Also at ]The University of Melbourne, Australia

Also at ]The Australian Nuclear Science and Technology Organisation (ANSTO)

# Design of a large energy acceptance beamline using Fixed Field Accelerator
optics

A. F. Steinberg  [<EMAIL_ADDRESS>R. B. Appleby  The
University of Manchester, Manchester, UK  Cockcroft Institute, Warrington, UK
J. S.L. Yap  S. L. Sheehy  [  The University of Melbourne, Parkville,
Australia

###### Abstract

Large energy acceptance arcs have been proposed for applications such as
cancer therapy, muon accelerators, and recirculating linacs. The efficacy of
charged particle therapy can be improved by reducing the energy layer
switching time, however this is currently limited by the small momentum
acceptance of the beam delivery system ($<$$\pm 1\text{\,}\mathrm{\char
37\relax}$). A ‘closed-dispersion arc’ with a large momentum acceptance has
the potential to remove this bottleneck, however such a beamline has not yet
been constructed. We have developed a design methodology for large momentum
acceptance arcs with Fixed Field Accelerator optics, applying it to a
demonstrator beam delivery system for protons at
$0.5$–$3.0\text{\,}\mathrm{MeV}$ ($\pm 42\text{\,}\mathrm{\char 37\relax}$
momentum acceptance) as part of the TURBO project at the University of
Melbourne. Using realistic magnetic fields, a beamline has been designed with
zero dispersion at either end. An algorithm has been devised for the
construction of permanent magnet Halbach arrays for this beamline with
multipole error below one part in $10^{4}$, using commercially available
magnets. The sensitivity to errors has been investigated, finding that the
delivered beam is robust in realistic conditions. This study demonstrates that
a closed-dispersion arc with fixed fields can achieve a large momentum
acceptance, and we outline future work required to develop these ideas into a
complete proof-of-principle beam delivery system that can be scaled up for a
medical facility.

††preprint: Physical Review Accelerators and Beams

##  I Motivation

Over the course of the past thirty years, charged particle therapy has become
a mature treatment modality, with over $100$ proton therapy facilities
worldwide and $300\,000$ patients treated up to $2022$ [1]. Charged particle
therapy is advantageous in many cases compared to conventional cancer
treatment with X-rays [2, 3, 4]: whereas an X-ray beam deposits dose following
an approximately exponential fall-off, protons (usually between
$70$–$250\text{\,}\mathrm{MeV}$) deliver a small entrance dose that rises
sharply at a point, then rapidly drops to zero [5]. The depth of this
characteristic ‘Bragg peak’ increases as a function of the beam energy. In
practice, the particle beam energy is varied to create a ‘spread-out Bragg
peak’ covering the treatment volume, giving many discrete ‘energy layers’ [6].

Realising the full potential of particle therapy has taken time, with most
improvements arising from technological advances. One example of a significant
change to enhance tumour conformality was the move away from using passively
scattered beams to cover the tumour volume to pencil-beam scanning, enabling
intensity modulated proton therapy [5]. In addition, the footprint of proton
therapy facilities has decreased, from the dual synchrotron rings with
$130\text{\,}\mathrm{m}$ circumference at HIMAC [7] down to recent proposals
for cyclotrons enabling single-room systems. In this time, treatment delivery
methods have also varied, from fixed beamlines to gantries which rotate around
a stationary patient to ensure the scanned beam can maximise dose to the
treatment volume while minimising it elsewhere.

Recent studies have suggested that proton therapy can be made more efficient
and effective by reducing the time taken to switch beam energies [8, 9, 10,
11]. This would reduce the impact of interplay effects, where patient motion
relative to the beam during treatment requires mitigation by methods such as
treatment gating while the patient holds their breath, beam rescanning, and
increasing treatment margins to improve dose uniformity [12, 13, 14]. By
lowering the energy layer switching time, the need for dose repainting may be
reduced [15], however this has not been well-studied experimentally as the
minimum time to switch energy layers is limited by the beam delivery system.
In addition, rapid energy switching would shorten treatment times, improve the
patient experience, and increase patient throughput, though overall treatment
times would still be dominated by patient setup [8]. A potential future
improvement to charged particle therapy may be realised if the beam delivery
time can be reduced even further, such that a dose
$\gtrsim$$40\text{\,}\mathrm{Gy}$ is delivered in
$\lesssim$$100\text{\,}\mathrm{ms}$. Studies suggest that the ‘FLASH effect’
in this regime may enable dose escalation and hypofractionation without
further damage to healthy tissue [16, 17], however current beam delivery
systems have insufficient energy acceptance to enable active beam scanning
with energy layers at FLASH dose rates. In proton therapy, this has lead to
‘shoot through’ or transmission beam studies which lose the conformality
provided by the Bragg peak [18]. As such, enabling rapid beam delivery by
minimising the energy layer switching time is a clear goal toward improved
particle therapy systems in future, however new technologies will be required.

Current beam delivery systems typically have a momentum acceptance $\leq
1\text{\,}\mathrm{\char 37\relax}$, [19] limited by their linear magnetic
fields and small beampipe aperture: to vary the beam energy, all the magnets
in the beamline must have their fields ramped synchronously as the beam
momentum changes, which gives rise to the energy layer switching time. An
ideal beamline would be able to deliver any clinical proton beam energy
without adjustment: for the case of $70$–$250\text{\,}\mathrm{MeV}$ protons,
this requires a momentum acceptance greater than $\pm
32.8\text{\,}\mathrm{\char 37\relax}$. Such a beam delivery system would
enable energy variation as rapid as the proton source can provide. If the
momentum acceptance can be increased to $\pm 40\text{\,}\mathrm{\char
37\relax}$ (giving a maximum energy of $330\text{\,}\mathrm{MeV}$), the
beamline would also be suitable for proton CT [20]. All energies must enter
and exit the beam delivery system at the same point in the transverse plane to
ensure that no magnet adjustments are required: the beamline must comprise a
‘closed-dispersion arc’.

A closed-dispersion arc with a very large energy acceptance would have many
applications beyond medicine, in cases where beam transport over a large
momentum range without varying the magnet settings would be useful. One
example is muon accelerators, where rapid acceleration is vital to prevent
beam losses due to decay: without dispersion-free regions, the transverse size
of the RF may prove prohibitively large. Another application is in
recirculating linacs, where the energy gain per turn is sufficiently large
that the magnet ramping speed becomes a limiting factor: separation and
recombination of the different energy beams could be performed without ramping
the magnets, enabling continuous operation. This was demonstrated by CBETA
[21] using a dedicated splitter section. A more ambitious proposal might be
for a future collider, where regions with zero dispersion are vital to
maximise luminosity: this is achieved using dedicated splitter and combiner
magnets for eRHIC [22]. A closed-dispersion arc would enable beam transport of
a continuous energy spectrum, unlike the cases of CBETA and eRHIC which are
limited to discrete energies.

At the University of Melbourne, the Technology for Ultra-Rapid Beam Operation
(TURBO) project seeks to demonstrate technologies for removing the energy
layer switching time bottleneck associated with the beam delivery system. This
will be achieved using a scaled-down beamline at low energies with an
equivalent momentum range to a clinical beam delivery system. By using Fixed
Field Accelerator (FFA) optics, where the magnetic field varies spatially
rather than temporally, it is possible to deliver the full range of energies
much more rapidly than is currently possible. Though there have been many
proposals for medical FFAs, this would be the first demonstration of an
accelerator for medical applications using fixed fields including zero
dispersion regions.

##  II Fixed Field Accelerators with Zero-Dispersion Regions

We begin with an overview of the core concepts of FFAs, including the design
rationale, tracking codes, and examples that have been constructed. This leads
into a summary of FFA proposals with zero-dispersion regions. These ideas are
then synthesised and developed into our new design methodology, as applied to
the TURBO project.

###  II.1 Review of FFAs

In a particle accelerator, the transverse bending strength is proportional to
the local magnetic field and the focusing strength is proportional to its
gradient. Both are inversely proportional to the beam momentum, so they are
kept constant in most accelerators by ramping up magnetic fields synchronously
with the rising beam momentum. The result is that all energies follow the same
closed-orbit around the accelerator, but the momentum acceptance is small,
usually less than $\pm 1\text{\,}\mathrm{\char 37\relax}$, and the energy
switching time is limited by magnet hysteresis. An alternative would be to
vary the magnetic field in space, with each energy having a unique trajectory:
this is the core concept behind FFAs. This scheme allows for rapid
acceleration and a large momentum acceptance, at the expense of larger and
more complex magnets.

Fixed Field Accelerators with sector magnets were first proposed in 1956,
around the advent of the strong-focusing synchrotron [23, 24]. FFAs have some
advantages over synchrotrons or cyclotrons, such as their large energy
acceptance and rapid acceleration capabilities. However, difficulties with the
beam dynamics and complicated magnet designs have meant that cyclotrons,
synchrotrons, and linacs have dominated since the $1960$s. More recently, FFAs
have experienced a resurgence with the advent of computer-aided design
techniques, leading to FFA proposals for muon colliders [25], LHC successors
[26], and often, as the main accelerators for medical facilities [27, 28, 29].
They have also been proposed for beam transport and delivery for medical
applications, the subject of this work.

In general, the beam dynamics in an FFA is a function of momentum, as the
focusing strength varies with energy. In addition, the particle trajectory is
a function of momentum, meaning each energy has a unique closed orbit and
experiences a different magnetic field. It is possible to enforce a phase
advance that is constant by definition by having a magnetic field that follows
a ‘scaling law’ [30]. Today, there are many variations of the ‘scaling law’,
depending on whether the beam moves in the horizontal or vertical plane as
momentum varies, and whether the beamline is straight or follows a circular
path: the best known is for the horizontal arc, where the magnetic field in
the horizontal midplane is

$B=B_{0}\left(\frac{r}{r_{0}}\right)^{k}\mathcal{F}\left(\theta\right),$ (1)

where $r_{0}$ and $B_{0}$ are a reference radius and field respectively, the
field index $k$ gives the power law dependence of the field, and $\mathcal{F}$
describes how the magnetic fringe and body fields vary around the machine
circumference. The scaling law ensures constant tunes but requires that all
closed orbit trajectories are scale enlargements of one another, restricting
the accelerator optics. This leads to a large difference between the
trajectories of low and high energy beams (known as orbit excursions). As a
consequence, the dispersion in a scaling FFA is a nonzero constant. Strictly
speaking, the scaling law is only valid where the accelerator has sector-
shaped magnets following the curvature defined in Equation 1: if rectangular
magnets are used, the accelerator can only be approximately scaling. However,
the difference between exactly and approximately scaling FFAs has been shown
to be negligible where the curvature is sufficiently low [27]. FFAs which do
not adhere to the field in Equation 1 are generally referred to as
‘nonscaling’.

Most accelerator modelling codes are not suitable for modelling FFAs, as they
assume small angles and energy variations to simplify beam tracking. There are
several specialised codes that are able to reproduce accurate dynamics for
FFAs, each with different capabilities and weaknesses. In this work, Zgoubi
[31] is used for particle tracking, as it enables sophisticated control over
the lattice parameters and collective effects can be neglected in this case.
The Python interface Zgoubidoo [32] is used to simplify the simulation inputs,
enabling the use of optimisation and analysis tools more sophisticated than
those included in a pure Zgoubi environment.

Several FFAs have been successfully constructed and operated. In Japan, a
scaling FFA complex has been in operation for many years [33, 34], used for
basic accelerator science experiments as well as investigations into
Accelerator Driven Subcritical Reactors, and more recently including energy
recovery with an internal target for radioisotope production [35]. Outside of
Japan, there have been two nonscaling FFAs: EMMA [36] demonstrated that rapid
resonance crossing in nonscaling FFAs was not detrimental to beam quality, and
CBETA [21] showed that the fast acceleration and large momentum acceptance
achievable by FFAs works well for Energy Recovery Linacs.

At present, there are four future FFA projects other than TURBO, at various
stages of development. In the UK, FFA options are being investigated for a
future neutron spallation source as part of the ISIS-II programme, and a high-
intensity proton driver FFA will be demonstrated with the Front End Test Stand
(FETS) at RAL [37]. At Jefferson Lab, the CEBAF upgrade project may use an FFA
to increase the maximum energy from $12\text{\,}\mathrm{GeV}$ to
$22\text{\,}\mathrm{GeV}$ by reusing the same tunnel space but increasing the
energy acceptance in the recirculating arcs [38]. For medical applications, a
heavy ion research facility is being investigated in America [39], proposing
to use an isochronous ‘racetrack’ FFA, and an FFA with spiral sectors has been
proposed for the second stage of the LhARA accelerator complex [40] to
accelerate protons up to $127\text{\,}\mathrm{MeV}$ for radiobiological
research. The principal focus of these projects is on beam acceleration,
whereas the TURBO project is investigating beam transport and delivery in more
detail.

###  II.2 Closed-Dispersion Arc Considerations

One of the major challenges with using an FFA arc for particle therapy is that
the particle trajectory is a function of energy, unlike in synchrotrons where
the ramping fields ensure that the ideal particle path does not change. To
enable rapid energy variation, all particle trajectories must converge (in
both position and angle) at either end of the beamline. In the linear matrix
formalism, the position of a particle in the horizontal plane with momentum
deviation $\delta$ is given by [41]

$\begin{pmatrix}x\\\ x^{\prime}\\\
\delta\end{pmatrix}_{2}=\begin{pmatrix}R_{11}&R_{12}&D\\\
R_{21}&R_{22}&D^{\prime}\\\ 0&0&1\end{pmatrix}_{(2|1)}\begin{pmatrix}x\\\
x^{\prime}\\\ \delta\end{pmatrix}_{1},$ (2)

where $(D,D^{\prime})$ is the dispersive element. To comprise a ‘closed-
dispersion arc’, both the dispersion and its derivative must be zero at both
ends of the beamline.

There are many dispersion suppressor schemes, including half or missing-bend
optics [42] where the transition to a zero-dispersion section can be completed
smoothly by insertion of matching cells: although these methods have been
developed for synchrotrons, there is no reason why the first order results
should not also apply to FFAs. In the half-bend scheme, the requirements are
that: the bend angle of the suppressor is half that of the normal optics; the
lattice is periodic; $D^{\prime}$ is zero at the interface between the normal
and suppressor optics; and the dispersion suppression region must have a phase
advance of $(2n+1)\pi$, where $n$ is an integer.

Large energy acceptance beam transport systems with fixed fields have been
explored previously. If the beam to be transported is composed of several
discrete energies, it is possible to use splitter magnets which use dipoles
and quadrupoles to separate discrete energies and manipulate path lengths and
lattice parameters to allow for specific matching for each one. This was
successfully demonstrated in CBETA [21] and has been proposed for CEBAF [43],
however it would not be suitable for particle therapy as the energy range is
continuous. Another technique employed by CBETA to bring the beam dispersion
to zero is to create an ‘adiabatic transition’, where the dipole gradient
around an arc is gradually reduced following a sigmoid curve, such that the
final lattice has no bending and the overall beam has almost zero dispersion.
This has been proposed for particle therapy with a potential design [44],
however the need for the bending strength variation to be slow along the beam
path length requires a long beamline with many unique magnets, which would be
more expensive and potentially unreliable in a clinical environment.

Other methods of creating a closed-dispersion arc have been devised, that do
not use adiabatic transitions or discrete splitter arcs. Where beamline
footprint reduction is paramount, the large bending angle per magnet has been
found to require the addition of sextupolar and higher order multipoles to
ensure strong focusing over the full range of momenta [45, 46]. In these
cases, to keep the beam excursion range small, the momentum acceptance is
generally limited to less than $\pm$$15\text{\,}\mathrm{\char 37\relax}$.
Conversely, in designs where the bending strength is reduced, combined-
function dipole/quadrupole magnets with fine-tuned length, strength, and edge
angles can transport all energies [47, 48, 49]. However, such delicate fine-
tuning may be sensitive to errors, and the difficulty of building a lattice
with many unique magnets and a high packing factor has not been explored. The
use of scaling FFA optics has also been considered for beam delivery at a
medical facility [50, 51], where the upsides of the straightforward beam
optics are counteracted by limitations imposed by the scaling law, such as
large beam excursions. By combining nonscaling multipolar fields with low-
dispersion beamlines, the energy acceptance can be increased while limiting
the transverse beam excursion range [52].

With a scaling FFA, the constant tune for all energies appears to make the
design of a dispersion suppressor with a large energy acceptance no different
to the synchrotron case. However, particle trajectories in a scaling FFA are
constrained by the natural dispersion which is required to keep the focusing
strength constant over the full range of energies: even though the optics
imply zero dispersion, the closed-orbit trajectories do not. Previous work has
indicated that scaling FFA optics can provide a good starting point for a
dispersion suppressor [52], however it is limited by the chromatic aberrations
arising from particles not following the periodic trajectories. By breaking
the scaling law, the effectiveness of the dispersion suppressor may be
improved [53], although it can lead to severe distortions in the final beam
where the CS parameters are not correctly matched.

We define momentum acceptance $\alpha_{p}$ as

$\alpha_{p}=\frac{p_{\text{max}}-p_{\text{min}}}{p_{\text{max}}+p_{\text{min}}},$
(3)

such that the reference momentum $p_{0}$ is given by the middle of the
momentum range, and related to the maximum and minimum momenta by

$\displaystyle p_{\text{min}}$ $\displaystyle=p_{0}(1-\alpha_{p}),$
$\displaystyle p_{\text{max}}$ $\displaystyle=p_{0}(1+\alpha_{p}).$

As only protons are considered in this study, the rigidity and momentum
acceptances are equivalent. For the TURBO project, the momentum acceptance is
$\pm 42\text{\,}\mathrm{\char 37\relax}$.

##  III Halbach Arrays without Custom Magnets

Fixed Field Accelerators utilise magnets where the fields do not need to vary
in time. These magnets are usually combined function, as FFAs require a high
packing factor to achieve sufficient focusing. FFAs thus break both the
assumption of separated-function magnets and of ramping fields, in contrast to
most accelerators where the separated function magnetic fields of dipoles,
quadrupoles and higher order correctors must be ramped simultaneously as the
particle energy increases. In all cases, if the required fields are below
$1.8\text{\,}\mathrm{T}$ this can be achieved with electromagnets, otherwise
superconducting magnets may be required. An advantage of using static magnetic
fields is to enable a third option: so long as the required fields are
sufficiently low, permanent magnets can be used to create the required
multipoles. Here we further consider the use of commercially available
permanent magnets to reduce overall costs and waste, while allowing rapid
prototyping and design.

Permanent magnet arrays have been employed in accelerators for several decades
[54]. With the advent of rare earth magnets in the 1980s, Halbach proposed
that correctly oriented blocks of permanent magnet material could be used to
produce solenoidal, helical, and multipolar fields [55]. They are often used
in light sources, wigglers and undulators made with permanent magnets can be
moved together or apart to tune the synchrotron radiation [56]. Halbach
quadrupoles are also used for linacs, where the energy at a given point does
not change [57, 58]. A recent innovation for conventional synchrotrons is a
tunable permanent-magnet: the ZEPTO project allows for varying the field
strength of permanent magnets by moving the magnetic material relative to an
iron yolk [59]. For Fixed Field Accelerators, magnet arrays can be constructed
without requiring momentum-dependent tunability, allowing for more specialised
permanent magnet designs.

By arranging blocks of permanent magnet material, it is possible to produce
arbitrary combined function multipoles. This was trialled with CBETA [60],
which required combined function dipole/quadrupole magnets in some sections of
the return arc. For each of these magnets, $16$ custom trapezoidal wedges of
NdFeB were used to approximate the desired field: each wedge required a unique
size and magnetisation direction, which dominated the cost of the magnet
arrays. It was found that multipole errors arising from assembly could be
corrected with thin iron rods, inserted into the magnet bore following a
correction algorithm. A similar permanent magnet solution is being explored
for the CEBAF upgrade [61].

A different method of designing permanent magnet arrays is being investigated
for the TURBO project. Rather than using expensive custom permanent magnet
pieces, commercially available magnet blocks can be combined to produce the
desired fields to sufficient accuracy. As well as lowering costs, using many
identical magnetic blocks ensures that all the magnetic material can be
readily reused for other projects, which would not be possible if each magnet
required custom-designed pieces. Although this method requires a larger total
volume of magnetic material – due to the gaps in the magnet introduced by the
segmentation – we estimate that the total cost is an order of magnitude less
than for custom blocks. One concern is that the field quality of the magnet
array may be worse than with custom magnets: in Section V, we show that the
field quality is sufficient for the demonstrator beamline where only a single
pass through each magnet is necessary.

We demonstrate the magnet design algorithm with a modified version of the
magnet denoted ‘BDT1’ in [62], which was developed for the CBETA return arc.
This is a combined function dipole/quadrupole with strengths of
${-}0.1002\text{\,}\mathrm{T}$ and
$11.1475\text{\,}\mathrm{T}\mathrm{/}\mathrm{m}$. The aperture radius of the
array used here is approximately $45\text{\,}\mathrm{mm}$, with a good field
region with a $15\text{\,}\mathrm{mm}$ radius. The magnets used for CBETA were
$122\text{\,}\mathrm{mm}$ long, however we have shortened our example to
$100\text{\,}\mathrm{mm}$ to match the requirements of the TURBO project. The
individual magnet blocks are modelled as
$$12.7$\times$12.7$\times$100\text{\,}\mathrm{m}\mathrm{m}^{3}$$ cuboids with
a remanent field $B_{r}$ of $1.3\text{\,}\mathrm{T}$, as this matches
commercially available products such as those available from [63]. Magnet
modelling was performed using the Python package Magpylib [64], which allows
for fast optimisation but does not include demagnetisation in the presence of
strong B-fields.

###  III.1 Magnet Array Design Methodology

To generate a magnetic multipole of order $N$ with idealised bulk permanent
magnet material, the continuously-varying magnetisation in plane-polar
coordinates $(\rho,\phi)$ follows

$\vec{M}=M_{0}\left[\cos\left(N\left(\phi-\phi_{0}\right)\right)\hat{\rho}+\sin\left(N\left(\phi-\phi_{0}\right)\right)\hat{\phi}\right],$
(4)

where reference angle $\phi$ is $\pi/2$ or $3\pi/2$ for normal multipoles, or
a multiple of $\pi$ for skew multipoles [55]. The field strength inside the
magnet bore is determined by the inner and outer radii of the cylinder
($r_{i}$ and $r_{o}$ respectively) and the magnet remanence field $B_{r}$. For
infinitely long N-poles, the field inside the bore is given by

$\displaystyle\vec{B}$
$\displaystyle=B_{r}\ln\left(\frac{r_{o}}{r_{i}}\right)\hat{y}$ $\displaystyle
N=1,$ (5) $\displaystyle\vec{B}$
$\displaystyle=B_{r}\left(\frac{\vec{x}+\vec{y}}{r_{i}}\right)^{N}\frac{N}{N-1}\left[1-\left(\frac{r_{i}}{r_{o}}\right)^{N}\right]$
$\displaystyle N\geq 2,$ (6)

which suggests that for fixed cylinder dimensions, multipole strength
decreases with increasing order.

Figure 1: Combining dipole and quadrupole fields to produce a combined-
function magnet. The B-field is shown within the dashed red circles, and the
ideal magnetisation vectors are given by the exterior grey streamlines. The
equipotentials of the magnetisation vectors are shown in blue: for pure
N-poles they are circular, but for the combined-function case they are
distorted. The distorted streamlines and contours in the combined-function
magnet are from addition of the magnetisation vectors.

Using Equations 4–6, the magnetisation required to produce any individual
N-pole can be calculated. However, it is not immediately apparent how to
combine the magnetisation vectors to create an arbitrary combined-function
multipole. From the design of superconducting magnets [65], it is known that
the field for a multipole of order $N$ is produced inside a longitudinal
$\cos(N\phi)$ current distribution. In addition, the field outside this
current ring corresponds to the magnetisation that would produce the N-pole,
due to the magnetic field reciprocation theorem [66, 67]. As such, by
superposing a series of $\cos(N\phi)$ current rings, we can simultaneously
produce any combined-function magnetic field and the ideal continuously
varying magnetisation that would produce it. From this we can derive a layout
of permanent magnet segments to produce the correct magnetisation. An
illustrative example is shown in Fig. 1.

Our design procedure is as follows: we first use the cosine-theta current
rings to produce the internal fields, normalised to the desired strengths; we
then apply linear superposition of magnetic fields to combine the fields
outside the rings, which is the dual of the desired magnetisation. In Fig. 1,
we see that the magnetisation follows Equation 4 for the N-poles, and is a
linear combination for the combined-function magnet. The equipotential lines
indicate the outer distance of magnetic material required to produce a
magnetic field of a desired strength. An advantage of this method over just
using Equation 4 is that it calculates both the field and the magnetisation
simultaneously, allowing both to be used in optimisation. The next challenge
is to find an arrangement of permanent magnets which approximates the ideal
solution found here.

The goal of the optimisation algorithm is to reproduce a set of multipoles
using an array of discrete magnet blocks. This is quantified by the field
quality of a magnet, giving the difference between the design and achieved
multipoles. Field quality is defined by the Fourier expansion of the radial
magnetic field at a radius $r$ [68],

$\displaystyle B_{n}$
$\displaystyle=\frac{1}{\pi}\int_{-\pi}^{\pi}B_{r}(r,\phi)\sin(N\phi)d\phi,$
(7) $\displaystyle A_{n}$
$\displaystyle=\frac{1}{\pi}\int_{-\pi}^{\pi}B_{r}(r,\phi)\cos(N\phi)d\phi,$
(8)

where the $B_{N}$ and $A_{N}$ correspond to the normal and multipole
components respectively. In many cases, these are normalised relative to a
‘main’ multipole component (usually quadrupole). The stray multipole terms in
an accelerator are usually required to be less than one part in ${10}^{4}$ of
the main component.

For both optimiser stages described here, we define $B_{N,\text{opt}}$ as the
goal multipoles for some magnet, and $B_{N,i}$ as the multipoles from a
possible solution. The objective function $f$ to be minimised is given by

$f=\sqrt{\sum_{N}\left(B_{N}-B_{N,\text{opt}}\right)^{2}+\left(A_{N}-A_{N,\text{opt}}\right)^{2}},$
(9)

which is constrained by keeping all magnets outside the beampipe, and to
prevent intersections. Multipoles are calculated at the longitudinal midpoint
of the magnet at the outer radius of the good field region
$(r=R_{\text{good}})$.

In cases where the discrete magnet blocks are not able to fully reproduce the
desired set of multipoles, it may be preferred to use the midplane magnetic
field as the optimisation objective. This is particularly useful where the
horizontal beam excursion is much larger than the vertical size of the beam,
as is often the case in FFAs. The multipole error inside the good field region
may increase if the optimisation objective is the midplane field, however the
greatest error is beyond the vertical beam extent. The midplane field error to
be optimised is

$\left.\sum_{i=0}^{p}\bigg{(}B_{y}(x_{i})-B_{y,\text{opt}}(x_{i})\bigg{)}^{2}\right|_{\shortstack{y=0\\\
z=0}},$ (10)

where ${x_{i}}$ spans the good field region, with $x_{0}=-r_{\text{good}}$,
$x_{p}=r_{\text{good}}$. The horizontal B-field can be neglected as the
enforced symmetry of the magnet arrangements keeps skew multipoles small,
allowing us to exclude $B_{x}$ in this calculation.

The optimal arrangement of magnetic blocks, to first order, will follow the
contour lines shown in Fig. 1, oriented such that their magnetisation follows
the local field direction. Our algorithm approximates this by placing magnet
blocks in concentric layers, and removing any that lie outside the ultimate
contour defined by some ‘reference potential’. An optimisation algorithm – in
this case, simulated annealing – then varies the radii of the layers and the
value of the reference potential, minimising the objective function. The
number of magnets per layer is maximised to ensure the strongest possible
B-field. The first step in the optimisation process is shown in Fig. 2.

Figure 2: The first stage in the magnet optimisation algorithm, where the
variables are the radii for magnet placement and the outermost contour from
Fig. 1. Magnets are transparent if they are outside the reference contour:
they are excluded from the field calculation. The good field region and
reference radius for multipole calculations are shown.

Although this initial optimisation is usually good at finding an approximate
solution for the magnet placement, it is not always successful. One issue can
be attempting to use too few or too many magnet layers: if there aren’t enough
magnet layers, it will not be possible to make a strong enough magnetic field
inside the magnet bore; if there are too many rings, the algorithm execution
time can become prohibitive, and the extra search space makes it difficult to
converge on a good result. Assuming the dipole term is dominant, the
approximate number of magnet rings can be calculated using Equation 5, and can
be fine-tuned if the algorithm fails to converge.

The second optimisation step searches for a minimum in the vicinity of the
solution from the first stage. For every magnet, the horizontal and vertical
positions can be varied, as well as the rotation angle. This significantly
increases the number of optimisation variables, and as such, the second
optimisation stage takes longer to execute than the first. An example array
from the second stage of optimisation is shown in Fig. 3. The number of
variables is reduced by noting the midplane symmetry of the B-fields required
for accelerator applications: only the magnets in the ‘top-half’ of the magnet
array need to be optimised, as the bottom half can be found by mirroring
magnets through the array midplane and reflecting each magnet about the
vertical axis. In addition, the magnet positions and orientations are
constrained to prevent non-physical overlaps, with some extra space given to
ensure that the magnet mount will be feasible from an engineering perspective.

Figure 3: The second magnet optimisation step. The result from the first
stage, used as the initial points, is shown in the background. The good field
region (shaded blue region) and reference radius (dashed blue circle) are
indicated for comparison with Fig. 2.

After the second algorithm stage, the solution can be quite different to the
first arrangement as shown in Fig. 2, due to the gaps introduced between the
magnets. Despite this, we see in Fig. 4 that the magnitude of the error
multipoles has gone down significantly during the second optimisation stage.
It is not possible to say whether the solution displayed in Fig. 3 is globally
optimal given our constraints, as the parameter space of possible magnet
arrangements is too large to fully explore.

Figure 4: Pole strengths after both optimisation stages, relative to the ideal
quadrupole value. The second stage brings all undesired multipoles below one
part in $10^{4}$. Only normal poles are shown, as skew poles are negligible
due to symmetry.

The magnetic field produced by this magnet array is explored further in Fig.
5. Even though the longitudinal magnet edges only extend to $\pm
50\text{\,}\mathrm{mm}$, the fringe fields extend far beyond this limit, with
a small residual field up to $100\text{\,}\mathrm{mm}$ beyond the magnet ends.
These long fringes are a consequence of the large inner bore radius relative
to the magnet length: in a closed-dispersion arc with long magnets or a small
magnet bore, the fringe fields could be significantly reduced. In the
transverse plane at the magnet centre, the desired multipolar fields are
accurately reproduced, although the field error grows at larger radii.
Interestingly, we see in Fig. 5e) and 5f) that the smallest error is at a
radius of approximately $15\text{\,}\mathrm{mm}$: this corresponds to the
radius of the ‘good field region’ set earlier, suggesting that the judicious
choice of the good field region is important to ensure the optimisation
routine is effective.

Figure 5: Magnetic fields achieved with our optimised magnet array. The black
and red circles in the lower plots show the magnet array inner radius and the
internal reference radius respectively. (a, b) Magnetic field strength in the
horizontal midplane along the longitudinal axis. (c, d) Magnetic field in the
transverse plane measured at the centre of the magnet. (e, f) Error in
achieved magnetic field at the magnet centre, relative to the ideal field.

###  III.2 Modelling Magnets for Tracking Studies

During the beam optics design phase, we must construct a representation of the
magnetic field that is suitable for rapid calculations and optimisation.
Though it is possible to use a full 3D fieldmap in Zgoubi, it is not practical
for all design simulations as each iteration would require creation and
optimisation of a new magnet array. In addition, particle tracking through a
full fieldmap in Zgoubi is generally slower than using in-built components,
due to the extra computation required for interpolation.

Magnetic elements in Zgoubi can be modelled using multipole coefficients,
specified by the field at the magnet inner bore of each contributing
multipole. To account for the magnetic field beyond the magnet edges, Zgoubi
uses the Enge-type fringe fields [69]. Zgoubi also has options available to
modify the fringe fields as a function of the horizontal offset from the
magnet centre, however this is not required for this model as the magnet bore
is cylindrical. The Enge fit works well in our case, as seen in Fig. 5a),
confirming that the Enge model is suitable for Halbach arrays. The Enge
coefficients are found with a least-squares fit, measured along the
longitudinal axis of the magnet. Although the fitting parameters will be
different for each magnet, it is reasonable to assume that they are all
approximately the same for the initial modelling of the TURBO beamline and
varying them in subsequent iterations.

As the magnet packing factor for the TURBO beamline will be large, the fringe
fields will overlap between magnets. This is not possible using most Zgoubi
elements, however overlapping fringes are an option when the DIPOLES element
is used. Although this is intended for modelling sector magnets, it was
demonstrated for the PAMELA project [70] that the sectors could be
‘straightened’ by scaling up the effective radius, and reducing the effective
angular extent by the same amount. This enables the use of straight magnets
with overlapping fringe fields in Zgoubi, so long as no more than $5$ magnets
are required per section. For the TURBO project, we employ this method to
model cells with three magnets, assuming that the fringe field overlap between
cells is negligible.

###  III.3 Field Sensitivity to Magnet Block Errors

It is important to consider how errors in the individual magnet blocks
contribute to field imperfections in the overall array. These errors can be
from misplacements of the individual magnets in their mount, or from
misalignments of the magnetisation axis from the expected direction. Very
large errors, such as a full $90\text{\,}\mathrm{\SIUnitSymbolDegree}$
rotation of an individual magnet block, should be discovered during
fabrication and as such only small errors that are difficult to correct are
discussed here.

We define a Figure of Merit (FoM) based on Equation 9, modified to

$\sqrt{\sum_{N=1}^{15}\left(b_{N}-b_{N,\text{mag}}\right)^{2}+\left(a_{N}-a_{N,\text{mag}}\right)^{2}},$
(11)

where $(b_{N},a_{N})$ represent the normalised $N$-order normal and skew
multipoles from a given configuration, and
$(b_{N,\text{mag}},a_{N,\text{mag}})$ are found from our stage 2 output from
Section III.1. In this case, all the values are normalised to the quadrupole
component of the ideal magnet. Multipole values are normalised in the same way
as in Fig. 4, such that the normal quadrupole coefficient $b_{2}=$ ${10}^{4}$.
For CBETA, the average FoM was $41.09$ before tuning [62], although values
over $100$ were recorded in some cases. Our FoM is expected to be slightly
larger than the CBETA values, as their calculation does not include the dipole
contribution, but the two are otherwise directly comparable.

The most probable error in magnetisation is due to misalignment of the
physical axes of the magnets and the magnetisation direction. So far, it has
been assumed that the magnetisation is exactly aligned with the magnet,
however errors may be introduced. Errors in the magnetisation are more
difficult to detect than magnet misalignments, which should be discovered by
visual inspection. To study the impact of magnetisation errors, we assume that
the the error is entirely in the transverse plane (i.e. no longitudinal
magnetisation), and that the overall remanant field $B_{r}$ is not changed.
The error angle standard deviation is varied up to
$1\text{\,}\mathrm{\SIUnitSymbolDegree}$. Each individual magnet is assigned a
normally distributed error, and $1000$ simulations are performed for each
error magnitude.

Figure 6: Error in magnetisation angle. The Figure of Merit is defined by
Equation 11. Studying magnetisation errors up to $1\text{\,}\mathrm{\char
37\relax}$, the FoM varies linearly with error magnitude.

In Fig. 6, we see that the FoM grows linearly with the error magnitude, with a
gradient of $49.8(2)\text{\,}\text{/}\mathrm{\SIUnitSymbolDegree}$. If our
magnet tuning regime has the same requirements and success as CBETA’s, this
suggests that magnetisation errors up to
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$ would be acceptable, in the absence
of other errors.

The other error that may arise will be due to misalignments of the magnet
blocks, in both position and angle. These errors will depend on the precision
that can be reached with our magnet mounts: here we set the maximum position
error in the horizontal plane to $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
and rotation about the longitudinal axis
$1\text{\,}\mathrm{\SIUnitSymbolDegree}$. These values may be difficult to
reach if the magnet mounts are 3D printed, but should be feasible if a
standard machining approach is used i.e. milled aluminum. In a separate study,
we studied the impacts of both position and orientation errors to the FoM: we
found that the contribution from the position errors (up to
$0.1\text{\,}\mathrm{mm}$) is smaller than from the rotational errors (up to
$1\text{\,}\mathrm{\SIUnitSymbolDegree}$). The FoM increases linearly with the
errors in all cases, suggesting that no single type of error will dominate the
final FoM.

In practice, all the different errors considered here will be present, as well
as others including temperature-dependent field changes and demagnetisation
due to radiation damage. The impacts of irradiation and temperature variation
on accelerator magnets has been summarised in [71]. The magnet arrays will
also require good alignment relative to the beamline, which should be achieved
using standard metrology techniques. It is expected that the short length of
this single-pass beamline should ensure that the errors will not significantly
degrade beamline performance: this hypothesis is explored in Section V.

###  III.4 Limitations of Halbach Arrays without Custom Magnets

Although the magnet array creation algorithm presented here is sufficient for
our first-order design, it has some limitations in its current form. Some of
these are due to the specific implementation of the magnet modelling code, and
may be solved by using a more complete program such as the finite element
analysis code OPERA. One such issue is that the algorithm does not include
demagnetisation effects, which may be an issue as there will be strong fields
over some of the magnet blocks, although NdFeB is usually robust against
demagnetisation. In addition, the required calculation times can become large
when several rings are required for the first optimisation stage, but this may
be alleviated with a more efficient implementation or more powerful computing
resources.

Though the design algorithm performs well, there are opportunities for further
improvement. At present, it only allows for one type of magnet for the ring,
although it would be possible to use larger magnet blocks to produce the bulk
of the field, and smaller blocks to fill in the spaces in between. The
requirement for only one type of block makes some magnets infeasible: an
example is the CBETA ‘BDT2’ magnet [62], where the space between the array
inner radius and the outermost contour for stage 1 of the algorithm may be
smaller than the size of magnet chosen. This may be countered by including
more than one type of magnet block, although this is not compatible with the
current implementation of the first stage of the optimisation algorithm.

This method of producing permanent magnet arrays works well for the TURBO
project, where the ability to reuse magnetic material and reduce costs is
desirable. However, a different conclusion may be drawn for accelerators with
other requirements, such as a more restrictive error tolerance or strict
limitations on the transverse magnet size: in such cases, a design method more
like the CBETA trapezoidal wedges may be preferred.

##  IV Large Momentum Acceptance Beamline Design

Closed-dispersion arcs with a large momentum acceptance for medical
applications have been proposed in the past, however one has never been
constructed. We present a method for designing such an arc, which could be
applied to a particle therapy beamline or other facilities. We use our design
methodology on the TURBO technology demonstrator arc, using magnetic fields
that could be achieved by the approach in Section III. Realistic field errors
are then considered in Section V. Although a possible matching section, beam
diagnostics, and the beamline end station constitute the final overall design
of TURBO, they are beyond the scope of this study.

TURBO must demonstrate transport for $0.5$–$3.0\text{\,}\mathrm{MeV}$ protons,
a momentum acceptance of $\pm 42\text{\,}\mathrm{\char 37\relax}$ (a rigidity
range of $0.1$–$0.25\text{\,}\mathrm{T}\text{\,}\mathrm{m}$). The arc must
have a nonzero bending angle to represent realistic beamline requirements,
while also fitting within the existing infrastructure in the Pelletron lab at
the University of Melbourne. The constraints on the arc design are given in
Table 1.

Table 1: TURBO beamline constraints, from the geometric requirements of the University of Melbourne Pelletron lab, cost of parts, expected feasibility, and safety concerns Parameter | Value | Units | Explanation
---|---|---|---
Bend Angle | $30$ | $\mathrm{\SIUnitSymbolDegree}$ | Must be nontrivial for this demonstrator arc, constrained by space
Overall Footprint | $2$ | $\mathrm{m}^{2}$ | Limited by available space in the Pelletron lab in Melbourne
Beampipe Radius | $31.75$ | $\mathrm{mm}$ | Matches existing Pelletron beampipes
Magnet Bore Radius | $35$ | $\mathrm{mm}$ | Larger than the beampipe radius, with extra space
Magnet Length | $100$ | $\mathrm{mm}$ | Matches commercially available permanent magnets
Max B-Field | $0.6$ | $\mathrm{T}$ | Readily achieved by Halbach arrays, and safer than larger field

The TURBO beamline will use beam pipes with DN63LF flanges, which have a
$63.5\text{\,}\mathrm{mm}$ inner diameter and slightly larger outer size: this
motivates a maximum beam excursion range of $\pm 25\text{\,}\mathrm{mm}$ (i.e.
$50\text{\,}\mathrm{mm}$ total), and a magnet inner bore radius of
$35\text{\,}\mathrm{mm}$. As discussed in Section III, permanent magnet
Halbach arrays will be used to produce the required B-fields. These magnets
will be rectangular for ease of fabrication. The maximum magnetic field is set
to $0.6\text{\,}\mathrm{T}$, which should be achievable by the permanent
magnets: from Equation 5, we find that the outer radius for the magnetic
material will be at least $56\text{\,}\mathrm{mm}$, although it is expected to
be larger due to the additional multipoles and the gaps between magnet blocks.
The magnet ends are modelled using Enge-style fringe fields. For the initial
arc design, all magnet models in Zgoubi use the same Enge fit coefficients:
$C_{0}=$$-3.55\text{\times}{10}^{-02}$, $C_{1}=$$4.17$, and
$\lambda=$$7\text{\,}\mathrm{mm}$.

The closed-dispersion arc for the TURBO project will comprise four cells in
total. The beam will enter the first cell with both dispersion and its
derivative at zero, and must be brought back to the same condition by the end
of the beamline. This implies that the dispersion function must be symmetric,
with a turning point between the second and third cells. Due to this symmetry,
the optics of the third and fourth cells must be mirrored from the second and
first respectively, simplifying the arc design. Using the same magnet blocks
as in Section III, we assume that each magnet is $10\text{\,}\mathrm{cm}$
long. The drift spaces are set as $7.5\text{\,}\mathrm{cm}$ within the cells
and $11.25\text{\,}\mathrm{cm}$ between them, preventing extreme overlap of
fringe fields and providing space for diagnostics. Each lattice cell is made
up of three magnets on a single straight girder: this is modelled in Zgoubi
using the methods described in Section III.2. The arc bends a total of
$30\text{\,}\mathrm{\SIUnitSymbolDegree}$, with each cell starting and ending
with a $3.75\text{\,}\mathrm{\SIUnitSymbolDegree}$ rotation of the reference
axis. An extra $5\text{\,}\mathrm{cm}$ drift is added to at the start and end
of the arc, representing the additional space that may be included after
matching and before the end station: it is likely that this drift length will
change in the final version of the TURBO beamline. The layout of the arc,
including trajectories for a scaling FFA case, is shown in Fig. 8.

###  IV.1 Dispersion Suppression with a Scaling FFA

As discussed in Section II.2 our initial lattice is an FFA following the
scaling law. As our case has two cells to suppress dispersion and two to
restore it, each one must have a phase advance of $\pi/2$, and the overall arc
has a phase advance of $2\pi$. In addition, as we do not need to match to some
other periodic lattice, we do not need to halve the bending strength of our
suppressor.

A scan of possible working points must be performed to select a scaling FFA
suitable for a closed-dispersion arc. For a single cell, the horizontal phase
advance must be close to $\pi/2$, which leaves some freedom in the strength of
the vertical focusing. The results of the scan of working points for the TURBO
geometry is shown in Fig. 7. The horizontal bending strength $B_{0\text{F}}$
and the $k$-index are varied independently. The value of $B_{0\text{D}}$ is
chosen to ensure that the closed orbit passes through the centre of the first
magnet in the transverse plane. To first order, the horizontal tune is mostly
a function of $k$. The vertical tune rises with increasing $B_{0\text{F}}$. In
this case, only the first stability region (i.e. horizontal tunes less than
$0.5$) is explored, although previous studies have shown that increasing the
horizontal focusing into a second stability region can reduce the orbit
excursion [72] at the cost of tighter magnet tolerances.

Figure 7: Working point scan for the scaling FFA lattice used as the first
iteration of the closed-dispersion arc, by varying the $k$-index and $B_{0}$
for the focusing magnets. $B_{0}$ for the defocusing magnets is determined by
ensuring the closed orbit of a reference energy starts at the origin. The
cross marks the selected working point.

The phase advance per cell, $k$ value and F/D ratio must be chosen carefully,
even prior to full optimisation. To keep the orbit excursion of the scaling
FFA small, strong horizontal focusing is required: $k$ must be maximised.
However, too large a $k$ value will result in weak vertical focusing, leading
to a large vertical beta function which risks increasing the vertical beam
size beyond the magnet aperture. In addition, the magnetic fields should not
exceed the limit of $0.6\text{\,}\mathrm{T}$: this limitation excludes much of
the high vertical tune space, as well as anything beyond the first stability
region. As a compromise between all these requirements, a lattice is chosen
with $B_{0\text{F}}=$$0.325\text{\,}\mathrm{T}$,
$B_{0\text{D}}=$$-0.5449\text{\,}\mathrm{T}$, and $k=$ $85$. The closed orbits
and Courant-Snyder (CS) parameters of this lattice are shown in Fig. 8.

(a) Particle closed orbits through four triplet cells, with the beam rigidity
relative to the reference indicated (b) Particle trajectories when all momenta
are injected at the origin.

(c) Beta functions for the reference momentum along the closed orbit.

Figure 8: As the phase advance for the arc is close to $2\pi$, it is difficult
to determine the closed orbit and CS parameters for the full arc with four
cells: instead, the calculation is performed for one cell, which is not at
such a disruptive resonance.

Even though the tune and CS parameters of the scaling lattice match the
requirements for a closed-dispersion arc for the reference momentum, we see in
Fig. 8 that injecting all energies at a single point does not bring them back
to the same position by the end of the arc. In Fig. 9, we show the phase space
map for a selection of momenta, where it is clear that the origin is not
inside the stable phase space volume for the full range of energies. This can
be understood as an amplitude-dependent (and thus momentum-dependent)
variation in the beam dynamics. This is counter to the usual operation of
scaling FFAs, where phase advance is not a function of energy as particles
follow closed orbits. In addition, the excursion becomes quite large in the
centre of the lattice, with the lowest energy passing well beyond the expected
magnet aperture. As such, this lattice is not an effective closed-dispersion
arc: optimisation is required, breaking the scaling law.

Figure 9: Stable horizontal phase space of five momenta in a single cell
scaling FFA cell, measured at the start of the lattice. Closed orbit variation
with momentum can be seen. The phase space is distorted at large amplitudes as
the working point is close to the $\nicefrac{{1}}{{4}}$ resonance, chosen to
meet the closed-dispersion condition.

###  IV.2 Optimisation of the Closed-Dispersion Arc

An optimisation routine on the multipoles of the FFA lattice is performed,
aiming to find a nonscaling FFA that comprises a viable closed-dispersion arc
using similar techniques to those in [53]. In this case, there are $30$ free
parameters: each magnet has five multipoles (dipole–decapole) and there are
six unique magnets, with the second half of the arc mirroring the first.
During this optimisation, the fringe field profile is kept constant.

Once an optimal solution is found for the arc optics, the methods described in
Section III are used to attempt to design magnets that can produce the
required fields and bore size. However, these magnet designs will not
perfectly match those used in the arc optimisation, where differences in the
fringe fields are the greatest contributing factor. As such, the optimisation
is iterative: the arc and magnet routines are performed sequentially until
they converge on a solution that meets the criteria for both the optics and
magnet design.

We use a multi-objective genetic algorithm for the optimisation of the arc,
allowing for all objectives to be achieved without imposing an a-priori
weighting. We use the RNSGA-II implementation in pymoo [73], as this allows
for the addition of reference points. These reference points are required to
ensure that the solutions converge towards the useful part of the solution
space: for example, by adding a reference point in the solution space to keep
the dispersion at the beamline midpoint just below the beampipe radius, we can
ensure that solutions are physically feasible without attempting to bring the
central dispersion down further at the expense of the other objectives.

The optimisation objectives are to:

  * •

reduce the residual dispersion at the end of the arc

  * •

reduce the dispersion at the midpoint of the arc

  * •

match the output CS parameters to their values at the start of the arc

For this study, tracking is performed in Zgoubi with $31$ evenly spaced
rigidities to ensure the solution is valid over the full energy range. We
implement the objectives as

$\displaystyle f_{1}=\sum$
$\displaystyle\left[x_{i}(s=s_{\text{end}})\right]^{2},$ $\displaystyle
f_{2}=\sum$ $\displaystyle\left[x_{i}^{\prime}(s=s_{\text{end}})\right]^{2},$
$\displaystyle f_{3}=\sum$
$\displaystyle\left[x_{i}(s=s_{\text{mid}})\right]^{2},$ $\displaystyle
f_{4}=\sum$ $\displaystyle\left[x_{i}(s=s_{\text{mark}})\right]^{2},$
$\displaystyle f_{5}=\sum$
$\displaystyle\left[\beta_{i}(s=s_{\text{end}})-\beta_{\text{ref}}\right]^{2},$
$\displaystyle f_{6}=\sum$
$\displaystyle\left[\alpha_{i}(s=s_{\text{end}})-\alpha_{\text{ref}})\right]^{2},$

where $i$ indexes the $31$ rigidities measured, and the subscripts ‘mid’ and
‘end’ refer to the midpoint and end of the beamline respectively. In Zgoubi, a
marker was placed between the first and second cells to ensure the dispersion
at the midpoint was not minimised at the expense of the dispersion at other
points in the arc, as would be the case for a strong focus in a collider: this
location is noted by the subscript ‘mark’. We set the injected beam CS
parameters to $\beta_{\text{ref}}=$ $3\text{\,}\mathrm{m}$ and
$\alpha_{\text{ref}}=$ $0$ for both the horizontal and vertical profiles.
These CS parameters are assumed to be energy-independent, which can be ensured
by collimation. We also set the emittance $\varepsilon$ to
$0.2\text{\,}\mathrm{mm}\text{\,}\mathrm{mrad}$, although this has no impact
on the optimisation.

In each iteration of the beamline design, the optimisation algorithm was
allowed to investigate approximately $100\,000$ potential solutions before
termination: for example, $102\,907$ configurations evaluated in the final
iteration. The final optimised beamline is shown in Fig. 10, with the orbit
excursion shown separately in Fig. 11. Details of the magnet parameters are
given in Table 2.

Magnet | Dipole [$10^{-1}$$\mathrm{T}$] | Quad. [$10^{0}$$\mathrm{T}\mathrm{/}\mathrm{m}$] | Sext. [$10^{1}$$\mathrm{T}\mathrm{/}\mathrm{m}^{2}$] | Oct. [$10^{2}$$\mathrm{T}\mathrm{/}\mathrm{m}^{3}$] | Dec. [$10^{3}$$\mathrm{T}\mathrm{/}\mathrm{m}^{4}$] | C0 | C1 | Offset [$\mathrm{mm}$]
---|---|---|---|---|---|---|---|---
F1 | $2.876$ | $5.331$ | $7.240$ | $3.937$ | $1.646$ | $-0.0422$ | $4.33$ | $0.0$
D1 | $-4.322$ | $-11.63$ | $-14.31$ | $-8.358$ | $-1.538$ | $-0.0313$ | $3.89$ | $-3.5$
F2 | $2.425$ | $9.685$ | $4.867$ | $1.867$ | $32.50$ | $-0.0295$ | $3.85$ | $-3.5$
F3 | $2.828$ | $5.590$ | $8.023$ | $3.540$ | $2.264$ | $-0.0426$ | $4.34$ | $-2.5$
D2 | $-4.418$ | $-9.760$ | $-7.937$ | $-8.390$ | $-11.35$ | $-0.0303$ | $3.88$ | $-2.5$
F4 | $2.861$ | $6.235$ | $6.953$ | $3.878$ | $4.314$ | $-0.0384$ | $4.46$ | $0.0$
Table 2: Parameters of the magnets in the optimised TURBO arc. By offsetting
the magnets horizontally, we ensure that the good field region covers the beam
excursion at all points. Coefficients $C_{i}$ correspond to the standard Enge
formula, assuming that the fringes are the same at the entrance and exit. In
all cases, the magnet bore diameter $\lambda$ is $7.0\text{\,}\mathrm{cm}$.
Figure 10: Trajectories in the optimised arc. By comparison with Fig. 8b), we
see that the dispersion at the arc midpoint is now small enough to keep all
energies within the beampipe. Dispersion at the end is also significantly
reduced. Figure 11: Orbit excursion in the optimised arc, with the approximate
beampipe indicated. Colours correspond to momenta labelled in Fig. 10.
Discontinuities arise from rotation of the reference axis between cells: paths
in real space are smooth.

In this study, the bottleneck optimiser objective was minimisation of the
dispersion at the marker and midpoint. For the basic scaling lattice shown in
Fig. 8, the orbit excursion range at the midpoint is $11\text{\,}\mathrm{cm}$,
which is more than double the beampipe diameter. If the excursion range at
this point were allowed to be so large, lattices could be found more readily
with a small CS parameter mismatch and minimal residual dispersion at the end
of the beamline. In our final lattice, the maximum excursion range is less
than $5\text{\,}\mathrm{cm}$, and is approximately $4\text{\,}\mathrm{cm}$ at
the centre of the beamline: we find that any attempt to further reduce the
excursion has a severe detrimental impact on the CS parameter mismatch,
motivating the use of reference points in the optimisation algorithm to keep
solutions in the physically useful region of the possible solution space.

###  IV.3 Beamline Performance

For the TURBO project, the beamline needs to be considered from both an
accelerator and clinical perspective. From an accelerator point of view the
evolution of the CS parameters and the dispersion $\mathcal{H}$-function are
of interest. Conversely, a clinical perspective focuses more on the beam size,
ellipticity [74], and spot position variation at the end of the beamline. In
both cases, these vary as a function of energy, which is not usually a
significant factor for clinical systems. The accelerator and medical
parametrisations are related: for example, the size of the beam at the end of
the beamline is given by $\sqrt{\beta\varepsilon}$.

In Fig. 12, we see the CS parameters are not periodic over the length of the
arc, and the centre of the arc is not a symmetry point. This can be understood
as a mismatch between the initial values of $\alpha$ and $\beta$ that are
implied by the beam optics, and the injected parameters which are energy-
independent. The dispersion $\mathcal{H}$-function is not periodic for the
same reason. However, the $\mathcal{H}$-function returns to zero at the end of
the arc, indicating that the overall beamline cancels out any introduced
dispersion at its end. The beta functions through the arc are kept small,
although there is some distortion of the lowest energy at large amplitudes due
to the nonlinear B-fields.

Figure 12: Evolution of the beta functions parameters and the dispersion
$\mathcal{H}$-function. Though only five momenta are displayed, the CS
parameters vary smoothly between them. Colours are the same as in Fig. 10.

The delivered beam spot quality is of paramount importance for clinical
applications, and this demonstrator beamline indicates what might be
achievable with a closed-dispersion arc. According to the AAPM TG 224 QA
guidelines and recommendations for proton therapy [75], a scanned pencil beam
should have less than $\pm 10\text{\,}\mathrm{\char 37\relax}$ spot size
variation with the spot position deviating less than $1\text{\,}\mathrm{mm}$.
Though this is not directly applicable in our case as the recommendation is
relative to each individual energy, the AAPM guidelines still provide a useful
indicator for clinical performance given there are currently no protocols:
there are no clinical beamlines with a large energy acceptance. The variation
in spot size at the end of our beamline is shown in Fig. 13.

Figure 13: Final spot size as a function of momentum deviation. Spots for five
demonstrative momenta are shown ($10^{4}$ particles each). Beam centroid
deviation is within $\pm 0.2\text{\,}\mathrm{mm}$

At the end of the beamline, we see that there is some variation in spot size
and divergence, which is more pronounced at low momenta: as focusing strength
is inversely proportional to momentum, this enhanced sensitivity is to be
expected. The beam centre shift is below the AAPM recommended threshold, which
is encouraging. It is expected that much of the variation with the delivered
beam could be mitigated by the delivery nozzle as it plays a significant role
in beam shaping, collimating, and steering for clinical treatment [76].
Collimation in both planes as a function of momentum could be used to further
reduce the variation in beam spot size and shape without redesigning the
beamline optics [77, 78]: although this is standard practice for conventional
X-ray radiotherapy, ‘dynamic collimation’ is not commonplace for proton
therapy. It is also not known how much scattering in air would remove
differences in the beam spot shape: these points should be addressed in
greater detail in future studies.

##  V Robustness of the TURBO Arc

In practice, there are many errors which will impact the performance of the
closed-dispersion arc. Some errors can be compensated, such as multipole
errors which will be corrected as far as possible before beamline
installation, and alignment errors that can be reduced by adjusting beamline
elements. However, no matter how closely all the beamline parameters are
controlled, it is inevitable that some discrepancy will exist between the
ideal design presented in Section IV and the beamline that is ultimately
constructed.

For a closed-dispersion arc, there are two key performance indicators. The
more straightforward metric is variation in beam spot position due to errors,
away from the ideal case: simulations shown in Fig. 13 indicate that the beam
spot varies by up to $\pm 0.2\text{\,}\mathrm{mm}$ with momentum, which we use
as our threshold for acceptable beam spot position variation. To evaluate the
extent of beam spot shape distortion due to errors, we use the mismatch factor
$\mathcal{F}$ [79] between an intended set of CS parameters
$(\beta_{0},\alpha_{0},\gamma_{0})$ and those that are transported in practice
$(\beta,\alpha,\gamma)$, given by

$\mathcal{F}=\frac{1}{2}\left(\beta_{0}\gamma+\beta\gamma_{0}-2\alpha\alpha_{0}\right).$
(12)

We use the mismatch factor $\mathcal{F}$ to quantify the magnitude of beam
errors: if there is no mismatch (i.e. the CS parameters are exactly their
intended values), $\mathcal{F}=$ $1$, otherwise it is larger. The studies here
evaluate $\mathcal{F}-1$ at the end of the arc. We only investigate at
$\mathcal{F}$ in the horizontal plane.

For all the error studies here, $500$ configurations were evaluated for $10$
different error magnitudes. Errors are assigned using a normal distribution
truncated at three standard deviations to remove unrealistic outliers. The
five representative energies used in the previous section were transported
through each of the $5000$ configurations. Bootstrapping [80] is used to find
the approximate confidence intervals in all cases.

###  V.1 Injected LaTeX Error: Invalid UTF-8 byte sequenceBeam Parameters

The beam waist ($\alpha=0$) is designed to be at the start of the closed
dispersion beamline. However, fluctuations in the injected beam profile or
positioning errors may shift the beam waist, impacting the output beam
characteristics: this has implications in a clinical beamline or other
applications. For the case of TURBO, the Pelletron output is known to have
time-dependent fluctuations [81]. As such, we study the mismatch parameter as
a function of beam waist displacement to provide a baseline understanding of
the relative sensitivity of the arc, to different errors.

The results of applying input beam waist displacement errors up to
$1\text{\,}\mathrm{cm}$ are in Fig. 14, which shows the output mismatch due to
injection errors from a shift in the beam waist. We find that the mismatch
factor at the beamline end grows with the square of the beam waist error,
which is the same as the growth in $\mathcal{F}$ from an extra drift: this
suggests that the final mismatch is proportional to the injected mismatch. As
such, we conclude that the TURBO beamline design is robust against injection
errors despite the highly nonlinear fields. We also see that there is not a
significant energy dependence, which further suggests that the highest and
lowest energies are not more sensitive to the input, despite the large
excursions from the beamline centre.

Figure 14: CS parameter mismatch as a function of beam waist offset distance,
with $95\text{\,}\mathrm{\char 37\relax}$ confidence intervals.
$<\mathcal{F}-1>$ at the output is proportional to the $<\mathcal{F}-1>$ at
the input, which goes as the square of the beam waist error.

One goal of the TURBO project will be to confirm that a closed-dispersion arc
with nonlinear fields is as robust against injection errors as this modelling
suggests. A variable matching section of four quadrupoles would be sufficient
to move the beam waist between experiments. To induce more rapid beam
fluctuations, a dynamic collimator would be required. To determine the beam
phase space both before and after the arc, a standard pepperpot device would
be sufficient. Studies are ongoing for the beam manipulation and diagnostic
systems for the TURBO demonstrator beamline.

###  V.2 Multipole Errors

Multipole errors are a key driver of beam distortion and instability in all
accelerators, and as such the usual standard is to keep these errors less than
one part in $10^{4}$. As discussed in Section III, the TURBO beamline
demonstration should achieve this, and any residual errors can be further
corrected using methods such as those discussed in [62]. Other errors arise
from temperature-dependent fluctuations in magnetic remanence $B_{r}$, or
warping of the magnet mount over time. For this study, we assign errors with a
standard deviation up to $1\text{\,}\mathrm{\char 37\relax}$ to investigate
the robustness of the arc: magnet studies in Section III suggest that an error
up to $0.1\text{\,}\mathrm{\char 37\relax}$ is expected for the TURBO
beamline. Errors are applied individually to each multipole of each magnet in
the arc, assuming that each error is independent.

In Fig. 15, we see that the closed-dispersion arc shown here is sensitive to
errors, particularly for the lowest momentum. The beam spot centre experiences
a small shift for most energies, with a much larger error at the lowest
energy. It is interesting to note that the beam spot shift is not symmetric,
even though the multipole errors are: taking the lowest energy as an extreme
example, for large multipole errors the final beam spot can shift by several
$\mathrm{mm}$. The beam spot shift asymmetry can be understood by looking at
the trajectory of the lowest momentum in Fig. 11: due to the large negative
excursion, multipole errors tend to push the beam further out from the
reference axis. At the $0.1\text{\,}\mathrm{\char 37\relax}$ multipole error
that can be achieved for the TURBO arc, the beam spot centre shift in Fig. 15
is smaller than the intrinsic offsets from the beamline, meeting our design
requirements.

Figure 15: Beam centroid displacement and mismatch factor at the end of the
beamline as a function of multipole error. In the extreme case of error values
of 1%, the lowest energy would likely impact the beampipe before being
transported to the end.

The mismatch factor error is several orders of magnitude larger than we
predict for beam waist errors. At points in the arc where the CS beta function
is small, we expect the multipole error sensitivity to be enhanced. In
particular, the point where $\beta_{x}$ is small near the start of the arc
(around $0.3\text{\,}\mathrm{m}$ in Fig. 12) will be particularly susceptible,
as even small errors here will be compounded and amplified by the end of the
arc. Once again the lowest energy appears particularly sensitive,
corresponding to it having the smallest value of $\beta_{x}$ at this point.

From these results, magnet multipole inaccuracies are shown to dominate the
delivered beam errors, so it is important to consider possible mitigations.
One option is to mount each magnet on a translating stand with several degrees
of freedom, providing some correction to multipole errors through harmonic
feed-down. Although this correction technique proved effective for orbit
corrections in CBETA, it would drive up the overall cost of the beamline.
Alternatively, the multipole correction method applied before installation
could be tuned to provide extra compensation to lower energies, which may
reduce the mismatch seen here for the lowest momentum at the expense of
increased mismatch in other cases.

###  V.3 Alignment Errors

Magnet misalignments are another common source of error in accelerators. In
many cases, sophisticated laser systems and girder assembly enable alignment
to sub-$0.1\text{\,}\mathrm{mm}$ precision [82]. For the TURBO beamline,
vibrations in the room and misalignments on the girder may be more difficult
to mitigate. In addition, the relative positions of the beampipe, the magnets,
and the Pelletron itself may shift over the course of a day as the building
itself moves by a small amount. With proper isolation of the equipment from
the surroundings and careful maintenance, it is expected that most
misalignments can be kept below $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$:
the impacts of these errors on the beamline outputs are shown in Fig. 16. To
facilitate the simulation of misalignment errors, we use the Zgoubi DRM
parameter, which enables offsets in the horizontal plane.

Figure 16: Beam spot and mismatch error due to magnet translation in the
horizontal plane. The vertical axes and colour scale are identical to Figure
15.

Qualitatively, the impact of misalignments appears similar to the multipole
errors in Fig. 15, with a similar shape to the mismatch factor curve and the
highest sensitivity for the lowest momentum: this similarity is to be
expected, as the highly nonlinear magnetic fields convert any position offset
to a feed-down multipole error. The performance of the arc is not
significantly degraded, even where errors get up to around
$0.1\text{\,}\mathrm{mm}$, which suggests that the feed-down multipole error
will not be the limiting factor for beamline utility. Although only horizontal
misalignments have been analysed, it is expected that full 3D translation and
rotation would not be significantly different.

##  VI Discussion

We have devised a method of designing closed-dispersion arcs using FFA optics,
enabling transport of a large range of momenta without adjusting magnetic
fields. This is achieved by using multipoles up to decapole order to provide
sufficient beam steering and focusing over the full momentum range. Applying
this idea to a scaled-down prototype for the TURBO project, we have designed a
beamline to transport $0.5$–$3.0\text{\,}\mathrm{MeV}$ proton beams ($\pm
42\text{\,}\mathrm{\char 37\relax}$ momentum acceptance) produced by the
Pelletron at the University of Melbourne. In addition to a basic optics
design, we have produced a method to design suitable Halbach arrays for the
arc using low-cost commercial permanent magnets, and error studies have been
performed to better understand the beamline performance under realistic
operating conditions.

Permanent magnets will be used for the TURBO beamline as field ramping is not
required, and they can produce fields in excess of $0.5\text{\,}\mathrm{T}$
over a large aperture. Previous FFA designs with permanent magnets required
custom wedges of magnetic material to make a Halbach array with the desired
field, however these wedges are only suitable for a single use application and
may be expensive to manufacture. For this project, we have found a method to
design Halbach arrays using a single block shape with multipole errors below
one part in $10^{4}$, by approximating the required magnetisation that would
be used in the ideal case and optimising the positions of the individual
blocks. By using a single type of magnet block, we hope to use commercially
available permanent magnets inserted in a simple 3D printed or aluminum-milled
housing, which should significantly reduce costs, and allows for rapid
prototyping if the magnets must be redesigned. The magnetic material can also
be reused for future projects, reducing waste and overall costs over the
beamline lifetime.

For the beamline itself, we have presented a generic method for designing
closed-dispersion arcs with Fixed Field Accelerator optics that is able to
transport a large range of momenta, as applied to the TURBO project. Our
method begins with an approximately scaling FFA triplet cell with a phase
advance of $\nicefrac{{$\pi$}}{{2}}$, with four cells to introduce and remove
dispersion. We show that a scaling FFA is not sufficient to remove dispersion
over such a large range of momenta, and requires a large beam excursion at the
midpoint of the arc. The symmetry of the arc requires that the second half
mirror the first: as we go up to decapole order multipoles, there are 30
optimiser variables. To solve this, a genetic algorithm with reference points
is used to explore the solution space.

This algorithm produced a suitable lattice for the TURBO beamline, meeting our
key design requirements with a maximum field below $0.6\text{\,}\mathrm{T}$
and an excursion range below $50\text{\,}\mathrm{mm}$ while fitting inside our
required beamline geometry. In the optimisation process, we found that there
is a trade-off between matching the CS parameters, and returning all
trajectories back to the origin. In this case, there may simply not be
sufficient degrees of freedom to match all energies perfectly: in the future,
a design going beyond decapole terms could be considered to give better
matching. In addition, the number of magnets in the arc could be increased,
either by reducing the drift lengths or by increasing the total length of the
arc. However, both of these potential solutions raise further challenges, such
as an increased B-field overlap between adjacent magnets, stricter spacial
constraints, and increased costs. Despite the observed optics mismatch, the
variation in beam spot size and shape as a function of energy is acceptable,
and may be further mitigated using methods such as dynamic collimation if
required.

Using a scaling FFA as the basis for a closed-dispersion arc provides a good
starting point for further optimisation, as it ensures stable optics exist
along closed orbits for all momenta. However, there are disadvantages to this
choice of initial optics. The scaling law determines the dispersion in the
initial lattice, increasing the difficulty of later optimisation. In addition,
our optimisation does not allow the polarity of any magnets to change, so the
reverse bending from the ‘D’ magnets remains significant. As such, the overall
arc is larger than would be necessary for a lattice with minimal reverse
bending, and the magnetic fields required throughout the arc are larger than
would be otherwise required. A future study could investigate the design of a
closed-dispersion arc that does not use a scaling optics as the initial
solution, however it may be difficult to find a suitable optics given the
large possible solution space.

It should be noted that the beamline design presented here does not use the
realistic model created with the method described in Section III, as the
design iterations did not converge. The limiting factor is the large fringe
field extent: the fringe fields vary significantly with each magnet iteration,
as they are strongly dependent on the positioning of the magnet blocks. In
addition, we find that attempts to reduce the residual dispersion further than
is achieved here are generally not successful, due to the contributions of
overlapping fringe fields counteracting one another. This could be mitigated
by reducing the fringe field extent and variability, either by reducing the
magnet aperture or by the introduction of field clamps [83, 84]. These options
should be investigated to improve the performance of the final TURBO beamline
design.

Our initial error study of the TURBO lattice has suggested that closed-
dispersion arcs are robust against realistic misalignments and multipole
errors, which can be managed by standard mitigation techniques. The highly
nonlinear fields and large beam excursions make all energies susceptible to
these errors, although the lowest was found to be particularly sensitive for
the lattice shown here as it spends the longest time at the largest excursion.
This may be countered by increasing the relative weighting of low rigidity
beams during the optimisation process. Further work is required to fully
investigate the effectiveness of error mitigation techniques, but the initial
tolerance study shown here suggests that it will be possible to ensure good
beam delivery for all momenta.

##  VII Conclusions

To act as a closed-dispersion arc over a large range of momenta, the TURBO
beamline will be a nonlinear FFA lattice suitable for the beam available at
the University of Melbourne Pelletron lab, transporting proton beams between
$0.5$–$3.0\text{\,}\mathrm{MeV}$. This momentum range is equivalent to
$46$–$250\text{\,}\mathrm{MeV}$ protons when scaled up to the clinical range.
By covering a greater momentum acceptance than is required for standard proton
therapy, TURBO will also demonstrate possible extensions to proton CT and ion
therapy. The arc has been designed such that all energies are overlayed at the
start and the end, without significant distortion of the output beam spot.
Initial magnet designs have also been completed, using commercially available
blocks of permanent magnet material to reproduce the required magnetic fields.
This study is supplemented by analysis of the errors that may arise from
realistic magnet misalignments, finding that the beamline is robust in
realistic conditions, however further optimisation may be required to ensure
good performance.

The ultimate goal of the TURBO project is to design and demonstrate a large
energy acceptance for the energy range relevant to charged particle therapy.
The low energy demonstrator discussed here has explored many of the challenges
that will be faced by a clinical closed-dispersion arc, such as ensuring the
beam excursion is sufficiently small and maintaining good beam quality in
spite of errors. The magnets shown here would not be well-suited to a higher
energy, as they would not be able to produce sufficient field to keep the
beamline compact: instead, it is expected that superconducting curved canted
cosine theta magnets would be able to provide the strong beam steering and
focusing required [85, 86, 45]. In addition, these magnets are able to produce
a highly nonlinear magnetic field over a large aperture, a key requirement
shown in this work. Investigations will be required to ensure the field
quality is sufficient, in particular due to the fringe fields associated with
these magnets, and this will form part of the full-scale beamline design at a
later stage.

This study is a first investigation of a large energy acceptance arc for the
proof-of-principle TURBO beamline, and there are many avenues for further
investigation. This work also opens up the possibility of applying a large
energy acceptance closed-dispersion arc as an insertion as part of a ring. By
providing a design method for feasible permanent magnets, an initial lattice
design, and an analysis of likely machine errors, we have shown that a closed-
dispersion arc at scaled-down energies will be able to demonstrate viability
of large momentum acceptance beamlines for charged particle therapy.

##  VIII Acknowledgements

The authors are grateful for the support of Hannah Norman and Paarangat
Pushkarna, whose feedback, comments, and advice were of great assistance.
Funding for the contributors on this project is through the Universities of
Manchester and Melbourne Cookson Scholarship, the Laby Foundation, and ANSTO.

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