94 Beam-based Correction and Optimization for Accelerators
of the quadrupole strength. The beam distribution at point 1 can be given in
terms of the beam emittance and the Courant-Snyder parameters by
Σ 1 =
σ
2
1
σ 12
σ 12 σ
2
1
=
β 1 −α 1
−α 1 γ 1
.
(4.2)
Σ 1 is not affected by the quadrupole strength. From Eq. (4.2), the measured
rms beam size at point 2, σ
2
2 , is related to the integrated quadrupole gradient, k = K 1 L q (with quadrupole gradient K 1 and length L q ), and the beam
parameters at point 1 by
σ
2
2 = (L
2 σ
2
1 )k
2 + (2Lσ
2
1 + 2L
2 σ 12 )k + (σ
2
1 + 2Lσ 12 + L
2 σ
2
1 ).
(4.3)
By fitting σ
2
2 to a quadratic function of k and identifying the coefficients, the
second order moments at point 1 can be determined, which in turn can be
used to calculate the emittance and Courant-Snyder parameters.
The quadrupole scan method can determine the beam profile and optics
functions at one location of a one-pass system. The measured optics functions
can be used to calculate the required quadrupole strength adjustments to meet
the desired optics matching conditions. It is not suitable for rings because
in a ring the beam profile at the entrance of the quadrupole is changed by
its own strength and is thus not fixed. Equilibrium beam sizes in electron
storage rings can be measured with pinhole cameras that image the beam
profile in a dipole magnet through synchrotron radiation, which can be used
to determine the beta functions at the radiation source points (up to a scaling
constant). However, because beam profile measurements are available only at
few locations, this method cannot be used to measure the global linear optics.
4.1.2 Sampling linear optics with beam orbit
Another way to sample the linear optics is through the beam trajectory or
orbit. The transfer matrix directly relates the phase space coordinates of a
single particle at two locations through
X 2 = M 21 X 1 ,
(4.4)
where X = (x, x
)
T or (y, y
)
T . The BPMs measure the average position of all
particles in the beam, i.e., the beam center.
When the beam is kicked, in some cases all particles in the beam move with
the beam center, without changing the phase space distribution around the
beam center. In such cases, the beam center behaves like a single particle. For
example, one such case is when the strength of a corrector magnet is changed
in a transport line or a storage ring.
The beam may also be kicked by a kicker or a pinger in a storage ring during
only one pass. After the kick all particles in the beam will start to oscillate.
Because the betatron tunes of the particles may differ, due to amplitude or
chromatic detuning, the particles will gradually move out of phase. The beam
of the quadrupole strength. The beam distribution at point 1 can be given in
terms of the beam emittance and the Courant-Snyder parameters by
Σ 1 =
σ
2
1
σ 12
σ 12 σ
2
1
=
β 1 −α 1
−α 1 γ 1
.
(4.2)
Σ 1 is not affected by the quadrupole strength. From Eq. (4.2), the measured
rms beam size at point 2, σ
2
2 , is related to the integrated quadrupole gradient, k = K 1 L q (with quadrupole gradient K 1 and length L q ), and the beam
parameters at point 1 by
σ
2
2 = (L
2 σ
2
1 )k
2 + (2Lσ
2
1 + 2L
2 σ 12 )k + (σ
2
1 + 2Lσ 12 + L
2 σ
2
1 ).
(4.3)
By fitting σ
2
2 to a quadratic function of k and identifying the coefficients, the
second order moments at point 1 can be determined, which in turn can be
used to calculate the emittance and Courant-Snyder parameters.
The quadrupole scan method can determine the beam profile and optics
functions at one location of a one-pass system. The measured optics functions
can be used to calculate the required quadrupole strength adjustments to meet
the desired optics matching conditions. It is not suitable for rings because
in a ring the beam profile at the entrance of the quadrupole is changed by
its own strength and is thus not fixed. Equilibrium beam sizes in electron
storage rings can be measured with pinhole cameras that image the beam
profile in a dipole magnet through synchrotron radiation, which can be used
to determine the beta functions at the radiation source points (up to a scaling
constant). However, because beam profile measurements are available only at
few locations, this method cannot be used to measure the global linear optics.
4.1.2 Sampling linear optics with beam orbit
Another way to sample the linear optics is through the beam trajectory or
orbit. The transfer matrix directly relates the phase space coordinates of a
single particle at two locations through
X 2 = M 21 X 1 ,
(4.4)
where X = (x, x
)
T or (y, y
)
T . The BPMs measure the average position of all
particles in the beam, i.e., the beam center.
When the beam is kicked, in some cases all particles in the beam move with
the beam center, without changing the phase space distribution around the
beam center. In such cases, the beam center behaves like a single particle. For
example, one such case is when the strength of a corrector magnet is changed
in a transport line or a storage ring.
The beam may also be kicked by a kicker or a pinger in a storage ring during
only one pass. After the kick all particles in the beam will start to oscillate.
Because the betatron tunes of the particles may differ, due to amplitude or
chromatic detuning, the particles will gradually move out of phase. The beam