A Wave-based Approach to Solve Linear Integral Equations

Metamaterials have been explored as an interesting tool to transfer the well-known computational capabilities of electromagnetic waves into the realm of nano-photonics [Silva, Monticone, Castaldi, Galdi, Alu, and Engheta, Science 343, 6167 (2014)]. Along with performing the basic mathematical tasks such as differentiation and integration on the spatial profile of the wave, here we propose a more desired functionality for wave-based computation, namely solving integral equations. Our proposal is based on creating a feedback path for the wave, comprising the kernel integral operator. As the wave travels along this path, the solution to the integral equation is generated in the steady state. An implementable configuration is proposed including the mechanisms to excite and probe the system and the final output. The arbitrary kernel operator is designed in the form of a medium with inhomogeneous permittivity, and single-mode waveguides are employed as feedback lines. We will also discuss the potential advantages of this approach in terms of power consumption, capacity, and speed.