Transcript Effect of AC Resistive Wall on SASE Analytic Treatment

Effect of AC RW Wake on SASE
- Analytical Treatment
Z. Huang, G. Stupakov
• see SLAC-PUB-10863, to appear in PRST-AB
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
Introduction
 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
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
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)
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
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
(z)
2) A small correction in growth rate |1| << |0| that gives rise
to a sizeable change in radiation power at undulator end.
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
Comparison with simulation
 Linear energy change  = , cold beam, seeded at 0
gain energy
Power growth rate difference
for different  with respect to
a constant-parameter FEL
lose energy
=2kuz
 For a variable-parameter FEL, slightly above resonance
has a larger growth rate since energy modulation is
immediately accompanied by gain in radiation power
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
SASE power
 Integrate all frequencies to obtain SASE power
maximum power
optimal energy gain or taper SASE rms bandwidth
 P/(Pbeam) vs. fractional
energy loss in units of 
at =2kuz = 8
Theory (curve)
Cold beam simulation (symbol)
=
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
Optimal energy gain or taper
• Maximum SASE power occurs for a small energy gain
(better than a constant-parameter SASE!)

1-D Cold beam simulation
re (resonant to e-beam)
back in sync
rc radiation freq.
rc
z
re
out of sync
=2kuz
• Optimal energy gain is about  = 2() over saturation length
(140 keV/m for LCLS) with about twice as much power
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
3D studies
 Compare with GENESIS (similar results from GINGER)
 Power enhancement ~ 2 when energy gain 2 at saturation
4
2
 Power as a function of  is Gaussian with RMS =
FWHM ≈ 4 (~4 at saturation)
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
AC resistive wall wake
 Assume a sinusoidal wake energy change for the bunch
core (from s=-30 m to 0 m, wake=30 m period)
1 nC bunch shape
Current spike enhance
wake loss amplitude
Bane &
Stupakov
A ~ 6 for Cu
at Zsat = 90 m
A ~ 3 for Al
s
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
Average power in the bunch core
 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
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)
April 7. 2005
LCLS FAC
2 taper
Al (round pipe)
no taper
Cu (round pipe)
Z. Huang
[email protected]
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
April 7. 2005
LCLS FAC
Z. Huang
[email protected]
Summary
 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 ~1012 xray photons, comparable to the 1 nC output (see P.
Emma’s talk)
April 7. 2005
LCLS FAC
Z. Huang
[email protected]