  Coherent synchrotron radiation (CSR) induced microbunching instability has
been one of the most challenging issues in the design of modern accelerators. A
linear Vlasov solver has been developed [1] and applied to investigate the
physical processes of microbunching gain amplification for several example
lattices [2]. In this paper, by further extending the concept of stage gain as
proposed by Huang and Kim [3], we develop a method to characterize the
microbunching development in terms of stage orders that allow the quantitative
comparison of optics impacts on microbunching gain for different lattices. We
find that the microbunching instability in our demonstrated arcs has a
distinguishing feature of multistage amplification (e.g, up to 6th stage
amplification for our example transport arcs, in contrast to two-stage
amplification for a typical 4-dipole bunch compressor chicane). We also try to
connect lattice optics pattern with the obtained stage gain functions by a
physical interpretation. This Vlasov analysis is validated by ELEGANT [4]
tracking results with excellent agreement.
