Coherent Synchrotron Radiation and Beam Interaction · curvature in BC magnets R 0 ~10 m N 250...

Coherent Synchrotron Radiationand Beam Interaction

3rd ARD ST3 workshopMartin Dohlus16th July 2015

FEL process

beam preparation: bunch compressor

1 m

without self‐interaction

1 mm

with self‐interaction

1 ‐ one particle

2 ‐ near field

3 ‐multiple particles

4 ‐ circular motion & shielding

5 ‐ general trajectories

6 ‐ projected model

7 ‐ bunch compressor

8 ‐ other forces / effects

9 ‐ transverse effects

one particle

one particle

instantaneouslyhttp://www.shintakelab.com/en/enEducationalSoft.htm Radiation 2D Simulator Free Download

one particle

Lienert‐Wiechert

)(trs

)(0 trrttc s

Enc

B

Rnc

nn

Rn

nqEt

0

3

0223

0

0

1

114

)(trs

0q

trs 0q

tr ,(observer)R

n

charge on trajectory0q

fields

retarded time

radiation term

one particle

near and far

)(tr )( ttr

0cRtT

R

0ctR

tcnR 01

n

dVceTrSdtP r

tt

t

1rad ),(

V

),(12rad TrSnRddP

with Poynting flux in far zone

6

22

0

00 14 n

nn

Rq

cnS

from radiation term

one particle

power loss

226

00

20

rad 6

cqP

0

0

2

44

0

02024

00

20

rad 66 Rcq

cqP

2

00

2026

00

20

rad 66

cq

cqP 5

2rad

cos1sin

ddP

22

22

3rad

cos1cossin1

cos11

ddP

linear acceleration

circular motion

nn , , ))

||||0 vEqeffective longitudinal field (self effect)

near field

circular motion

near field

Rnc

nn

Rn

neqEeE 3

0223||

0

0||||

114

longitudinal component on arc R0,test particle at s=vt, source particle at s=v(t+)

rEc

qE

20

0||

sgn4

c

cE

E ||

singular part as for linear motionresidual part

for R0 = 1m, = 10

30 cRc

c

with 200

40

4 RqEc

c

r

EE

residual part

tt

rRnc

nneqcRn

neqE

3

0

||0

02223

||

0

0

14sgn

14

term 1 term 2

term 1

term 2

both terms contribute to near‐interaction!

near field

c

r

EE

c

small angle approximation of residual part

)12)(4(

)8(4

324

0 22

2

220

40

,

R

qE sar

rad

sarsar PEE

qc

200 ,,

0

242 33

0

cRwith

34

power loss of one particle: far field radiation

near field

effective self field

c

c

sar

EE ,

power loss of two particles

001 rr EEcqP

002 rcEqP

head particle

tail particle

4

2

fully coherent incoherent

PPsingle

1total power loss vs. distance

near field

c

100 particles: total power loss vs. distance (first‐last)

PPsingle

1

2

100

1002

22

1) fully coherent2) energy independent3) cool beam4) incoherentx) transition

1 2 3 4

near field

c

multiple particles

multiple particles

superpositionfor instance all (N) particles on the same trajectory )()(, trtr ss

trBtrB

trEtrE

,,

,,

0

0

random time delayprobability distribution of delay: Np ,,, 21

independent delay: 021 ,,, pp N

(delay is not independent for systems with longitudinal dispersion + self effects!)

20

20 1~ iPNNNiFS dttitpiP exp00with

“coherent”white

one particle “collective term”

expectation of spectral power density (in principle)

multiple particles

spectral power density (loss) for circular motion

rr EE ,

,

0,0, rrr EcqEcqP

power loss of all particles

impedance of residual part

deiZqE irr

20

20

20 1Re

iPNNNiZcqS r

0

~21

dSdSPnotation

c

cr ZZ Re

multiple particles

impedance of residual part

dxxK 35839S

0

3

23Rc

c

ccr ZiZ

SRe with

critical frequency

10

dSand

c

s

exp83.0

3 148.0

91

0

0

RZZc

S

circular motion & shielding


radiated power of Gaussian bunch in circular motion

30

0 R

2

4

0

020

0 6 RNcqP

0NPP

with


energy independent regime

WP

m

0

nC 1q

N

m 100 R

= 500

= 1000

43

0

1 N

30

0 R

with

shape dependent

Gaussian shape: 343200

0202

CSR1 R

cqxNP with 0279.06465 3123 x

~1⁄


energy independent regime 43

0

1 N

30

0 R

with

f.i. LCLS 2009

mm 83.0bunch mm 19.0bunch μm 22bunch

m 10~0Rcurvature in BC magnets

N 250 pC/q0

643

34

0

bunch

108

102.2 10

N

66 101.3 1011

coherent “” incoherent


incoherent regime0

43

N 30

0 R

with

shape independent

20

4

0

020

0ISR 6 RNcqPP

3043

RNi

f.i. HERA‐E

mm 310

50050000

bunch

100

N

R

f.i. PETRA

mm 1310

20012000

bunch

100

N

R

mm 13.0i

mm 7.3i


a simple shielding model = parallel conducting planes

0Re , shcrZ

30

, 3hRcshc

h

path

real part of impedance is short‐circuited below a cutoff frequency

20

20 1Re

iPNNNiZcqS r

spectral power density:


a simple shielding model = parallel conducting planes

0PP

0

without shielding

with shieldingcoherent part

210

23

, 51Rh

shc

general trajectoriesand transients

general trajectories and transients

llL 221

L

30

3 20 for 24 RllRLo

retarded particles, seen from head particleseen from the head (1) before, (2) in and (3) after a 90 degree bending magnet

(1) (2) (3)

nm 10 cm, 29m 10 1000, μm, 10 300 RlLRl o

“overtaking length”

m 201000 μm, 10 Ll for instance:


longitudinal field in the center of a Gaussian spherical bunchthat travels through a bending magnet

m 100 R

μm 20r

nC 1q

note: (all contributions cancel for the center of the distribution)rEE 2rer

03432

0233

1231

q

REc

beamline coordinate


transient CSR field, injection of a Gaussian bunch

bunch coordinate / head tail

parametero

i

LLx ˆ

iL3 2

024 RLo with


transient CSR field, ejection of a Gaussian bunch


parameter

o

d

LLx ˆ

3 2024 RLo and

dL


some remarks

sloppy notation: “residual” part (of longitudinal E‐field) is called CSR‐field

for free space: interaction by CSR‐field (CSR‐interaction) is tail‐to‐head interaction

significant part of interaction is over long distance weak sensitivity on transverse offset 1D model

energy independent model for

closed environment (metallic chambers):tail‐to‐head and head‐to‐tail interaction

influence of finite conductivity

3 2024 RLo

R

oe

se

n

1D model (follows)

30 R


vertical offset dependency: Gaussian bunch in circular motion


||EEc m 10 1000, μm, 100 0 R

: y = 1mmx x x: y = ____ y = 0

∆ 4.6mm

∆ 2

shielding parameter


closed environment, long range

projected model

projected model

phase space

Esyyxx

ZXtransverse coordinates (local system)

longitudinal coordinates (bunch coordinate ~ time, energy)

beamline coordinate

ZsvF

ZfdZd

,00000

,

CSR

ext

XX

projected model, equation of motion

particle index

external forces

collective longitudinal CSR‐self‐force

SC XF

collective SC‐self‐force

line charge density

projected model

Zs,

ideal trajectory

particles

dxZxKZxsZsF ,,,

collective longitudinal self‐force

projected particles to ideal trajectory (neglect all coordinates but s)generate continuous function (s,Z) by binning and smoothing techniques

CSR kernel

xdZxKZxxsZsF ~,~~,~,

in principle

in practice, with retarded source position x~

projected model

kernel function

no transverse forcesno transverse beam dimensionslocal rigid bunch approximationonly residual part

??? add collective SC forces

approximations

R

oe

se

n

observer particle

retarded source particle

Rsne

ReeeenZxK sosos

11,~4 222

energy independent approximation with 0 ,1 2

sZs ,

partial compensation of transverse effects, see SLAC‐PUB‐7181

projected model

implementations

(incomplete list)

Elegant only one magnetCSRtrack projected model, alternatively 2.5D modelImpact‐T + collective SC forcesBmadGPT

bunch compressor

bunch compressor

4 magnet bunch compressor

in principle

growth of emittanceand energy spread

bunch compressor

“typical” beam dimensions in a “typical” bunch compressorexample: benchmark BC from CSR workshop 2002http://www.desy.de/csr/csr_workshop_2002/csr_workshop_2002_index.html

bunch compressor

beamline coordinate / m beamline coordinate / m

mm56r

longitudinal dispersion

mmx

μmy

μmz

μm 20μm 200

rms bunch dimensions

compression

usually (but not always) the bunch is short after magnet3

“typical” beam dimensions in a “typical” bunch compressor

bunch compressor

simple model for emittance growth in last magnet

assumption: (1) neglect all self effects before last magnet(2) represent total energy loss in chicane by discrete loss E before magnet

growth of emittance ref020 EErms

with ,0 emittance before / after magnet beta function at magnet (lattice) deflection angle

rmsE energy spread of particle bunch(slice or full bunch)

refE energy of particle bunch

depends weak on energy (energy independent CSR regime)

therefore: small; focus of lattice function in last magent high

tt syyxxsyyxxX EEE

rmsE

Eref

bunch compressor

again the “typical” bunch compressorcompression 600 A (1 nC) 6 kA at 5 GeV

rms energy spread of bunch

mean energy of bunch

energy of particle in bunch center

CSR induced energy change along BC

the rms energy is created essentially: end of magnet 3, drift m3m4 and magnet 4

rough estimation of steady state field in magneto

c LIZEEˆ1 0

and transient in drift SL

sIZEo

5.02

1 0

0 rmsE

bunch compressor

projected emittance and slice emittance

222 xxxx naNa 1

with

projected emittance: use all particles:

slice emittance: use particles of a certain slice: few percent growth

0 2 4 6 8 10 12 14 161

1.2

1.4

1.6

S /m

γε x [μm]

Bend−plane normalized emittance

0 2 4 6 8 10 12 14 161

1.2

1.4

1.6

S /m

γε x [μm]

Bend−plane normalized emittance

x 1.52 m

0 = 1.00 m

from P. Emma

initial emittance:

again the “typical” bunch compressor

bunch compressor

growth of slice emittance (in principle)

x / m top view: a bit before last magnet directly after last magnet

slicein the middle of the bunch

z / m z / m

x / m

last magnet

blue: particles in one slice before the chicane,they are not in the same slice in the chicane, …

smallconstz

… but they come to the same slice after the chicane

in the chicane these particles may be in different slices and may observe differentlongitudinal fields (, even with the projected CSR model)

bunch compressor

growth of slice emittance, the “typical” bunch compressor

μmz

beamline coordinate / m

rms bunch length

rms length of an initial slice

GeV 5ref E

bunch compressor

growth of slice emittance, the “typical” bunch compressor

μmz

beamline coordinate / m

rms bunch length

rms length of an initial slice

MeV 500ref Esame compressor, but low energy version:

other forces / effects

rough estimation / scaling


space charge zz

IZqE 2

02

0|| 3

14

rz for

CSR, circular motionoLIZE 0

||1

3 2024 zo RL

30 Rz

CSR, after magnet(distance S) SL

IZEo

5.02

1 0||

zS 22

0

0|| 8 Za

IZEz

resistive wall wake 3

0

2

ch 2 ZaSz

z

cqI2

(round pipe, radius a)

for Gaussian bunch with peak current


μm 100nC1

z

q

steady statealusteel

mm 10a

μm 1m 10twiss

130 MeV500 MeV


μm 100nC1

z

q

m 100 R

steady state

S = 1 m

steady statealusteel

mm 10a

μm 1m 10twiss

130 MeV500 MeV

compression work

)()(2

),(2 rzrz

rgzgqzr

)(21)(2

),(0

rzz

r rgzgrqzrE

),(),( 0 zrEczrB r

zR /

rzmetot

RqWWW 5.1

ln4 0

23

2



compression work

R = 1 cmz = 200mr = 100mq = 1 nC

z = 20m

Wtot = 0.107 mJ Wtot = 1.065 mJWtot = 0.958 mJ

simple example:

R0 = 10m, L = 0.5m , z = 20m P = 375 kW, P L/c0 = 0.625mJ

for comparison: steady state CSR energy loss in magnet

transverse effects

see: Transverse Effects of Microbunch Radiative InteractionY. Derbenev, V. Shiltsev, SLAC‐PUB‐7181

transverse effects

example: the “typical” bunch compressor (5 GeV case)methods for tracking: projected model

full field, approach (1)full field, approach (2) “greens”

EE0K

energy loss of one “typical” particle

transverse phase space after chicane

“on axis particles” middle slice

transverse effects

equation of horizontal motion

EE

EE xFKxxnKx

2

self effects

external fields: RzK 1 inverse curvature

to first order: chref

CSRch2

EEEE

xFKKxnKx

zn external focusing quadrupole field index

energy: CSRchref EEEE

chirp

horizontal CSR force: CSRCSR0 yxx vBEqF

transverse effects

EE xFKxnKx

2

EsF ExF

EE0K

test particle particle with 00000syyxx schref EEE0 andexample: the “typical” bunch compressor (5 GeV case)

A

eRvdtedqAe

dtdqvAeq

dtAdvAeq

AvtAeqF

xxx

x

xx

||

000

0

0

transverse effects

compensation

vAt

qΦqdtd

00E KΦqK 0E

BvEeqF xx

0

||eAvΦ

weak

compensation

some literature

E. Saldin, E. Schneidmiller, M. Yurkov: Radiative Interaction of Electrons in aBunch Moving in an Undulator. NIM A417 (1998) 158‐168.

M. Borland: Simple method for particle tracking with coherent synchrotronradiation. Phys. Rev. Special Topics ‐ Accelerators and Beams, Vol. 4, 070701(2001).

Y. Derbenev, J. Rossbach, E. Saldin, V. Shiltsev: Microbunching Radiative Tail‐Head Interaction. TESLA‐FEL 95‐05, September 95.

E. Saldin, E. Schneidmiller, M. Yurkov: On the Coherent Radiation of anElectron Bunch Moving in an Arc of a Circle. NIM A398 (1997) 373‐394.

J. Murphy, S. Krinsky, R. Glukstern: Longitudinal Wakefield forSynchrotron Radiation, Proc. of IEEE PAC 1995, Dallas (1995).

R. Li, Self‐Consistent Simulation of the CSR Effect. NIM A429 (1998) 310‐314.

Y. Derbenev, V. Shiltsev: Transverse Effects of Microbunch Radiative Interaction. (1996)SLAC‐PUB‐7181.

G. Bassi,T. Agoh, M. Dohlus, L. Giannessi, R. Hajima, A. Kabel, T. Limberg, M. Quattromini: Overview of CSR codes. NIM A 557 (2006) 189–204.

G. Bassi, J. Ellison, K. Heinemann, R. Warnock: Microbunching instability in a chicane: Two‐dimensional mean field treatment. Phys. Rev. ST Accel. Beams 12, 080704. (2009)

T. Agoh, K. Yokoya: Calculation of coherent synchrotron radiation using mesh. Phys. Rev. ST Accel. Beams 7, 054403. (2004)

D. Sagan, G. Hoffstaetter, C. Mayes, U. Sae‐Ueng: Extended one‐dimensional methodfor coherent synchrotron radiation including shielding. Phys. Rev. ST Accel. Beams 12 (2009) 040703

Coherent Synchrotron Radiation and Beam Interaction · curvature in BC magnets R 0 ~10 m N 250...

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