Parametric sum envelope instability of periodically focused intense beams

The envelope instability is a second order parametric resonance with the periodic focusing and known to appear in space charge dominated beams near 90° phase advance per focusing period. We show in 2d approximation that space charge may also induce parametric “sum envelope instabilities” leading to simultaneous growth of envelopes or skew angles as well as emittances. This can happen by two-plane envelope coupling or by exciting a skew (“odd”) mode in an otherwise fully uncoupled linear lattice. At resonance, the two individual phase advances are split more or less symmetrically away from 90°, and exponential growth occurs. Results from perturbation theory are compared with full envelope models, particle-in-cell simulations, and smooth approximation stopband calculations, all showing very good agreement for realistic space charge parameters.