Suppression of Microbunching Instability based on optimized velocity bunching in linac-driven FELs

The microbunching instability (MBI) driven by beam collective effects can cause significant electron beam quality degradation in advanced x-ray free electron lasers. Typically, multiple stage magnetic bunch compressors used to generate high peak current electron beam will dramatically amplify the microbunching instability. In this paper, by redesigning the solenoid elaborately and adopting a dual-mode buncher cavity with the third harmonic mode used to correct the RF curvature, it is potential for the electron beam to be further compressed in velocity bunching (VB) process. Therefore, a VB plus one bunch compressor could be a promising alternative scheme to achieve moderate peak current beam for high-repetition-rate X-ray FELs to suppress the additional MBI gain due to multi-stage magnetic bunch compressors.


Introduction
The microbunching instability driven by beam collective effects degrades the electron beam quality significantly in linacs for advanced X-ray FELs [1,2,3,4,5].Typically, two stage or multiple stage magnetic bunch compressors are used to provide high peak current electron beam.However, in the bunch compression process, the microbunching instability will be intensely amplified and degrade FEL performances.Therefore, suppression of the MBI is crucial for linac-driven FELs.The most widely applied method is using a laser heater to increase the uncorrelated energy spread of the electron beam against the instability [2].However, it could reduce the performances of the seeded FELs.Several novel methods based on transverse-tolongitudinal coupling and reversible slice energy spread growth by means of coupled devices were proposed to suppress the microbunching instability [6,7,8,9].However, only theoretical calculations and simulation results were presented in the above schemes and not applied in real machine.
Magnetic bunch compressor (BC) inherently amplifies the microbunching instability due to coherent synchrotron radiation (CSR) [10] and conversion of energy modulation induced by longitudinal space charge (LSC) [11].A natural idea to restrain the microbunching gain is to use one stage magnetic BC, but there will be more pressure to maintain stability against timing jitter and peak current jitter [12].Alternatively, a hybrid mode (VB plus one BC) may be a promising method to increase the bunch current based on bunch compression in velocity bunching.The principle of velocity bunching is that the low-energy beam just emitted from the photocathode is injected into the RF cavity at the zero-acceleration phase, resulting in an energy chirp where particles at the tail of the beam are faster than those at the head, thus creating beam compression during the acceleration process [13].Therefore, velocity bunching inherently mitigates the initial density modulation and may have a lower microbunching instability gain [14].
In this paper, we study the VB at the photoinjector section.The injector buncher parameters are scanned for further bunch compression.Meanwhile, the transverse emittance is controlled by adjusting the strength and position of the solenoids.In order to overcome the limitations of RF compressor performances resulting from RF nonlinearities, a dual-mode buncher cavity with the 3rd harmonic field [15] is proposed.Comparison of the theoretical MBI gain for the hybrid mode (VB plus one BC) scheme with the normal two-stage compression scheme shows the potential of the former in suppressing the microbunching instability at short-wavelength region.

The transverse and longitudinal dynamics optimization in VB
The method based on velocity bunching was proposed to increase the peak current of electron beam more than two decades ago [13].The principles of VB have been demonstrated in previous study with detailed formula derivation [14].Indeed, by properly choosing the injection phase and the acceleration gradient, the electron beam can be compressed in VB process.
Shanghai High-Repetition-Rate XFEL and Extreme Light Facility (SHINE), which is the first high-repetition-rate hard X-ray FEL facility based on superconducting linac in China, is being constructed.The schematic layout of photoinjector is shown in Fig. 1.The 216.7 MHz VHF gun is followed by the buncher, the single and eight 9-cell superconducting cryomodules.Three sets of solenoids are placed along the beamline to preserve the ultra-low transverse emittance.

Single-cavity cryomodule Multi-cavity cryomodule
Figure 1.The schematic layout of the SHINE injector section.
Here we consider the typical parameters of the SHINE photoinjector beamline where the beam charge is 100 pC and peak RF field is 30 MV/m in the VHF gun.The RF phase and peak RF field of Buncher is set to −60 • (relative to an acceleration on crest) and 2 MV/m, respectively to provide a compression factor of 3 compared to the RMS bunch length at the exit of the gun.
According to the principle of VB, the compression degree of the electron beam in VB is highly dependent on the RF phase and the acceleration gradient of the Buncher.These parameters could be optimized with the simulation code ASTRA [16] for further compression.By scanning the RF field acceleration gradient of the Buncher, we investigate the effect of parameter α (α = eE 0 mc 2 k , where E 0 is the peak RF field, k is the RF wave number) on the beam compression.The simulation results are shown in Fig. 2. It is indicated that a minimum bunch length is obtained for E 0 of 5.4 MV/m, where the compression factor is 20 compared to the RMS bunch length at the exit of the gun.Therefore, the beam current generated in this case is sufficient to replace the first stage magnetic bunch compressor.However, since VB operates in the ultra-low energy phase, the strong space charge force will result in a large emittance growth and the nonlinear beam longitudinal phase space distortion.Therefore, more optimization need to be performed for the transverse and longitudinal dynamics in VB.The theoretical and numerical study of the beam dynamics in VB showed that the transverse emittance growth can be corrected by proper-matching solenoid [17].We added a set of solenoid behind Solenoid-2 with the optimized solenoids strength based on the simulation code ASTRA to preserve the beam transverse emittance.The field strength and position of solenoids for typical case and optimized case are listed in Table 1.The simulation results of the transverse emittance evolution along the longitudinal position for two cases are shown in Fig. 3, where one can see for optimized solenoid case there is a significant decrease in the normalized transverse emittance from 2.950 mm • mrad to 0.637 mm • mrad.Previous studies have illustrated that the RF nonlinearities induced during velocity bunching will result in the longitudinal Emittance growth [13].To solve this problem, a dual-mode buncher cavity (with the fundamental 1.3 GHz mode and 3rd harmonic 3.9 GHz mode) is presented to compensate for the RF curvature at the SHINE injector [15], where the two types of RF modes are supplied by respective power source.Therefore, we introduced the dual-mode buncher cavity to obtain a more linear longitudinal phase space.After parameter optimisation, the simulation results of the longitudinal emittance and the slice current is shown in Fig. 4 and Fig. 5. Compared to the normal buncher without the 3rd harmonic mode, a more symmetric current profile at the exit of the photoinjector and lower longitudinal emittance are achieved with the harmonic mode applied.Eventually, beam with the RMS bunch length of 0.25914 mm is generated from the injector, where the compression factor is 16 compared to the RMS bunch length at the exit of the gun.

Microbunching gain in photoinjector and linac
Microbunching instability in VB has been studied in [14], which illustrated that the initial current and energy modulations are suppressed in velocity bunching process.Assuming for simplicity that the beam has a flat current distribution and Gaussian energy distribution with a sinusoidal energy modulation at the entrance of buncher section, the distribution is expressed as where h is the energy chirp, k i and ∆γ are the wave number and amplitude of the energy modulation, respectively.Rewriting the particle position z after the VB section and integrating Eq.1 over δ γ , one obtains the density modulation induced by the energy modulation at the entrance of the VB section, as where b = 1/ 2kαγ 2 0 sin ψ ∞ , sin ψ ∞ is the exit phase of particles and C is the compression factor.The gain in density modulation induced by LSC at linac section can be derived in the same way [2], described as where I 0 is the bunch current at the entrance of linac, I A ≈ 17045A is the Alfvén current, Z 0 = 377Ω is the free space impedance, k f is the wave number of the modulation after the where k is the modulation wave number, r b is the beam transverse radius, γ is the electron energy in units of its rest mess mc 2 , and K 1 is the modified Bessel function of the first kind.Following the parameters in SHINE photoinjector and linac section listed in Table 2, the final MBI gain for the typical scheme 'VB plus 2-stage BC' and the optimized scheme 'VB plus one BC' discussed in last section are shown in Fig. 6, which illustrates that the microbunching gain is completely suppressed at the short wavelength region (0-500 µm) for the optimized VB plus one BC scheme.

Conclusion
In this paper, microbunching instability suppression based on velocity bunching plus one BC scheme is demonstrated.The transverse emittance growth and the longitudinal phase space distortion are controlled and corrected by the optimized solenoid and a dual-mode buncher cavity.The theoretical analysis shows that the scheme is feasible to suppress the microbuching gain at short-wavelength region.More detailed studies including start-to-end simulations will be carried out in future work.

Figure 2 .
Figure 2. Simulation results of the RMS bunch length versus the RF field acceleration gradient of the Buncher.

Figure 3 .
Figure 3. Simulation results of the transverse emittance evolution along the longitudinal position for typical solenoid case and optimized solenoid case.

Figure 4 .
Figure 4. Simulation results of the slice current for Buncher with the 3rd harmonic mode case and without the 3rd harmonic mode case.

Figure 5 .
Figure 5. Simulation results of the slice current for Buncher with the 3rd harmonic mode case and without the 3rd harmonic mode case.

Figure 6 .Table 2 .
Figure 6.The theoretical microbunching gain spectrum at the exit of SHINE linac for the typical scheme 'VB+2*BC' and the optimized scheme 'VB+1*BC'.λ 0 is the initial uncompressed modulation wavelength at the entrance of Buncher.

Table 1 .
Solenoids field strength and position for typical case and optimized case Z(k) is the free-space LSC impedance per unit length, and is given as