## 2 NBDSs  
Defect states of THz waves in defect-containing periodically corrugated metal tubes can arise not only in the traditional Bragg gap (created by the interaction of two identical transverse modes) but also in a non-Bragg gap (caused by a more complexed resonance). Based on a previous theoretical investigation [39], the non-Bragg resonance refers to the interaction between the different guided transverse modes. Like Bragg resonances, non-Bragg resonances can cause band splitting and form a frequency-forbidden band. However, the characteristics of a non-Bragg forbidden band are completely different from those caused by Bragg resonances. When a defect is introduced into the periodically corrugated metal tube, the defect states can be found to create a transmission peak in the non-Bragg forbidden band. The defect-containing periodically corrugated gold wave that has been used in the present study is shown in Figure 1. It comprises a periodically corrugated metal tube with a gold thickness of 0.5 \(\upmu\)m and a high-density polyethylene (HDPE) of 10 \(\upmu\)m thickness. The outer layer of the tube is made of the HDPE, and the inner layer is deposited (i.e., sputtered) with gold. The thickness of the gold layer has been chosen to be 0.5 \(\upmu\)m, which is much larger than the skin depth of gold (of around 0.7 THz). Moreover, \(r_{0}\), \(\Lambda\), and \(\Delta r\) represent the mean radius of the inner part of the tube, the wall period, and the corrugation amplitude, respectively. \(r\) and \(z\) denote a cylindrical coordinate system. A straight tube with a length, \(L\), of 0.8\(\Lambda\), and a radius, \(R\), of 0.9\(r_{0}\), was inserted into the middle of the perfect waveguide. The composition of this structure has earlier been proven to show low loss and dispersion in the THz range, and the refractive index of HDPE is 1.5 [40]. Due to the periodicity of the structure, all dispersion curves (for different transverse modes) were folded back to the first Brillouin zone, where they intersect with each other. Bragg and non-Bragg resonances did also occur in this Brillouin zone, and related frequency gaps appeared. The non-Bragg resonances were created by interactions between the different transverse modes, while the Bragg ones were caused by the same modes in the waveguide [39]. The trends in the dispersion curves, and the frequencies of the resonances, could be predicted by calculating the reference line in the first Brillouin zone:  
\[f=\frac{c}{2\pi}\sqrt{\frac{{k_{r}^{(m)}}^{2}}{r_{0}^{2}}+\left(\beta+\frac{2 \pi n}{\Lambda}\right)^{2}} \tag{1}\]  
where \(f\) is the frequency, \(c\) is the velocity of light, \(k_{r}^{(m)}\) is the \(m\)th zero of the zero-order Bessel function [\([k_{r}^{(m)}\), \(m=1\), \(2\), \(3\), \(4\)... ] = \(\{2.4048\), \(5.5201\), \(8.6537\), \(11.7915\)... ]], and \(\beta\) (\(-\eta\)/\(\Lambda\) \(\leq\)\(\beta\)\(\leq\)\(\eta\)/\(\Lambda\)) is the reference propagation constant. To guarantee the formation of a non-Bragg gap near 0.7 THz, the wall period, \(\Lambda\), and the mean radius, \(r_{0}\), were set to 353 \(\upmu\)m and 395.1 \(\upmu\)m, respectively (according to Equation (1)). The corrugation amplitude, \(\Delta r\), of 39.5 \(\upmu\)m was then approximately 10% of the mean radius, \(r_{0}\). The 2D axisymmetric structure was constructed by using the COMSOL Multiphysics software, with finite element method (FEM) simulations at room temperature (\(T\) = 293 K). The transmission spectrum of the waveguide is shown by a blue dashed line in Figure 2a. The periodic waveguide has a Bragg gap in the range 0.415-0.560 THz, and a non-Bragg gap in the range 0.610-0.715 THz. For the situation with a straight tube inserted into the middle of the perfect waveguide, the resulting transmission spectrum is demonstrated by a red solid line in Figure 2a. As shown in Figure 2a, a transmission peak appears in each forbidden band, and the Bragg and non-Bragg gaps are split by these induced defect states.