Deeply Subwavelength Localization with Reverberation-Coded Aperture

Accessing subwavelength information about a scene from the far-field without invasive near-field manipulations is a fundamental challenge in wave engineering. Yet it is well understood that the dwell time of waves in complex media sets the scale for the waves’ sensitivity to perturbations. Modern coded-aperture imagers leverage the degrees of freedom (d.o.f.) offered by complex media as natural multiplexor but do not recognize and reap the fundamental difference between placing the object of interest outside or within the complex medium. Here, we show that the precision of localizing a subwavelength object can be improved by several orders of magnitude simply by enclosing it in its far field with a reverberant passive chaotic cavity. We identify deep learning as a suitable noise-robust tool to extract subwavelength localization information encoded in multiplexed measurements, achieving resolutions well beyond those available in the training data. We demonstrate our finding in the microwave domain: harnessing the configurational d.o.f. of a simple programmable metasurface, we localize a subwavelength object along a curved trajectory inside a chaotic cavity with a resolution of $\lambda /76$ using intensity-only single-frequency single-pixel measurements. Our results may have important applications in photoacoustic imaging as well as human-machine interaction based on reverberating elastic waves, sound, or microwaves.

Figure 1

(a) Conventional CA. A wavefront is scattered by an object and then multiplexed across different masks onto a single-pixel detector. (b) Use of a complex medium’s spectral d.o.f. (color coded) as CA. Wavefronts are scattered by an object and then propagate through a chaotic cavity such that spatial information is multiplexed across different frequencies captured by a single-pixel detector. (c) Reverberation-coded aperture: same as (b) except that object and wave source are inside the chaotic cavity.

Figure 2

(a)–(c) Electric field magnitude in 2D semianalytical simulations for (a) free space, a chaotic cavity with (b) $Q=263$, and (c) $Q=556$. All maps use the same color scale. (d) CDF of the dwell time magnitude $|\tau |$ distribution in the three cases. Vertical dashed lines indicate the corresponding mean values. (e) CDF of the parametric derivative magnitude $|\chi |$ (with respect to the object position) in the three cases. Vertical dashed lines indicate the corresponding mean values. (f) SVs ${\sigma }_i$ of $\mathbf{T}(X,f)$ (without any noise) for the three considered cases. (g) Average localization error $\epsilon$ in terms of the smallest utilized wavelength ${\lambda }_{\min }$ as a function of the absolute magnitude of the measurement noise for the three cases. Vertical dashed lines indicate the corresponding signal magnitudes. The horizontal black line indicates the training data resolution.

Figure 3

(a) Experimental setup: a 3D complex scattering enclosure contains a subwavelength metallic object on a precision turntable, two monopole antennas, and a programmable metasurface in the vicinity of one antenna. The enclosure’s ceiling can be open ($Q=40$), partially covered ($Q=92$), or fully covered ($Q=252$) with metal. See Supplemental Material [48] for technical details. The inset shows the standard deviation ${\zeta }_a$ of $|S_{12}|$ over 100 random metasurface configurations and identifies the chosen operating frequency $f_0$. (b) Dependence of mean of ${\zeta }_a$ over object positions on $Q$. (c) Sketch of $S_{12}(f_0)$ distribution for 100 random metasurface configurations. (d) SV spectra of $\mathbf{T}(X,c)$ for the three considered cases. (e) Average localization error $\epsilon$ in terms of the utilized wavelength ${\lambda }_0$ as a function of SNR. The horizontal black line indicates the training data resolution.