Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

Effect of AC RW Wake on SASEEffect of AC RW Wake on SASE

- Analytical Treatment- Analytical Treatment

Z. Huang, G. StupakovZ. Huang, G. Stupakov

• see SLAC-PUB-10863, to appear in PRST-AB

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 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

IntroductionIntroductionIntroductionIntroduction

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

FEL theory with slowly varying parametersFEL theory with slowly varying parametersFEL theory with slowly varying parametersFEL 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)

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

WKB solutionWKB solutionWKB solutionWKB solution1) 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

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

Comparison with simulationComparison with simulationComparison with simulationComparison 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

2kuz

gain energy

Linear energy change = cold beam, seeded at 0

Power growth rate difference for different with respect toa constant-parameter FEL

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

SASE powerSASE powerSASE powerSASE power Integrate all frequencies to obtain SASE power

P/(Pbeam) vs. fractional energy loss in units of at

Theory (curve)Cold beam simulation (symbol)

=2kuz = 8

=

maximum power optimal energy gain or taper SASE rms bandwidth

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

Optimal energy gain or taperOptimal energy gain or taperOptimal energy gain or taperOptimal 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

2kuz

z

reresonant to e-beam)

rc radiation freq.

re

rc

back in sync

out of sync

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

3D studies3D studies3D studies3D 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

4

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 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 wakeAC resistive wall wakeAC resistive wall wakeAC resistive wall wake

s

A ~ 6 for CuA ~ 3 for Al

Bane &Stupakov

at Zsat = 90 m

1 nC bunch shape Current spike enhance wake loss amplitude

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

Set undulator taper to change resonant energy by 2~0.1% over saturation length zsat=90 m (referred as 2 taper) Evaluate average saturation power over the bunch core

Average power in the bunch core Average power in the bunch core Average power in 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)

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

Al wake from recent measurements Al wake from recent measurements Al wake from recent measurements 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: <P> = 7.4 GW• fit model: <P> = 7.5 GW• model 2: <P> = 7.1 GW

Z. HuangZ. Huang

LCLS FACLCLS FAC zrh@slac.stanford.eduzrh@slac.stanford.edu

April 7. 2005April 7. 2005

SummarySummarySummarySummary

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)

Operating LCLS at 200 pC significantly reduces AC wake amplitude and allows for effective taper to reach ~10101212 x-ray photons, comparable to the 1 nC output (see P. Emma’s talk)