## 2 Design of the Hollow-Core Bragg Fiber  
The 3D view of the proposed THz Bragg fiber is shown in Figure 1. The cross section of it is shown in Figure 2. The hollow-core Bragg fiber consists of a hollow core and eleven alternating cyclical layers of photosensitive resin and air with the same thickness. In the cladding region, the layers of photosensitive resin with high RI and air with low RI are radially arranged. The radius of the core is 2.5 mm, and the thickness of layers is 0.5 mm. In the proposed Bragg fiber, the defect layer is introduced by the first layer of air filled with water at different temperatures. Figure 1: 3D view of the proposed fiber where terahertz (THz) radiation impinges at the center of the fiber. In the Bragg fiber, when the THz wave propagates from the hollow core to the cladding, the THz wave is confined in the core due to the Bragg diffraction produced by the periodical cladding. The defect can be considered as a lossy Fabry-Perot cavity. Depending on the phase condition at the core-cladding boundary and the anti-resonant reflection condition in the cladding, the resonant frequency can be expressed by the following equation [28]  
\[f=c/\left(d_{w}\sqrt{n_{w}{}^{2}-n_{c}{}^{2}}+d_{p}\sqrt{n_{p}{}^{2}-n_{c}{}^{2}}\right) \tag{1}\]  
where \(n_{w}\) and \(n_{p}\) are RI of the defect layer and photosensitive resin, \(d_{w}\) and \(d_{p}\) are the corresponding thicknesses, and \(n_{c}\) is RI of the core material. In this paper, the core material is air (\(n_{c}=1\)). Equation (1) shows that the resonant frequency is shifted to the lower frequency with an increase in RI of the defect layer. It indicates that the transmission characteristic can be changed by the RI of the defect layer. Therefore, the transmission characteristic could be controlled through the water with different temperatures filled into the defect layer. The complex RI of liquid water at different temperatures can be deduced by the dielectric constant. Depending on the double Debye mode, the complex dielectric constant of liquid water can be expressed by [29]  
\[\varepsilon(\omega)=\varepsilon_{\infty}+\frac{\varepsilon_{s}-\varepsilon_{1 }}{1+i\omega\tau_{D}}+\frac{\varepsilon_{1}-\varepsilon_{\infty}}{1+i\omega \tau_{1}} \tag{2}\]  
where \(\varepsilon_{s}\) is the static dielectric constant at low frequency, \(\varepsilon_{\infty}\) is the ultimate dielectric constant at high frequency, \(\varepsilon_{1}\) is the transitional dielectric constant between the slow and fast relaxation process, \(\tau_{D}\) is the slow relaxation time constant decaying from \(\varepsilon_{s}\) to \(\varepsilon_{1}\), and \(\tau_{1}\) is the fast relaxation time constant decaying from \(\varepsilon_{1}\) to \(\varepsilon_{\infty}\). The fundamental description of the complex dielectric constant related to the frequency is [30]  
\[\varepsilon(\omega)=\varepsilon^{\prime}(\omega)-ie^{\prime\prime}(\omega) \tag{3}\]  
where \(\varepsilon^{\prime}(\omega)\) and \(\varepsilon^{\prime\prime}(\omega)\) are the real and imaginary parts of the complex dielectric constant, respectively. Then, the complex RI can be easily calculated by the following relationships [30]  
\[\overset{\wedge}{n}(\omega)=n(\omega)-ik(\omega), \tag{4}\]  
Figure 2: Cross section of the proposed THz Bragg fiber. \[n(\omega)=\left[\frac{\sqrt{\varepsilon^{\prime}{(\omega)}^{2}+\varepsilon^{\prime \prime}{(\omega)}^{2}}+\varepsilon^{\prime}{(\omega)}}{2}\right]^{1/2}, \tag{5}\]  
\[\alpha(\omega)=\frac{4\pi f}{c}\left[\frac{\sqrt{\varepsilon^{\prime}{(\omega)} ^{2}+\varepsilon^{\prime\prime}{(\omega)}^{2}}-\varepsilon^{\prime}{(\omega)} }{2}\right]^{1/2}, \tag{6}\]  
Figure 2.0  
where \(n(\omega)\) is the real part of the complex RI, which represents RI, \(k(\omega)\) is the imaginary part of the complex RI, which represents extinction coefficient, \(\alpha(\omega)\) is the absorption coefficient, which represents absorption loss, \(f\) is the frequency, and \(c\) is the speed of the light wave in the vacuum.