Abstract—Self-consistent nonlinear theory of beam-wave interaction in complex-shaped gyrotron cavities is developed. The theory combines the generalized telegrapher’s equations and mode-matching technique for gradual and abrupt transitions, respectively. As an example, a complex cavity of a second- harmonic 0.4-THz gyrotron is considered. For this gyrotron, the results of beam-wave interaction modeling are utilized as a check on accuracy of the simplified approach used in previous research. I. INTRODUCTION IGH power and efficiency of continuous-wave gyrotrons in the millimeter, sub-millimeter, and terahertz (THz) ranges have led to their wide application in heating and diagnostics of plasma, spectroscopy, material processing, and deep-space communication [1]. Complex-shaped cavities with gradual and abrupt structural variations are often considered as a means for further improvement of gyrotron performances. Examples are sectioned cavities [2], [3], tapered cavities [4] and quasi-regular cavity with a short irregularity [5]. Contrary to conventional uniform cavities, these cavities, however, can exhibit unwanted conversion of the operating mode into spurious radial modes [6], [7]. Such mode conversion can reduce the strength of beam interaction with the operating mode and impairs the output mode purity of the gyrotron cavity. In modeling of the beam-wave interaction in complex- shaped gyrotron cavities, the mode conversion is often neglected or estimated by the approximate methods. In the fixed-field (cold-cavity) approximation, the effect of mode conversion on starting currents of modes of gyrotron cavities with gradual and abrupt transitions was investigated in [8]. This approximation, however, was found to be inaccurate for high- order axial modes and transitions between them. Such modes find application in gyrotrons with broadband continuous frequency tuning [9]. The self-consistent single-mode theory is widely used in beam-wave interaction modeling for uniform gyrotron cavities [10], [11]. There are several coupled-mode approaches intended to generalize this theory for cavities with gradual or abrupt transitions. For cavities with gradual transitions, one should mention the approaches based on the generalized telegrapher’s equations [12] and linearized electron motion equations [13]. A simplified theoretical approach was applied in [14]-[16] to study the beam-wave interaction in gyrotron cavities with radial steps. This approach assumes no beam interaction with spurious modes and neglects reflection and conversion of these mode in the tapered cavity sections. In this study, the self-consistent nonlinear theory of beam-wave interaction is extended to cylindrical gyrotron cavities with transitions of arbitrary shape. The theory combines the generalized telegrapher’s equations and mode-matching technique for gradual and abrupt transitions, respectively. II. RESULTS Consider beam-wave interaction in a cylindrical gyrotron cavity with metallic walls of the finite conductivity σ and radius ( ) z R . Using the same procedure as in [7], [13], one can derive the system of field equations for propagated TE and TM modes in gradually tapered cavity sections () , 4 ~    * ⊥ ω ⋅ ∇ ω π - + = z S k z i i ki i i ki k dS j i I T V T V dz d e () . 4   * ω ⊥ π - Γ + = z S k k k i i ki k dS c V I T I dz d e j (1) which are subject to outgoing-wave boundary conditions on the cavity ends. These equations are known as generalized telegrapher’s equations. The motion of electrons satisfies the following equations: ( ) ( ), Re ϑ θ ⊥ β + - = i r z z e H E v e p dz d ( ) ( ), Re ϑ θ ⊥ ⊥ β - β + =         ω - ω + ϑ i z z r z z H e H H E v ne v n dz d p (2) ( ) ( ) ϑ ⊥ β - - = i r z z z e H E v e p dz d Re with initial conditions at the cavity input. At the cavity steps, the amplitudes of propagating and evanescent TE and TM modes are found by the mode-matching technique [14]-[16]. As numerical example, we consider a complex-cavity second-harmonic 0.4-THz gyrotron powered by the electron beam with current 3 . 0 = b I A, voltage 15 = b V kV, radius 915 . 0 = b r mm, and pitch factor = α 1.5. In the previous study [15], performance of this gyrotron was investigated by the simplified approach [14]. The influence of mode conversion on the starting current and frequency of the operating mode is shown in Figs. 1(a) and 1(b), respectively. The results of the simplified approach [14] are also depicted for comparison. It can be seen that, despite a number of assumptions, the simplified approach yields rather accurate results. This argues for reliability of findings in [15]. A good agreement between simplified and advanced approaches can also be seen in Figs. 2, which depicts the field amplitudes of the basic radial modes for = 0 B 7.2 T. Fig. 3 shows the result of the simplified and advanced approaches for the spatial distribution of the azimuthal electric Aleksandr Maksimenko 1,2 , Vitalii Shcherbinin 1,2 , Manfred Thumm 1 and John Jelonnek 1 1 Institute for Pulsed Power and Microwave Technology, Karlsruhe Institute of Technology (KIT), Kaiserstr. 12, 76131 Karlsruhe, Germany 2 National Science Center “Kharkiv Institute of Physics and Technology” (KIPT), Akademicheskaya St. 1, 61108, Kharkiv, Ukraine Nonlinear Theory of Beam-Wave Interaction in Gyrotron Cavities with Gradual and Abrupt Transitions H 2023 48th International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz) | 979-8-3503-3660-3/23/$31.00 ©2023 IEEE | DOI: 10.1109/IRMMW-THz57677.2023.10299300 Authorized licensed use limited to: KIT Library. Downloaded on September 19,2025 at 10:42:33 UTC from IEEE Xplore. Restrictions apply.