Consideration on the BPM alignment tolerance in X-ray FELs

The effects of the trajectory error on the FEL amplification process are studied analytically and numerically in order to estimate the alignment tolerance of the beam position monitor (BPM) installed in the undulator line of the SASE-based FEL. Analytical investigations show that an angle error on the electron beam disturbs the FEL amplification process by two mechanisms: one is a decrease in radiation efficiency, and the other is smearing of microbunch. Simple formulae to denote these two effects are derived, and a critical angle is introduced as a criterion for the single kick error to be allowed. Then, numerical simulations are performed to estimate the tolerance of the BPM alignment to be required practically. These are in good agreement with the results of the analytical investigations