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

figure from FLS2006

1. Introduction and Parameters

LPA matching LSCAstage

LSCAstage

LSCAstage

undulatorradiator

matching to FODO latticeexample (without SC):

to FODO lattice with

deg 71

m 1

pL

length ~ few meters

extreme optics !!!from x = y = 0 x = y = 0.7 mm

to ~ 0.5 m

not investigated now

1. Introduction and Parameters

LPA matching LSCAstage

LSCAstage

LSCAstage

undulatorradiator

mL

μms

LINAC coordinate

bunch coordinate

bunch current / A bunch current / Achargedensity

1. Introduction and Parameters

LPA matching LSCAstage

LSCAstage

LSCAstage

undulatorradiator

f.i. = 70 nm

temporal structure of the radiation pulse

-5 -4 -3 -2 -1 0 1 2 3 4 50

1000

2000

3000

4000

5000

6000

7000

8000

bunch current / A

μms

colors correspond tothree different shots

undulator10 periods; period length 2.7 cm; wavelength = 70 … 200 nm (tunable gap)

2. Longitudinal Microbunching Instability

chicanes in FODO structure

4 magnet chicane: create longitudinal dispersion

2 cm

2 cm

2 cm

14 cm

LSCA stage

LSCA stages

μmyx

μmx

μmyFODO structure: keep transverse beam dimensions

~ 40 cm

path-length difference ~ energy deviation

ref56

Rs with R56 ~ 10 µm

L

2. Longitudinal Microbunching Instability

111 2exp1Re zimIsI 222 2exp1Re zimGIsI

sI1 sI2

linear gain

1

1

2

principle

coasting beam and compression: 121 ICII

modulated beam:

wavelength of modulation C12

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

rms out / eV

rms in / eV

scan: rms out versus rms in

Z /m

x /m

y /m

r11

r11 from end of LH undulator