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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

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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

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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)

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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

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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 2kuz2kuz gain energy Linear energy change = cold beam, seeded at 0 Power growth rate difference for different with respect to a constant-parameter FEL

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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

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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 2kuz2kuz z re resonant to e-beam) rc radiation freq. re rc back in sync out of sync

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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) 22 44

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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

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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)

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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

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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)

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