# A Multi-Constrained Optimization Method for THz Backward Wave Oscillators
## 2 Problem Description  
A THz BWO mainly consists of a cathode, an anode, an SWS, an output structure and a collector, as shown in Figure 1[17]. The electromagnetic wave on the SWS has a phase velocity that is slowed down to be approximately equal to the drifting velocity of the electrons so that the beam-wave interaction can occur under the Cerenkov radiation scheme. A corrugated rectangular waveguide SWS (CRWSWS) is under observation and shown in Figure 2.  
Figure 3 shows the \(\omega\)--\(k_{z}\) diagram of the CRWSWS [7], which describes the relationship between the frequency \(\omega\) and the wave-number \(k_{z}\). According to [PERSON]'s theorem, an electromagnetic mode propagating on a CRWSWS consists of infinite space harmonics so that its dispersion curves (as shown in Figure 3) repeatedly expand along the wave-number axis. The straight lines in Figure 3 correspond to the electron beams with different beam voltages. The slopes of the lines are proportional to the beam velocities, which can be derived from the beam voltages. When the beam voltage is given, the operating frequency can be determined by the intersection of the corresponding beam line and the dispersion curve of the lowest electromagnetic mode. As a result, the SWS determines the relationship between the beam voltage and the operating frequency. Before optimizing the SWS of the device, the operating frequency range and expected beam voltage are commonly given.  
Figure 1: Schematic of a BWO with beam voltage \(V\), beam velocity \(v_{0}\), and phase velocity \(v_{\text{ph}}\).  
Figure 3: Dispersion curves of the electromagnetic modes in CRWSWS [7].  
Figure 2: CRWSWS of a THz BWO [7]. \(L\) is the grating period; \(s\) is the gap width between teeth; \(h\) is the tooth height; \(w\) is the waveguide width; \(b\) is the waveguide height; \(N\) is the SWS period number. The light gray part shows the position and size of the electron beam that has a beam thickness in \(y\) direction and a beam width in \(x\) direction. The height of the beam tunnel is \(b\)-\(h\).  
When the position of the electron beam and the geometrical parameters of the SWS are given, the working parameters such as the beam current and the beam voltage of the device can be determined with theoretical models [7]. As shown in a previous study, the beam voltage and beam current are constrained due to mode competition [7]. When the device operates at the maximum beam voltage, the corresponding beam line should intersect only with the first backward space harmonic (\(n=-1\)) of the TE\({}_{x10}\) dispersion curve to avoid the mode competition between electromagnetic modes. This constraint on the beam voltage can be quantitatively described as [7]  
\[\Delta U\equiv U-U_{1}<0, \tag{1}\]  
where \(U_{1}\) is the beam voltage corresponding to the beam line (Line 2 in Figure 3) that intersects with the low-frequency cut-off point of the \(-1\) harmonic of the TEx10 dispersion curve and \(U\) is the beam voltage. The beam current should be smaller than the minimum starting current of the second lowest oscillation mode in the operating frequency range to avoid mode competition between oscillation modes [7].  
In summary, the specific constraints considered in the optimization problem and the optimization objective (the fourth item) are listed below:  
1. The oscillation frequency is centered at a given GHz, ranging from \(f_{\min}\) to \(f_{\max}\);
2. The maximum beam voltage (denoted as \(V_{m}\)) at \(f_{\max}\) is \(\sim\)16 kV and \(\Delta U<0\) (the beam line has no intersection with the first backward harmonic of higher-order electromagnetic modes);
3. The current density (denoted as \(J\)) is \(\sim\)500 A/cm\({}^{2}\) and the beam current should not exceed the minimum starting current of the second-lowest oscillation mode in the operating frequency range;
4. The minimum output power (denoted as \(P\)) in the operating frequency range is maximized and the acceptable relative error for each constraint is 10%.