Another important feature of these diagrams is the dependency of the bandwidth of the stopbands on the twist angle. As we introduce

Figure 7.5 Imaginary part of the propagation constant for a twist angle equal to (a) 10, (b) 20, (c) 30, (d) 40, (e) 50, (f) 60 degrees.

a nonzero twist angle, the bandwidth in which modes 1 and 4 are evanescent starts to increase. As the twist angle increases between 0 and 90°, the lowest frequency of the stopband at about kd = 0.8 decreases, while the highest frequency of the stopband increases. As a result, the bandwidth increases from 258 THz at zero twist angle to 366 THz at 60° twist. For larger twist angles, the appearance of complex modes complicates the propagation phenomena. This will be discussed later in the chapter. Furthermore, a second stopband for modes 2 and 3 emerges for nonzero twist angles. Both the lowest and the highest frequencies of this second stopband increase as the twist angle increases. Nevertheless, the bandwidth of this stopband also increases by increasing the twist angle. At 60° twist angle, the second bandwidth reaches 217 THz.

Figure 7.6 depicts the aforementioned stopbands as a function of the twist angles. The yellow and cyan regions corresponds to the stopband for modes 1 and 4, and modes 2 and 3, respectively. The two stopbands change differently with the twist angle. For twist angles larger than ~50°, the two stopbands merge, where the structure starts to exhibit a complete stopband with no propagating modes and thus can act as a stopband filter. For example, with a 60° twist angle, the twisted metamaterial passes modes 1 and 4 from 263 THz to 375 THz, while stopping modes 2 and 3. Below 263 THz, all modes propagate, and from 375 THz to 480 THz, there are no propagating modes in the structure.

Figure 7.6 Stopband of the modes of the twisted metamaterial.

When the twist angles are larger than ~63°, complex modes appear in the dispersion diagram of the twisted metamaterials. The propagation constants of complex modes have nonzero real and imaginary parts simultaneously. The propagation constants form a complex conjugate pair and their negative counterparts (i.e., p, p*, -p, -p*). The dispersion diagrams for 70°, 80°, and 90° twist angles are shown in Fig. 7.7. The complex modes are clearly seen at these twist angles. Interestingly, the bandwidth of the complex modes also increases as the twist angle increases. Although these complex modes cannot be excited directly by plane waves in free space, they may affect the evanescent spectrum and the local density of states in the proximity of a slab of twisted metamaterials.

Figure 7.7 (a), (c), (e) Real and (b), (d), (f) imaginary parts of the propagation

constant of the twisted metamaterial with (a), (b) 70, (c), (d) 80 and (e), (f) 90 degrees twist angles.

The propagation constants found from the above analysis are different in nature from the propagation constant in a homogenized medium or a periodic array without rotation. In addition to a phase shift due to the propagation constant that is defined in Eq. (7.4), the fields also experience a rotation. Nevertheless, it is possible in some cases, discussed in the next section, to find a simple relationship between the propagation constant of the waves inside the twisted metamaterials and the propagation constant in a periodic structure.