1. Beam Modulation due to Longitudinal Space Charge Zhirong Huang, SLAC Berlin S2E Workshop 8/18/2003

2. 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

3. 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

4. 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 process IE

5. 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

6. 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) ignore in the linacLSCIn a drift

7. 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”

8. 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

9. 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, (courtesy of J.H. Wu)

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

11. 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)

12. SDL microbunching experiment E z E z E zz E z 65 MeV Energy spectrometer (W. Graves, T. Shaftan et al.) X (E) profile

13. 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 energy modulation

14. Energy deviation = chirp + sinusoidal modulation or magnification Total charge Enhancement of horizontal modulation Energy profile

15. 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

16. Beam size and It’s Effect on the modulation Beam size in the zero-phasing linac is varied (courtesy of T. Shaftan)

17. IR measurements Wavelength, um Bolometer signal, uVs >40 um >100 um >160 um Filters: (T. Shaftan)

18. 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)