Elastic computational metasurfaces for subwavelength differentiations

Analog computations enable information processing with negligible energy costs and massively parallel architectures, but currently are limited to process macroscale waveforms with characteristic lengths much larger than the operating wavelength ${\lambda }_0$. We explore here, in contrast, the differentiation of subwavelength waveforms by using an elastic computational metasurface. We find that the numerical aperture of metasurface governs the threshold of the characteristic length of waveforms, below which the metasurface outputs an identical differentiated pattern. Remarkably, for a subwavelength waveform below the threshold, the metasurface can locate the source because the differentiated pattern is of cylindrical wavefronts centered at the source, which can be harnessed to detect single or multiple subwavelength-scaled scatterers. The detectability reaches a deep subwavelength of $0.12{\lambda }_0$, and the localization error stays smaller than ${\lambda }_0$. Our work elucidates the physical image of subwavelength differentiations, which may promote promising applications in nondestructive testing, signal processing, and computational acoustics.

Figure 1

Macroscale and subwavelength analog differentiations by using metasurfaces. (a) For macro analog differentiation, the characteristic length of the input waveform is much larger than the operating wavelength ${\lambda }_0$. Its typical application is edge detection, where the sizes of scatterers are generally hundreds of ${\lambda }_0$. (b) For subwavelength differentiation, the characteristic length of the input waveform is less than the diffraction limit ${\lambda }_0/2\mathrm{NA}$. One may find applications in detecting a subwavelength tiny scatterer.

Figure 2

Elastic computational metasurface for the second-order spatial differentiation. (a) Dual-layer configuration of the metasurface. The enlarged view shows the unit cell, which consists of only one cascaded zigzag subunit. (b) Transmitted amplitude $\left|t\right|$ and phases $\angle t$ of the optimized metasurfaces. The solid lines are exact values.

Figure 3

Subwavelength differentiation by using an elastic computational metasurface. (a) Photograph of the fabricated elastic computational metasurface. The enlarged view shows that the point source is placed ahead of the metasurface along its middle line at a distance of $2{\lambda }_0$. (b) Simulated output wave field under illumination of the input signal in (c) impinges on the metasurface. The characteristic length of incident waveform is $0.2{\lambda }_0$. (d) The tested wave field for a point source to stimulate the input waveform. The solid box marks the region for experimental measuring. (e) Output amplitude and phase distributions on the dashed lines in (b) and (d) that are behind the metasurface at a distance of $5{\lambda }_0$. (f) Output amplitude profiles with different characteristic lengths of the input waveforms ranging from $0.2{\lambda }_0$ to ${\lambda }_0$. Note that (b) and (d) display the out-of-plane displacement fields that are normalized by the maximal magnitudes of differentiated waves.

Figure 4

Subwavelength scatterer detection and localization by using a computational metasurface. (a) Schematic of the detection principle. (b) Simulated plane wave field perturbed by a subwavelength circular cavity with diameter $d=0.4{\lambda }_0$. (c) The counterpart of (b) with an elastic computational metasurface behind the cavity at a distance of $2{\lambda }_0$. (d) Simulated resulted wave field for the metasurface filtering out the incident plane wave. (e) Localization of cavities at different positions. (f) Variations of the scattering cross section $\overline{\gamma }$ with the cavity diameter $d$. Note that (b)–(d) display the out-of-plane displacement fields normalized by the magnitudes of incident waves. To clearly display the differentiated scattered waves, the scale of (c) is decreased to ±0.05.