## 2 Modeling, Fabrication, and Experimental Characterization of THz FSSs
During the simulations, we designed and analyzed a 6 mm length by 6 mm width sample, which was positioned between the THz source and the detector section. Figure 2a,b illustrates the transmission characteristics obtained with the simulation (green line). To validate our simulation and experimental results and evaluate the filter response of both types of THz FSSs, we used the well-known numerical modeling reported by [41]. It was demonstrated that the model can accurately predict the transmission characteristics of FSS, consisting of circular apertures with triangular lattices. The transmission coefficients \(T\) can be obtained as [41]:  
\[T=\frac{1}{1-j[A+B\text{tanh}(\beta l)]}-\frac{1}{1-j[A+B\text{coth}(\beta l)]} \tag{2}\]  
where \(\beta\) is the phase constant and \(l\) is thickness of the sample. To calculate the parameters \(A\) and \(B\) as functions of hole spacing and hole size, respectively, the following Equations (3) and (4) can be applied [39,41]:  
\[A=12\left(\frac{4}{3}\left(\frac{\lambda}{d}\right)^{2}-1\right)^{\frac{1}{2}} \left[\frac{J_{1}^{\prime}\left(\frac{4\pi a}{\sqrt{3}d}\right)}{1-\left( \frac{4\pi a}{1.841\sqrt{3}d}\right)^{2}}\right]^{2}-\frac{12}{\left(\frac{4}{ 3}\left(\frac{\lambda}{d}\right)^{2}-1\right)^{\frac{1}{2}}}\left[\frac{J_{1} \left(\frac{4\pi a}{\sqrt{3}d}\right)}{\frac{4\pi a}{\sqrt{3}d}}\right]^{2} \tag{3}\]  
\[B=0.33\left(\frac{d}{a}\right)^{2}\left(\left(\frac{0.293\lambda}{a}\right)^{2} -1\right)^{\frac{1}{2}} \tag{4}\]  
The circular holes' centers are separated by a distance of \(d\), the radius of the circular hole is denoted by \(a\), and \(\lambda\) stands for the wavelength. \(J_{1}\) and \({J_{1}}^{\prime}\) are the first kind of Bessel function and the derivative, respectively. The phase constant can be obtained as follows:  
\[\beta=\frac{2\pi}{\lambda}\left(\left(\frac{0.293\lambda}{a}\right)^{2}-1 \right)^{\frac{1}{2}} \tag{5}\]  
Figure 2: Normalized transmission as a function of frequency in the case of a single layer of THz FSS, (**a**) printable electronics-based; (**b**) 3D printed. The above mathematical model is useful to accurately anticipate the transmission characteristics of a THz FSS without substrate, which is the case of the THz FSS made using 3D printing. On the other hand, for an effective prediction of the PE-based THz FSS's transmission characteristics, we ought to take into consideration the effect of the PET substrate. To this end, we replaced the \(\lambda\) with \(\lambda_{2}\) in Equations (3) and (5) and defined as \(\lambda_{2}=\lambda\times(n/n_{2})\), where \(n\) and \(n_{2}\) are the refractive indices of air (1.00) and PET (1.74 at 0.5 THz) [42], respectively. As such, the calculated transmission characteristics for the two types of FSSs were evaluated using Equation (2). Figure 2a,b illustrates the transmission characteristics obtained with the mathematical model (black dotted line) and shows the calculated peak transmission frequencies for the PE and 3D-printed FSSs at 247 and 370 GHz, respectively. From the calculation, it is clear that both FSSs behave like band-pass filters (BPF). The experimental measurements and simulation results demonstrated that the peak transmission frequency of the PE-based THz BPF is at a lower frequency region as compared to the 3D-printed THz BPF due to the PET substrate. The peak transmission frequency for a single PE-based THz filter was determined to be 228, 247, and 235 GHz from simulation, calculation, and measurement, respectively. Similarly, the peak transmission frequency of a single 3D-printed THz filter obtained by simulation, calculation, and measurement are 360, 370, and 365 GHz, respectively. In Figure 2, we can see from experiments that the 3D-printed THz BPF provides a sharper transmission with a higher quality factor than the PE-based THz BPF. In addition, the stopband is exactly the same as the calculated value and more pronounced than that obtained with the PE filter.