A quantum graph approach to metamaterial design

We are examining the phenomena of refraction and reflection at the interface of two distinct metamaterials. Each metamaterial has its own dispersion curve, which necessitates a reevaluation of the boundary conditions. Unlike the continuous case, where boundary conditions are typically expressed in terms of wave functions and their normal derivatives, we must ensure that the fluxes and wave functions remain consistent across the boundary region.

At the interface, the wave vector component ( \kappa_Y ) is preserved. As a plane wave approaches this interface, it experiences bending according to the corresponding ( \kappa_X ) value on the right side. This interaction illustrates the complexities involved in analyzing wave behavior in metamaterials with differing properties.