Subwavelength Position Sensing Using Nonlinear Feedback and Wave Chaos

We demonstrate a position-sensing technique that relies on the inherent sensitivity of chaos, where we illuminate a subwavelength object with a complex structured radio-frequency field generated using wave chaos and nonlinear feedback. We operate the system in a quasiperiodic state and analyze changes in the frequency content of the scalar voltage signal in the feedback loop. This allows us to extract the object’s position with a one-dimensional resolution of $\sim \lambda /10000$ and a two-dimensional resolution of $\sim \lambda /300$, where $\lambda$ is the shortest wavelength of the illuminating source.

Figure 1

(a) Experimental setup. Nonlinear circuit consisting of a transistor-based (BFG620) NLE, low-noise amplifiers (LNA, Mini-Circuits ZX60-4016E and Picosecond Pulse Labs 5828-108), and a low-pass filter. A bias voltage $V_{\mathrm{bias}}$ tunes the nonlinearity. The NLE is housed in a shielded aluminum box, but otherwise there is no active stabilization to control the thermal or vibrational effects of the environment. The output voltage $V_{\mathrm{out}}$ is measured with a 8-GHz-analog-bandwidth $40\mathrm{\text{−}}\mathrm{GS}/\mathrm{s}$ oscilloscope (Agilent DSO80804B). Feedback passes through a cavity with a dielectric object ($2\mathrm{cm}\times 4\mathrm{cm}$ water-filled container) that is positioned in 2D using Thorlabs (LTS150) and Zaber Technologies (TLSR150B) translation stages. (b) Cavity pulse response. Here, the nonlinear circuit is removed to examine just the linear response of the cavity and antennas. We inject a 0.1 ns EM pulse of amplitude 1.5 V into the TX antenna and measure at the RX antenna. The time at which the radiation arrives at the RX antenna is a measure of the path length of EM energy through the cavity.

Figure 2

Bifurcation Diagram. We store and analyze the time series of $V_{\mathrm{out}}$ at each object position along a 1D path in the $x$ direction. The local maxima of each time series is plotted as a function of object position $x$.

Figure 3

(a) Temporal evolution of a typical quasiperiodic state used for position sensing. (b) Logarithmic power spectral density (PSD) of this signal.

Figure 4

Frequency shifts $\Delta f_1$ and $\Delta f_2$ of the QP state as the object translates along the (a) $x$ direction, where the fit from Eq. (1) yields $a_1=2.8\mathrm{mm}/\mathrm{kHz}$, $a_2=-8.7\mathrm{mm}/\mathrm{kHz}$, $c_0=27,113.0\mathrm{mm}$, $c_1=-439.2$ and $c_2=1.5{\mathrm{mm}}^{-1}$, and (b) along the $y$ direction, where the fit from Eq. (2) yields $b_1=0.8\mathrm{mm}/\mathrm{kHz}$, $b_2=-0.5\mathrm{mm}/\mathrm{kHz}$, $d_0=-73452.0\mathrm{mm}$, $d_1=242.0$, and $d_2=-2.0{\mathrm{mm}}^{-1}$. (c) Frequency shifts $\Delta f_1(x,y)$ and $\Delta f_2(x,y)$ for object translations ($x$, $y$) in a $5\mathrm{mm}\times 5\mathrm{mm}$ grid of positions. The planar fits from Eqs. (3, 4) yield ${\alpha }_1=-84.68\mathrm{kHz}/\mathrm{mm}$, ${\alpha }_2=-15.20\mathrm{kHz}/\mathrm{mm}$, ${\beta }_1=-56.74\mathrm{kHz}/\mathrm{mm}$, ${\beta }_2=-14.75\mathrm{kHz}/\mathrm{mm}$, ${ϵ}_1=11.72\mathrm{MHz}$ and ${ϵ}_2=2.43\mathrm{MHz}$. The measured determinant $|{\alpha }_1{\beta }_2-{\alpha }_2{\beta }_1|=386{\mathrm{kHz}}^2/{\mathrm{mm}}^2$ with an error of $5.8{\mathrm{kHz}}^2/{\mathrm{mm}}^2$.