214 Beam-based Correction and Optimization for Accelerators
from the vertical dispersion as well as a significant fraction of the emittance
coming from linear coupling.
An indirect measure of the vertical emittance can be obtained through
the dependence of the Touschek scattering beam loss on the vertical beam
size. Touschek scattering is the process of electron-electron collisions in the
beam bunch that results in a large energy loss or gain on the two colliding
electrons and the subsequent loss of both particles to the momentum aperture. Beam loss due to Touschek scattering is the main mechanism that limits
the beam lifetime of low emittance, high charge beams in storage rings. The
Touschek scattering rate is proportional to the density of the electrons in the
configuration space,
1
τ T
∝
I b
σ x σ y σ z
,
(8.2)
where τ T is the Touschek lifetime, I b is the bunch current, and σ xyz are the
beam sizes in the three dimensions. Since σ x and σ z are mostly not affected
by skew quadrupole variations, any change of the Touschek loss rate by skew
quadrupoles must come through a change of the average inverse vertical beam
size,
1
σy , throughout the ring.
The Touschek loss rate can be measured by observing the beam current
change, ∆I, over a short period of time, ∆t, for a beam with a high bunch
charge. For a Touschek loss dominated beam, we have ∆I = −
I∆t
τ T
. Hence,
the objective function may be defined as
f (x) =
I
2
0 ∆I
I 2 ∆t
∝ −−
1
σ y
,
(8.3)
where I 0 is a reference current. Note the minimization of the objective function
is equivalent to minimizing the vertical emittance.
Vertical emittance minimization with beam-based optimization has been
applied to the SPEAR3 storage ring in both simulation and experiments [57,
115]. This problem is ideal for testing online optimization algorithms as it is a
real-life application with sufficient challenges in the number of knobs, the noise
level, and the parameter space complexity. In the following both simulation
and experimental tests are discussed.
8.3.1 Simulation
With a storage ring lattice model, the equilibrium distribution of the electron
beam can be obtained by calculating the one-turn transfer matrix with radiation damping included and the radiation-induced diffusion matrix integrated
throughout the ring and solving for a self-consistent second order moment
matrix [90]. The horizontal and vertical emittances can be readily calculated
from the second order moment matrix. This calculation has been implemented
in the lattice modeling code Accelerator Toolbox (AT) [114], which is used in
this simulation.
from the vertical dispersion as well as a significant fraction of the emittance
coming from linear coupling.
An indirect measure of the vertical emittance can be obtained through
the dependence of the Touschek scattering beam loss on the vertical beam
size. Touschek scattering is the process of electron-electron collisions in the
beam bunch that results in a large energy loss or gain on the two colliding
electrons and the subsequent loss of both particles to the momentum aperture. Beam loss due to Touschek scattering is the main mechanism that limits
the beam lifetime of low emittance, high charge beams in storage rings. The
Touschek scattering rate is proportional to the density of the electrons in the
configuration space,
1
τ T
∝
I b
σ x σ y σ z
,
(8.2)
where τ T is the Touschek lifetime, I b is the bunch current, and σ xyz are the
beam sizes in the three dimensions. Since σ x and σ z are mostly not affected
by skew quadrupole variations, any change of the Touschek loss rate by skew
quadrupoles must come through a change of the average inverse vertical beam
size,
1
σy , throughout the ring.
The Touschek loss rate can be measured by observing the beam current
change, ∆I, over a short period of time, ∆t, for a beam with a high bunch
charge. For a Touschek loss dominated beam, we have ∆I = −
I∆t
τ T
. Hence,
the objective function may be defined as
f (x) =
I
2
0 ∆I
I 2 ∆t
∝ −−
1
σ y
,
(8.3)
where I 0 is a reference current. Note the minimization of the objective function
is equivalent to minimizing the vertical emittance.
Vertical emittance minimization with beam-based optimization has been
applied to the SPEAR3 storage ring in both simulation and experiments [57,
115]. This problem is ideal for testing online optimization algorithms as it is a
real-life application with sufficient challenges in the number of knobs, the noise
level, and the parameter space complexity. In the following both simulation
and experimental tests are discussed.
8.3.1 Simulation
With a storage ring lattice model, the equilibrium distribution of the electron
beam can be obtained by calculating the one-turn transfer matrix with radiation damping included and the radiation-induced diffusion matrix integrated
throughout the ring and solving for a self-consistent second order moment
matrix [90]. The horizontal and vertical emittances can be readily calculated
from the second order moment matrix. This calculation has been implemented
in the lattice modeling code Accelerator Toolbox (AT) [114], which is used in
this simulation.