a1
&RKHUHQW
a1 ,QFRKHUHQW
)UHTXHQF\7+]
FIGURE 10.5
Qualitative comparison of the spectrum of coherent synchrotron
radiation in comparison with the spectrum of incoherent SR.
3RZHUDX

advanced beam manipulation, cooling, damping and stability 189
and its bunch length is σ. For certain frequencies ω of synchrotron radiation, its wavelength ≈ c/ω can be longer than
the bunch length σ; the beam at these frequencies will radiate
coherently, as illustrated in Fig. 10.4.
FIGURE 10.4
Incoherent radiation (left) and coherent radiation (right).
The two regimes can thus be defined as c/ω < σ — incoherent radiation — and c/ω > σ — coherent radiation.
For frequencies where c/ω is longer than σ, the beam radiates coherently, increasing radiated power proportionally
to the square of the number of particles involved, i.e., as N 2 .
As the typical bunch population N is ∼ 10 9 to ∼ 10 10 , this
effect increases the radiated power tremendously.
Assuming that the spectrum of incoherent SR is given by
I (ω), the CSR spectrum can be approximated as
r
(
2
)1
ωσ
I b (ω) = I (ω) N 1 + N exp −
(10.11)
c
A typical spectrum described by the above equation is shown
in Fig. 10.5.
10.1.3 CSR effects on the beam longitudinal phase space
We will now make a back-of-the-envelope estimate 1 of the
force that CSR exerts on the beam. We will use an approxiFI
Tw
1 Following derivations from Ya. Derbenev et al., Tesla-FEL Report 1995–
be
05.
GURE 10.6
o-particle model of the
am and its field.