Linear optics measurement and correction - I 95
center will move toward the closed orbit as particles populate along ellipses
centered on the closed orbit, even though the individual particles are still
oscillating with large amplitude. This phenomenon is called decoherence. The
beam center for a beam undergoing decoherence does not behave like a single
particle. However, the decay of the apparent motion of the beam center takes
tens to hundreds of turns. Orbit data taken during this period can still be seen
as representing the motion of a single particle. Therefore, orbit data taken with
both orbit corrector changes or one-pass kicks can be used to sample the linear
optics.
Since BPMs can only directly measure the position coordinates, x or y,
not the angle coordinates, Eq. (4.4) usually cannot be used to measure the
transfer matrix. However, if two BPMs are separated by a beam line of which
the transfer matrix is known, then the angle coordinates at the two BPMs can
be calculated using the position coordinates and the transfer matrix between
the two BPMs,
x
1 =
x 2 − M 11 x 1
M 12
, x
2 = M 21 x 1 + M 22 x
1 ,
(4.5)
where M 11 , M 12 , M 21 , and M 22 are the four elements of the transfer matrix
from BPM 1 to BPM 2. For the special case when the beam line between the
two BPMs is a drift space, the transfer matrix is known with a high accuracy.
In this case, we have
x
1 = x
2 =
x 2 − x 1
L
,
(4.6)
where L is the distance between the BPMs.
If the phase space coordinates can be determined at two locations (with a
pair of BPMs for each location) using the above method, the transfer matrix
between the two locations can be determined from Eq. (4.4). In principle,
only two different orbits are needed to determine the four matrix elements.
In reality, more orbit data can be used to achieve a higher accuracy under
random noise in the data. The orbit data with N orbit measurements can be
arranged in the form of
X 1 =
x 1 (1) x 1 (2) · · · x 1 (N )
x
1 (1) x
1 (2) · · · x
1 (N )
, X 2 =
x 2 (1) x 2 (2) · · · x 2 (N )
x
2 (1) x
2 (2) · · · x
2 (N )
,
(4.7)
for BPMs 1 and 2, respectively. Using the least-square method to fit the matrix
elements, it is straightforward to show that the transfer matrix can be found
with
M 21 = X 2 X
T
1 (X 1 X
T
1 )
−1 .
(4.8)
Because of random errors, the transfer matrix obtained with Eq. (4.8)
is not strictly symplectic. An equivalent symplectic transfer matrix can be
obtained from M using a procedure given in Ref. [45]. The symplectic matrix
center will move toward the closed orbit as particles populate along ellipses
centered on the closed orbit, even though the individual particles are still
oscillating with large amplitude. This phenomenon is called decoherence. The
beam center for a beam undergoing decoherence does not behave like a single
particle. However, the decay of the apparent motion of the beam center takes
tens to hundreds of turns. Orbit data taken during this period can still be seen
as representing the motion of a single particle. Therefore, orbit data taken with
both orbit corrector changes or one-pass kicks can be used to sample the linear
optics.
Since BPMs can only directly measure the position coordinates, x or y,
not the angle coordinates, Eq. (4.4) usually cannot be used to measure the
transfer matrix. However, if two BPMs are separated by a beam line of which
the transfer matrix is known, then the angle coordinates at the two BPMs can
be calculated using the position coordinates and the transfer matrix between
the two BPMs,
x
1 =
x 2 − M 11 x 1
M 12
, x
2 = M 21 x 1 + M 22 x
1 ,
(4.5)
where M 11 , M 12 , M 21 , and M 22 are the four elements of the transfer matrix
from BPM 1 to BPM 2. For the special case when the beam line between the
two BPMs is a drift space, the transfer matrix is known with a high accuracy.
In this case, we have
x
1 = x
2 =
x 2 − x 1
L
,
(4.6)
where L is the distance between the BPMs.
If the phase space coordinates can be determined at two locations (with a
pair of BPMs for each location) using the above method, the transfer matrix
between the two locations can be determined from Eq. (4.4). In principle,
only two different orbits are needed to determine the four matrix elements.
In reality, more orbit data can be used to achieve a higher accuracy under
random noise in the data. The orbit data with N orbit measurements can be
arranged in the form of
X 1 =
x 1 (1) x 1 (2) · · · x 1 (N )
x
1 (1) x
1 (2) · · · x
1 (N )
, X 2 =
x 2 (1) x 2 (2) · · · x 2 (N )
x
2 (1) x
2 (2) · · · x
2 (N )
,
(4.7)
for BPMs 1 and 2, respectively. Using the least-square method to fit the matrix
elements, it is straightforward to show that the transfer matrix can be found
with
M 21 = X 2 X
T
1 (X 1 X
T
1 )
−1 .
(4.8)
Because of random errors, the transfer matrix obtained with Eq. (4.8)
is not strictly symplectic. An equivalent symplectic transfer matrix can be
obtained from M using a procedure given in Ref. [45]. The symplectic matrix