3.3.7 Brightness of a Bend Magnet Source
In order to evaluate the brightness of a bend magnet source, some assumptions need
to be made about the electron beam source size and divergence (the emittance). In
most calculations, people assume that the contribution of the curvature of the
electron trajectory to the source size is negligible and that the angular spread in
electron trajectories is small compared to the natural opening angle of synchrotron
radiation, σ ψ . One is thus left with three terms: the betatron oscillations, the
dispersion for electrons with different energies, and finally σ r , the contribution of
diffraction to the apparent source size. The first two terms have been discussed in
Chap. 2, and the last term is given by:
σ r ¼
λ
4πσ ψ
ð3:24Þ
The effective x and y source dimensions are then, as detailed by Hulbert and
Weber, or Kim[75]:
X
x
¼ ε x β x
|{z}
betatron
þ η
2
x σ
2
E
|ffl{zffl}
dispersion
þ σ
2
r
|{z}
diffraction
0
B
@
1
C
A
1=2
X
y
¼ ε y β y þ σ
2
r þ
ε
2
y þ ε y γ y σ
2
r
σ 2
ψ
! 1=2
ð3:25Þ
Thus, if we use the angular density of flux from Eq. 3.16, then the in-plane
brightness of a bend magnet source is given by:
B bm ¼
d
2 F bm
dθdψ j ψ ¼ 0
2π
P
x
P
y
ð3:26Þ
Fig. 3.6 Left: vertical divergence (σ) of bend magnet radiation (in units of γσ) as a function of energy
E with respect to the critical energy E c . Middle: relative strengths of horizontal and vertical electric
field components as a function of the observation angle from a bend magnet source. Right: degree of
circular polarization of bend magnet radiation, P 3 , as a function of the vertical angle γψ, for photon
energies below (Ε/Ε c ¼ 0.1), equal to (Ε/Ε c ¼ 1), or greater than (Ε/Ε c ¼ 10) the critical energy
3.3 Bend Magnet Radiation: The Details
47