A quantum graph approach to metamaterial design

This slide presents a simple extension of our previous discussion. We observe two bands: the first band and the second band. By varying the dimensions in one direction, we set ( l_x = 2 ) along the x-direction and ( l_y = 1 ) along the y-direction. The first band remains similar to previous observations, while the second band exhibits a saddle structure. This saddle is critical as it facilitates negative refraction within that band.

During the presentation, please feel free to ask questions at any time. Your inquiries provide valuable feedback and help maintain engagement.

Next, we will explore wave propagation through this material, moving beyond infinite plane waves. We can analyze wave solutions based on a specific set of wave vectors, defined by ( \kappa_Y ) and ( \kappa_X ). For a fixed wave number ( K ), ( \kappa_X ) and ( \kappa_Y ) are interconnected through a dispersion relation. By fixing ( \kappa_Y ) and ( K ), ( \kappa_X ) becomes determined. This represents a specific solution within the mesh. By superimposing this solution with an appropriate coefficient and utilizing a Gaussian form for the parameters, we can generate a beam.