## 3 Thermal Sensing  
In the development of a temperature sensor, one must consider the thermal expansion of the waveguide material (Au in this case). Thus, when the temperature is changed, the geometry of the proposed waveguide will also change. Consequently, the THz response of the waveguide material will vary by changing the temperature. Thus, the frequency of the NBDS will shift according to the varying temperature, which can be directly used to sense temperature changes. With the linear expansion  
Figure 2: (**a**) Comparison of Bragg and non-Bragg stop-bands, with and without defects: red solid line = transmission curve for the waveguide with defects; blue dashed line = transmission curve for the waveguide without defects. (**b**) Radial distributions for the Bragg and non-Bragg defect states: red squares = normalized magnetic field strengths for the NBDS; blue circles = normalized magnetic field strengths for the Bragg defect state. The blue and red solid lines show the best fitting curves. coefficient of Au, \(\alpha\), the period, \(\Lambda\), and mean radius, \(r_{0}\), of the proposed waveguide, the following temperature dependencies can be obtained:  
\[\Lambda(T)=(1+\alpha\Delta T)\Lambda \tag{3}\]  
\[r_{0}(T)=(1+\alpha\Delta T)r_{0} \tag{4}\]  
where the temperature difference, \(\Delta T\), is the deviation from room temperature, 293 K.  
In addition, the traditional Drude model must be modified because of the temperature dependency of the THz response. The well-known Drude conductivity formula is shown in Equation (5). \[\sigma(0)=\frac{\omega_{p}^{2}}{\omega_{\tau}}\varepsilon_{0} \tag{5}\]  
where \(\sigma(0)\) is the DC conductivity, \(\varepsilon_{0}\) is the dielectric constant in vacuum, \(\omega_{p}\) is the plasma frequency, and \(\omega_{\tau}\) is the damping frequency. Furthermore, \(\omega_{p}\) and \(\omega_{\tau}\) are defined as  
\[\left\{ \begin{array}{l}\omega_{p}=\sqrt{\frac{q(-e)^{2}}{\varepsilon_{0}m}} \\ \omega_{\tau}=\frac{1}{\tau}\end{array} \right. \tag{6}\]  
where \(q\) is the number of conducting electrons per unit volume, \(e\) is the charge of the electron, \(m\) is the electron mass, and \(\tau\) is the relaxation time [37]. Moreover, the electric conductivity, \(\sigma(\omega)\), can be given by  
\[\sigma(\omega)=\frac{\sigma(0)}{1-\mathrm{i}\omega\tau} \tag{7}\]  
where the circular frequency \(\omega\) = 2\(\pi\)_f_. The complex dielectric constant in the Drude model, \(\varepsilon=\varepsilon_{1}+\mathrm{i}\varepsilon_{2}\), is also expressed as  
\[\left\{ \begin{array}{l}\varepsilon_{1}=1-\frac{\omega_{p}^{2}}{\omega^{2}+ \omega_{\tau}^{2}}\\ \varepsilon_{2}=\frac{\omega_{p}^{2}\omega_{\tau}}{\omega^{3}+\omega\omega_{ \tau}^{2}}\end{array} \right. \tag{8}\]  
where \(\omega_{p}=1.37\times 10^{4}\) THz, and \(\omega_{\tau}\) = 40.7 THz (for Au) [38]. For a temperature close to room temperature, the Drude model of the dielectric constant can be modified to become dependent on the temperature. At first, assume that Equations (5)-(8) are valid at room temperature, \(T_{0}\) = 293 K. It is then possible to study the temperature dependency as a function of the variation in \(T\), \(\Delta T\) (when close to room temperature). For isotropic materials, the thermally induced expansion (in volume) yields the temperature-dependent electron density, \(q/(1+3\alpha\Delta T)\). This is due to the small value of the expansion coefficient, \(\alpha\). The temperature-dependent plasma frequency, \(\omega_{p}\) (\(T\)), can therefore be calculated by using Equation (9). \[\omega_{p}(T)=\frac{\omega_{p}}{\sqrt{1+3\alpha\Delta T}} \tag{9}\]  
According to the quantum theories of electrons for pure metals, the damping frequency, \(\omega_{\tau}\), is linearly dependent on the temperature in the high-temperature range (i.e., over the Debye temperature). For this high-temperature range, the temperature-dependent damping frequency, \(\omega_{\tau}(T)\), will be linearly dependent on \(\omega_{\tau}\):\(\omega_{\tau}(T)\propto\omega_{\tau}\).