LCLS Conceptual Design Report - Stanford Synchrotron Radiation ...

is proportional to
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
z
dx
∆ E = e∫ Ex⎡⎣x, y, z′ , t( z′ ) ⎤⎦
( z′ ) dz′
(8.13)
dz
0
z − iκ ⎢z′ + ∫ I
1 ( z′′ ) dz′′ x
+ ∫ I
1 ( z′′ ) dz′′
y
⎥
⎢ 0 0 ⎥
pz′
F = I1x( z ) e ⎣ ⎦
∫ ′ u⎡⎣x( z′ ) , y( z′ ) , z′ ⎤⎦e
dz′
(8.14)
0
⎡ z′ z′
2 2
⎤
where κ = k/(2γ 2 ). For the LCLS the breaks between undulator segments are relatively short and
produce a phase shift of an integer multiple of the x-ray wavelength. Therefore it is close to the
homogeneous undulator case, described analytically in Ref. [8]. Then the fundamental eigenmode
is close to the Gaussian beam with almost flat wavefronts
x
2
+ y
2
−
2σ
2
∝ r
(8.15)
u e
The relative reduction of the maximum energy gain caused by different field imperfections is
calculated using Eq. (8.14) and F is expanded near the “ideal” state, by different kinds of
“imperfections”.
8.2.5.2 Trajectory Straightness
y:
Using Eq. (8.15) at z = NL at the end of the N th undulator segment, one can expand F in x and
⎧ ⎡ z z
2 2
⎤ ⎫
⎪NL − iκ ⎢z+ ∫ I
1 ( z′ ) dz′ x
+ ∫ I
1y(
z′ ) dz′
⎥ ⎪
⎪
F | I1( z) e 0 0 e
pz ⎪
≈ ⎨
⎢ ⎥
∫ x ⎣ ⎦ dz⎬
−
⎪ 0
⎪
⎪ ⎪
⎩ ⎭0
⎡ z z
2 2
⎤
( ) ( )
1 NL − iκ⎢z+ ∫ I
1
z′ dz′ x
+ ∫ I
1
z′ dz′
y
⎥
I
2 2
1 ( z) e 0 0 e
pz
x
2
∫
⎢
x ⎣ ⎥⎦
⎡ ( z) + y ( z) ⎤dz
2σ ⎢ ⎥
r 0
⎣ ⎦
(8.16)
For the “ideal” case (the curly brackets in Eq. (8.16)), the slow part of the expression under the
integral is almost constant. Therefore, Eq. (8.16) leads to
F − F 0
F 0
≈− x 2 + y 2
2 , (8.17)
2σ r
U N D U L A T O R ♦ 8-11