Beam Modulation due toLongitudinal Space Charge

Zhirong Huang, SLAC

Berlin S2E Workshop

8/18/2003

• 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)

Introduction

LSC Impedance• For a round, parallel electron beams with a uniform transverse cross section of radius rb, 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

Space Charge Oscillation

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

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

Space Charge Oscillation II• Density and energy modulation in a drift at distance s

• At a very large , plasma phase advance (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)

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Including Acceleration• beam energy r(s) increases in the linac. Generalize the momentum compaction R56’(! 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 , R56’=0, b(k,s)=b0(k,s), beam is “frozen”

Comparison with Parmela

• 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)

• Energy Modulation

LSC 3-D Model• LSC impedance is r-dependant, which leads to decoherence

• We have

• Impedance at arbitrary radial coordinate r from a -ring with unit charge and radial coordinate a is

• Convolution with a Parabolic distribution,

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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

• 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

Parmela simulations (C. Limborg)

SDL microbunching experiment

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(W. Graves, T. Shaftan et al.)

X (E) profile

Long. Phase Space Distortion

• Small modulation gets projected to large modulation• Energy modulation can be induced by LSC in the zero-phasing section if c/ » L (length of the section, ~15 m)

• rf zero phasing energy spectrum is sensitive to beam energymodulation

Energy deviation = chirp + sinusoidal modulation

or

magnification

Total charge

Enhancement of horizontal modulation

Energy profile

• Define “gain” = x modulation amplitude/current modulation

I0=300 A, =130, rb=600 m Gm >> l

(Z. Huang, T. Shaftan, SLAC-PUB-9788, 2003)

• zero-phasing images are dominated by effects of energy modulation instead of current modulation

Beam size and It’s Effect on the modulationBeam size and It’s Effect on the modulation

0 50 100 150 200 250 300 350 4000

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1401

0 50 100 150 200 250 300 350 4000

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1402

0 50 100 150 200 250 300 350 4000

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1403

Beam size in the zero-phasinglinac is varied (courtesy of T. Shaftan)

0 50 100 150 2000

50

100

150

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250 a

IR measurements1 2

Wavelength, um

Bolometer signal, uVs

>40 um >100 um >160 umFilters:

(T. Shaftan)

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)