A quantum graph approach to metamaterial design

The two-dimensional scattering matrix at the interface exhibits intriguing properties. Singularities can occur in both the denominator and numerator of the matrix.

To determine the conditions for full transmission at the interface, we analyze the parameters ( K L_\mu ) and ( \kappa Y ). The condition for achieving full transmission is that ( \cos(K L_\mu) ) must equal ( \cos(\kappa Y L) ). This relationship is illustrated in the accompanying graph, where the x-axis represents ( K L_u ) and the y-axis represents ( \kappa Y ).

The black lines in the graph indicate regions of full transmission, while the red line represents a parameter related to the length connecting the next to nearest neighbors. By varying ( \mu ), we effectively modify the system. For a fixed ( K L_\mu ), the graph shows different values of ( \kappa ).