1. Two Longitudinal Space Charge Amplifiers and a Poisson Solver for Periodic Micro Structures Longitudinal Space Charge Amplifier 1: Longitudinal Space Charge Amplifier 2: Poisson Solver for Periodic Micro Structures: Longitudinal Space Charge Amplifier driven by a Laser-Plasma Accelerator Generation of Attosecond Soft X-RAY Pulses in a Longitudinal Space Carge Amplifier 300 MeV, 200 nm, ~ 3kA 1200 MeV, 5 nm, 1.2 kA Approach Example

2. Longitudinal Microbunching Instability courtesy P. Emma no laser heater LCLS: final long. phase space at 14 GeV (simulation) from unwilling effect to radiation source

3. Longitudinal Space Charge Amplifier driven by a Laser-Plasma Accelerator 1. Introduction and Parameters 2. Longitudinal Microbunching Instability M.Dohlus E. Schneidmiller M.Yurkov C. Henning F. Gruener 2.1. One Dimensional Model 2.2. Three Dimensional Model 2.3. Effects from Coherent Synchrotron Radiation

4. LPAmatching LSCA stage LSCA stage LSCA stage undulator radiator LPA = laser plasma accelerator LSCA = longitudinal space charge amplifier SC = space charge CSR = coherent synchrotron radiation few meters~ 8 m~ 0.3 m controlled LSC instabilityshot noise Longitudinal Space Charge Amplifier driven by a Laser-Plasma Accelerator

5. 1. Introduction and Parameters LPAmatching LSCA stage LSCA stage LSCA stage undulator radiator LPA = laser plasma accelerator 250E6 electrons slice energy spread waist correlated spread is neglected in the following (small compared to SC induced correlation) parameter set for the following investigations very compact electron sources ultra-high field gradients routinely: length few fsec, charge few 10 pC energy ~ GeV figure from FLS2006

6. 1. Introduction and Parameters LPAmatching LSCA stage LSCA stage LSCA stage undulator radiator matching to FODO lattice example (without SC): to FODO lattice with length ~ few meters extreme optics !!! from  x =  y = 0  x =  y = 0.7 mm to   ~ 0.5 m not investigated now

7. 1. Introduction and Parameters LPAmatching LSCA stage LSCA stage LSCA stage undulator radiator LINAC coordinate bunch coordinate bunch current / A charge density

8. 1. Introduction and Parameters LPAmatching LSCA stage LSCA stage LSCA stage undulator radiator f.i. = 70 nm temporal structure of the radiation pulse bunch current / A colors correspond to three different shots undulator 10 periods; period length 2.7 cm; wavelength = 70 … 200 nm (tunable gap)

9. 2. Longitudinal Microbunching Instability chicanes in FODO structure 4 magnet chicane: create longitudinal dispersion 2 cm 14 cm LSCA stage LSCA stages FODO structure: keep transverse beam dimensions ~ 40 cm path-length difference ~ energy deviation with R 56 ~ 10 µm

10. 2. Longitudinal Microbunching Instability linear gain 1 1 2 principle coasting beam and compression: modulated beam: wavelength of modulation

11. 2.1. One Dimensional Model Longitudinal Oscillations periodic density modulation - microscopic scale bunch coordinate self field periodic energy modulation energy t0t0 t 0 + T/4 t 0 + T/2 t 0 + T

12. 2.1. One Dimensional Model Longitudinal Oscillations periodic microscopic distribution  micro modulation bunch coordinate LINAC coordinate  L / m SpSp S p is LINAC length for a complete longitudinal oscillation is wavelength of micro modulation (bunch coordinate) charge density plasma oscillation: density modulation is converted to energy modulation and vice verse LSCA stage should be short compared to S p /4 … cannot be realised with our parameters

13. 2.1. One Dimensional Model Bunch Lengthening, Macroscopic Effects longitudinal phase space bunch current vs. bunch coordinatepeak bunch current vs. linac coordinate smooth distribution  no microscopic effects self field bunch gets longer, particle gains energy figures: 6 x (1.2 m channel + discrete R 56 )

14. 2.1. One Dimensional Model Linear Multi Stage Model with Bunch Lengthening model: linearized working point for middle of bunch working point depends on LINAC coordinate! full bunching and saturation if N p = particles per LINAC coordinate current  generation of higher harmonics use non-linear model !

15. 2.2. Three Dimensional Model 3d particle tracking external fields = dipoles + quadrupoles self field = quasi stationary field (of uniform motion) rest frame transformation + Poisson solver numerical parameters longitudinal / transverse resolution = 10 nm / 5 µm length / step width of tracking ~ 8 m / 0.5.. 2 cm setup FODO lattice: 90 deg, period = 40 cm, quadrupole length = 2cm chicanes: length = 14 cm, magnet length = 2cm, R56  11 µm 6 LSCA cascades, each with 3 FODO periods, chicane in last half-period particles 40 pC  250E6 electrons 250E6 particles: realistic shot noise initial condition: “periodic solution” for FODO lattice with SC, (optimized)   r  10 µm

16. LINAC coordinate bunch coordinate bunch current / A 2.2. Three Dimensional Model 250E6 particles 6 LSCA cascades LSCA cascade chicane

17. 2.2. Three Dimensional Model

18. after 4 cascades after 6 cascades bunch current / A spectrum wavelength

19. 2.3. Effects from Coherent Synchrotron Radiation after 3 cascades after 4 cascades uses transient CSR-impedance in arcs and drifts and SC-impedance Numerical simulation with CSRtrack no significant effect

20. Generation of Attosecond Soft X-Ray Pulses in a Longitudinal Space Charge Amplifier 1. Parameters 2. Setup M.Dohlus E. Schneidmiller M.Yurkov 3. Simulation

21. 1. Parameters FLASH Parameters Energy Charge Peak Current Slice Energy Spread Slice Emittance (norm) 1.2 GeV 100 pC 1 kA 150 keV 0.4 µm Simulation Parameters real shot noise → macro particles  electrons short part of buch; length = 2 µm → 6.7 pC → 42E6 particles longitudinal resolution  2 nm

22. 2. Setup LSCA cascades FODO-lattice, period = 1.4 m, = 1.4 m  0  40 µm 3 standard cascades: 2 FODO periods + chicane with R 56  50 µm, length / cascade = 3.5 m modified cascade: 2 FODO periods + modulator undulator + chicane with R 56  7.1 µm, compression C  10  0 /C  4..5 µm Modulator short pulse laser: L  800 nm, duration  5 fs (FWHM), W  3 mJ amplitude  20 MeV (existing TiSa laser: 35 fs, W <  50 mJ) undulator: 2 periods, B  1.4 T; u  10 cm, (1.2 GeV)  proposed attosecond scheme at FLASH total length  15 m

23. 3. Simulation

25. Poisson Solver for Periodic Micro Structures problem: „space charge field“ self field of a particle distribution that is nearly in uniform motion full modelperiodic model

26. 1. Approach Lorentz transformation electrostatic problem PDE → solve equation system implement open boundary integral equation → use particle-mesh method + fast convolution

27. 1. Approach periodic source distribution integral diverges for finite observer positions = a technical problem that can be solved periodic kernel → modified periodic kernel

28. 2. Example parasitic heating after LCLS laser heater

29. 2. Example numerical parameters period particles/period longitudinal mesh, dz transverse mesh 800 nm (in z-direction) 1E6 800 nm / 50 = 16 nm  dz = 4 µm (about 380 lines) beam and setup parameters from cpu time5 min

30. z /m primary heating 2keV → 5 keV 11 m

31. z /m primary heating 2keV → 5 keV 15.5 m

32. 17.5 m z /m primary heating 2keV → 5 keV

33. Z /m primary heating 2keV → 5 keV growth of rms energy spread and modification of energy spectrum

34. rms out / eV rms in / eV scan: rms out versus rms in

35. Z /m  x /m  y /m r 11 r 11 from end of LH undulator