  Coherent synchrotron radiations (CSR) emitted by a high-brightness electron
beam during transport in a bending magnet is a double-edged sword in electron
accelerators. While CSR contributes to a stronger radiation field than the
incoherent radiation, it simultaneously leads to degradation of the electron
beam quality. Specifically, CSR effects manifest in increases of the beam
energy spread and the projected emittance, and amplification of the
microbunching instability. This paper presents analytical formulas for the
CSR-induced microbunching instability gain and for the induced emittance growth
in an arbitrary achromatic four-bend chicane with inclusion of both the
steady-state and transient CSR effects. The analytical formulas are compared
and show good agreement with Vlasov calculations and particle tracking
simulations. The obtained analytical formulas are then applied to evaluate the
CSR effects in the design of a general achromatic four-bend bunch compressor
chicane, providing a quick estimate on the microbunching gain and the induced
emittance growth. From the widely adopted symmetric C-shape chicane to a
non-symmetric S-shape chicane, our analytical formulas offer insight into the
evolution of the microbunching gain and the emittance growth with the
variations of design parameters. In comparison to particle tracking simulations
currently employed for CSR effect analyses, the analytical formulas presented
in this paper significantly reduce the evaluation time, enabling systematic
study of parametric dependencies with inclusion of CSR effects within specified
design parameter ranges.
