## Two-stage bunch compression with resistive impedance approximation of coherent synchrotron radiation

##### Abstract

Two-stage bunch compression, where each stage compresses a chirped beam in a magnetic chicane, may be utilized in the design of a driver for a free-electron laser (FEL). For the high currents required of an FEL, compressor performance may be adversely affected by the wake of coherent synchrotron radiation (CSR). When the CSR wake is dominated by edge radiation downstream of the chicane magnets, it may be approximated as the wake from resistive impedance. For a typical bunch-compressor chicane magnet, this approximation applies for wavelengths exceeding 1 micron. For a preliminary two-stage bunch compressor design that compresses by a factor of twenty, longitudinal tracking with resistive impedance approximates the longitudinal phase space obtained from three-dimensional tracking with computed CSR. By approximating CSR as resistive impedance, longitudinal CSR effects may be studied analytically, and tracking with “CSR” may be performed without modeling the transverse dynamics. With this approximation, we develop formulas that describe when part of the bunch becomes upright in phase space, producing a large current spike and energy depression. The formulas correctly predict good performance when Gaussian bunches with peak current of 50 A and rms bunch length zσ of 1 mm are compressed, and that the bunch tail becomes upright in phase space when parabolic bunches with peak current of 50 A and zσ of 0.4 mm are compressed. The formulas also show how the compressor design may be modified to prevent the upright bunch behavior with the shorter bunch length. We obtain analytic formulas for the jitter in the bunch arrival time resulting from bunch-to-bunch variation in current, energy and chirp. The formulas determine the permitted variations that give arrival-time jitter less than 15 fs in the preliminary design. Bunch-to bunch peak current variation should be less than 6.5 A, which is 13% of the peak current. The bunch-to-bunch energy variation should be less than 5.6 keV, which is 5105.4−× times the bunch energy when entering the compressor. The bunch-to-bunch chirp variation should be less than 7103.3× eV/m, which is 3% of the design chirp. Analytic small-signal gain formulas for microbunching describe the output current and energy modulations that result from given input current and energy modulations, including the suppression at short wavelengths by the bunch’s energy spread. Large output modulations of the bunch current may be caused by relatively small input energy modulations. Heating the uncompressed bunches by a laser heater so that the energy spread is 5-10 keV is predicted to suppress microbunching for initial wavelengths less than 100 microns. In this case, initial energy modulations at wavelengths around 100 microns should be smaller than ~500 eV to prevent large current modulations in the compressed bunches.