1516 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 34, NO. 4, AUGUST 2006 Eigenvalues and Ohmic Losses in Coaxial Gyrotron Cavity Zisis C. Ioannidis, Olgierd Dumbrajs, and Ioannis G. Tigelis, Member,IEEE Abstract—The authors present the mathematical analysis for the calculation of the dispersion relation, the field distributions, and the ohmic losses for TE m,p modes in an infinite coaxial waveguide with a longitudinally corrugated insert. The method employed is based on an appropriate eigenfunction expansion, and its main advantage is the very fast convergence with a few spatial harmonics. The analysis is properly extended to include tapered cavities with varying, in respect to the z-coordinate, outer and/or inner radius. Numerical results are presented for several tapered cavity geometries and compared with already published methods. Index Terms—Coaxial cavity, coaxial gyrotron, eigenvalues, ohmic losses. I. I NTRODUCTION I N THE development of high-power high-frequency gyro- trons capable of continuous-wave operation, the main tech- nological constraint is the resonator heating caused by the ohmic losses of the generated microwave power on the con- ductive walls of the cavity [1], [2]. To keep the walls’ thermal loading within the acceptable limit of 2–3 kW/cm 2 , it is essen- tial to increase the resonator volume, resulting in high-order mode operation. As the size of the resonator increases, the eigenfrequency spectrum becomes more dense, leading to the possibility of multimode oscillation that in turns can evoke de- terioration of the coherence and the directivity of the generated radiation, give rise to trapped modes, and decrease the gyrotron efficiency [3]. A quite effective means in mode selection is the appropriate choice of the radius of a thin annular electron beam correspond- ing to the maximum value of the coupling impedance between such a beam and a rotating nonsymmetric TE m,p mode, ensur- ing this way the strong interaction between the mode and the beam [4], [5]. Considering the selection of whispering TE m,p modes with different and relatively small radial indexes, this method has been proven to be quite effective [6]. However, for the problem under consideration, this method is ineffective because the operating frequency and the operating mode index are large. Manuscript received January 3, 2006; revised March 1, 2006. This work was supported by the Association Euratom-Hellenic Republic. Z. C. Ioannidis and I. G. Tigelis are with the Electronics Laboratory, Applied Physics Division, Department of Physics, National and Kapodistrian University of Athens, 157 84 Athens, Greece (e-mail: zioanni@phys.uoa.gr; itigelis@phys.uoa.gr). O. Dumbrajs is with the Department of Engineering, Physics and Mathemat- ics, Helsinki University of Technology, Euratom-TEKES Association, 02015 Helsinki, Finland, and also with the Institute of Solid State Physics, University of Latvia, Riga, LV-1063, Latvia (e-mail: dumbrajs@users.csc.fi). Digital Object Identifier 10.1109/TPS.2006.876518 Coaxial cavities have been proposed to provide additional means in mode selection of higher order modes [7]. The in- sertion of a conductive wall into a hollow resonator brings on deformation in the eigenvalue spectrum that may rarefy the spectrum of the competing modes provided that the geometrical characteristics of the insert are properly selected. Moreover, if the radius of the insert is smaller than the caustic radius of the operating mode but close to the caustic radius of the unwanted parasitic modes with larger indexes, one can disturb the properties of the latter without serious changes in the frequency and the Q-factor of the operating mode by modifying the shape of the insert [8]. The selective properties of a coaxial cylindrical resonator can be further enhanced by the introduction of longitudinal corrugations on the inner conductor. Choosing corrugation geo- metric characteristics properly, the eigenvalue curves χ(C = R cav /R in ) for parasitic modes become monotonic, whereas the eigenvalue curve for the operating mode retains its minimum. As a result, the quality factors of parasitic modes become significantly smaller than the quality factor of the operating mode, but their starting currents increase. This means that intro- ducing corrugations parasitic modes will be easier suppressed by the operating mode during the mode competition. Such structures have been studied extensively using the simplified surface impedance model (SIM) [8], [9], which is based on the assumption that the width of the corrugations is smaller than the wavelength of the oscillations, and therefore, the effect of the corrugations is characterized by a surface impedance. However, as the azimuthal index of the operating mode in- creases, this assumption is only marginally valid. Another technique to treat this problem is the singular integral equa- tion (SIE) method, which has been used to study various RF structures [10]–[12]. In this paper, the problem in question is solved by the method that has been used recently to calculate the eigenvalue spectrum in circular cylindrical waveguides [13] and reentrant cavities [14]. In particular, in Section II, the fields are expanded in terms of spatial eigenfunctions, and the application of the appropriate boundary conditions at the interfaces leads to a homogeneous system of equations for the unknown expansion coefficients. Nontrivial solutions demand that the determinant of this system is nullified and then the dispersion relation is obtained, and for every pair of axial wavenumber and eigenfrequency, the field distributions can be calculated everywhere. In Section III, we present numerical results for several coaxial geometries including tapered gyrotron cavities, and we compare our results concerning the eigenvalue spectrum and the ohmic losses on the conductive walls with those of previously published works [8], [15], [16]. 0093-3813/$20.00 © 2005 IEEE