Coherent Normal Modes of Colliding Beams

The coherent normal modes of the nonlinear colliding beam system are calculated using averaging methods to find an equivalent infinite system of linear coupled oscillators. The condition of a consistent, stationary state leads to an eigenvalue equation for the characteristic frequencies and transverse exitation distributions of the modes. Results are presented for modes along the narrow axis of a Gaussian ribbon beam. The tune split is related to L/√(I+I-) for different values of I+/I- and of σ+/σ- .