Equation (7.11) describes the relation between the propagation constant of the twisted metamaterial and that of a periodic structure, which enables us to compare them with full-wave numerical simulations. Let us consider a twisted metamaterial with the PEC rod inclusions and d = 100 nm as an example. The twist angle is 60°. A periodic structure with a supercell composed of three rods has been numerically studied using CST to obtain its dispersion diagram. The results of this full-wave calculation are compared with the results of the above stretch and fold procedure in Fig. 7.8. For simplicity, we only plot the real part of the propagation constant, since our full-wave simulations cannot compute evanescent modes. It is seen that the analytical solutions agree very well with the full-wave calculations, except that the numerical simulation was also unable to obtain the complex modes present in the structure. Note that unlike the dispersion diagram of a twisted metamaterial, the dispersion relation of the periodic structure starts at zero, as expected.

The dispersion diagrams of Fig. 7.8a start at zero frequency with two modes (modes 2 and 4) propagating in the forward direction. As the propagation constant of the second mode crosses п/d' (i.e., p'd' = 180°), it becomes evanescent since this corresponds to /3d = 0. Mode 4 enters from the other side of the dispersion diagram at -180° after crossing p'd' = 180°, which is not shown here. Similarly, at this frequency (kd ~ 0.6), mode 1 reaches p'd' = -180° and enters from the +180° side of the dispersion diagram. Starting at kd ~ 0.8, mode 1 becomes evanescent and we see a stopband up to kd ~ 1.0. The full-wave calculations have a similar behavior, although the exact numbers are slightly different. The small bandgaps at kd ~ 0.65, 1.15, and 1.4 are due to the fact that we have neglected the thickness of the metasurfaces in our analysis.

Figure 7.8 The dispersion diagram of a twisted metamaterial with super cell periodicity consisting of three twisted unit cells in the direction of propagation. Dispersion diagram obtained from (a) analytical results, and (b) numerical simulations.

At higher frequencies (kd > 1), more discrepancies between full- wave calculations and our analytical solution are caused by the fact that the homogenized surface impedance can no longer describe the exact behavior of the metasurfaces. This is expected since at higher frequencies, the transverse periodicity of the metasurfaces is getting comparable to the free-space wavelength. In addition, at higher frequencies, the difference between rotating the entire metasurfaces and rotating individual inclusions becomes more pronounced; the former is considered in our analytical solution, while the latter is considered in our full-wave numerical simulations [85].