Beam dynamics topics 35
Introducing the transformation ψ(s) = νφ(s), and using ds = βdψ =
βνdφ, the closed orbit, Eq. (2.11) can be rewritten as
y co (s) =
ν
β(s)
2 sin πν
φ+2π
φ
θ(ξ)β
3
2 (ξ) cos ν(π + φ − ξ)dξ.
(2.12)
Because the function θ(φ)β
3
2 (φ) is periodic (with the period 2π), it can be
Fourier expanded
θ(φ)β
3
2 (φ) =
∞
n=−∞
f n e
inφ ,
(2.13)
with the Fourier coefficients f n given by
f n =
1
2πν
θ(s)β
1
2 (s)e
−inφ(s) ds =
1
2πν
k
θ k β
1
2
k e
−inφ k ,
(2.14)
where in the last equation discrete dipole kicks are assumed, with a kick
angle θ k = θ(s k )∆s k . The Fourier coefficients in Eq. (2.14) are called integer
stopband integrals. Inserting Eq. (2.13) into Eq.(2.12), we obtain the Fourier
expansion of the closed orbit deviation
y co (s) =
β(s)
n
ν
2 f n
ν 2 − n 2 e
inφ(s) ,
=
β(s)
f 0 +
∞
n=1
2ν
2 |f n |
ν 2 − n 2 cos(nφ(s) + χ n )
,
(2.15)
where we have used f n = f
∗
−n = |f n |e
iχn . The closed orbit can be approximated with only a few Fourier terms with n near the betatron tune, ν. Keeping
the leading term only, we have an approximation
y co (s) ≈
β(s)
ν|f [ν] |
ν − [ν]
cos([ν]φ(s) + χ [ν] ),
(2.16)
where [ν] is the integer closest to ν.
Eq. (2.16) shows that, unless the orbit kicks are specifically arranged, the
closed orbit deviation in a ring accelerator tends to appear as an amplitude
modulated sinusoidal function of the phase advance, with the number of peaks
around the circumference given by the integer part of the betatron tune. It also
indicates that if the betatron tune is close to an integer, a small perturbation
could cause very large orbit deviations.
Orbit errors of the beam in accelerators generally need to be corrected.
Depending on the purpose of the accelerator, the precision requirement for
orbit correction may vary. The orbit at the interaction points of colliders or at
the photon beamline source points of light sources needs to be held constant
with a high precision at the sub-micron level for hours.
Introducing the transformation ψ(s) = νφ(s), and using ds = βdψ =
βνdφ, the closed orbit, Eq. (2.11) can be rewritten as
y co (s) =
ν
β(s)
2 sin πν
φ+2π
φ
θ(ξ)β
3
2 (ξ) cos ν(π + φ − ξ)dξ.
(2.12)
Because the function θ(φ)β
3
2 (φ) is periodic (with the period 2π), it can be
Fourier expanded
θ(φ)β
3
2 (φ) =
∞
n=−∞
f n e
inφ ,
(2.13)
with the Fourier coefficients f n given by
f n =
1
2πν
θ(s)β
1
2 (s)e
−inφ(s) ds =
1
2πν
k
θ k β
1
2
k e
−inφ k ,
(2.14)
where in the last equation discrete dipole kicks are assumed, with a kick
angle θ k = θ(s k )∆s k . The Fourier coefficients in Eq. (2.14) are called integer
stopband integrals. Inserting Eq. (2.13) into Eq.(2.12), we obtain the Fourier
expansion of the closed orbit deviation
y co (s) =
β(s)
n
ν
2 f n
ν 2 − n 2 e
inφ(s) ,
=
β(s)
f 0 +
∞
n=1
2ν
2 |f n |
ν 2 − n 2 cos(nφ(s) + χ n )
,
(2.15)
where we have used f n = f
∗
−n = |f n |e
iχn . The closed orbit can be approximated with only a few Fourier terms with n near the betatron tune, ν. Keeping
the leading term only, we have an approximation
y co (s) ≈
β(s)
ν|f [ν] |
ν − [ν]
cos([ν]φ(s) + χ [ν] ),
(2.16)
where [ν] is the integer closest to ν.
Eq. (2.16) shows that, unless the orbit kicks are specifically arranged, the
closed orbit deviation in a ring accelerator tends to appear as an amplitude
modulated sinusoidal function of the phase advance, with the number of peaks
around the circumference given by the integer part of the betatron tune. It also
indicates that if the betatron tune is close to an integer, a small perturbation
could cause very large orbit deviations.
Orbit errors of the beam in accelerators generally need to be corrected.
Depending on the purpose of the accelerator, the precision requirement for
orbit correction may vary. The orbit at the interaction points of colliders or at
the photon beamline source points of light sources needs to be held constant
with a high precision at the sub-micron level for hours.