Transverse Mode Coupling in a Bunched Beam

Since the fast transverse instability was observed in PETRA and, more recently, in PEP, this instability has become the problem of great importance for the design of large storage rings. The reason is that this instability imposes the serious limitation on a stored current; as soon as the current exceeds the threshold, the bunch distribution begins to blow up vertically or horizontally faster than the synchrotron-oscillation, and beam particles are lost. Short range fields created by the beam in rf cavities are considered to play an important role. The fast instability has been explained in terms of mode-coupling in the longitudinal phase space or "head-tail turbulence" by many authors. The purpose of this paper is to derive a formalism of the mode-coupling theory for a Gaussian bunch, starting from Sacherer\´s integral equation. The problem is reduced to solving the determinant equation for eigenfrequencies of coherent oscillations. Eigenfunctions are expressed as superposition of the linearly independent functions where higher radial modes and azimuthal modes are combined and rearranged into "hybrid modes". The theory is applied to explanation of the fast instability observed in PEP and to a preliminary estimate of the threshold current in the TRISTAN electron-positron storage ring. The transverse broad-band impedances of rf cavities in PEP and TRISTAN are calculated with the program code TBCI and characterized by a single low-Q resonator impedance. A satisfactory agreement with PEP experimental result is obtained.