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
Figure 8.33 The relative energy loss of the LCLS beam at the end of the undulator as a function of
position within the bunch.
The properties of the synchronous mode in the case of rectangular corrugation of the wall
were studied in Ref. [27]. In this paper, the wall roughness was modeled by axisymmetric
periodic steps on the surface of height δ, width g, and period p. All three parameters were
assumed much smaller than the pipe radius b. The model gives for the frequency ω0 of the mode
2 p
ω0
= c , (8.53)
δ bg
and for the longitudinal wakefunction of the point charge
Zc 0 w( s) = cos 2 ( ω0s/
c)
, (8.54)
π b
Surprisingly, the amplitude of the wake in this approximation does not depend on the
roughness properties at all. These results however are valid if kp 1
. Eq. (8.53) shows that
when δ becomes very small, the parameter k increases and eventually kp becomes comparable to
unity. Hence, this model becomes invalid in the limit δ → 0 . The results of computer
simulations that confirm the predictions of this model can be found in Refs. [36,37].
To take into account the effect of the shallowness of the roughness a different model was
developed in Ref. [28]. In this model the roughness was treated as a sinusoidal perturbation of the
wall with h0κ 1
. It was found that, indeed, under certain conditions, a low-frequency
synchronous mode with κ 1can
propagate in this system. The longitudinal wake generated
by this mode is given by
2Zc
0 w( s) = Ucos 2 ( ω0s/
c)
, (8.55)
πb
where the dimensionless factor U and the frequency of the mode ω0 depend on the parameter
3
r ≡ h0 bκ/2.
The plot of these two functions is shown in Figure 8.34. In the limit h 0 → 0
4
the frequency ω0 approaches κc
/2,
and U ≈ r /32.
For large values of r, ω0 ≈ 2 c/ h bκand
U → 1/2 .
U N D U L A T O R ♦ 8-61