Electromagnetic waves solve partial differential equations

Amazing stuff!

"Waveguide-based structures can solve partial differential equations by mimicking elements in standard electronic circuits. This novel approach, developed by researchers ... could boost efforts to use analogue computers to investigate complex mathematical problems. ...

Unlike in previous setups, however, Pacheco-Peña’s team exploited a grid-like network of parallel plate waveguides filled with dielectric materials. This structure behaves like a network of interconnected T-circuits, or metatronic elements, with the waveguide junctions acting as sampling points for the PDE solution, ... “By carefully manipulating the impedances of the metatronic circuits connecting these points, we can fully control the parameters of the PDE to be solved,”  ...

“This might also allow the waveguide-based structure to be integrated with silicon photonics or plasmonic devices.” ..."

From the abstract:

"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."

Electromagnetic waves solve partial differential equations – Physics World

Solving partial differential equations with waveguide-based metatronic networks (open access)

Fig. 1 Transmission line schematic representation of metatronic loaded network to solve PDEs.