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 u(x, y, z+L) = u(x, y, z), L is the system period, k is the design value of the undulator
radiation fundamental harmonic wavevector, and p is the complex growth rate. Then the electron
longitudinal motion equations are:
dE
= eE
dz
dt 1
=
dz v
z
x
dx
dz
where e and E = γmc 2 are the electron charge and energy, t is the moment of time when the
electron passes the longitudinal coordinate z, and the electron velocity vz can be expressed
through the electron energy and angles:
2 2
⎡ ⎛dx ⎞ ⎛dy ⎞ ⎤
⎢12⎜ ⎟ ⎜ ⎟ ⎥
2 2 2
1 1 1 1 1
≈ + + +
vz c ⎢⎣ γ ⎝dz ⎠ ⎝dz ⎠ ⎥⎦
The angles dx/dz = α and dy/dz can be calculated from the measured magnetic field B using the
trajectory equations:
d ⎛ dx ⎞
⎜ ⎟ = −
dz ⎝ dz⎠
e
d ⎛ dy ⎞
⎜ ⎟ =
dz ⎝ dz⎠
e
()
γmc 2 B y z
()
γmc 2 B x z
(8.8)
(8.9)
(8.10)
The LCLS undulator line includes beam position monitors and angle steering at the section ends.
For ideal steering, the trajectory displacement at the segment ends is zero for the equilibrium
(beam centroid) trajectory. Therefore, for the undulator specification, the solution of Eq. (8.10)
with x(0) = x(L) = y(0) = y(L) = 0 is chosen. It is convenient to introduce the corresponding
“corrected” dimensionless first field integrals I1 x = - γ dx/dz and I1 y = γ dy/dz:
z z z′
e ⎡ 1
⎤
I1x( z) = 2 ⎢ Bx( z′ ) dz′ − Bx( z′′ ) dz dz ⎥
mc ⎣ L
0 0 0
⎦
∫ ∫∫ ′′ ′ (8.11)
z z z′
e ⎡ 1
⎤
I1y( z) = 2 ⎢ By( z′ ) dz′ − By( z′′ ) dz dz ⎥
mc ⎣ L
0 0 0
⎦
The maximum particle energy variation
8-10 ♦ U N D U L A T O R
∫ ∫∫ ′′ ′ (8.12)