1. X-band FEL beam dynamics issues
A. Latina (CERN)
for the X-band FEL and CLIC collaboration
Many thanks for their inputs to
A. Aksoy (Ankara), A. Charitonidis (NTUA),
S. Di Mitri (Elettra), D. Schulte (CERN)
LCWS2014 – Oct 9, 2014 – Belgrade

2. Table of contents Introduction Beam dynamics issues
X-band and FEL
Beam dynamics issues
Design optimisation procedures
Longitudinal
Transverse
Sensitivities
Static and dynamic imperfections
Phase space profiles
Conclusions and to-do work

3. X-band and XFEL
Normal conductive X-band (12 Ghz) offers several advantages w.r.t. traditional S-band, or C-band. E.g.:
High-gradient acceleration (>100 MV/m already achieved)
Higher repetition rate (kHz)
Power efficiency
Ultimately, these advantages offer: reduced costs, compactness, affordability
For instance, if one imagines an LCLS-like machine based on X-band technology, the length of both the accelerator and the undulators could be about 250 meters
This figure has to be compared with the length of the actual LCLS, which is about 1230 meters including 1100 meters of accelerator and 130 meters of undulators.

4. X-band and XFEL X-band acceleration has some drawbacks:
Technical difficulties, due to novelty of this technology
Geometric wakefields scale as:
These impart:
-> Transverse deflection
-> Longitudinal energy chirp
Transverse WF can lead to instabilities
Longitudinal WF induce energy spread, but can be exploited

5. Transverse constraints
D. Schulte
Small a/λ: strong lattice. Simplified wakefield model (K. Bane)
Stability of beam with initial jitter
requires to stay above red line
Emittance growth with 100um tolerances:
We need dispersion free steering or
CLIC-style alignment for FEL

6. Machine layout
Main subsystems: RF gun, injectors (two boosters), a laser heater, two linacs, two bunch compressors, spreaders (doglegs).
Typical input bunch parameters in Booster 2: 125 MeV, 200 pC beam charge, 2.5 ps RMS duration (750 micron), 0.15 m rad transverse normalized emittance (slice and projected), 15 keV RMS uncorrelated energy spread
Typical target parameters: 6 GeV, ~25 fs duration (~7 μm bunch length), sliced energy spread: <0.01%

7. RF gun & Injector optimization
A. Aksoy
Laser
Booster -1
Booster -2
X-band structure
75 cell, ~0.9 m, 65MV/m
3 GHz Photo Cathode RF (PHIN) Gun
2.6 Cell, 100 MV/m
3 GHz Traveling wave structures (PSI type)
120 cell, ~4 m, max 18 MV/m
Optimization goal  minimum projected emittance,
minimum sliced emittance
minimum bunch length
Laser parameters
Laser pulse length?
Laser spot size?
Gun parameters
Gradient-phase?
Bunch charge?
Solenoid fields and positions?
Injector structure
Gradient?
Position of structure?
Phase of structure?

8. Injector Optimisation
A. Aksoy
Acceptable projected emittance has been observed
Sliced emittance can be optimized with mismatch parameter..

9. Linacs Design Beam dynamics Issues: Design: Single-bunch wakefields
Longitudinal space-charge and CSR (micro-bunching instability)
Geometric and chromatic aberrations
ISR induced emittance growth
Bunch profile
Static errors:
Beam break-up instability
Emittance growth due to misalignments
Dynamic errors:
Longitudinal stability

10. Longitudinal Linac Optimisation
Must take into account: longitudinal short-range wakefields, RF curvature, CSR, ISR, sensitivity, …
X-band structures have a/λ = 0.125, and a gradient G=70 MV/m
Start from Booster 2: there is a total of 10 degrees of freedom
Booster2: S0.V, S0.Phi, X0.V, X0.Phi
Linac1: X1.V, and X1.Phi
Linac2: X2.V, and X2.Phi
BC1: R56
BC2: R56
We use the simplex.

11. Semi-analytical Optimisation
Degrees of freedom can be reduced using some analytical considerations. E.g. BC1:
1) S-band RF curvature linearisation: Taylor-expand Ef to vanish the 2nd-order terms:
2) Analytically compute R65s in order to find R56s.
3) Fix intermediate parameters, such as:
Bunch length after BC1: from 55 to 70 μm
Energy after BC1 and BC2: e.g. 250/300 MeV, and 2.5 GeV

12. Longitudinal Linac Optimisation
A fast C++ 1d code has been developed for optimization:
RF nonlinearities,
Wakefields
Chicanes’ T566
are taken into account
It is integrated within Octave, to access available optimization toolboxes
It allows fast tracking of the longitudinal phase space:
simplex optimisation of a merit function

13. Optimisation procedure
Total of 10 degrees of freedom:
Booster2: S0.V, S0.Phi, X0.V, X0.Phi
Linac1: X1.V, X1.Phi
Linac2: X2.V, X2.Phi
BC1: R56, BC2: R56
Optimize BC1 and BC2 independently (**)
BC1:
S0.Phi = between 0 and 45 degrees
X0.Phi = between 150 and 220 degrees
S0.V, X0.V chosen to achieve the desired intermediate energy, and to linearize bunch
R56 is computed to achieve the desired intermediate bunch length, within ±10% (*).
BC2:
X1.Phi = between 0 and 30 degrees
R56 is computed for full compression, and varied within ±10%
X2.Phi = between -15 and 30 degrees
(*) Full compression not required.
(**) Alternative: fix R56 to optimal values and optimise both linacs and BCs

14. Optimisation procedure
Initial curvature from S-band injector is taken into account
Merit function to be optimised:
Achieve desired intemediate and final bunch lengths (55 / 65 um, and 7 um)
Sliced energy spread is the theoretical miminum
Longitudinal distribution is as flat / symmetric as possible
Target parameters:
Final energy: 6 GeV
Total energy spread < 2%
Sliced energy spread: < 0.01 %

15. Solution to the Optimisation (I)
S0.Phi = deg ;
S0.V = GV;
X0.Phi = deg ;
X0.V = GV ;
X1.Phi = deg ;
X1.V = GV ;
X2.Phi = 5.5 deg ;
X2.V = GV ;
BC1.R56 = m
BC2.R56 =
Intermediate_blength = 56 um
intermediate_energy = 300 MeV
intermediate_energy2 = 2.5 GeV
final_bunch_length = 9.6 um
final_energy = 6 GeV
final_espread = 0.02 %
final_slice_espread = 0.01 %
total_RF_length = m

16. Transverse Linac Optimisation
Check bunch compression
Definite energy sections and R56 of BCs
(include longitudinal Wakefield and 2nd order effects at BCs)
Define lattice
Optimize cost, number of structures per module et…
Optimize lattice
(include longitudinal & transverse Wakefield, 2nd order effects at BCs and CSR)
Check longitudinal tolerances
Optimize R56 and energy error tolerances
Check Transverse tolerances
Optimize lattice and alignment tolerances

17. 1st optimisation criteria: the wake effect
A. Aksoy
Strong wake field plays main role on bunch charge distribution and its length along beamline.
Longitudinal wake potential changes linearity on chirp
uniform bunch distribution and no tail is preferred
Transverse wake potential causes deflection along bunch
Longitudinal wake potentials experienced by bunch (which has σ=100 um) in x-band structure
Transverse wake potentials experienced by bunch (which has σ=100 um) in x-band structure

18. Transverse beam dynamics
A. Aksoy
However the transverse deflection is proportional
FODO type of lattice is proposed. In order to get minimum deflection. We propose different number of structures per one FODO cell
The most critical section is linac 2 since the energy is low and bunch length is long

19. Transverse deflection in Linac 1
Plots show the transverse deflection of coordinate (x) and angle (x’) of slices along the bunch in linac 1 for a Gaussian bunch.
σz= 100µm
σz= 150µm
For compression of lattices and bunch profile we check
A. Aksoy

20. The amplification on Linac 1
The amplification for different bunch charge distributions on a lattice that has FODO cell with 6 structure per cell and 16 structure per cell
The uniform charge distribution has lowest amplification.
In order to get lower amplification factor than 1.5 we need to have bunch length σz < 70 µm
A. Aksoy

21. The amplification on Linac 1
The amplification of uniform charge distribution on different type of lattices
In order to get lower amplification factor than 1.5 we are allowed to use the FODO lattice with 6 or 8 structure per cell
A. Aksoy

22. The lattice
We have proposed FODO type of lattice on which 8 structures located in one cell
Linac 1: 40 x-band strcuture, phase 25 degre
Linac 1: 80 x-band strcuture, phase 3 degre
BC1 R56= m
BC2 R56= m

23. Bunch compression

24. Sensitivity Studies
A. Charitonidis
Theoretical / Analytical formulation (in progress):
Arrival time jitter
Final bunch length variation due to phase jitter
Compression factor (BC1)

25. Longitudinal coherent errors
We have checked only linac coherent phase error (≡ beam phase error) and gradient error.
A. Aksoy
ΔΦ < 10 fs required (bunch length is ~25 fs)

26. Solution to the Optimisation (II)
S0.Phi = deg ;
S0.V = GeV ;
X0.Phi = deg ;
X0.V = GeV;
X1.Phi = deg ;
X1.V = GV ;
X2.Phi = deg ;
X2.V = GV ;
BC1.R56 = m ;
BC2.R56 = m ;
intermediate_blength = 70 um
intermediate_energy = 250 MeV
intermediate_energy2 = 2.5 GeV
final_bunch_length = um
final_energy = 6 GeV
final_espread = 0.05 %
final_slice_espread = %
total_RF_length = m

27. Fermi @ ELETTRA Upgrade
Existing FEL is based on injector for synchrotron (FERMI)
Upgrade with X-band to increase beam energy for FEL

28. Emittance
S. Di Mitri
Normalized RMS projected emittance at the entrance of BC2 (left) and linac exit, as function of the linac-to-beam RMS misalignment and the horizontal betatron function in the BC1 (left) and BC2 (right). The left-hand plot assumes an initial emittance of 0.15 m, the lower, 0.5m.

29. Conclusions & further work
Beam dynamics poses some known issues
Semi-analytical / numerical tools are being developed
Optimisation procedures
Preliminary designs have been produced
We are trying to reduce the R56 of bunch compressors in order to relax longitudinal tolerances..
Include analytical sensitivities in optimisation procedure
We need to derive tolerances
Phase stability is alerting
We have several unknown parameters
Tolerance of bunch arrival time
Tolerance for bunch length variation
Tolerance for energy jitter..
Transverse jitter?
Tools for FEL simulation are ready