Beam dynamics topics 45
Using the transfer matrix decomposition in Eq. (1.68) for the unperturbed
horizontal and vertical transfer matrices, the off-diagonal blocks of T can be
written as
χM 2 WN 1 = χ
β x β y B x,2 R(ψ x,2S )WR(ψ y,S1 )B
−1
y,1 ,
χN 2 WM 1 = χ
β x β y B y,2 R(ψ y,2S )WR(ψ x,S1 )B
−1
x,1 ,
where β x,y are the beta functions at the skew quadrupole, matrices B and R
are as defined in Eq. (1.60), and the subscripts indicate the plane and the location. The composition of the upper-right off-diagonal block describes the propagation of the vertical coordinates from point 1 to the skew quadrupole, the
application of the horizontal kick to the particle according to its y-coordinate,
and the subsequent propagation of the horizontal kick to point 2. Similarly,
the lower-left block represents the component of vertical motion at point 2
that comes from the initial horizontal motion at point 1.
When there are multiple skew quadrupoles in the beam line, the transfer
matrix can be calculated by concatenating the transfer matrices of all sections
and that of the skew quadrupoles. If the coupling strengths, χ
β x β y , of the
skew quadrupoles are weak, the higher order terms resulting from the changes
of skew quadrupole kicks by the effects of other skew quadrupoles can be
neglected. In such cases, the new transfer matrix may be written as the unperturbed transfer matrix plus a summation of the perturbations of all skew
quadrupoles in the beam line, i.e.,
T ≈ T 0 +
l
χ l T 2S l W 4 T S l 1 ,
(2.56)
where l indicates the l’th skew quadrupole. To first order in χ l , the skew
quadrupoles only change the off-diagonal blocks.
Using the coupled transfer matrix, T, the impact of skew quadrupole
components to the beam motion can be described. The non-zero off-diagonal
blocks will cause vertical motion for a particle initially launched on the horizontal plane, and vice versa.
In a circular accelerator, the coupled one-turn transfer matrix can be used
to track the beam motion for multiple turns. With linear coupling, the horizontal and vertical betatron oscillations will show up in the motion observed
on both planes. Figure 2.5 shows the coupled motion observed on the x and
y coordinates in a ring for a particle launched with initial offsets of 0.1 mm
in both planes. The Fourier spectrum of the x motion includes a component
with the vertical tune, while the y spectrum includes the horizontal tune.
Linear coupling also causes the closed orbit to be coupled between the two
transverse planes. With linear coupling, a kick on the horizontal plane causes
orbit deviations not only in the horizontal plane, but also in the vertical plane.
Similarly, a vertical kick causes both horizontal and vertical orbit deviations.
The orbit deviations in the other plane are small if the coupling is weak.
Using the transfer matrix decomposition in Eq. (1.68) for the unperturbed
horizontal and vertical transfer matrices, the off-diagonal blocks of T can be
written as
χM 2 WN 1 = χ
β x β y B x,2 R(ψ x,2S )WR(ψ y,S1 )B
−1
y,1 ,
χN 2 WM 1 = χ
β x β y B y,2 R(ψ y,2S )WR(ψ x,S1 )B
−1
x,1 ,
where β x,y are the beta functions at the skew quadrupole, matrices B and R
are as defined in Eq. (1.60), and the subscripts indicate the plane and the location. The composition of the upper-right off-diagonal block describes the propagation of the vertical coordinates from point 1 to the skew quadrupole, the
application of the horizontal kick to the particle according to its y-coordinate,
and the subsequent propagation of the horizontal kick to point 2. Similarly,
the lower-left block represents the component of vertical motion at point 2
that comes from the initial horizontal motion at point 1.
When there are multiple skew quadrupoles in the beam line, the transfer
matrix can be calculated by concatenating the transfer matrices of all sections
and that of the skew quadrupoles. If the coupling strengths, χ
β x β y , of the
skew quadrupoles are weak, the higher order terms resulting from the changes
of skew quadrupole kicks by the effects of other skew quadrupoles can be
neglected. In such cases, the new transfer matrix may be written as the unperturbed transfer matrix plus a summation of the perturbations of all skew
quadrupoles in the beam line, i.e.,
T ≈ T 0 +
l
χ l T 2S l W 4 T S l 1 ,
(2.56)
where l indicates the l’th skew quadrupole. To first order in χ l , the skew
quadrupoles only change the off-diagonal blocks.
Using the coupled transfer matrix, T, the impact of skew quadrupole
components to the beam motion can be described. The non-zero off-diagonal
blocks will cause vertical motion for a particle initially launched on the horizontal plane, and vice versa.
In a circular accelerator, the coupled one-turn transfer matrix can be used
to track the beam motion for multiple turns. With linear coupling, the horizontal and vertical betatron oscillations will show up in the motion observed
on both planes. Figure 2.5 shows the coupled motion observed on the x and
y coordinates in a ring for a particle launched with initial offsets of 0.1 mm
in both planes. The Fourier spectrum of the x motion includes a component
with the vertical tune, while the y spectrum includes the horizontal tune.
Linear coupling also causes the closed orbit to be coupled between the two
transverse planes. With linear coupling, a kick on the horizontal plane causes
orbit deviations not only in the horizontal plane, but also in the vertical plane.
Similarly, a vertical kick causes both horizontal and vertical orbit deviations.
The orbit deviations in the other plane are small if the coupling is weak.