11institutetext: University of Oxford, UK

# Fixed-Field Alternating-Gradient Accelerators

S.L. Sheehy

###### Abstract

These notes provide an overview of Fixed-Field Alternating-Gradient (FFAG)
accelerators for medical applications. We begin with a review of the basic
principles of this type of accelerator, including the scaling and non-scaling
types, highlighting beam dynamics issues that are of relevance to hadron
accelerators. The potential of FFAG accelerators in the field of hadron
therapy is discussed in detail, including an overview of existing medical FFAG
designs. The options for FFAG treatment gantries are also considered.

###### keywords:

ffag, medical, accelerator, hadron therapy

##  0.1 Preface

These notes are broken down into two main parts. The first gives a general
introduction to Fixed-Field Alternating-Gradient (FFAG) accelerators (often
referred to simply as FFAGs) including an overview of the basic transverse
dynamics of both the so-called ‘scaling’ and the ‘non-scaling’ variety, and a
brief comparison with other machines. In the second part, I discuss the
motivations for considering FFAGs for medical applications, focusing on
charged-particle therapy (hadron therapy). I then introduce a number of FFAG
designs specifically aimed at medical applications, including developments of
FFAG gantries for medical use.

The FFAG accelerator is a class of circular accelerator that combines
properties of both the cyclotron and the synchrotron. It uses a magnetic field
which is constant in time, hence the ‘fixed-field’, together with an increased
focusing strength achieved using the ‘alternating-gradient’ principle [1]. The
RF acceleration scheme is usually variable-frequency, but in some specific
instances a fixed-frequency system is possible.

Many accelerator physicists have difficulty when first encountering FFAGs, as
they may have learned about only one example of an FFAG machine and therefore
approach the topic with some preconceptions. Many questions arise, such as
‘aren’t the magnets very large?’, ‘have any of them actually been built?’, and
‘isn’t the FFAG only good for large muon beams?’ The difficulty with these
questions is that the modern FFAG is not really a single type of machine. This
would be like assuming that a synchrotron can only be a light source, rather
than a hadron collider or a medical accelerator. For the record, the answers
to the questions are ‘sometimes’, ‘yes’, and ‘not only’. Of course, for
synchrotrons it is easy to see that the application can change the layout,
design, and properties of an accelerator. This is perhaps even more true in
the case of the FFAG.

Starting with the idea that FFAGs are just accelerators which have both a
fixed field and alternating-gradient focusing produces a large spectrum of
designs. Some designs may have purely linear (quadrupole) magnets and fixed-
frequency RF acceleration, whereas others have large-aperture magnets which
produce a complex variation of the field with radius and a variable-frequency
RF system. Most FFAGs have a very large dynamic aperture. This flexibility of
FFAG design has only emerged in roughly the last 15 years and the field
continues to be a rich source of novel developments.

From these lecture notes you should be able to answer questions like those
above, and describe the various types of FFAG accelerators and some of the
design principles behind them. I hope you will also learn of the potential and
some of the challenges in utilizing these machines in medical applications.

##  0.2 Fixed-field alternating-gradient accelerators

###  0.2.1 A historical perspective

The concept of an FFAG accelerator is not new. This type of accelerator was
invented in the 1950s and 1960s at the same time as the synchrotron was being
developed. Much of the early work in developing FFAGs was carried out at the
Midwestern Universities Research Association (MURA), but only electron FFAG
accelerators were constructed at the time [2]. It is interesting also to note
that in Soviet terminology the FFAG is known as a ‘ring phasotron’ [3]. It was
not until the 1990s that interest in this type of accelerator re-emerged in
Japan, with a particular focus on what such machines could offer as hadron
accelerators. For a little more on the history, see Ref. [4] and the
references therein.

A particular area of recent interest in the field of FFAGs is their potential
for high-intensity operation, because of their high repetition rate, large
acceptance, simpler and cheaper power supplies, and flexibility of the RF
acceleration system. High intensity may be required for some medical
applications (such as radioisotope production), but for hadron therapy the
beam currents are typically low. For this reason, we shall not discuss high
intensities here but instead focus on the advantages in terms of high
repetition rate, flexibility, and simple magnet power supplies. The impact of
these qualities will be discussed in Section 0.4.

###  0.2.2 The original or ‘scaling’ FFAG

In 1943 Marcus Oliphant described the idea of the synchrotron as follows:

> Particles should be constrained to move in a circle of constant radius thus
> enabling the use of an annular ring of magnetic field …which would be varied
> in such a way that the radius of curvature remains constant as the particles
> gain energy through successive accelerations.

He intended that the magnetic field should be varied temporally and the beam
should always follow the same annulus [5]. However, in principle there is no
reason why the annulus may not change radius and the field vary spatially
rather than temporally. This is the fundamental idea behind the FFAG. A large
variation of the field with radius will constrain the change in radius of the
orbits; this can lead to a larger field increase with radius and more compact
orbits than in a cyclotron. This is the original type of FFAG, which we now
call ‘scaling’.

To introduce the scaling FFAG, it is useful to review the four main different
types of circular accelerator, which we can classify by the magnetic field
they use to guide the particles [3]. These are the following.

  1. 1.

Fixed-field constant-gradient accelerators, including conventional cyclotrons,
synchrocyclotrons, and microtrons.

  2. 2.

Pulsed-field constant-gradient accelerators, which includes weak-focusing
synchrotrons and betatrons.

  3. 3.

Pulsed-field alternating-gradient accelerators, which are the well-known AG
synchrotrons.

  4. 4.

Fixed-field alternating-gradient accelerator, otherwise known as FFAGs.

The fourth variety, FFAG accelerators, were proposed independently in the
early 1950s by Ohkawa in Japan [6], Symon _et al._ in the United States [7],
and Kolomensky in Russia [8]. Symon _et al._ proposed:

> A type of circular accelerator with magnetic guide fields which are constant
> in time, and which can accommodate stable orbits at all energies from
> injection to output energy.

This relies on introducing sectors with a reversed magnetic field into a
cyclotron-like machine, producing strong focusing throughout the energy range.
The field may rise rapidly with radius such that the orbits are relatively
compact over a large energy range. It is possible to accelerate both light
particles (electrons) and heavier particles (hadrons) to relativistic energies
with this method. The time-independent magnetic field means that the
repetition rate can be much higher than in a pulsed-field alternating-gradient
machine (i.e., synchrotron), as the RF modulation can be on a much shorter
time-scale than the modulation of a magnetic field. The consequences of this
for medical applications will be discussed later.

The field is arranged in such a way that the increase in gradient with
momentum results in the beam experiencing the same focusing independent of
radius. This means that the betatron tunes are constant for all orbits. This
constant focusing (or constant betatron tune) is ensured if two conditions are
met. First, the field index $k$ must be constant, where we can define $k$ in
terms of the bending radius $\rho$, the vertical magnetic field $B_{y}$, and
its derivative in the horizontal direction $x$:

$k=-\frac{\rho}{B_{y}}\frac{\partial B_{y}}{\partial x}.$ (1)

Therefore we require

$\frac{\partial k}{\partial p}\bigg{|}_{\theta=\mathrm{const.}}=0.$ (2)

The second requirement is that the shape of the particle orbits remains
constant as the size of the orbits ‘scales’ with energy, such that each
higher-energy orbit is a geometrically similar enlargement of the lower-energy
orbits as described by the following equation, derived by Kolomensky [3]:

$\frac{\partial}{\partial
p}\left(\frac{\rho_{0}}{\rho}\right)\bigg{|}_{\theta=\mathrm{const.}}=0.$ (3)

If the field meets these two conditions, the FFAG is referred to as being of
the ‘scaling’ variety.

To satisfy these requirements, we use a magnetic field that increases with
radius. The particular shape of the field is given by the $r^{k}$ law of the
following equation, which describes the increase in field with radius $r$ with
respect to a reference radius $r_{0}$, where the field increase is
characterized by the field index, $k$:

$B_{y}=B_{0}\left(\frac{r}{r_{0}}\right)^{k}.$ (4)

The field is shown in Fig. 1. It should be clear that for a given value of the
field index $k$, the field at a smaller radius not is only lower but also has
a lower gradient. As the momentum increases and particles move to higher
radius, the value of the confining field increases along with the field
gradient. Of course, this is the field for only one of the two alternating-
gradient types, the ‘F’ or focusing type. The field for the ‘D’ (defocusing)
type has the opposite sign, as expected.

Figure 1: An example of the characteristic scaling law for different values of
the field index $k$, with $r_{0}=10$ m and $B_{0}=1.0$ T. Figure 2: Radial-
sector FFAG layout with reverse bend

The fact that the different sectors in the scaling FFAG have reverse-polarity
magnetic fields means that the length of the orbit and the mean radius of the
machine are necessarily larger than if there were no reverse fields, as shown
in Fig. 2. This is necessary to ensure stability in both planes and is the
main disadvantage of the scaling FFAG. This can be partly or fully overcome,
however, as the maximum magnetic-field value can be higher than that in a
synchrotron with a time-varying field, which can help to constrain the machine
size. In addition, there are no separate-function magnets, so provided
sufficient space is left for injection, extraction, accelerating cavities,
vacuum ports, and so forth, the machine may be made compact as the main
magnets perform all the required functions simultaneously. The FFAG layout in
Fig. 2 is called a ‘radial-sector’ layout as the faces of the magnets lie
along radial lines from the machine centre.

Figure 3: Spiral FFAG layout, removing the need for reverse bends

There is in fact a second type of FFAG which does away with the need for the
reverse-polarity field, known as the ‘spiral’ FFAG. This is formed from a
succession of hills and valleys of field distributed in the azimuthal
direction, where the magnets have a spiral angle with respect to the beam as
shown in Fig. 3. In this case the beam does not enter the magnet exactly
perpendicular to the face of the magnet, and thus edge focusing results. This
removes the need for the opposing reverse bending magnet, while maintaining
strong focusing. In the spiral FFAG, the focusing can also be made independent
of momentum.

The field in a spiral FFAG in general has the form

$B(r,\theta)=B_{0}\left(\frac{r}{r_{0}}\right)^{k}F(\vartheta),$ (5)

where we call $\vartheta$ a generalized angle. This is related to the usual
azimuth by

$\vartheta=\theta-\tan\zeta\ln\frac{r}{r_{0}}.$ (6)

The value of $\zeta$ is the angle between the field spiral and the radial
direction, and $\tan\zeta$ is constant. As before, $B_{0}$ and $r_{0}$ are
constants. The function $F(\vartheta)$ is an arbitrary periodic function with
period $\vartheta_{0}=2\pi/N$, and $N$ is the number of periods. In the
radial-sector scaling FFAG that we discussed earlier, all the orbits enlarge
and remain geometrically similar, but in the spiral FFAG they also turn about
the centre. This can be seen in an example spiral FFAG design for the RACCAM
project in Section 0.4.

In terms of beam dynamics, it is useful to compare the scaling FFAG with a
synchrotron. Modern synchrotrons employ the principle of alternating-gradient
or ‘strong’ focusing [9, 10], in which alternating focusing and defocusing
magnets lead to much stronger focusing forces in the transverse plane than in
constant-gradient weak-focusing synchrotrons. This alternating-gradient
focusing is also employed in the FFAG. The transverse beam dynamics in the
FFAG is therefore much the same as in the synchrotron, at least for a single
orbit or energy, in the sense that we may discuss beta functions, dispersion,
and so forth. The difference is that in this case the field is highly
nonlinear and these transverse optics functions may vary with radius.

##  0.3 The non-scaling FFAG

The non-scaling FFAG allows the strict scaling laws applied in the original
scaling FFAG to be relaxed. The idea of violating the strict scaling law of
the FFAG occurred to Kent Terwilliger and Lawrence W. Jones in the 1950s [11],
but such a machine was never pursued. Two of the main disadvantages of the
original FFAG are the highly nonlinear magnetic field required and the large
aperture of the magnets and RF arising from the shift of the orbit with
energy, which can be up to the order of 0.5–1.0 m. The non-scaling FFAG arose
from the question “what if we violate the scaling law?” Or, more specifically,
“What if we take a line tangent to the scaling law in Fig. 1, such that the
field is linear with radius?” This radical idea led to the linear non-scaling
FFAG and was proposed in the 1990s [12].

###  0.3.1 Linear non-scaling FFAG

The linear non-scaling FFAG is so called because it uses only up to linear
focusing elements, that is, quadrupole and dipole fields. When only
quadrupoles and no higher-order multipoles are used for focusing, the beam
shifts outward with acceleration because of dispersion and sees a reduced
level of focusing. This is really like considering a synchrotron where we do
not ramp the magnets with time. In the scaling FFAG, we got around this by
varying the gradient with the momentum and by making the beam pipe wider to
allow for the orbits moving. But in the linear non-scaling FFAG we ignore the
scaling law and any focusing issues for now, which allows us to increase the
gradient and reduce the dispersion function even further to reduce the shift
of the orbit with momentum. To achieve this, a linear non-scaling FFAG lattice
may use normal bending with a defocusing ‘D’ quadrupole and reverse bending
with a focusing ‘F’ quadrupole, and may (or may not, depending on the design)
change at high momentum to use the ‘D’ quadrupole for reverse bending and the
‘F’ for normal bending, as described in Ref. [4].

One must then ask what happens to the beam dynamics in such an accelerator.
One consequence is that the orbits no longer ‘scale’, so they are no longer
geometrically similar at different energies. The orbits can be made much more
compact than in the scaling FFAG. However, the most dramatic difference is
that the betatron tunes are no longer constant with energy. This may seem
surprising if you have worked on (almost) any other type of accelerator, as
the betatron tunes are usually designed to be kept constant. In the linear
non-scaling FFAG, they vary dramatically throughout the acceleration cycle,
crossing not just high-order betatron resonances but also integer resonances.

In theory, if the acceleration is fast enough, the beam may be able to cross
betatron resonances before they have time to build up, and therefore any
amplitude growth effects may be mitigated. How fast this crossing needs to be
depends on imperfections and alignment errors in the machine, and clearly
necessitates a fast acceleration rate. In fact, the linear non-scaling FFAG
was proposed in the context of muon acceleration, where very fast acceleration
before the muons decay is an absolute requirement. The many questions
surrounding the dynamics of such a machine led to the construction of the
first non-scaling FFAG accelerator, known as EMMA.

###  0.3.2 The EMMA non-scaling FFAG

EMMA is the Electron Machine for Many Applications, a 10–20 MeV electron
accelerator constructed at Daresbury Laboratory to demonstrate the technology
of the linear non-scaling FFAG accelerator. The accelerator is shown in Fig.
4, and measures just over 5 m in diameter. Despite the name, EMMA is in fact a
demonstrator machine; the primary purpose of its commissioning in 2010–2011
was to investigate some of the key dynamics issues and technical issues
involved in realizing the technology. This being demonstrated, it aimed to
show that the non-scaling FFAG was a viable technology for use in ‘many
applications’ ranging from medical use and muon acceleration to high-power
proton accelerators.

Figure 4: The EMMA non-scaling FFAG at the STFC Daresbury Laboratory, UK

A few key points of interest, of relevance to medical accelerators, were
learned from the EMMA experiment.111EMMA also demonstrated a number of other
interesting accelerator physics concepts, including a novel acceleration
mechanism known as ‘serpentine acceleration’. Most importantly, EMMA
demonstrated the successful operation of a linear non-scaling FFAG, bringing
this technology into reality. Next, EMMA demonstrated that fast crossing of
integer betatron resonances is possible if the beam is accelerated quickly
enough. However, EMMA accelerates in just 10 turns from injection to
extraction and it is not practicable to accelerate a hadron beam so quickly
over the energies required for medical use. We shall return to this point in
our discussion of hadron FFAG design studies. More information about EMMA can
be found in Refs. [13, 14].

##  0.4 FFAGs for hadron therapy

In this section we turn to addressing the specific idea of FFAG accelerators
for medical applications, in particular hadron therapy. State-of-the-art
hadron therapy (proton and light-ion therapy) centres must provide beams for
patient treatment with unprecedented precision, flexibility, and reliability
within a hospital environment. This can pose challenges for present
accelerator technology, in terms of both beam requirements and considerations
of machine size and cost. Existing technological options may meet present
requirements, but one should ask whether they are the optimal solution for the
future of this treatment modality. Can we do better?

In accelerator terms, the beam energies required for therapy are relatively
low compared with the multi-TeV energy range required for high-energy physics
experiments. Proton energies up to 250 MeV are required for treatment, and up
to 330 MeV if online proton radiography is incorporated. For heavier ions,
full-body-treatment-energy C6+ is equivalent in magnetic rigidity to protons
with an energy of around 1.2 GeV. These energies can be achieved with a number
of different types of accelerator, including synchrotrons, cyclotrons, and
linear accelerators. Because of the limited space available in a clinical
environment, a circular accelerator is usually chosen as the main accelerator
for a facility.222However, recent developments such as the ‘Cyclinac’ design
are being studied for this application [15]. There are a number of possible
choices of circular accelerator. Most existing or planned facilities use the
established technology of either the synchrotron or the cyclotron.

Cyclotrons have a limited energy range, as it is difficult to maintain
vertical focusing at high energy and the magnets become increasingly unwieldy
as the energy and thus machine size grow. They are also limited by their fixed
extraction energy. For hadron therapy, this means in practice that the beam
energy must be adjusted between the accelerator and the patient using an
energy degrader. Although a cyclotron can deliver ample dose to the patient,
the use of energy degraders and passive scattering systems has led to concerns
about activation and radiation protection [16]. State-of-the-art systems ought
to use active scanning systems and active energy variation to deliver the
best-quality treatment.

Synchrotrons operate with a pulsed beam; however, most synchrotrons have a
slow cycle rate, on the order of 1 Hz. Rapid-cycling synchrotrons have been
proposed for this application, with rates up to around 50 Hz. Their main
advantage is their flexibility, particularly in terms of energy reach and easy
variable-energy extraction. However, the pulsed nature of these machines makes
scanning rather slow. This problem has been partly overcome by using a stable
slow extraction of the beam, although the energy variability is still limited
by the repetition rate of the synchrotron. In addition, synchrotrons are
generally much larger than cyclotrons.

This discussion leads us to ask where the FFAG accelerator might enter the
picture. With a fixed magnetic field, the repetition or cycle rate can be much
higher than in a synchrotron, up to the range of kHz, limited only by the
speed of the RF system. With variable-energy extraction, this high repetition
rate could enable slice-to-slice energy variation without limiting the dose
rate for full 3D conformal irradiation, or even 3D tumour motion tracking
during treatment. Such a machine is not limited in energy range by its
dynamics like a cyclotron, and so can easily reach the energies required for
therapy. If fast extraction can be achieved at any energy, one may even
conceive of a machine which operates with different ion species on a pulse-by-
pulse basis, using some pulses for imaging and some for treatment while using
the same acceleration and beam delivery system. Of course, this requires that
the extraction line, beamlines, gantry, diagnostics, and quality assurance are
able to handle such flexibility. In essence, the FFAG has the potential to
remove some of the limitations of existing technologies and provide a
flexible, relatively compact alternative.

###  0.4.1 The KEK 150 MeV FFAG

The first prototype FFAG for medical applications was a 150 MeV proton
accelerator at KEK in Japan. This was one of the first hadron FFAGs ever to be
built and is a radial-sector scaling FFAG using a 12-sector DFD triplet
lattice structure. The orbits vary from an average radius of 4.47 m at 12 MeV
to 5.2 m at 150 MeV at the centre of the F magnet. This machine was as much
about proof of technology as it was about suitability for medical
applications. A number of key innovations were made which addressed some of
the technology challenges of the scaling FFAG, including wide-aperture
magnetic-alloy RF cavities for acceleration [17] and the development of a
‘return-yoke-free’ main magnet triplet, which eased beam injection and
extraction by allowing traversal through the side of the main magnets. A very
similar machine for studies of accelerator-driven systems is also in operation
at Kyoto University Research Reactor Institute, and the original KEK machine
is now in use at Kyushu University. The machine is shown in Fig. 5 along with
the cyclotron injector and transport line.

Figure 5: The KEK 150 MeV scaling FFAG, Japan

###  0.4.2 The RACCAM project

As discussed earlier, one issue with the radial-sector scaling FFAG is that it
has a relatively large circumference because of the reverse bends, but the
spiral FFAG does not have this issue. The RACCAM project, which ran from 2006
to 2008 in France, was a university and industry collaboration to design and
prototype a spiral scaling FFAG for medical applications. It combined some key
benefits, including variable-energy fast extraction and multiport extraction
to different beamlines. The machine was designed to accelerate beams over a
variable energy range from as low as 5.55 MeV to a maximum of 180 MeV with a
repetition rate of at least 100 Hz. The design principles of the machine can
be found in Ref. [18] and the layout and orbits for varying energies can be
seen in Fig. 6.

Figure 6: (a) Machine layout and (b) orbits in the RACCAM FFAG

As part of the RACCAM project, a suitable spiral FFAG magnet was designed in
iteration with beam dynamics studies, and a prototype magnet was fabricated
and measured. Details of the design can be found in Ref. [19]. Along with the
machine study and the technical design, a lot of work was undertaken in an
attempt to optimize the facility itself. The collaboration carried out a study
on how best to utilize multiple treatment rooms taking advantage of a machine
with multiple extraction ports [20].

###  0.4.3 Linear non-scaling FFAG designs

Around the time the linear non-scaling FFAG concept was invented, a number of
designs to apply this concept to hadron therapy arose. One example which was
studied in detail was designed by Keil _et al._ [21] and consists of three
concentric rings, to produce full-energy protons and carbon ions for hadron
therapy.

This is the simplest type of non-scaling FFAG, the linear version, consisting
of F and D quadrupoles, providing alternating-gradient (F/D) strong focusing.
In this lattice, the use of F and D doublet magnets is proposed, in which a
radial offset between the F and D quadrupoles provides the dipole bending
field, removing the need for separate dipole magnets, as in the EMMA design.
The lattice of the second ring, which covers the 31–250 MeV proton energy
range (and the equivalent for carbon ions), comprises 48 FD doublet cells and
the ring radius is around 6.9 m. The layout of the lattice cell can be seen in
Fig. 7 and the optics in Fig. 8.

Figure 7: Layout of one cell of the second (middle) ring from centre to centre
of the D magnets in two adjacent cells Figure 8: Beta functions and dispersion
in the Keil _et al._ non-scaling FFAG for hadron therapy

As we now know, the betatron tune in a linear non-scaling FFAG changes during
acceleration; the variation of the tune in this lattice is shown in Fig. 9.
This changing tune will cross integer and half-integer betatron resonances,
where any errors in the lattice will result in small kicks to the beam which
will build up with each subsequent turn, potentially damaging the beam
quality. Acceleration in this case occurs in roughly 1000 turns, which is far
slower than the 10-turn acceleration in EMMA.

Figure 9: Variation of betatron tune with acceleration. The cell tune is
shown; the machine tune can be found by multiplying by the number of cells,
which is 48 in this ring, resulting in the tunes crossing around 10 integers
in each plane.

Detailed studies were carried out on the sensitivity of such a design to
alignment errors, and a comparison was made with an alternative design by
Trbojevic [22]. The results showed that alignment tolerances in such a machine
may be as tight as 10 microns, an extremely stringent requirement. One may
improve things by optimizing the lattice for the application so as to cross
fewer resonances, but it is likely that resonance crossing will remain an
issue [23, 24]. The topic of resonance crossing in non-scaling FFAGs is an
ongoing area of active research [25]. In addition, one only needs to note the
density of such a lattice to realize that injection and extraction and the
lack of long straight sections may be problematic in an operational machine.
However, the advantages of the linear non-scaling FFAG may be realized if the
machine is made non-symmetric with long straight insertions. The principle may
also be used to optimize the treatment gantry rather than the accelerator
itself, an option which we shall discuss later.

###  0.4.4 PAMELA: Particle Accelerator for MEdicaL Applications

As part of the CONFORM (COnstruction of a Non-scaling FFAG for Oncology,
Research and Medicine) project, which included the construction of the EMMA
non-scaling FFAG, a design study for a hadron non-scaling FFAG for hadron
therapy was undertaken from 2007 to 2011. Known as the PAMELA project, this
was a design study aimed at applying the concept of a non-scaling FFAG to
hadron therapy. The design principle was based on two rings to cover the full
energy range of both protons and carbon ions up to an equivalent magnetic
rigidity of 6.7 T m, that is, 440 MeV/nucleon for C6+ ions (Fig. 10). The
machine was designed to operate with a very fast repetition rate of 1 kHz.

The starting point for the study was a linear non-scaling FFAG. However, the
concern about beam deterioration from resonance crossing discussed in the
previous section was significant enough to motivate studies of alternative
designs of FFAGs for medical use. This resulted in a non-linear FFAG design.
In each ring, the design strategy resulted in a variation of the total
betatron tunes with acceleration which was well within half an integer (i.e.,
$\Delta\nu_{\mathrm{total}}=n_{\mathrm{cell}}\,\Delta\nu_{\mathrm{cell}}<0.5$).
It also maintained some advantageous non-scaling properties such as small
orbit excursion and simpler magnets which are easy to align.

The steps of the design were as follows:

  1. 1.

The designers started with a scaling law based on a radial-sector FFAG with an
FDF triplet lattice.

  2. 2.

The second stable region of Hill’s equation was used, with a horizontal phase
advance per cell greater than 180${}^{\degree}$. This allowed a larger field
index to be used, resulting in a reduction in the orbit excursion by a factor
of roughly five compared with the first stability region [26].

  3. 3.

The field was decomposed into its multipole components and the expansion was
truncated at some low order, usually octupole or decapole, to simplify the
magnets.

  4. 4.

The shape of the magnets was changed to rectangular rather than sector-shaped,
and aligned on a straight line rather than along an arc to simplify
construction.

  5. 5.

The field profile was finally optimized such that the variation in betatron
tune was minimized throughout acceleration.

Figure 10: Layout of the PAMELA proton ring lattice showing the low-energy
(inner) and high-energy (outer) orbits. The axis units are metres.

The resulting machine was studied up to engineering design level, including a
novel superconducting magnet design [27]; studies of RF cavities; engineering
design of the cryostats, injectors, injection and extraction magnets, and
beamlines; and a first look at a gantry system [28]. The layout of the
injectors and two main rings is shown in Fig. 11.

Figure 11: Layout of the PAMELA facility

###  0.4.5 NORMA: NORmal conducting Medical Accelerator

The NORMA study followed on from the PAMELA design to address some of the
issues that arose during that study and to iterate the design. For example, it
was felt that the PAMELA machine might have been closer to implementation if
normal-conducting magnets had been used instead of the novel superconducting
design. The NORMA design also extended the energy range of a single ring to
provide beams up to 350 MeV which could travel through a patient to enable
proton radiography. Although the design was for protons only (unlike PAMELA,
which was for both protons and carbon ions), it provided detailed studies and
optimization based on dynamic aperture [29].

The cell structure (Fig. 12) in this design was very similar to that of
PAMELA, using an FDF triplet. The NORMA study also made further innovations by
introducing two long straight sections almost 5 m long for injection,
extraction, and acceleration by creating a racetrack-shaped machine, shown in
Fig. 13.

Figure 12: Layout of the NORMA cell, showing the magnetic field throughout the
cell and trajectories of low-energy (inner) to high-energy (outer) orbits.
Figure 13: Layout of the NORMA racetrack FFAG

###  0.4.6 Arbitrary-edge-angle non-scaling FFAG designs

Contemporary with the PAMELA design, an alternative approach by Johnstone and
Koscielniak used wedge-shaped quadrupoles to achieve control over the betatron
tunes with acceleration [30]. In this case both the path length through the
quadrupoles and the contribution of edge focusing were designed to be a
function of the beam momentum. However, over the energy ranges considered, the
orbit shift was relatively large, of the order of 1 m. More information on
these designs can be found in Refs. [31, 32].

##  0.5 FFAG gantry designs

One of the most promising application areas for the FFAG is in the beam
delivery system from accelerator to patient. Existing treatment gantries are
large, expensive, and slow to provide variable-energy beams to the patient, as
one must wait for the magnetic field to be adjusted before a different energy
can be accepted through the transport line. In particular, for carbon ions,
the existing gantry at the Heidelberg Ion Therapy centre is both large and
heavy, weighing in at 630 t, where 135 t of that is the magnets and the
remainder is the supporting structure to allow rotation around the patient.

An FFAG gantry would be able to have a large energy acceptance with a single
magnet setting and no limit on rapid variation of energy. This means that if
the machine could provide energy variability at a rate of 1 kHz, the beam
delivery system and gantry would be able to transport this to the patient
treatment room.

A transport line and gantry design based on similar optics was developed in
the context of the PAMELA project [33, 35, 34]. However, in the PAMELA-type
design the aperture is still relatively large. Noting that the linear non-
scaling FFAG can have an extremely compact aperture over a large momentum
range has motivated significant development work on the use of this technique
for gantries. Resonance crossing and injection or extraction are obviously not
issues in single-pass gantries. The major advantages that could be realized
using this technology are reductions in the size, cost, and complexity of
treatment gantries.

A design exists for a superconducting non-scaling FFAG gantry for carbon ions
[36, 37, 38] which may reduce the weight of the magnetic components of such a
design from 135 t to around 2 t. There are also designs for compact proton
FFAG gantries utilizing either compact magnets or even permanent magnets [39,
40].

##  0.6 Summary

I hope you have gained from these notes a basic understanding of FFAG
accelerators and their potential for use in medical applications. Here we have
focused on hadron therapy applications but, of course, in the high-intensity
regime FFAGs also hold promise for other medical applications such as
radioisotope production and boron neutron capture therapy. Listed throughout
are many references which will provide the reader with a starting point for
learning more about the application of FFAG accelerators.

## Acknowledgements

The author would like to thank members of the FFAG community who have
contributed valuable references, images, and discussions. In particular, I
would like to thank S. Tygier, F. Méot, S. Machida, and D. Kelliher.

## References

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# Hertz-linewidth and frequency-agile photonic integrated extended-DBR lasers

Anat Siddharth  Laboratory of Photonics and Quantum Measurements, Swiss
Federal Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Center of Quantum Science and Engineering, EPFL, CH-1015 Lausanne, Switzerland
Alaina Anastasio  OxideMEMS Lab, Purdue University, 47907 West Lafayette, IN,
USA  Grigory Lihachev  Laboratory of Photonics and Quantum Measurements, Swiss
Federal Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Center of Quantum Science and Engineering, EPFL, CH-1015 Lausanne, Switzerland
Junyin Zhang  Laboratory of Photonics and Quantum Measurements, Swiss Federal
Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Center
of Quantum Science and Engineering, EPFL, CH-1015 Lausanne, Switzerland  Zheru
Qiu  Laboratory of Photonics and Quantum Measurements, Swiss Federal Institute
of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Center of
Quantum Science and Engineering, EPFL, CH-1015 Lausanne, Switzerland  Scott
Kenning  OxideMEMS Lab, Purdue University, 47907 West Lafayette, IN, USA  Rui
Ning Wang  Laboratory of Photonics and Quantum Measurements, Swiss Federal
Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Center
of Quantum Science and Engineering, EPFL, CH-1015 Lausanne, Switzerland  Sunil
A. Bhave  OxideMEMS Lab, Purdue University, 47907 West Lafayette, IN, USA
Johann Riemensberger<EMAIL_ADDRESS>Laboratory of Photonics and
Quantum Measurements, Swiss Federal Institute of Technology Lausanne (EPFL),
CH-1015 Lausanne, Switzerland  Center of Quantum Science and Engineering,
EPFL, CH-1015 Lausanne, Switzerland  Tobias J. Kippenberg
<EMAIL_ADDRESS>Laboratory of Photonics and Quantum Measurements,
Swiss Federal Institute of Technology Lausanne (EPFL), CH-1015 Lausanne,
Switzerland  Center of Quantum Science and Engineering, EPFL, CH-1015
Lausanne, Switzerland

Recent advances in the development of ultra-low loss silicon nitride (Si3N4
)-based photonic integrated circuits have allowed integrated lasers to achieve
a coherence exceeding those of fiber lasers [1], and enabled unprecedentedly
fast (Megahertz bandwidth) tuning using monolithically integrated
piezoelectrical actuators [2]. While this marks the first time that fiber
laser coherence is achieved using photonic integrated circuits, in conjunction
with frequency agility that exceeds those of legacy bulk lasers, the approach
is presently compounded by the high cost of manufacturing DFB, as required for
self-injection locking [2, 3], as well as the precise control over the laser
current and temperature to sustain a low noise locked operation. Reflective
semiconductor optical amplifiers (RSOA) provide a cost-effective alternative
solution, but have not yet achieved similar performance in coherence or
frequency agility, as required for frequency modulated continuous wave (FMCW)
LiDAR [4], laser locking in frequency metrology [5] or wavelength modulation
spectroscopy for gas sensing [6]. Here, we overcome this challenge and
demonstrate an RSOA-based and frequency agile integrated laser that can be
tuned with high speed, good linearity, high optical output power, and turn-key
operability while maintaining a small footprint. This is achieved using a
tunable extended distributed Bragg reflector (E-DBR) in an ultra-low loss 200
nm thin Si3N4 platform with monolithically integrated piezoelectric actuators
[7, 8]. We co-integrate the DBR with a compact ultra-low loss spiral resonator
to further reduce the intrinsic optical linewidth of the laser to the Hertz-
level - on par with the noise of a fiber laser - via self-injection locking.
The photonic integrated E-DBR lasers operate at 1550 nm, feature up to 25 mW
fiber-coupled output power in the free-running and up to 10.5 mW output power
in the self-injection locked state. The intrinsic linewidth is 2.5 kHz in the
free running state and as low as 4 Hz in the self-injection locked state. In
addition, we demonstrate the suitability for FMCW LiDAR by showing laser
frequency tuning over 1.0 GHz at up to 1 MHz triangular chirp rate with a
nonlinearity of less than 0.4% without linearization by modulating Bragg
grating using monolithically integrated Aluminum Nitride (AlN) piezoactuators.

Figure 1: Concept and device. (a) Schematic illustration of the hybrid
integrated extended-DBR (E-DBR) laser. An RSOA is butt-coupled to the piezo-
optomechanical chip comprising Bragg gratings of Si3N4 and AlN as
piezoelectric material with Mo as top and bottom metal electrodes. (b) Voltage
tuning of the center frequency of the $\kappa$=0.6 cm-1 grating. (c) Vertical
stress distribution in the Si3N4 -AlN photonic chip on applying 1 V across the
Mo electrodes. (d) Simulated coupling strength of the uniform second-order
Bragg gratings. The plot on the right shows the variation of peak reflectivity
of such gratings with changes in coupling strength and the length of the
grating. (e) False-colored Scanning Electron Microscope (SEM) images of the
top and (f) cross-section view of the MEMS-Photonic chip. Figure 2:
Performance of laser with external distributed Bragg reflector (E-DBR). (a)
Photograph of the photonic chip (D116_02_F9_C9) comprising of the 8 mm long
E-DBR with monolithically integrated AlN as piezoactuator. The inset shows a
zoomed-in image of the E-DBR with electrodes of the piezoactuator fabricated
via the ’Pull-back’ process. (b) Measured reflection of 2nd-order Bragg
gratings with different post distances and simulated grating strength. (c)
Schematic measurement setup for laser frequency noise measurement by
heterodyne beat spectroscopy with an FP-cavity stabilized and filtered ECDL.
(d) Top: Laser power as a function of RSOA injection current. Bottom:
Heterodyne frequency tuning of laser with injection current modulation. (e)
Frequency noise of E-DBR lasers with different reflection gratings. (f)
Optical spectrum of E-DBR laser featuring more than 60 dB side mode
suppression ratio. Figure 3: Frequency tuning of the E-DBR laser. (a) Photo of
optical setup used for E-DBR laser tuning. (b) Heterodyne beat spectrogram of
laser tuning using an optical microheater showing mode-hop free tuning range
over 40 GHz. (c) Optical spectrum of laser for different microheater powers.
(d) Measurement setup for response measurement of piezoelectric tuning. The
external laser (ECDL) is tuned to the side of the grating reflection peak and
the reflected light is isolated using an optical circulator and the tuning
response is recorded using a vector network analyzer (VNA). (e) Piezoelectric
tuning response of the DBR grating. The displacement field of the first FBAR
mode at 400 kHz is plotted in the inset. (f) Heterodyne beat spectrograms of
laser tuning applying 250 Vpp (peak-to-peak) triangular waveforms at 10 kHz
(left), 100 kHz (middle), and 500 kHz (right) across the AlN actuator. Figure
4: Noise characterization of the E-DBR laser self-injection locked to a high-Q
800 MHz spiral cavity. (a) Photo of the photonic chip (D128_01_F7_C11_1_01)
comprising of a 6.5 mm long E-DBR and a compact and ultra-low loss spiral
resonator with 800 MHz free-spectral range(FSR). The inset shows the pulley
coupler for coupling light to the high-Q spiral resonator. (b) Optical cavity
loss rate histogram of the spiral resonator. (c) Single sideband power
spectral density of frequency noise of E-DBR laser in the unlocked and the
self-injection-locked regimes. Inset: Heterodyne beat note spectrum (RBW: 100
kHz). (d) Relative intensity noise (RIN) of E-DBR laser in the free-running
(blue) and the self-injection locked regime (red). (e) Optical spectrum of
E-DBR laser in the self-injection locked regime achieving more than 65 dB SMSR
(RBW: 0.02 nm).

##  I Introduction

The use of narrow linewidth lasers has become widespread due to their multiple
applications in fields such as metrology [9], optical sensing [10], microwave
photonics [11], addressing atomic and molecular quantum systems such as atomic
clocks [5], quantum computers [12], and coherent communications for metro- and
long-haul interconnects. The ability to tune and control laser frequency is a
prerequisite for laser locking in numerous quantum physics experiments, fast
wavelength switching in telecommunications [13], and frequency-modulated
continuous-wave LiDAR [4, 14]. While silicon-based integrated lasers have
become more compact and commercially used in data center transceivers, their
phase noise performance so far remains inferior to continuous-wave fiber
lasers. E-DBR lasers with feedback circuits implemented in photonic integrated
circuits demonstrated significant progress in recent years [15, 16, 17, 18].
The low propagation loss and low defect densities Si3N4 integrated photonic
circuits [19] have enabled novel functionalities, traveling wave parametric
amplifiers [20], Erbium-doped amplifiers [21] or Erbium lasers [22] along with
meter long spiral resonators that have led to a new class of integrated lasers
that can, for the first time, surpass the coherence of fiber lasers in terms
of phase noise [3, 23]. Low loss Si3N4 PICs have also been integrated
monolithically with piezoelectric actuators - using both Aluminum Nitride
(AlN) and Lead Zirconate Titanate (PZT) [7, 24, 2, 25], allowing for flat,
fast (> 1 MHz), low-power, low hysteresis and linear tuning, resulting in
energy-efficient phase shifters [26], frequency-agile low-noise lasers [2],
fast-tunable soliton frequency combs [7] or non-reciprocal devices [27].

Here, we demonstrate a high-power ultra-low noise laser with rapid non-thermal
and highly linear tuning capability based on the monolithic integration of AlN
piezoelectric thin films on top of an Si3N4 waveguide Bragg grating. We
achieve a GHz-level tuning range at up to 1 MHz triangular chirp rate with an
RMS nonlinearity of less than 0.4$\%$, limited by the available electronics
and without the need for signal predistortion [28] at a fiber-coupled output
power of 25 mW. We also demonstrate that the E-DBR laser can be self-injection
locked to an ultra-low loss spiral resonator with a small free-spectral range
of 800 MHz on the same photonic chip with a footprint of only 20 mm2 and
achieve an intrinsic linewidth of less than 4 Hz and a RIN of -160 dBc/Hz at
100 kHz offset and output power of more than 10 mW.

##  II E-DBR Design and Characterization

As illustrated in Figure 1 the E-DBR laser is comprised of a reflective
semiconductor optical amplifier (RSOA) and a Si3N4 PIC with a monolithically
integrated AlN-based piezoelectric actuator. A Bragg grating with low coupling
strength $\kappa$ is needed to attain a narrowband reflection for E-DBR
operation [29].

We fabricate 8 mm long E-DBR gratings in a 200 nm thick subtractive Si3N4
platform with a 2.5 $\mu m$ wide waveguide. The grating is formed by circular
Si3N4 posts placed on either side of the waveguide [30]. We designed the Si3N4
Bragg gratings as second-order gratings because of the resolution limits of
our DUV lithography system. We set the duty cycle to 35% presenting a strong
second-order resonance with grating period $\Lambda$=997.1 nm and post
diameter 350 nm as shown in the SEM images in Figure 1(e,f). We analyzed the
Fourier components of the circular grating posts in the second order period to
calculate the coupling strength of the forward and backward components in the
grating as a function of the distance between the waveguide and post sidewalls
(cf. Supplementary Information). The coupling strength of the designed Si3N4
gratings with its peak reflectivity for different grating lengths is shown in
Figure 1d. The size and spacing of the grating posts are kept constant as no
apodization is required in weakly reflecting E-DBR gratings to suppress side
maxima to the level of the chip facet reflections. To make the PIC grating
voltage-tunable, we monolithically integrate piezo-electrical actuators based
on a 1 $\mu$m AlN thin film [24, 7] with top and bottom electrodes formed from
100 nm Molybdenum (Mo) films. The voltage tuning of the reflection from the
Bragg gratings is shown in Figure 1b and the vertical stress distribution on
applying 1 Volt to the AlN piezoactuator is shown in Figure 1c. The AlN
piezoactuators with Molybdenum (Mo) top and bottom electrodes are placed on
top of a 3.5 $\mu$m SiO2 cladding to minimize the metal-induced optical loss.

Figure 2b depicts the optical measurement of the grating reflection ranging
between 4% and 12%, which we measured using a lensed fiber and external cavity
diode laser (ECDL). The input coupling between the chip and the RSOA is
facilitated by a 4.5 $\mu$m wide and 300 $\mu$m long linear horn taper. The
taper angle of 12.4∘ is chosen to match the emission angle of the RSOA
accounting for refraction according to Snell’s law. The overlap integral of
the laser and the waveguide mode is larger than 80$\%$ (cf. Supplementary
Information). The insertion loss from lensed fiber to the 500 nm wide inverse
taper is estimated as 1 dB. The fabrication process of the Si3N4 -AlN PIC is
described in the Methods section and the fabrication process to fabricate low-
capacitance bond pads is explained in the Supplementary Information.

##  III E-DBR laser operation and noise

The RSOA is mounted on a 3D piezoelectrical translation stage and butt-coupled
to the Si3N4 PIC as shown in Figure 3a with a nominal maximal output power of
60 mW. As the drive current is increased above the threshold of 50 mA, lasing
at the grating wavelength of 1549.5 nm occurs. The laser power and frequency
tuning curves as a function of the drive current are depicted in Figure 2d. As
the current is increased to 350 mA, the laser power increases up to 25 mW with
a side-mode suppression ratio in excess of 60 dB (cf. Figure 2f), and
intermediate mode hops are observed at drive currents 110 mA, 200 mA, 240 mA,
278 mA, and 320 mA. The laser frequency tunes over the 15 GHz width of the
grating, highlighting the function of the RSOA as an intracavity phase
shifter. Hence, no additional thermal phase shifter is required on the Si3N4
chip to operate the laser. We measure the laser frequency noise power spectral
density (single-sided PSD, in units of Hz2/Hz) using heterodyne beat
spectroscopy with an external-cavity diode laser (ECDL) that is locked to and
filtered by a free-space Fabry-Pérot (FP) cavity with 1.5 GHz FSR and a
linewidth of 70 kHz (cf. Figure 2c). For the gratings with strength 1 cm-1 and
0.61 cm-1, we measured nearly identical frequency noise and intrinsic
linewidth of 3.45 kHz and for the grating with strength 0.4 cm-1, we measured
an intrinsic linewidth of 2.5 kHz.

##  IV Laser frequency tuning with integrated piezoactuator

Next, we characterize the frequency tuning characteristics of the RSOA-based
E-DBR lasers. The first laser grating is configured with Ti/Pt thermal heaters
that are fabricated atop SiO2 cladding, with a 2.5 $\mu$m offset from the
waveguide. This allows for slow but wideband tuning of the E-DBR reflection
peak and the laser power. We increase the power on the thermal microheater up
to 446.3 mW and measure up to 0.5 nm ($>$ 40 GHz) mode-hop-free laser tuning
(cf. Figure 3b,c). The heater efficiency can be further improved by optimizing
the placement of metal heaters with respect to the waveguide. The laser
operates in a longitudinal mode across the full tuning range. For fast
frequency modulation, we fabricated a Si3N4 gratings PIC with AlN
piezoactuators. We measured the small-signal frequency response of the
gratings using an auxiliary laser and a vector network analyzer (VNA). Figure
3d shows the setup of the actuation response measurement. In this measurement,
the actuation voltage derived from a vector network analyzer (VNA) is applied
to the actuator, and a laser is frequency-tuned to sit on the side of the
reflection peak, to measure the response. The reflected light is separated via
an optical circulator (CIRC) and detected on a photodiode (PD). We find that
the frequency response is flat up to the 400 kHz frequency of the fundamental
bulk mode of the 1 cm by 5 mm PIC. The inset of Figure 3(e) shows finite
element simulations of the bulk modes of the photonic chip, that match the
observed actuation resonance. At higher frequencies, the density of observable
flexural and bulk modes increases and at 17 MHz the first bulk acoustic mode
appears. The increasing mode density of the Si3N4 photonic chip towards higher
frequencies limits the flat modulation bandwidth. The excitation of spurious
mechanical modes of the chip can be avoided using the differential drive of
two piezo actuators in opposite phase [2] or by using irregular polygonal
shapes for the photonic chips [31]. Appropriate photonic packaging can also
greatly dampen the flexural and bulk modes of the chip. These methods can
extend the actuation bandwidth of the laser to greater than 10 MHz [32] up to
the first bulk acoustic resonance of the chip. However, for periodic actuation
with sinusoidal or triangular waveforms, the actuation of the narrow flexural
and bulk modes can also be averted by choice of actuation frequency such that
the (harmonic) spectra of the drive frequency do not overlap with such
mechanical modes. In Figure 3(f), we demonstrate high-speed tuning of the
E-DBR laser and show the time-frequency pattern $f_{\mathrm{het}}$ determined
by short-time Fourier transform of the heterodyne beat note of the E-DBR laser
and the reference laser with triangular modulation frequencies of 10 kHz, 100
kHz, and 500 kHz. The bottom panel row depicts the deviation of the measured
frequency tuning pattern from a perfect triangular frequency modulation with
RMS deviations ranging from 0.3% to 1% without predistortion compensation.

##  V Self-injection locking of E-DBR laser

In order to further reduce the optical linewidth of the E-DBR laser, we can
use the very low waveguide propagation loss of the 200 nm thin Si3N4 platform
and fabricate a high-Q microresonator on the same substrate as the grating and
utilize self-injection locking (SIL) to drastically reduce the optical
linewidth [2, 33, 34]. The waveguide width of the spiral resonator is 5 $\mu
m$ to decrease the field overlap with the rough sidewalls and mitigate
scattering loss [35]. The spiral resonator is comprised of three archimedean
spirals connected with curvature and curvature-derivative continuous splines
[36] and coupled to the bus waveguide using pulley couplers as shown in Figure
4a. We eliminate higher order modes in the resonators by inserting 1.5 $\mu m$
wide tapering sections in the central S-bend of each spiral. The median
intrinsic cavity linewidth $\kappa_{0}/2\pi$ of the Si3N4 cavity is 18.2 MHz,
which corresponds to a linear loss of 2.5 dB/m. Since the laser linewidth
narrowing factor, i.e. the ratio of the free-running laser linewidth to the
linewidth of the injection-locked laser is proportional to the square of the
loaded quality factor of the resonator mode to which the laser is self-
injection-locked[37], the high loaded Q of the Si3N4 spiral microresonator
(cf. Figure 4d) can significantly reduce the optical linewidth and improve the
side mode suppression ratio (SMSR), which increases from 60 dB to 65 dB (cf.
Figure 4e). We operate the spiral microresonator in the undercoupled
($\kappa_{ex}$=2 MHz) regime to minimize the power loss of the E-DBR laser
system in the locked state and hence achieve an output power in the fiber of
10.5 mW compared to 15 mW in the unlocked state. The loaded quality factor of
the resonance used for self-injection locking is 10 $\times$ 106. Locking is
achieved by tuning the RSOA drive current to tune the laser within the 10 GHz
FWHM reflection bandwidth of the grating. The dense mode spacing of the long
spiral resonator ensures frequency overlap between the grating and the
resonator. We measure the frequency noise power spectral density (PSD) of the
self-injection locked E-DBR laser by employing the same technique as described
previously (cf. Figure 2c). Here, the gratings are 6.5 mm long and have
coupling strength of 0.6 cm-1 and the frequency noise PSD before (red trace)
and after (blue trace) self-injection locking to the spiral cavity is shown in
Figure 4b. As can be seen, the laser frequency noise is decreased by more than
20 dB over the whole frequency range, resulting in a white noise floor below
1.5 Hz2/Hz and an intrinsic linewidth of about 3.8 Hz. The inset shows the
narrowing of the heterodyne beatnote spectrum upon self-injection locking. The
grey trace shows the frequency noise PSD of a commercial fiber laser and thus
verifies that the self-injection locked E-DBR laser reaches fiber laser
coherence (cf. Supplementary Information). The relative intensity noise (RIN)
spectrum of the laser in the locked and unlocked state is presented in Figure
4(c). Fabricating both the grating and the optical microresonator on the same
chip improves the relative stability of both elements and the stability of the
laser overall. We measure laser intensity noise close to the shot noise
limited regime for frequencies larger than 100 kHz, which reaches a RIN of
-160 dBc/Hz regardless of operation in the self-injection locking or free-
running regime of the E-DBR.

## Conclusion

Our E-DBR laser demonstrates the ability to manufacture frequency-agile lasers
without relying on expensive high-power DFB laser diodes by transferring the
frequency selective element of an ultra-low noise SIL laser from the III-V
chip to the Si3N4 PIC with increased fabrication tolerance due to the low
index and higher order E-DBR grating. We achieve high-power operation with
output powers up to 25 mW and GHz-level tuning range and up to 500 kHz rate.
By self-injection locking of the E-DBR laser to an ultra-low loss spiral
resonator, we achieve an intrinsic linewidth of 3.8 Hz at an unprecedented
output power of 10.5 mW for hybrid lasers with a total footprint of the
combined Si3N4 gratings and spiral resonator of merely 20 mm2, which is
considerably smaller than previous demonstrations using Si3N4 with thickness
below 100 nm [3, 23]. Placing the grating and the spiral resonator on the same
substrate and in close proximity should also greatly improve the temperature
stability of the combined system, where SIL lasers have hitherto required
temperature control in the millikelvin range for stable operation [2, 1]. In
summary, our laser system is an ideal solution for applications in frequency
metrology, fiber sensing, or spaceborne LiDAR, where high power, exceptional
coherence, and fast frequency modulation capabilities are required.

## Methods

###  V.1 Fabrication of the Si3N4 -AlN photonic chips

The Si3N4 photonic chips are fabricated with a custom subtractive waveguide
manufacturing process based on 248 nm DUV stepper lithography with fluoride-
based dry reactive ion etching chemistry. Si3N4 thin films of 200 nm are
procured externally and waveguide cores are etched using a fluoride-based dry
reactive ion etching chemistry. After etching we grow a 10 nm thin layer of
Si3N4 to conformally cover the waveguides using low-pressure chemical vapor
deposition to reduce sidewall roughness [38] and absorption by surface located
defect states induced in the etching [39]. A high-temperature anneal (11 h,
1200∘) is conducted before a 3 $\mu$m top oxide cladding is deposited. The AlN
actuators are fabricated with the same process used in Ref.[8]. The electrodes
are fabricated using a ’Pull-back’ process described in the Supplementary
Information. After chip release, a low temperature (30 min,300°C)) annealing
cycle is conducted to remove long-lived defect states induced by deep-UV
exposure.

###  V.2 Relative intensity noise measurement (RIN)

The laser relative intensity noise was measured using direct photodetection of
the laser output at 1550 nm by an impedance-matched photodetector (PD,
XPDV2120RA, responsivity $\gamma_{PD}=$0.63 A/W). In the measurement, the
input power to the photodetector from the E-DBR and the E-DBR SIL laser was
recorded as 15 mW and 10.5 mW, respectively. The output of the photodetector
was amplified by an RF amplifier (MITEQ AM-1676) to lift the detected noise
above the displayed average noise level (DANL) of the electrical spectrum
analyzer (ESA, R&S, FSW44). The gain response $G(f)$ of the RF amplifier was
characterized using a vector network analyzer with $-45$ dBm output power to
avoid gain saturation. The total noise power spectrum density (PSD) at the PD
output is given by

$\Delta P(f)=\frac{P_{\mathrm{N}}(f)}{G(f)\cdot\mathrm{RBW}},$ (1)

where $P(f)$ is the noise power measured by ESA when the amplified signal is
sent to the PD. The shot noise $S_{\mathrm{shot}}(f)=2h\nu\mathrm{\bar{P}}$ is
calculated based on the average output power. The laser RIN is then given by

$\mathrm{RIN}=10\log_{10}\frac{\Delta
P(f)}{I_{\mathrm{av}}^{2}R_{\mathrm{load}}}.$ (2)

RIN at higher frequencies can be measured using an RF amplifier with a broader
bandwidth, or without using an RF amplifier when the photocurrent is
sufficiently high to overcome the DANL and the shot noise.

###  V.3 Fabry-Pérot filtering cavity

The cavity is assembled using the 10CV00SR.70F High-Performance Concave
SuperMirrorsTM from Newport. The mirrors are glued to the ZERODUR ultra-low
expansion (ULE) glass spacer with the coating at the inner side such that they
would form a stable symmetric optical cavity. The ULE glass is used to ensure
the thermal stability of the cavity length because the cavity length
resolution could be very high. A piezo (Piezosystem Jena HH1-2515-07) is also
integrated into this assembly in order to tune the cavity resonance frequency
by changing the cavity length, and will eventually limit the cavity thermal
stability. The concave mirrors with the piezo form a very planar cavity with
Finesse $\mathcal{F}$, FSR 2.14 GHz, and linewidth
$\Delta\nu_{1/2}=\mathrm{FSR}/\mathcal{F}=71.4$ kHz. The goal is to have a
narrow linewidth, high transmission cavity at 1550 nm to lock the laser and
filter the high-frequency noise.

Author Contributions: A.S., J.R. simulated and designed the devices. R.N.W.,
Z.Q fabricated the device. A.A. fabricated integrated actuators. A.S. and G.L.
characterized the devices. A.S. did the experiments and analyzed the data with
the help of J.R., J.Z. and G.L. A.S., J.R., and G.L. wrote the manuscript with
input from all authors. T.J.K., J.R., and S.B. supervised the project.

Funding Information and Disclaimer: This publication was supported by Contract
W911NF2120248 (NINJA LASER) from the Defense Advanced Research Projects Agency
(DARPA), Microsystems Technology Office (MTO), as well as the Swiss National
Science Foundation (SNSF) through grant number 211728 (BRIDGE). A.S.
acknowledges support from the European Space Technology Centre with ESA
Contract No. 4000135357/21/NL/GLC/my and J.R. acknowledges support from the
Swiss National Science Foundation under grant no. 201923 (Ambizione).

Acknowledgments: The authors thank Guanhao Huang for assembling the Fabry-
Pérot filtering cavity. The chip samples were fabricated in the EPFL center of
MicroNanoTechnology (CMi), and in the Birck Nanotechnology Center at Purdue
University.

Data Availability Statement: The code and data used to produce the plots
within this work will be released on the repository Zenodo upon publication of
this preprint.

Correspondence and requests for materials should be addressed to T.J.K.

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# Cross-level Contrastive learning and consistency constraint for semi-
supervised medical image segmentation

###### Abstract

Semi-supervised learning (SSL), which aims at leveraging a few labeled images
and a large number of unlabeled images for network training, is beneficial for
relieving the burden of data annotation in medical image segmentation.
According to the experience of medical imaging experts, local attributes such
as texture, luster and smoothness are very important factors for identifying
target objects like lesions and polyps in medical images. Motivated by this,
we propose a cross-level contrastive learning scheme to enhance representation
capacity for local features in semi-supervised medical image segmentation.
Compared to existing image-wise, patch-wise and point-wise contrastive
learning algorithms, our devised method is capable of exploring more complex
similarity cues, namely the relational characteristics between global and
local patch-wise representations. Additionally, for fully making use of cross-
level semantic relations, we devise a novel consistency constraint that
compares the predictions of patches against those of the full image. With the
help of the cross-level contrastive learning and consistency constraint, the
unlabelled data can be effectively explored to improve segmentation
performance on two medical image datasets for polyp and skin lesion
segmentation respectively. Code of our approach is available here.

Index Terms—  Medical Image Segmentation, Semi-supervised Learning,
Contrastive Learning, Consistency

##  1 Introduction

The optimization of segmentation models based on convolutional neural networks
(CNN) is very data-hungry, which requires a large number of carefully
annotated images. Semi-supervised learning, which requires limited amount of
annotated data and large amount of unlabeled data, can help to reduce the
workload of annotation. In this topic, the reasonable exploration of unlabeled
data is critical for preventing the segmentation models from overfitting with
scarce annotated data.

A classical method for involving in unlabeled images is the self-labeling
algorithm [1], which attempt to automatically allocate pseudo labels for
unlabeled images with self-learned models. The performance of this kind of
methods is dependent to the quality of pseudo labels. Besides, they may be
severely influenced by the confirmation bias since the self-labeling strategy
is difficult to rectify the incorrect predictions. The other typical manner
for taking advantage of unlabeled images is the self-supervised learning [2]
which is effective visual representations from unlabelled data with upstream
pretext tasks. These pretext tasks such as relative position prediction [2]
and rotation prediction [3] usually can be easily annotated and have
underlying relations to the understanding of images. They can be exerted to
enhancing the representation capacity of segmentation models. Owing to the
high interpretability, contrastive learning attracts extensive research
interest and brings about significant progress in self-supervised learning.

Fig. 1: The motivation and the core idea of this paper. We apply contrastive
learning or consistency constraint between features or predictions obtained
from the full image and local patches, thus encouraging the network to learn
the local characteristics in semi-supervised medical image segmentation.

Contrastive learning (CL) algorithms [4, 5, 6] usually regard similar image
pairs as positive instances and dissimilar image pairs as negative instances.
During training, positive instances are encouraged to get closer to each other
while negative instances are constrained to be mutually exclusive in the
latent feature space. The majority of CL approaches in image classification
tasks follow the pipeline as SimCLR [5], by applying contrastive metric
learning constraints on image-level embeddings. However, for pixel-level dense
prediction tasks like image segmentation, such image-level CL approaches
ignore the spatial relation information of the image. For exploring spatial
relation information, [7] and [8] construct positive instances with pairs of
points under different augmentations but having the same actual locations, and
select the negative instances according to semantic inconsistency. [9]
combines both image-level and point-level CL, and relies on the correspondence
between different views of the same image to sample positive instances for the
point-level CL. Similarly, [10] exploits both image-level and point-level CL
to enhance the representation capacity for semi-supervised medical image
segmentation. However, as far as we know, existing CL methods merely compare
features within the same level, namely constraining image-level, patch-level
or point-level features. The more complicate cross-level similarity cues
remains under-explored.

In medical image analysis, local characteristics such as textures and
smoothness are critical factors for the identification of the target objects
like polyps, tumors and organs. These features are gradually diluted as the
forward propagation reaches deeper layers in CNN models. Aiming to strengthen
the representation capacity on these local features, we propose a cross-level
contrastive learning scheme for semi-supervised medical image segmentation.
Provided a full image, a U-Net [11] is employed for extracting the global
dense featuremap and deriving of dense predictions. For constraining the
receptive field within a limited space, the full image is decomposed into
small patches which are successively fed into the CNN model for generating the
local dense featuremap. Afterwards, for each point in the global dense
featuremap, the point with the same location in the local dense featuremap is
regarded as the positive sample, and a set of points at other locations are
randomly selected from the global dense featuremap as negative samples. Our
proposed cross-level contrastive learning scheme is implemented based on the
above tuple formulation strategy. Furthermore, for fully taking advantage of
the cross-level relation knowledge from the perspective of semantic content,
we apply a consistency constraint between dense predictions of the full image
and local patches. The semantic consistency between the two prediction levels
can help to promote the intra-class compactness for both global and local
dense features. We evaluated our approach on two publicly available medical
image segmentation datasets including a polyp segmentation dataset (Kvasir-SEG
[12]) and a skin lesion segmentation dataset (ISIC 2018 [13]). Elaborate
experiments show that our approach outperforms existing semi-supervised
medical image segmentation methods significantly. Main contributions of this
paper are summarized as follows.

  * 1)

We devise a cross-level contrastive learning algorithm to enhance the
representation capacity for local features in semi-supervised semantic
segmentation.

  * 2)

A cross-level consistency constraint is devised to transmit the
representational similarity to the final prediction and explore the semantic
relation across levels.

  * 3)

Experiments on poly and skin lesion segmentation tasks indicate that our
proposed method derives of evidently better performance that state-of-the-art
semi-supervised segmentation methods.

Fig. 2: The overview of our approach for semi-supervised medical image
segmentation.

##  2 Method

This paper is targeted at tackling the semi-supervised medical image
segmentation task. Suppose we have a medical image dataset which is
constituted by $N$ images $X_{L}=\\{\mathcal{X}_{i}\\}_{i=1}^{N}$ with pixel-
wise annotations $Y_{L}=\\{\mathcal{Y}_{i}\\}_{i=1}^{N}$, and $M$ images
$X_{U}=\\{\mathcal{X}_{i}\\}_{i=N+1}^{N+M}$ without annotations. The goal is
to learn a model with strong generalization capacity on unseen testing images.
The framework of our approach is shown in Fig. 2, which is composed of three
losses including a supervised loss, a dense cross-level contrastive loss, and
a dense cross-level consistency constraint.

###  2.1 Dense Cross-level Contrastive Learning

Different with many existing works that extract representations just in image
level or patch level, we propose a novel dense cross-level contrastive loss.
In our approach, backbone takes both global image and local patches as input,
extracting features to compare, thus sufficiently exploring the
representational information of respective details in the image.

For input image $\mathcal{X}\in\mathbb{R}^{H*W*3}$, we use a U-Net structure
backbone $f$ to extract the feature map $\mathcal{Z}=f(\mathcal{X})$.
Meanwhile, for each input image $\mathcal{X}$, we decompose one image into
$n*n$ patches
${x}_{i}\in\mathbb{R}^{\frac{H}{n}*\frac{W}{n}*3},i\in[1...n*n]$. Thereafter,
we use the same backbone to extract a local feature map for each patch,
${z}_{i}=f({x}_{i})$. Both global feature map $\mathcal{Z}$ and local feature
maps $\\{{z}_{i}\\}$ are fed into a same projection head $p$, constituted by
three convolutional layers, resulting to features
$p(\mathcal{Z})\in\mathbb{R}^{n*n*D}$ and $p({z}_{i})\in\mathbb{R}^{1*1*D}$.

We expect that the representations obtained from patches are similar to those
derived from the entire image. Therefore, we design a patch-image contrastive
learning loss. Specifically, we utilize a projection head to project feature
$\mathcal{Z}$ to $p(\mathcal{Z})$, where the perceptual fields of each pixel
in $p(\mathcal{Z})$ does not overlap on $\mathcal{Z}$. The feature map of the
entire image $p(\mathcal{Z})$ is decomposed into $n$*$n$ blocks, denoted as
$p(\mathcal{Z})_{i}\in\mathbb{R}^{1*1*D},i\in[1...n*n]$. For the two feature
sets, $\\{p({z}_{i})\\}$ and $\\{p(\mathcal{Z})_{i}\\}$, we use the following
formula to calculate the contrastive loss:

$\displaystyle\mathcal{L}_{\text{contrast}}(i)=$
$\displaystyle-\log\frac{\exp\left[p(\mathcal{Z})_{i}\cdot
p({z}_{i})/\tau\right]}{\exp\left[p(\mathcal{Z})_{i}\cdot
p({z}_{i})/\tau\right]+\sum\limits_{n\neq i}\exp\left[p(\mathcal{Z})_{i}\cdot
p(\mathcal{Z})_{n}/\tau\right]}$ (1)

where $p(\mathcal{Z})_{i}$ is a pixel of the projected feature map
$p(\mathcal{Z})$; $p(\mathcal{Z})_{i}$ and $p({z}_{i})$ are the feature
vectors in the same position $i$, and are regarded as the positive pair of
contrastive learning; $\tau$ is a scalar temperature hyperparameter. The loss
is averaged over all pixels in the feature map.

###  2.2 Dense Patch-image Consistency Learning

Contrastive learning is able to enhance the similarity between patches and
image at the feature level. In order to transmit the representational
similarity to the final prediction and further enhance the context-independent
prediction ability of the network, we apply a unsupervised consistency loss
between the complete prediction and partial predictions of an image.
Specifically, for global feature map $\mathcal{Z}$ and local feature maps
$\\{{z}_{i}\\}$ of the same image, we utilize a prediction head $h$, a single
convolutional layer with the convolutional kernel of 1*1., to generate the
segmentation predictions. We use mean-square error to measure the difference
between the predicted results:

$\displaystyle\vspace{-4mm}\small\mathcal{L}_{\text{consist}}=\sum\limits_{i}MSE(h(z_{i}),h(\mathcal{Z})_{i})\vspace{-4mm}$
(2)

where $h(\mathcal{Z})_{i}$ means the $i$-th patch having the same position as
$z_{i}$ in the prediction result of the entire image.

###  2.3 Overall

The overall loss function is formed by summing the supervised loss and the
unsupervised loss.

$\displaystyle\small\mathcal{L}_{all}=\mathcal{L}_{sup}+\alpha\mathcal{L}_{contrast}+\beta\mathcal{L}_{consist}$
(3)

For the supervised loss on labeled images, we adopt the combination of a Dice
loss and a cross entropy loss:

$\displaystyle\small\mathcal{L}_{sup}=\frac{1}{2}{\left({DICE}\left(h(\mathcal{Z}),\mathcal{Y}\right)+{CE}\left(h(\mathcal{Z}),\mathcal{Y}\right)\right)}$
(4)

The practical training process is consisting of two stages. In the first
stage, we use the supervised and contrastive loss to train the network
backbone, aiming to enable the network to extract generalized representations.
Namely, $\alpha$ and $\beta$ is set to 1 and 0 respectively. In the second
stage, we use consistency loss to further utilize cross-level relations in the
image and strengthen the network’s ability of capturing useful partial
information. Namely, $\alpha$ and $\beta$ is set to 0 and 1 respectively. The
overall training process has 300 epochs, including 100 epochs for the first
stage and 200 epochs for the second stage.

##  3 Experiments

We evaluate our proposed semi-supervised learning approach on two medical
segmentation datasets. Comparisons with existing state-of-the-art methods
including UAMT [14] and URPC [15] are conducted to showcase the efficacy of
our method.

###  3.1 Datasets

Kvasir-SEG dataset [12] is a polyp segmentation dataset, which contains 1,000
white-light colonoscopy images with pixel-level manual labels. We randomly
choose 60% images as the training set, 20% images as the validation set, and
the remaining 20% images as the testing set. Among the training set, we select
20% (120) images as labeled data and the other 80% (480) as unlabeled data.

ISIC 2018 dataset [13] is a skin lesion segmentation dataset. We use 2594 skin
images with pixel-level lesion labels. We randomly choose 60% images as the
training set, 20% images as the validation set, and the remaining 20% images
as the testing set. Among the training set, we select 10% (156) images as
labeled data and the other 90% (1400) as unlabeled data.

###  3.2 Implementation Details and Evaluation Metric

Our approach is implemented in Python with PyTorch library. We used NVIDIA
Tesla A100 GPUs for training and testing. The network is trained using Adamw
[16] optimizer with default parameters setting. Every training mini-batch is
consisting of four annotated images and four unannotated images. All images
are resized into 320*320. For fair comparison, the same backbone model,
supervised training loss, dataset division and epochs are used for all
methods.

We use mean absolute error (MAE), mean intersection over union (mIoU) and Dice
coefficient of foreground to evaluate the performance of each method. For each
experiment, we randomly select labeled images for 3 times and report the mean
and standard deviation.

###  3.3 Results

Table 1: Comparison of our approach with state-of-the-art semi-supervised medical image segmentation approaches on Kvasir-SEG dataset with 120 labeled images and 480 unlabeled images. Method | MAE | Dice | mIoU
---|---|---|---
U-Net | $8.16$ | $67.03\pm 0.87$ | $57.08\pm 0.64$
UAMT | $7.40$ | $69.87\pm 0.61$ | $59.43\pm 1.11$
URPC | $7.10$ | $70.83\pm 0.24$ | $61.50\pm 0.71$
ours (w/o consist) | $7.73$ | $70.37\pm 0.94$ | $59.73\pm 0.97$
ours (w/o contrast) | $7.77$ | $72.17\pm 1.46$ | $61.10\pm 1.56$
ours (all) | $6.63$ | $73.63\pm 0.25$ | $63.50\pm 0.16$

Table 1 shows the results of the semi-supervised segmentation algorithms on
Kvasir-SEG dataset. Compared with the baseline and other existing SOTA
methods, our approach achieves a significant and consistent improvement. In
addition, if we just use consistency loss without contrstive loss, or use
contrstive loss without consistency loss, all the evaluation metrics are
improved but not stable. The result of the ablation experiments demonstrates
that both of our proposed loss functions can significantly improve the
performance of the network. Table 2 shows the results in ISIC 2018 dataset.
Our approach also shows consistent effectiveness on ISIC 2018 dataset.

Table 2: Comparison of our approach with state-of-the-art methods on ISIC 2018 dataset with 156 labeled images and 1400 unlabeled images. Method | MAE | Dice | mIoU
---|---|---|---
U-Net | $8.03\pm 0.98$ | $79.2\pm 0.93$ | $70.77\pm 1.16$
UAMT | $7.13\pm 0.94$ | $80.70\pm 1.87$ | $72.16\pm 2.10$
URPC | $7.23\pm 0.78$ | $80.13\pm 2.21$ | $71.83\pm 2.19$
ours (all) | $6.90\pm 0.35$ | $82.47\pm 0.73$ | $74.00\pm 0.57$

###  3.4 Visualization

Figure 3 shows the visual segmentation results of our method on images of two
datasets. Three samples are presented for each dataset. It can be concluded
from the images that existing methods can effectively process images with
simple backgrounds and conspicuous objects However, these methods cannot fully
segment all polyp or skin lesion regions when the object and background look
similar, or suppress all background regions when the background is cluttered.
In contrast, our approach can robustly better segment out object regions from
background, even in complex cases.

Fig. 3: Qualitative results for different approaches. Images in the upper
three columns are from the Kvasir-SEG dataset, and images in the lower three
columns are from the ISIC 2018 dataset.

##  4 Conclusion

In this work, we investigate the problem of semi-supervised medical image
segmentation to reduce the human workload of labelling medical image data. We
propose a novel semi-supervised segmentation approach by introducing a cross-
level contrastive learning scheme and a cross-level consistency constraint.
Our framework can use unlabelled data by exploring the intrinsic relationships
across levels within the image. Extensive experiments on two public medical
datasets show that our approach outperforms state-of-the-art semi-supervised
learning methods consistently.

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## Acknowledgments

This work is supported in part by the Guangdong Basic and Applied Basic
Research Foundation under Grant No. 2020B1515020048, in part by the National
Natural Science Foundation of China under Grant No.61976250 and No.62003256,
in part by the Guangzhou Science and Technology Project under Grant
202102020633, and in part by the Guangdong Provincial Key Laboratory of Big
Data Computing, The Chinese University of Hong Kong, Shenzhen.

## Compliance with Ethical Standards

This is a numerical simulation study for which no ethical approval was
required.