# A Multi-Constrained Optimization Method for THz Backward Wave Oscillators
## 1 Introduction  
Terahertz radiation (0.3-3 THz) has great potential in a variety of applications; however, there is still a lack of compact and powerful THz radiation sources, which has become a bottleneck for the development of terahertz science and technologies [1; 2; 3]. Among different kinds of THz radiation source, backward wave oscillators (BWOs) are competitive in the 0.3-1 THz range and have attracted much attention in recent years [1; 4]. To fully explore the potential of a THz BWO, an optimization of the slow-wave structure (SWS) is essential [5]. Typically, a large amount of calculation is required during the optimization of the geometrical parameters of the SWS. The optimization design often takes several months if 3-D full-electromagnetic particle-in-cell simulations are required for obtaining accurate output performances. Very large amounts of possible combinations of the geometrical parameters make the optimization design of such devices very time-consuming. Sometimes, the design cycle can be reduced to several weeks when analytical models are applied for calculation of the output results [6; 7]. However, it is still hard to find the global optimum under specific constraints on the working parameters such as beam current, beam voltage and so on. Thus, an efficient and stable global optimization method is necessary for the optimization design of THz BWOs and other Cerenkov devices working in linear regions, no matter that the working performances are obtained by theoretical calculation or numerical simulation. As for the global optimization method, there is still no algorithm that can be proved mathematically to converge to the global optimum. To find the global optimum as far as possible, there are generally two types of method. The first type is the evolutionary algorithm [8], such as genetic algorithm (GA), particle swarm optimization (PSO) and differential evolution (DE). All evolutionary algorithms share the same procedures, i.e.,reproduction, mutation, recombination and selection, which are inspired by biological evolution. They also have different properties: EA converges more slowly than the others, but they have the best stability without error caused by noise; PSO has the lowest route length or number of generations; however, DE has the lowest calculation time due to the smaller requirement of the population size [9; 10]. The second type is the local search algorithm, together with the multi-start strategy [11]. Local search is a heuristic method that moves from solution to solution in the space of candidate solutions (the search space) by applying local changes until a local optimal is found. Local search can handle most problems with continuity and weak nonlinearity [11]. Even in the optimization of some complex systems, there is a trend toward the second type of method for better performances [12]. Which one is better for the optimization of THz BWOs? In previous studies on the optimization design of vacuum electronic devices, randomized optimization methods (the first type) such as genetic algorithms (GAs) were used in the optimization design of relativistic backward wave oscillators (RBWOs) [13; 14; 15]. Unlike non-randomized iteration optimization methods, randomized optimization methods choose a next move not only from the local solutions but also from randomly generated solutions. These'strange candidates' are the key factor for successful optimization of RBWOs because, in these high-power Cerenkov devices, the energy conversion takes place in non-linear regions and small changes of geometrical and working parameters can result in significantly different output performances. Non-randomized optimization methods have low efficiency in the optimization of such devices operating in nonlinear regimes.