A quantum graph approach to metamaterial design

This slide illustrates the typical profile of a beam, represented by the α curve. As the beam is broadened, it takes on a cone-like shape, while a narrower beam resembles a plane wave. We can also shift the beam in various directions by selecting different solutions along the dispersion curve, which is essential for defining group velocity.

In reflection and transmission scenarios, the beam propagates according to the group velocity, which is determined by the gradient of the dispersion curve, rather than the phase velocity. This distinction is particularly significant in the context of negative refraction.

It is important to note that the underlying structure consists of one-dimensional waveguides interconnected at vertices. Locally, energy propagates along these one-dimensional waveguides, while globally, the waves behave as if they are plane waves traveling through the material.