Self-modulation and chaos in delayed-feedhack mystron oscillators

Abstract

A general method for calculation of the start currents of modes of overmoded cavities in klystrons is described. The method is applicable to klystrons with an arbitraly geometry of electron beams and cavities with large transverse andior axial dimensions. When these dimensions are on the order of the wavelength or larger, the spectrum of mode frequencies is rather dense. Therefore, in addition to the desired mode, electrons can excite some parasitic modes. The conditions of self-excitation of various modes are formulated and applied to closed (gridded) and open (non-gridded) cavities. It is assumed that in closed cavities the field distribution can be described by sinusoidal functions, while for describing the field distribution in open cavities Hermite polynomials are used. In both cases, the coupling coefficients of the beam to the modes and corresponding real part of the gain function, which is analogous to the beam loading conductance taken with an opposite sign, can be calculated analytically.The formalism, in particular, is of immediate interest for sheet beam klystrons. As an example it was applied to one cavity of a sheet beam klystron. The cavity is of the extended interaction type, with 3 elements. Initial calculations with the real distribution indicate that the start current of the real cavity is higher than that evaluated with the use Non-stationary and chaotic generation of microwave radiation is important for numerous appications such as communication and radar systems, plasma heating, etc. In this paper, we summarize the results of investigation of nonlinear dynamics in klystron delayed feedback oscillators [I-31. Several models of oscillators with two or more cavities have been developed [I-21. Detailed theoretical analysis of selfexcitation conditions, system eigenmodes, self-oscillation conditions and basic features of stationary generation modes have been done. Self-excitation threshold has a form of discrete "oscillation zones'' that are 2n-periodic in the feedback phase parameter. Each zone corresponds to one eigenmode of an oscillator. Near the boundaries of two adjacent zones there is a region of bistahility and oscillation hysteresis where either of the two eigenmodes can survive as a result of a mode competition process.Numerical simulation of non-stationary processes has been carried out. Modeling of self-modulation regimes has revealed an existence of a variety of limit cycles of different shapes corresponding to regimes of periodical self-modulation. The studied systems demonstrate continuous complication of shapes of limit cycles and hard transitions between them. Latter is attended with a hopping of frequency of selfof Hermite polynomials, and thus the Hermite polynomials give a conservative estimate of the possibility of unwanted modes. Example calculations of the start currents will be given.Scenario of transition to chaos was studied in details. It depends on whether we are at the center of a zone or at its edge. In the center, period-doubling Feigenbaum scenario dominates. Near ...