NONLINEAR THEORY OF EXCITATION OF AN AXIALLY ASYMMETRIC WAKEFIELD IN DIELECTRIC RESONATOR* K. V. Galaydych # , G. V. Sotnikov ## Kharkov Institute of Physics and Technology, Kharkov, Ukraine I. L. Sheynman, LETI (ETU), Saint-Petersburg, Russia Abstract A nonlinear self-consistent theory of excitation of an axially asymmetric wakefield by relativistic electron bunches in cylindrical dielectric resonator with a vacuum channel for the charged particles transportation through the resonator is constructed. The formulated nonlinear theory allows investigating numerically the nonlinear effects such as increasing of the transverse bunch size, and head-tail beam breakup instability, which occurs if an electron bunch in the structure is misaligned. INTRODUCTION The dielectric wakefield accelerator is one of the modern trends of acceleration schemes, which can provide high-accelerating gradient for future colliders. But besides for high output energy of an accelerated bunches high demands are made on their quality, the same, for example, as low emittance. No loss of current under acceleration of the bunch are also desirable. This information about the bunch can not be obtained using assumption of the absence of reverse influence the excited field on the dynamics of electron bunches. In this paper we present nonlinear self-consistent theory of wakefield excitation in a dielectric–lined resonator by an electron bunches. The previous theoretical investigations on wakefield excitation in dielectric–lined structures, have been done for longitudinally unbounded structures [1]– [4]. In cited papers was noted, that it is necessary to taking into account the contribution of higher multipole modes to the total transverse field. A presented complete bunch–excited electromagnetic field includes all azimuthal modes, which allows calculating transverse wakefield in order to investigate bunch deflection problems. STATEMENT OF THE PROBLEM Consider cylindrical metallic resonator with inner radius b , partially filled with isotropic material with dielectric constant  , containing on-axis vacuum channel of radius a which allows charged particles to pass through. We suppose that the end walls of the resonator are closed by metal grids transparent for charged particles and nontransparent for an excited electromagnetic field. Consider an electron bunch, injected into the resonator and moving along a line parallel to the axis of the resonator. The electron bunches will be described in terms of macroparticles, therefore the charge density  and the current density j will be written as:     (), () (), R R p p p p p pV pV q t q t t            r r j v r r (1) where p q is the charge of the macroparticle, p r and p v are its time-dependent coordinates and velocity, respectively. The summation in Eq. (1) is carried out over the particles being in the resonator volume R V . FIELD SOLUTION We introduce the solenoidal t E t H and the potential Φ   l E fields defined as div( ) 0, div( ) 0, 0, rot      t t l E H E (2) which are given by Maxwell’s and Poisson equations: 4 , , rot rot c t c t c           t t t t H E E H j (3) ( Φ) 4     (4) The solenoidal t E and potential l E electric fields are mutually orthogonal [5] and satisfy the boundary conditions, making their tangential components vanish on the metal walls of the resonator. First we solve the equation (4) for the potential in the vacuum channel and dielectric. In cylindrical coordinate Eq.(4) rewrites as: 2 2 2 2 2 1 1 4 + + φ r r r r r z                     (5) Eq.(5) should be complemented by boundary conditions consisting in that the potential  on the resonator metal walls becomes zero ( 0) ( ) ( ) 0, z z L r b        (6) ___________________________________________ *The research is supported in part by STCU, project №. P522. #kgalaydych@gmail.com ##sotnikov@kipt.kharkov.ua