Linear Gain for the Microbunching Instability in an RF Compressor

Velocity (or rf) compression has been suggested as a technique for bunch compression complementary to the more established technique involving magnetic chicanes and represents an important research item being investigated at the SPARC test facility. One of the aspects of this technique still not sufficiently understood is its possible impact on the microbunching instability. The purpose of this report is to present the analytical framework for investigating this instability in rf compressors. We use methods similar to those successfully applied to magnetic compressors and derive some integral equations yielding the gain for the instability in linear approximation. The focus here is on the derivation of the relevant equations. Although examples of solutions to these equations are provided we defer a more comprehensive discussion of their implication to a future report. The present study is part of a larger effort for a more comprehensive investigation that eventually will include macroparticle simulations and experiments.