Technical Notes A novel representation for the static space-charge fields in waveguide excitation theory Kostyantyn Ilyenko a,n , Anatoliy Opanasenko b a Department of Electrical and Computer Engineering, University of New Mexico, MSC011100, Albuquerque, NM 87131, USA b National Science Center, “Kharkiv Institute of Physics and Technology” of National Academy of Sciences of Ukraine, Kharkiv 61108, Ukraine article info Article history: Received 17 January 2014 Accepted 24 January 2014 Available online 6 February 2014 Keywords: Kisunko–Vainshtein waveguide excitation theory Potential and rotational fields Green's function Evanescent eigen-modes Time-Fourier transform abstract In the present technical note, we obtain representations for static electric and magnetic fields induced by charged-particle current in a regular simply connected ideally conducting waveguide in the form of solutions to excitation equations of the Kisunko–Vainshtein waveguide excitation theory. Using generic expressions for evanescent (and propagating) waveguide modes, an equivalence of the developed expressions with Green's function representations of these fields is demonstrated as well as a correspondence between equations of excitation for non-static part of the potential electric field in the standard and Sovetov's variants of the Kisunko–Vainshtein theory and the present approach is established. & 2014 Elsevier B.V. All rights reserved. 1. Introduction Usually, static (∂/∂ t ¼ 0, ω ¼ 0) electric and magnetic fields are detached from non-static ð∂=∂t a0; ω a0Þ ones [1] and treated separately by the method of Greens function [2] when one considers extended interaction of charged-particle (e.g. electron) beams with electromagnetic waves in the framework of the Kisunko–Vainshtein waveguide excitation theory [3,4]. However, it may be convenient to shape those static fields into the form of solutions to equations of the Kisunko–Vainshtein waveguide excitation theory and subsequently treat them on equal footing together with the non-static fields. The rational for such a representation lies in the uniformity of utilization of expansion of the total excited electromagnetic field into the complete set of regular waveguide eigen-modes and realization of the fact that methods (see, e.g., [6]) developed for calculation of the dynamic (time-dependent) space-charge fields can be equally applied to their static counterparts (cf. [2,5]). 2. Governing equations We consider regular vacuum waveguide with ideally conduct- ing walls excited by a charged-particle current obeying the continuity equation ∂ϱ ∂t þ div j ! ¼ 0 ð1Þ where ϱð r ! ; t Þ and j ! ð r ! ; t Þ are the corresponding charge and current densities. Static ð∂=∂t ¼ 0; ω ¼ 0Þ electric and magnetic parts of the total electromagnetic field excited by a charged- particle beam in a regular waveguide are subject to the equations (for the sake of visualization we assume the time periodic process, which entails time-Fourier series expansion of all quantities, and explicitly write the superscript “p” whenever denoting potential parts of the excited electric field) div E ! p 0 ð r ! Þ¼ 4πϱ 0 ð r ! Þ; rot E ! p 0 ð r ! Þ¼ 0 ð2Þ rot B ! 0 ð r ! Þ¼ 4π c j ! 0 ð r ! Þ; div B ! 0 ð r ! Þ¼ 0: ð3Þ Here the subscript “0” denotes the zero-order (static) time-Fourier components of the involved quantities and ϱ 0 ð r ! Þ, j ! 0 ð r ! Þ (div j ! 0 ¼ 0, cf. Eq. (1)), E ! p 0 ð r ! Þ and B ! 0 ð r ! Þ are the corresponding time-Fourier components of beam charge and current densities, the potential part of the electric field and magnetic induction. In compact notations, modes of a regular simply connected ideally conducting waveguide can be written as follows [7,6] (upper and lower signs are held for the TE and TM modes, respectively; the asterisk n denotes complex conjugation): E ! TXjpr n;q ð r ! Þ¼ e ! TXjpr n;q ð r ! ? Þe ik 0 TX zjn;q z ; B ! TXjpr n;q ð r ! Þ¼ b ! TXjpr n;q ð r ! ? Þe ik 0 TX zjn;q z ; Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/nima Nuclear Instruments and Methods in Physics Research A http://dx.doi.org/10.1016/j.nima.2014.01.048 0168-9002 & 2014 Elsevier B.V. All rights reserved. n Corresponding author. E-mail address: k.ilyenko@gmail.com (K. Ilyenko). Nuclear Instruments and Methods in Physics Research A 745 (2014) 88–90