Dependency of a compact and analytic gain function for the linac with multiple bunch compressors on the initial density modulation

We suggest a compact, analytic, and practical gain function for the bunch compressors in the linac of an X-ray free electron laser which is composed of multiple bunch compressors. This is done by using the time evolution of distribution function in the electrons' phase space. The analytic gain function agrees well with the numerical calculation. In contrast to the available linear formulae the analytic gain function is not based on the assumption of small initial density modulation and can well explain its dependency on the initial density modulation which may be observed from the numerical calculation based on the integral equations. This analytic gain function should simplify the relevant numerical gain function while explaining the dependency that may be observed for the initial density modulation within the realizable range, which may be useful for designs of tools reducing the microbunching instability.