Nonlocal Scatterer for Compact Wave-Based Analog Computing

Analog computing based on wave interactions with metamaterials has been raising significant interest as a low-energy, ultrafast platform to process large amounts of data. Engineered materials can be tailored to impart mathematical operations of choice on the spatial distribution of the impinging signals, but they also require extended footprints and precise large-area fabrication, which may hinder their practical applicability. Here we show that the nonlocal response of a compact scatterer can be engineered to impart operations of choice on arbitrary impinging waves, and even to solve integro-differential equations, whose solution is observed in the scattered fields. The lack of strongly resonant phenomena makes the response robust, and the compact nature opens to scalability and cascading of these processes, paving the way to efficient, compact analog computers based on engineered microstructures.

Figure 1

A nonlocal scatterer for wave-based analog computing. A source function $f(s)$ modulates the incident fields; the solution $u(s)={\mathcal{L}}^{-1}f(s)$ is obtained by measuring the scattered (outgoing) fields around the scatterer.

Figure 2

Inverse-designed nonlocal scatterer to solve a Fredholm equation of type II of choice. (a) Target and inverted scattering matrix elements plotted in the complex plane (here $\alpha =1$). (b) Inverted relative permittivity profile. The inner dotted curve is the boundary of the control domain, the outer dashed curve represents the observation contour over which the scattered fields are sampled, and the dashed rectangular line represents the PML boundary in our simulations.

Figure 3

Wave-based analog computation of Fredholm equation of type II. (a) Real part of the incident field. (b) Imaginary part of the incident field. (c) Real part of the scattered field. (d) Imaginary part of the scattered field. (e) Input function $f^h$. (f) Solution $u^h$, reconstructed from the measured ${\mathbf{d}}^{+}$. The retrieved solution overlaps with the exact value $u^{\mathrm{ref}}$.

Figure 4

Inverse-designed nonlocal scatterer to solve a differential equation. (a) Inverted relative permittivity profile. (b) Solution $u^h$ corresponds to the input function (13). (c) Solution $u^h$ corresponds to the input function (15). The retrieved solution shows an excellent agreement with the exact value $u^{\mathrm{ref}}$ for both input functions. The inverted scattering matrix, incident wavefield, and scattered wavefield are shown in Ref. [27].

Figure 5

Solving a differential equation in the presence of noise. A uniformly distributed random noises are added to the frequency (10% of the operating frequency), the relative permittivity of the scatterer (10% of the mean value), and the scattered fields (20% of the mean value). (a) Real part of the scattered field with noise. (b) Imaginary part of the scattered field with noise. (c) Solution $u^h$ reconstructed from the measured ${\mathbf{d}}^{+}$.