Two Longitudinal Space Charge Amplifiers and a Poisson Solver for Periodic Micro Structures ongitudinal Space Charge Amplifier 1: ongitudinal Space Charge Amplifier 2: sson 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, ~ 3k 1200 MeV, 5 nm, 1.2 k Approach Example
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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:. - PowerPoint PPT Presentation
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Two Longitudinal Space Charge Amplifiersand 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 Amplifierdriven by a Laser-Plasma Accelerator
Generation of Attosecond Soft X-RAY Pulses in aLongitudinal Space Carge Amplifier
300 MeV, 200 nm, ~ 3kA
1200 MeV, 5 nm, 1.2 kA
ApproachExample
Longitudinal Microbunching Instability
courtesy P. Emma
no laser heater
LCLS:final long. phase space at 14 GeV(simulation)
from unwilling effect
to radiation source
Longitudinal Space Charge Amplifierdriven by a Laser-Plasma Accelerator
1. Introduction and Parameters
2. Longitudinal Microbunching Instability
M.DohlusE. Schneidmiller
M.YurkovC. HenningF. Gruener
2.1. One Dimensional Model
2.2. Three Dimensional Model
2.3. Effects from Coherent Synchrotron Radiation
LPA matching LSCAstage
LSCAstage
LSCAstage
undulatorradiator
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 Amplifierdriven by a Laser-Plasma Accelerator
1. Introduction and Parameters
LPA matching LSCAstage
LSCAstage
LSCAstage
undulatorradiator
LPA = laser plasma accelerator
μm 5.0μm 0.2
MeV 6.0MeV 300
μm 5.1pC 40
x
n
rms
av
||
y
q
250E6 electrons
slice energy spread
waist mm 7.0 , 0
correlated spread is neglected in the following(small compared to SC induced correlation)
kA 2.3ˆI
parameter set for the following investigations
very compact electron sourcesultra-high field gradientsroutinely: length few fsec, charge few 10 pC energy ~ GeV
2.1. One Dimensional ModelLongitudinal Oscillations
periodic density modulation - microscopic scale
bunch coordinateself field
periodic energy modulationenergy
t0
t0 + T/4
t0 + T/2
t0 + T
2.1. One Dimensional ModelLongitudinal Oscillations
periodic microscopic distribution micro modulation bunch coordinate
LINAC coordinate
L / m
μms
SpSp is LINAC length for a complete longitudinal oscillation is wavelength of micro modulation (bunch coordinate)
chargedensity
plasma oscillation: density modulation is converted to energy modulationand vice verse
LSCA stage should be short compared to Sp/4… cannot be realised with our parameters
2.1. One Dimensional ModelBunch Lengthening, Macroscopic Effects
longitudinal phase space
bunch current vs. bunch coordinate peak bunch current vs. linac coordinate
ref
smooth distribution no microscopic effects
self field
bunch gets longer, particle gains energy
figures: 6 x (1.2 m channel + discrete R56)μm 10r μm 1156 R
2.1. One Dimensional ModelLinear Multi Stage Model with Bunch Lengthening
model: linearized working point for middle of bunch
working point depends onLINAC coordinate!
full bunching and saturation if
3000 pNA
Np = particles per
LINAC coordinate
current
generation of higher harmonicsuse non-linear model !
2.2. Three Dimensional Model
3d particle trackingexternal fields = dipoles + quadrupolesself field = quasi stationary field (of uniform motion) rest frame transformation + Poisson solver
numerical parameterslongitudinal / transverse resolution = 10 nm / 5 µmlength / step width of tracking ~ 8 m / 0.5 .. 2 cm
setupFODO lattice: 90 deg, period = 40 cm, quadrupole length = 2cmchicanes: length = 14 cm, magnet length = 2cm, R56 11µm6 LSCA cascades, each with 3 FODO periods, chicane in last half-period
particles40 pC 250E6 electrons 250E6 particles: realistic shot noise initial condition: “periodic solution” for FODO lattice with SC, (optimized) r 10 µm
mL
μms
LINAC coordinate
bunch coordinate
bunch current / A bunch current / A
2.2. Three Dimensional Model
250E6 particles6 LSCA cascades
LSCAcascade
chicane
2.2. Three Dimensional Model
2.2. Three Dimensional Model
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 10-6
-0.04-0.02
00.020.04
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 10-6
0
2000
4000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-6
0
0.02
0.04
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 10-6
-0.04-0.02
00.020.04
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 10-6
0
5000
10000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-6
0
0.1
0.2
after 4 cascades after 6 cascades
bunch current / A
spectrum
wavelength
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
Generation of Attosecond Soft X-Ray Pulses in aLongitudinal Space Charge Amplifier
1. Parameters
2. Setup
M.DohlusE. Schneidmiller
M.Yurkov
3. Simulation
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
2. Setup
LSCA cascadesFODO-lattice, period = 1.4 m, <β> = 1.4 m 0 40 µm3 standard cascades: 2 FODO periods + chicane with R56 50 µm, length / cascade = 3.5 mmodified cascade: 2 FODO periods + modulator undulator + chicane with R56 7.1 µm,
compression C 10 0/C 4..5 µmModulatorshort pulse laser: L 800 nm,
duration 5 fs (FWHM), W 3 mJamplitude 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
3. Simulation
3. Simulation
Poisson Solver for Periodic Micro Structures
problem:
„space charge field“
self field of aparticle distribution that is nearly in uniform motion
full model periodic model
1. Approach
Lorentz transformation
electrostatic problem
0
2
V
VdV
rrr
041
PDE → solve equation systemimplement open boundary
integral equation → use particle-mesh method + fast convolution
kkjjiikjikji KqV ,,,,0
,, 41
),,cell(
,,1
kji
kji Vdr
K
1. Approach
periodic source distribution
n
pp nrrr pr
Vd
nV n
pp
rr
rr
041 integral diverges for
finite observer positions= a technical problem that can be solved
Vdn
Vn p
p
rrrr 1
41
0
periodic kernel → modified periodic kernel
pppkji kkjjiiKkjiqV r,,,,,4
1
0,,
2. Example
parasitic heating after LCLS laser heater
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 time 5 min
z /m
eV
z /m
mx
primary heating2keV → 5 keV
11 m
z /m
eV
z /m
mx
primary heating2keV → 5 keV
15.5 m
17.5 m
z /m
eV
z /m
mx
primary heating2keV → 5 keV
Z /m
eVrms
primary heating2keV → 5 keV
growth of rms energy spread and modification of energy spectrum