Inverse-Designed Optical Metagratings for Free-Space Integral Equations Solving

The world's ever-growing need for efficient computing has been driving researchers from diverse research fields to explore alternatives to the current digital computing paradigm. In this context, the field of all-optical analog processing is gaining significant attention opening new possibilities to overcome the bottlenecks of standard microelectronics [1–5]. We present a Si-based optical metastructure that solves Fredholm integral equations of the second kind in a fully analog fashion at optical frequencies. We exploit the analogy between the integral equation solving and the behavior of an optimized periodic metagrating coupled to a feedback system. At the foundations of this mapping lies the possibility of designing the S-matrix of a periodic structure by setting its periodicity (i.e., the number of input/output modes and hence the dimension of the S-matrix) and optimizing its unit cell (i.e., optimizing the coupling of light into the defined diffraction modes in amplitude and phase), as shown in Fig. 1a. Next, we show how the designed metastructure effectively solves the problem of interest and compare the metasurface-based solution to the ideal solution. We show that electron beam lithography and reactive ion etching provide the spatial resolution required to create a hardware representation of a predefined Kernel, with relatively small deviations between experiments and simulations. We optically characterize the output for different input signals showing good agreement with the ideal simulated response and use the data to retrieve the solution experimentally provided by the metadevice (see Fig. 1b). Fig. 1 A. Schematic of the proposed structure. A silicon metasurface is patterned on top of a sapphire substrate. The superstrate consists of $\text{SiO}_{2}$ with a semi-transparent au mirror on top. Scale-bar 500 nm b. Estimate of the solution to the integral equation obtained by the fabricated metadevice.