  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 in the form of the
Helmholtz wave equation. 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 impedance, 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 partial
differential equations for Dirichlet and open boundary value problems,
demonstrating how the proposed metatronic-based structure has the ability to
calculate their solutions.
