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
where k is the fundamental harmonic wavevector of the undulator radiation, and
k ⎡ z z
2
( )
2
⎤
L − i ⎢z+ I
1
z dz I
1
( z ) dz
2
2
∫ ′ ′
x
+ ∫ ′ ′
y
⎥
γ ⎢ 0 0 ⎥
= ( ) ⎣ ⎦
∫ (8.5)
A I z e dz
0
1y
The “reduction” is as compared with an ideal undulator, but in practice the comparison
can be with the best undulator, i.e., the one which gives the highest value of |A|. A 4%
intensity reduction corresponds to an increase in the power gain length by 1.1%.
• The calculated electron phase deviation from the design value must be less than 10° in
one segment. This phase is simply the electron-wave slippage:
ϕ = k
2γ 2 L + I L
L
⎡ 2
∫ 1x (z)dz + ∫ I1y
⎣
⎢
0
0
2 (z)dz
⎤
⎦
⎥
(8.6)
and the “design value” is an integer multiple of 2π. A 10° phase error causes an increase
in power gain length of 1.7%.
• The undulator median plane must be defined (and after that aligned) with an accuracy
better than 50 microns vertically. If the beam is off-axis vertically by 50 microns, the
beam will see a stronger undulator field, resulting in about 10° of additional phase
slippage.
Implicit in these tolerances is the need for the magnetic field strength to be uniform along the
length of the undulator line. If the magnetic field in one undulator segment is wrong by ∆B/B =
1.5×10 -4 , the resulting phase error will be 10°. This tolerance agrees with the result from
independent simulations performed using the code RON [2] to vary the strength of one undulator
segment. Then the change in the overall gain became significant when segment-to-segment
variations reached ∆B/B = 1.3×10 -4 . This translates into an error in the magnetic field strength of
1.7 G, or an error in the undulator segment magnetic gap by 1.2 µm.
8.2.5 Derivation of the Tolerances for the X-Ray FEL
The following is based on a simple picture of acceleration (or deceleration) of the electron by
the given radiation eigenmode.
8.2.5.1 Derivation of Basic Equations
According to the Floquet theorem, the wave field eigenmode in the periodic amplifying
system can be represented as:
x
( , , )
( − )
pz ik z ct
⎡ ⎤
E =ℜ
⎣
u x y z e e
(8.7)
⎦
U N D U L A T O R ♦ 8-9