Time-Domain PIC-Modeling of the Multiple Cavity Klystron Oscillator with Delayed Feedback V.V. Emelyanov, N.M. Ryskin Saratov State University, Saratov, Russia, 410012 Abstract: The results of time-domain numerical simulation of a four-cavity klystron oscillator with external delayed feedback using a 1.5D PIC-code are presented. Basic properties of oscillation regimes are studied. With the increase of the beam current self-modulation and transition to chaos via the sequence of period–doubling bifurcations are observed. The results are in good agreement with the theoretical predictions obtained previously for approximate model of the oscillator based on a system of time-delayed differential equations. Keywords: multiple cavity klystron; delayed feedback; self-excitation; self–modulation; chaos; period–doubling bifurcations. Introduction Multiple cavity klystrons are widely used as high-power amplifiers for communications, radar, particle accelerators, etc. [1,2]. A klystron amplifier can be transformed to a self- excited oscillator by applying an external time-delayed feedback. Such oscillators have been widely investigated theoretically and experimentally [3-6]. It has been shown that various non-stationary processes such as complex transients, self–modulation and transition to chaos are typical for such an oscillator. Therefore, the primary role for investigation of these phenomena is played by numerical simulation. However most of the numerical results have been obtained for simplified models based on time-delayed differential equations. Such models can describe quite well the qualitative picture of behavior of the device but often fail to predict output power, efficiency and other important characteristics. In this paper, we describe a PIC code for efficient time-domain simulation and investigate the basic properties of oscillation regimes and transition to chaos scenario. PIC-code and numerical results A code for time-domain numerical simulation klystron amplifiers and oscillators is developed. The equations of motion of the particles are solved by the well-known 1.5D “particle-in-cell” (PIC) method [7]. Electromagnetic fields of the cavities are found from the non-stationary excitation equations [1] which are solved by the “predictor–corrector” method of the second order of accuracy. The numerical method is similar to that described in [8] for coupled-cavity TWT simulation. We study a 1.5D problem, i.e., assume that the electron motion is one-dimensional but reduction of space-charge forces due to finite beam diameter is taken into account. A four-cavity S-band klystron with the parameters close to that one studied experimentally [3,4] is considered. Figure 1. The boundary of self–excitation (1) and the threshold curve of modulation appearance (2) Figure 2. Dependence of the average output power on the beam current at the center of generation area Figure 1 shows the boundaries of the self–excitation (1) and threshold of onset of self–modulation (2) on the “beam current − phase shift in the feedback circuit ” parameter plane. Above the self-modulation threshold the regime of single-frequency operation becomes unstable and multiple- frequency oscillation named self-modulation appears. Figure 2 shows the average output power of the oscillator vs. the electron beam current at  that corresponds to center of generation zone where the starting current is minimal [3-6]. In figure 2 the domains of different oscillation regimes are shown. In the domain I regime of single–frequency oscillation is established (see figure 3).