Solving partial differential equations with waveguide-based metatronic networks

Photonic computing has recently become an interesting paradigm for high-speed calculation of computing processes using light–matter interactions. Here, we propose and study an electromagnetic wave-based structure with the ability to calculate the solution of partial differential equations (PDEs) in the form of the Helmholtz wave equation, ∇2f(x,y)+k2f(x,y)=0, with k as the wavenumber. To do this, we make use of a network of interconnected waveguides filled with dielectric inserts. In so doing, it is shown how the proposed network can mimic the response of a network of T-circuit elements formed by two series and a parallel impedances, i.e., the waveguide network effectively behaves as a metatronic network. An in-depth theoretical analysis of the proposed metatronic structure is presented, showing how the governing equation for the currents and impedances of the metatronic network resembles that of the finite difference representation of the Helmholtz wave equation. Different studies are then discussed including the solution of PDEs for Dirichlet and open boundary value problems, demonstrating how the proposed metatronic-based structure has the ability to calculate their solutions.