CSR instability in a Bunch Compressor

The coherent synchrotron radiation of a bunch in a bunch compressor may lead to the microwave instability producing longitudinal modulation of the bunch with wavelengths small compared to the bunch length. It can also be a source of an undesirable emittance growth in the compressor. We derive and analyze the equation that describes linear evolution of the microwave modulation taking into account incoherent energy spread and finite emittance of the beam. Numerical solution of this equation for the LCLS bunch compressor gives the amplification factor for different wavelengths of the beam microbunching.