1. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005 Effect of AC RW Wake on SASE - Analytical Treatment Z. Huang, G. Stupakov see SLAC-PUB-10863, to appear in PRST-AB

2. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005  AC wake changes beam energy along undulator, cannot be compensated by undulator taper for the whole bunch  Effects on SASE performance evaluated with simulations  A general question: How is the FEL process affected by variable beam and undulator parameters (energy, taper…)?  Kroll-Morton-Rosenbluth (KMR) treatment of tapered undulator FELs only addresses saturation regime  We develop a self-consistent theory of variable-parameter FEL in the small signal regime to evaluate SASE performance under any wake and to optimize undulator taper IntroductionIntroduction

3. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005 FEL theory with slowly varying parameters  E-beam energy  c (z), undulator parameter K(z)  Resonant energy  r (z) corresponds to initial radiation 0  A high-gain FEL is characterized by  : relative gain bandwidth is a few , and radiation field gain length ~ u /(4  )  Relative change in beam energy w.r.t resonant energy (normalized to  )  Solved by WKB method when relative energy change per field gain length is smaller than  (satisfied for AC wake)

4. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005 WKB solution 1) A zeroth-order growth rate Im[  0 ( ,z)] = shifting the growth rate of a constant-parameter FEL Im[  c (  )] by  (z) due to changes in beam and undulator parameters 2) A small correction in growth rate |  1 | << |  0 | that gives rise to a sizeable change in radiation power at undulator end.  z 

5. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005 Comparison with simulation  For a variable-parameter FEL, slightly above resonance has a larger growth rate since energy modulation is immediately accompanied by gain in radiation power lose energy 2kuz2kuz gain energy  Linear energy change  =  cold beam, seeded at  0 Power growth rate difference for different  with respect to a constant-parameter FEL

6. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005 SASE power  Integrate all frequencies to obtain SASE power  P/(  P beam ) vs. fractional energy loss in units of  at Theory (curve) Cold beam simulation (symbol)  =2  k u z = 8  =  maximum poweroptimal energy gain or taperSASE rms bandwidth

7. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005 Optimal energy gain or taper Maximum SASE power occurs for a small energy gain (better than a constant-parameter SASE!) Optimal energy gain is about  = 2(  ) over saturation length (140 keV/m for LCLS) with about twice as much power 1-D Cold beam simulation 2kuz2kuz z   re  resonant to e-beam)  rc radiation freq.  re  rc back in sync out of sync

8. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005 3D studies  Compare with GENESIS (similar results from GINGER)  Power enhancement ~ 2 when energy gain 2  at saturation  Power as a function of  is Gaussian with RMS = FWHM ≈ 4   (~ 4  at saturation) 22 44

9. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005  Assume a sinusoidal wake energy change for the bunch core (from s=-30  m to 0  m, wake =30  m period) AC resistive wall wake s  A ~ 6  for Cu  A ~ 3  for Al Bane & Stupakov at Z sat = 90 m 1 nC bunch shape Current spike enhance wake loss amplitude

10. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005  Set undulator taper to change resonant energy by 2  ~0.1% over saturation length z sat =90 m (referred as 2  taper)  Evaluate average saturation power over the bunch core Average power in the bunch core Cu (round pipe) Al (round pipe) 2  taper no taper For small wake amplitude, 2  taper can double the saturation power over the no taper case, as found in 200 pC setup (see P. Emma’s talk)

11. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005 Al wake from recent measurements  From K. Bane’s talk, how these different models affect LCLS performance?  Average power over bunch core (30  m flat part), no taper nom. model: = 7.4 GW fit model: = 7.5 GW model 2: = 7.1 GW

12. Z. Huang LCLS FAC zrh@slac.stanford.edu April 7. 2005 SummarySummary  Analytical treatment can be used to estimate effects of arbitrary wake on SASE FELs (for a decent beam) and can be used to optimize the undulator taper  For LCLS at 1 nC, AC wake from Cu round pipe reduces the FEL power by a factor of 2 compared to AL round pipe (at least for the flat bunch core), in agreement with S2E simulation results (see W. Fawley’s talk) 10 12  Operating LCLS at 200 pC significantly reduces AC wake amplitude and allows for effective taper to reach ~10 12 x- ray photons, comparable to the 1 nC output (see P. Emma’s talk)