconstruction of the nonlinear sona at the linear port (inset shows the original incident pulse, for comparison). (d) There is no appreciable reconstruction of the linear sona at the nonlinear port. (e) Reconstruction of the nonlinear sona at the nonlinear port.

Figure 3

(a) Reconstruction of a single linear time-reversed sona at the linear port generated using a 50 ns Gaussian pulse. (b) Reconstruction of a sona at the linear port constructed by overlapping two copies of the sona with a 50 ns time delay (or 99.5% overlap of sonas), showing two distinct reconstructed pulses of lower amplitude. (c) Reconstruction of a sona at the linear port constructed using 16 overlapped sonas, with 50 ns time delays between copies. Sixteen reconstructed pulses are distinguishable above the noise or sidelobes. (d)–(e) Images of “LI” and “NL” encoded using the linear sona (d) and nonlinear sona (e), transmitted exclusively to the linear and nonlinear ports in the experimental setup, respectively.

Figure 4

The model network of transmission lines connected in a star-graph topology. For each line, a length ($L_{\mu }$), characteristic admittance ($Y_{c,\mu }$), and reflectivity (${\Gamma }_{\mu }$) of the unconnected end is defined. Ports 1 and 2 act as the points where energy is injected into, or extracted from, the graph.

Figure 5

(a)–(c) Time domain signals in the linear two-port star graph with $n=10$ lines. (a) Initial Gaussian pulse (50 ns, 3.8 GHz carrier) applied to port 1. (b) Resultant sona collected at port 2. (c) Reconstruction at port 1 from linear time reversal of sona in (b) generated in the model. (d)–(f) Experimental results obtained in the linear resonant cavity. (d) Initial Gaussian pulse (50 ns, 3.8 GHz carrier) applied to the transmitting port. (e) Resultant sona collected at the receiving port. (f) Reconstruction of the time-reversed sona in (e) recorded at the original transmitting port.

Figure 6

A plot of Eq. (11) (using $I_s=1\times {10}^{-12}\text{A}$, $Y_c=0.02\text{S}$, and $V_T=0.030\text{V}$), mapping the incoming voltage $V_{+}$ to the outgoing voltage $V_{-}$ after reflection from the model diode. The dotted blue line is $V_{-}=V_{+}$, showing the deviation from linearity of the diode (plotted in the red solid line).

Figure 7

(a) A low-amplitude Gaussian pulse $V_{-}$ (100 ns, 5 GHz carrier) reflected from the model diode. (b) A high-amplitude Gaussian pulse $V_{-}$ (100 ns, 5 GHz carrier) reflected from the model diode. Inset shows a schematic of the model for generating the pulse reflections in (a)–(c). (c) FFT of the wave form shown in (b). (d) A low-power Gaussian pulse (500 ns, 5.0 GHz carrier) reflected from a diode in a single-transmission-line experiment. (e) A high-power Gaussian pulse (500 ns, 5.0 GHz carrier) reflected from a diode in the experiment. Inset shows the experimental setup for generating the pulse reflections in (d)–(f). (f) FFT of the pulse shown in (e). Inset shows a magnified view of the second harmonic frequency.

Figure 8

Wave forms generated by the nonlinear time-reversal star-graph model. (a) The sona generated by a low-amplitude (0.25 V) Gaussian pulse (50 ns, 5 GHz carrier) injected at port 1, recorded at port 2. (b) The frequency spectrum (log FFT magnitude vs frequency) of the sona in (a), demonstrating the linearity of the network for the low-amplitude pulse. Inset shows a diagram of the star-graph network, with the diode terminating the longest line (no. 10) (c) The sona generated by a high-amplitude (5 V) Gaussian pulse (50 ns, 5 GHz carrier) injected at port 1, recorded at port 2. (d) The frequency spectrum (log FFT magnitude vs frequency) of the sona in (c), with harmonics of the initial pulse frequency generated by interaction with the model diode. (e) Reconstruction of the full time-reversed sona in (c) (plotted in red), measured at port 1, compared to the initial Gaussian pulse (blue, larger amplitude). (f) Reconstruction of the time-reversed nonlinear sona (filtered to select second harmonic) measured at the diode location (with the diode removed). (g) Wave form reflected from diode back into the network after the initial excitation. (h) Wave form reflected from diode back into the network after the initial excitation, filtered to select the second harmonic.