Beam Modulation due to Longitudinal Space Charge

Zhirong Huang, SLAC

Berlin S2E Workshop 8/18/2003

Introduction

• SDL microbunching observations through rf zero-phasing • LSC driven microbunching instability (TESLA-FEL-2003-02) • Injector modulation studies  Important to know beam modulation induced by LSC • Discuss methods to evaluate current and energy modulation in the linac • Discuss its impact on rf zero-phasing measurements • Do not discuss gain in bunch compressors (until Thursday)

LSC Impedance

• For a round, parallel electron beams with a uniform transverse cross section of radius r b , the longitudinal space charge impedance on axis is (cgs units) • Off-axis LSC is smaller and can increase the energy spread • Free space approximation is good when   /(2  ) << beam pipe radius

I

Space Charge Oscillation

• If there is a density modulation, space charge pushes particles from high density to low density, creating energy modulation in the process

E

• Energy modulation converts back to density modulation to complete space charge oscillation with frequency

Space Charge Oscillation II

• Density and energy modulation in a drift at distance s • At a very large  , plasma phase advance ( W s/c) << 1, beam is “frozen,” energy modulation gets accumulated (Saldin/Schneidmiller/Yurkov, TESLA-FEL-2003-02) • LSC acts like a normal impedance at high energies

Non-rigid beam

• At lower energies (in the injector…), beam is not rigid • Space charge simulations may be time-consuming and noisy at high frequencies • Linear evolution of high-frequency beam modulations can be described by the same integral equation for CSR microbunching (Heifets et al., PRSTAB-064401; Huang/Kim, PRSTAB-074401)

b

(

k

(

s

);

s

) kernel

K

(  

b

0 (

k

(

s

);

s

)  ,

s

) 

ik

(

s

)

R

56  0

s d



K

(  ,

s

)

b

(

k

(  );  ) (  

s

)

I

(  

I A

)

Z

(

k

(  ),  )    ,   ...) Landau damping In a drift LSC ignore in the linac

Including Acceleration

• beam energy  r (s) increases in the linac. Generalize the momentum compaction R 56 ’(  ! s) as the path length change at s due to a small change in  (not  ) at  : • The integral equation for LSC microbunching in the linac is • In a drift,  Space charge oscillation • For very large  , R 56 ’=0, b(k,s)=b 0 (k,s), beam is “frozen”

Comparison with Parmela

• Energy Modulation • Parmela simulations (C. Limborg) of a 3-m drift at 6 and 12 MeV (beam size changes due to optics and transverse SC) • Theory-1D: integral equation using average LSC impedance • Theory-3D takes into account transverse variations of LSC (J.H. Wu)

LSC 3-D Model

(courtesy of J.H. Wu) • LSC impedance is r-dependant, which leads to decoherence • We have

b

(

k

;

s

;

r

) and   (

k

;

s

;

r

) • Impedance at arbitrary radial coordinate

r

from a  -ring with unit charge and radial coordinate

a

is

Z ring LSC

(

R

,  )  

ik

2   (  

R

) 2

K

0 (  )

I

0 (

R

)   (

R

  ) 2

I

1  (  )  

I

0 (  ) 

I

2 (  ) 

K

0 (

R

)   • Convolution with a Parabolic distribution,

n

(

a

)  2 3 

r

2   1 

a

2 3 

r

2  

Comparison with Elegant

• Borland implemented 1-D LSC impedance in elegant • Current modulation at different accelerating gradients Elegant tracking (M. Borland) Analytical calculation

Injector Modulation Studies

• Assume 10% initial density modulation at gun exit at 5.7 MeV • After 67 cm drift + 2 accelerating structures (150 MeV in 7 m), LSC induced energy modulation Parmela simulations (C. Limborg) • LSC induced energy modulation in the LCLS injector is small at shorter wavelengths (<250  m), where the downstream gain is the highest • Density modulation at these wavelengths is also reduced

SDL microbunching experiment

(W. Graves, T. Shaftan et al.) 65 MeV Energy spectrometer X (E) profile E z E z E z z E z

Long. Phase Space Distortion

• rf zero phasing energy spectrum is sensitive to beam energy modulation • Small modulation gets projected to large modulation • Energy modulation can be induced by LSC in the zero phasing section if c/ W »  L (length of the section, ~15 m)

Enhancement of horizontal modulation

Energy deviation = chirp + sinusoidal modulation Total charge Energy profile or magnification

• Define “gain” = x modulation amplitude/current modulation I 0 =300 A,  =130, r b =600  m  G m >> l (Z. Huang, T. Shaftan, SLAC-PUB-9788, 2003) • zero-phasing images are dominated by effects of energy modulation instead of current modulation

140 120 100 80 60 40 20 0 0 140 120 100 80 60 40 20 0 0 Beam size and It’s Effect on the modulation 140 1 120 50 100 150 200 250 300 350 400 3 100 80 60 40 20 0 0 2 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 Beam size in the zero-phasing linac is varied (courtesy of T. Shaftan)

Bolometer signal, uVs 250 200 150 100 1

IR measurements

(T. Shaftan) 2 a 50 0 0 Filters: >40 um 50 >100 um 100 >160 um 150 Wavelength, um 200

Summary

• LSC induced modulation in the linac can be described by a modified integral equation that includes acceleration • Comparable energy modulation with Parmela simulations • Initial studies suggest that accumulated energy modulation at the end of the injector is small at the most dangerous modulation wavelengths for LCLS • Density modulation is reduced in the injector, but can be amplified by downstream bunch compressors… • Energy spectrum of a chirped beam is sensitive to beam energy modulation, which could be induced by LSC in the SDL linac (  means to measure energy modulation)