LCLS Conceptual Design Report - Stanford Synchrotron Radiation ...

L C L S C O N C E P T U A L D E S I G N R E P O R T
8.9 Wakefield Effects in the Undulator
8.9.1 Introduction
When the electron beam moves through the undulator it will excite longitudinal and
transverse wakefields due to the resistance and the discontinuities in the beam tube wall. Let us
assume that the wall geometry is cylindrically symmetric. Then the longitudinal (monopole)
wakefield will generate an energy loss and an increase in energy spread independent of the beam
orbit, and the transverse (dipole) wakefield will generate an emittance growth that does depend
on the orbit. It is, however, important to recognize that the forces due to the wakefields are
correlated with longitudinal position. Assuming the bunch is composed of many slices at different
longitudinal positions, the wakefields affect only the centroid values of the slices — i.e., the
average energy and the average position of the slices in, respectively, the longitudinal and the
transverse case. The distributions of the slices about their centroids are not affected.
The critical issues concerning the electron beam with respect to wakefield effects in the
undulator are:
• The absolute value of the maximum relative energy deviation of a bunch “slice”
(slippage length: ~0.5 µm) with respect to the mean of the whole bunch generated
over the length of the undulator at 14.3 GeV should be less than ~0.1%. This
tolerance is derived from GINGER simulations.
• The dilution of the “projected” emittance (emittance projected over the entire bunch)
should not exceed ~10%.
• The mean energy loss over the undulator, including radiation losses, will determine
the necessary taper of the magnetic fields of the undulator dipoles.
Since undulator wakefields have very little effect on the “slice” energy spread and the “slice”
emittance, these tolerances are not considered here.
In this report the longitudinal and transverse wakefield effects on the LCLS beam during its
time in the undulator are estimated to see how well these conditions are satisfied. Note that the
beam dynamics and wakefield concepts that are presented are thoroughly discussed, with
equations in [20].
8.9.2 Wakefield Induced Beam Degradation
In the longitudinal case, the wake function for a Gaussian bunch, from which the average
wake (also known as the loss factor) 〈Wz〉 and the rms deviation of the wake with respect to the
mean, (Wz)rms (the units are V/C/m) are derived, is first obtained. Then the wakefield induced
energy loss is given by
2
eNLWz
δ =− , (8.40)
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U N D U L A T O R ♦ 8-53