## 2 Design and Simulations  
In this part, a low-frequency THz PC waveguide with a iliac-shaped defect is designed, as shown in Figure 1a,b. The symmetrical petal hollow core is induced in the waveguide structure to broaden the operation frequency bandwidth. For efficiently guiding the terahertz wave, the structural parameters are optimized to design a suitable structure. The lattice structure of the periodically arranged unit cells consists of air holes on the \(o\)-\(xy\) plane, which are described by the following [28; 29]  
\[x(M,m) = aM\cos\left(\frac{2m\pi\tau}{6M}\right) \tag{1}\]  
\[y(M,m) = aM\sin\left(\frac{2m\pi\tau}{6M}\right) \tag{2}\]  
where \(a\) is the lattice constant, \(M\) is the number of the air hole rings, and \(m\) (\(1\leq m\leq 6M\)) is the number of the air holes in the \(M_{th}\) ring. The ilac-shaped resonant cavities are formed by four larger symmetrical air holes at the center, with the first, second, and third rings of the periodically arranged unit holes removed. Each resonant cavity consists of an intersection of one circular air hole with a diameter of \(D\) and one square air hole with a length of \(L\). The gap distance between four petals of the ilac-shaped resonant cavities is defined as \(g\). Initial values of the structure parameters are \(a=1000\)\(\upmu\)m, \(d=0.8\)\(a=800\)\(\upmu\)m, \(D=1500\)\(\upmu\)m,\(L=750\)\(\upmu\)m, and \(g=500\)\(\upmu\)m. Meanwhile, the depth of the air holes in the \(z\) direction is much larger than \(a\) (\(h>>a\)). In the simulation process, the optical characteristics of the THz PC waveguide were analyzed using COMSOL Multiphysics. The boundary condition was set to the scattering boundary condition to absorb energies. The physics-controlled mesh was applied to the model [29]. The process of the reflection and transmission characteristics of the proposed waveguide is shown in Figure 1c. The TE-polarized Gaussian form was injected into the waveguide from the left side and a monitor was set on the right side of the waveguide. Air holes were placed throughout the photosensitive resin background. In the simulation, the refractive index (RI) of the photosensitive resin used in the range of 0.1 to 0.5 THz were measured using the Terahertz Time-Domain Spectroscopy (THz-TDS) system (Menlo TeraSmart, Martinsried, Germany), which is shown in Figure 1d. The gap distance (g) is an important parameter that can change the photonic bandgap of the photonic crystal; in particular, the guidance of the terahertz wave in the waveguide is caused by the photonic bandgap mechanism. Therefore, in order to explore the detailed functionality of the waveguide, the influence of the gap distance (\(g\)) has been simulated from 100 \(\upmu\)m to 2000 \(\upmu\)m in steps of 100 \(\upmu\)m. The reflection map of the waveguide with different gap distances is shown in Figure 2a. As the gap distance increases from 100 \(\upmu\)m to 2000 \(\upmu\)m, the reflection loss performs a resonant dip in the frequency region of 0.2 to 0.3 THz. Correspondingly, as shown in Figure 2b, the transmission loss increases continually within 0.5 THz. For instance, when the gap distance is 1000 \(\upmu\)m, the reflection loss (S11) and transmission coefficient (S21) are shown in Figure 2c. The band-stop behavior of the proposed waveguide is analyzed by the reflection loss. The resonant frequency of the band-stop is approximately at 0.23 THz with a transmission loss of about \(-44\) dB. The transmission coefficient shows the low-pass behavior. To better understand the reflection and transmission effect, the simulated electric field patterns at different frequencies are analyzed in Figure 2d when \(g\) is 1000 \(\upmu\)m. It is observed that the THz wave can smoothly pass through this waveguide within 0.2 THz and it is reflected mostly higher than 0.3 THz. Figure 1: (**a**) The 2D structure of the THz PC waveguide. (**b**) The 3D structure of the THz PC waveguide. (**c**) The process of reflection and transmission characteristics of the THz PC waveguide.