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
A beam delay must also be accounted for in the rf phase of the following linac (L2 in this
case) as the beam path length increases through a chicane. The phase delay is described in Eq.
(7.13), where the rf phase lag is ∆φ. The L2 rf phase needs to be delayed with respect to the
chicane-off phase by d∆φ/dR56 ≈ π/λ ≈ 1.72°/mm (or 61.5° S-band with the nominal R56 of
−35.9 mm):
2π∆s4π ⎡ ⎛ θ B ⎞ ⎛ 1 ⎞⎤
π R
∆ ϕ = = 2L 1 1
56
⎢ B ⎜ − ⎟+∆L⎜ − ⎟⎥
≈ . (7.13)
λ λ ⎢⎣ ⎝ sinθB ⎠ ⎝ cosθB
⎠⎥⎦
λ
7.4.1.3 Coherent Synchrotron Radiation (CSR)
Introduction
For very short bunches, the coherent component of synchrotron radiation can be significant
and may dilute the horizontal emittance by generating energy spread in the dipoles. In this case,
however, the energy spread is mostly correlated along the bunch and is not a random effect. For
an rms bunch length, σz, dipole length, L B, bend radius, R (≈ L B/θ B), and N electrons per bunch,
the CSR-induced rms relative energy spread per dipole for a gaussian bunch under steady-state
conditions is [20]
Nr
0.22 eLB
σδ
≈ , (7.14)
23 43
γ R σz
where re is the classical electron radius and γ is the Lorentz energy factor. This is valid for a
dipole magnet where radiation shielding (see Figure 7.20) of a conducting vacuum chamber is
not significant; i.e., for a full vertical vacuum chamber height h which satisfies [25]
2 3 ( z R ) ≡ hc
h >> πσ . (7.15)
Figure 7.20 Coherent radiation steady-state wake per 2π bend (R = 19.4 m, σ z = 30 µm,
h c = 5.6 mm) for both shielded (dash: h/h c = 0.5) and unshielded (solid: h/h c >> 1)
coherent radiation of a gaussian bunch. Here the bunch head is at z > 0 and V > 0
represents energy loss.
A C C E L E R A T O R ♦ 7-37