Single-Mode Beam-Cavity Interaction In Relativistic Klystrons

The interaction between a modulated, intense electron beam and a single-mode rf cavity is discussed. A formalism is described which accounts for steady-state and transient beam loading, input/output waveguide coupling, and finite Q effects. A circuit equation is analyzed for shorttime behavior. Algorithms are presented for designing detuned structures to ensure longitudinal stability in relativistic klystron two-beam accelerators. 1 FUNDAMENTAL ELEMENTS AND DYNAMICS We are specifically concerned with the interaction of the beam with the fundamental monopole mode (TM010) in a single standing-wave (SW) idler or output cavity. We express the cavity electric field as a product of a timedependent mode amplitude with a spatial mode profile (indexed by ), ! E ( r ; t) = a (t) ! E ( r ): (1) The spatial profile of the mode is assumed to have the socalled ’Slater’ normalization, Z cavity dr ! E ( r ) ! E ( r ) = 1: (2) The other dynamical quantity is the current density representing the beam travelling through the cavity structure. We define a modal current density, J (t), by computing the overlap of the time-dependent current density with the spatial profile of the mode electric field, as in (2). We may write down an equivalent circuit equation describing the time evolution of the mode amplitude due to excitation by both the external rf current drive and the incoming waveguide mode, and losses from wall heating, beam loading, and coupling to the outgoing waveguide mode [1] [2], d dt2 + ! Qw d dt + ! 2 a (t) = (3)