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
j th kick to the i th BPM. The equation, as written here, also indicates N BPMs, N kicks and two
different momenta (k = 1,2).
⎡ m11 ⎤ ⎡ –1 0 0
⎢ ⎥ ⎢
⎢ m21 ⎥ ⎢ 0 –1 0
⎢ ⎥ ⎢
⎢ ⎥ ⎢
⎢
⎢
m ⎥ ⎢
N1 ⎥ = ⎢
0 0 –1
⎢ m12 ⎥ ⎢ –1 0 0
⎢ ⎥ ⎢
⎢ m22 ⎥ ⎢ 0 –1 0
⎢
⎢
⎥ ⎢
⎥ ⎢
⎣ ⎢ mN 2⎦
⎥ ⎢
⎣ 0 0 –1
P11( 1)
0 0 ⎤ ⎡ b1 ⎤
⎥ ⎢ ⎥
P21() 1 P22() 1 0 ⎥ ⎢ b2 ⎥
⎥ ⎢ ⎥
⎥ ⎢ ⎥
PN1() 1 PN 2() 1 PNN() 1
⎥ ⎢
⎥
b ⎥
⋅ ⎢ N ⎥
P11() 2 0 0 ⎥ ⎢ ∆B1 ⎥
⎥ ⎢ ⎥
P21() 2 P22() 2 0 ⎥ ⎢ ∆B2⎥ ⎥ ⎢ ⎥
⎥ ⎢ ⎥
PN1() 2 PN 2() 2 PNN() 2
⎥
⎦ ⎣ ⎢ ∆BN ⎦
⎥
(8.66)
There are a very large number of undulator poles along the undulator (3762) and therefore
too many to determine, so the BPM data are fitted to quadrupole magnet misalignments and BPM
offsets only. Therefore, any BPM readback sensitivity to energy change will be identified as
upstream quadrupole offsets, and the determined quadrupole misalignments (in the wiggle-plane)
will necessarily be biased in order to best cancel the net dipole error (i.e., real localized pole
errors plus quadrupole misalignment). The quality of this cancellation is examined in the
simulation section described below. The non-wiggle plane has no dipoles and therefore the
determined quadrupole positions in this plane will not be biased (unless pole roll errors or other
stray magnetic fields exist). This is a significant advantage for the energy scan technique because
all bend fields, without explicit knowledge of their source, are approximately canceled by biasing
the final quadrupole positions in order to best remove the trajectory’s sensitivity to energy
variations.
To explicitly write Eq. (8.66) in terms of quadrupole misalignments, ∆Bj is replaced with
the quadrupole magnet misalignment ∆xj, and Pij(k) is replaced with
j ji j ji
Pij(k) →[ 1 − Q11(k) ]R11(k) − Q21(k)R12
(k)
. (8.67)
Where Q j 11(k) and Q j 21(k) are the thick-lens transfer matrix elements across the j th quadrupole
magnet evaluated at the k th momentum, and R11 ji (k) and R12 ji (k) (=Cij) are the position-to-position
and angle-to-position, respectively, transfer matrix elements from the exit of the j th quadrupole to
the i th BPM, also evaluated at the k th momentum. Note, a thin lens quadrupole of pole-tip field B,
radius r, and length has a focal length f = rp/B e.
Then the right side of Eq. (8.67) reduces to –
Cij(k)/f, where the minus sign indicates that a horizontally focusing quadrupole (1/f < 0) displaced
in the positive direction (∆x > 0) will kick the beam in the positive direction.
In practice, the linear system of Eq. (8.66) is solved by imposing ‘soft-constraints’ on the
solutions to stabilize the system. The inclusion of the soft-constraints is equivalent to including
the additional known information that the quadrupole and BPM offsets are zero to within a
reasonable scale (e.g., ~1 mm). The constraints are not hard limits but rather weight the fit error
U N D U L A T O R ♦ 8-75