# A Multi-Constrained Optimization Method for THz Backward Wave Oscillators
## 4 Case Studies of THz BWOs
### Second Case  
For the second case, the same structure of SWS and same constraints listed in Sec. II have been applied again. The operation frequency range changes to 800-900 GHz, corresponding to a THz window in air. Following Equation (4), the beginning can be obtained  
\[\overset{\rightarrow}{s}_{\textit{case2,1}}=(L_{0},s_{0},h_{0},w_{0},N_{0}) \frac{1000}{850} \tag{11}\]  
The conductivity of the SWS is also 5.96 \(\times\) 10\({}^{7}\) S/m. The optimized geometrical parameters are \(L\) = 61 \(\upmu\)m, \(s\) = 33 \(\upmu\)m, \(h\) = 79 \(\upmu\)m, \(w\) = 250 \(\upmu\)m, \(b\) = 130 \(\upmu\)m and \(N\) = 147. The current density is 543 A/cm\({}^{2}\), the beam voltage ranges from 8 to 15.8 kV, and the guiding magnetic field is 1 T. The dispersion curves of the optimized SWS are shown in Figure 11a. The 16 kV beam voltage line has no intersection point with the dispersion curves of higher-order modes as shown in Figure 11a. The output powers of the optimized BWO are compared with the theoretical results as shown in Figure 11b. The theoretical output powers can estimate the simulated output powers with acceptable accuracy. The simulated time signal at the beam voltage of 12 kV and its spectrum are shown in Figure 12. The optimized BWO delivers stable output signal and the maximum efficiency is 0.52 mW/9.2 mA/15.8 kV = 0.35%, which is close to the maximum efficient of 0.34% in Ref. [22]. Considering that the beam size in cross-plane, and the current density and the beam voltage are the same, it is reasonable to obtain a similar total efficiency for the optimized BWO in 800-900 GHz. To check whether the optimized structure is the global optimum, two more optimizations are performed. The beginning geometrical parameters are  
\[ \begin{array}{l}\overset{\rightarrow}{s}_{\textit{case2,2}}=(L_{0},s_{0},b_ {0},h_{0},w_{0},N_{0})\frac{1000}{850}(1.2,0.8,1.2,0.8,1.2,0.8)\ \text{and}\\ \overset{\rightarrow}{s}_{\textit{case2,3}}=(L_{0},s_{0},b_{0},h_{0},w_{0},N_{0 })\frac{1000}{850}(0.8,1.2,0.8,1.2,0.8,1.2).\end{array}  \tag{12}\]  
The optimization results from three different beginnings are listed in Table 1. It can be observed that the three results are close to each other although they correspond to quite different beginnings. This demonstrates the validity of the proposed global optimization method. Figure 10: Time signal of the output power (**a**) and frequency spectrum of the time signal (**b**) obtained by numerical simulation of the optimized 900–1100 GHz BWO [7]. It takes only 16 steps for the searching algorithm to reach a LO as shown in Figure 13. Figure 13 shows that the optimization process has three durations (\(\leq\)11, 12-14, 15-16) corresponding to three different gradient directions. The initial solution has out-ranged working parameters and the optimization pulls the working parameters back within the constraints successfully. As relative errors are allowed, the optimized working parameters may exceed the expectation values. \begin{table}
\begin{tabular}{c c c c} \hline  & **Case2-1** & **Case2-2** & **Case2-3** \\ \hline \(d\) (\(\mu\)m) & 61 & 61 & 62 \\ \(s\) (\(\mu\)m) & 33 & 31 & 29 \\ \(b\) (\(\mu\)m) & 130 & 131 & 127 \\ \(h\) (\(\mu\)m) & 79 & 80 & 79 \\ \(w\) (\(\mu\)m) & 250 & 252 & 250 \\ \(N\) & 147 & 148 & 147 \\ \(V_{m}\) (kV) & 15.8 & 15.9 & 16.3 \\ \(J\) (A/cm\({}^{2}\)) & 543 & 499 & 542 \\ \(P\) (W) & 0.20 & 0.19 & 0.19 \\ \hline \end{tabular}
\end{table} 
Table 1: Optimization results of the 800–900 GHz BWO from three different beginnings. Three separate optimizations are denoted as Case2-1, Case2-2 and Case2-3, respectively. Figure 11: Dispersion curves (**a**) and output powers vs frequency (**b**) for the optimized BWO in 800–900 GHz. Figure 12: Time signal of the output power (**a**) frequency spectrum of the output signal (**b**) for the optimized BWO in 800–900 GHz. A comparison is made between the previous optimization method in Ref. [7] and the proposed method in this paper. In Ref.