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Two Longitudinal Space Charge Amplifiers and a Poisson Solver for Periodic Micro Structures

Two Longitudinal Space Charge Amplifiers and a Poisson Solver for Periodic Micro Structures. Longitudinal Space Charge Amplifier 1:. Longitudinal Space Charge A mplifier driven by a Laser-Plasma A ccelerator. 300 MeV , 200 nm , ~ 3kA. Longitudinal Space Charge Amplifier 2:.

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Two Longitudinal Space Charge Amplifiers and a Poisson Solver for Periodic Micro Structures

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

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

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

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

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

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

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

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

  9. 2. Longitudinal Microbunching Instability LSCA stages FODO structure: keep transverse beam dimensions 2 cm 2 cm 2 cm 14 cm ~ 40 cm 4 magnet chicane: create longitudinal dispersion path-length difference ~ energy deviation with R56 ~ 10 µm LSCA stage chicanes in FODO structure

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

  11. 2.1. One Dimensional Model Longitudinal Oscillations periodic density modulation -microscopic scale bunch coordinate t0 self field energy periodic energy modulation t0 + T/4 t0 + T/2 t0 + T

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

  13. 2.1. One Dimensional Model Bunch Lengthening, Macroscopic Effects longitudinal phase space smooth distribution  no microscopic effects self field bunch gets longer, particle gains energy figures: 6 x (1.2 m channel + discrete R56) bunch current vs. bunch coordinate peak bunch current vs. linac coordinate

  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! current LINAC coordinate full bunching and saturation if Np = particles per  • 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  11µm 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. 2.2. Three Dimensional Model 250E6 particles 6 LSCA cascades bunch current / A bunch current / A chicane LSCA cascade bunch coordinate LINAC coordinate

  17. 2.2. Three Dimensional Model

  18. 2.2. Three Dimensional Model after 6 cascades after 4 cascades bunch current / A spectrum wavelength

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

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

  21. 1. Parameters FLASH Parameters Energy 1.2 GeV Charge 100 pC Peak Current 1 kA Slice EnergySpread 150 keV Slice Emittance (norm) 0.4 µm Simulation Parameters real shotnoise → macroparticles electrons shortpartof buch; length = 2 µm → 6.7 pC → 42E6 particles longitudinal resolution 2 nm

  22. 2. Setup proposed attosecond scheme at FLASH total length  15 m LSCA cascades FODO-lattice, period = 1.4 m, <β> = 1.4 m  0  40 µm 3 standard cascades: 2 FODO periods + chicane with R56 50 µm, length / cascade = 3.5 m modified cascade: 2 FODO periods + modulator undulator+ chicane with R56 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) 

  23. 3. Simulation

  24. 3. Simulation

  25. Poisson Solver for Periodic Micro Structures problem: „spacechargefield“ selffieldof a particledistributionthatisnearly in uniform motion fullmodel periodicmodel

  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 beam andsetupparameters from numericalparameters period 800 nm(in z-direction) particles/period 1E6 longitudinal mesh, dz 800nm / 50 = 16 nm transversemesh  dz = 4 µm (about380lines) cpu time 5 min

  30. primaryheating 2keV → 5 keV z /m 11 m z /m

  31. primaryheating 2keV → 5 keV z /m 15.5 m z /m

  32. primaryheating 2keV → 5 keV z /m 17.5 m z /m

  33. primaryheating 2keV → 5 keV growthofrmsenergyspreadandmodificationofenergyspectrum Z /m

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

  35. x/m y/m Z /m r11 r11from end of LH undulator

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