CCAP-TN-ACCL-06
|  |  April 13, 2021

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Anomalous Beam Transport Through Gabor (Plasma) Lens Prototype

T. Nonnenmacher †,1, T.S. Dascalu †,1, R. Bingham 2,3, C.L. Cheung 1, H.T. Lau
1, K.R. Long 4,3, J. Pozimski 4,3, C. Whyte 2,

1. | Department of Physics, Imperial College London, Exhibition Road, London SW7 2AZ, UK
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2. | Department of Physics, SUPA, University of Strathclyde, 16 Richmond Street, Glasgow G4 0NG, UK
3. | STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot OX11 0QX, UK
4. | John Adams Institute for Accelerator Science, Imperial College London, London SW7 2AZ, UK
† | Corresponding author, Email<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> An electron plasma lens is a cost-effective, compact, strong-focusing
> element that can ensure efficient capture of low-energy proton and ion beams
> from laser-driven sources. A Gabor lens prototype was built for high
> electron density operation at Imperial College London. The parameters of the
> stable operation regime of the lens and its performance during a beam test
> with 1.4 MeV protons are reported here. Narrow pencil beams were imaged on a
> scintillator screen 67 cm downstream of the lens. The lens converted the
> pencil beams into rings that show position-dependent shape and intensity
> modulation that are dependent on the settings of the lens. Characterisation
> of the focusing effect suggests that the plasma column exhibited an off-axis
> rotation similar to the $m=1$ diocotron instability. The association of the
> instability with the cause of the rings was investigated using particle
> tracking simulations.

##  1 Introduction

One of the principal challenges that must be addressed to deliver high-flux
pulsed proton or positive-ion beams for many applications is the efficient
capture of the ions ejected from the source. A typical source produces protons
with kinetic energies of approximately 60 keV [1, 2, 3] and ions with kinetic
energies typically below 120 keV [4, 5]. At this low energy the mutual
repulsion of the ions causes the beam to diverge rapidly. Capturing a large
fraction of this divergent flux therefore requires a focusing element of short
focal length. Proton- and ion-capture systems in use today employ magnetic,
electrostatic, or radio frequency quadrupoles, or solenoid magnets to capture
and focus the beam [6, 7, 8, 2].

Laser-driven proton and ion sources are disruptive technologies that offer
enormous potential to serve future high-flux, pulsed beam facilities [9, 10,
11, 12, 13, 14, 15, 16]. Possible applications include proton- and ion-beam
production for research, particle-beam therapy, radio-nuclide production, and
ion implantation. Recent measurements have demonstrated the laser-driven
production of large ion fluxes at kinetic energies in excess of 10 MeV [17,
18, 19, 20]. The further development of present technologies and the
introduction of novel techniques [21, 22] makes it conceivable that
significantly higher ion energies will be produced in the future [23, 24, 13].
By capturing the laser-driven ions at energies two orders of magnitude greater
than those pertaining to conventional sources, it will be possible to evade
the current space-charge limit on the instantaneous proton and ion flux that
can be delivered. While in some situations the high divergence of laser-driven
ion beams can be reduced [25, 26], for the tape-drive targets proposed for
medical beams [20, 16] it necessary to capture the beam using a strong-
focusing element as close to the ion-production point as possible.

An attractive approach to providing the strong-focusing element required to
capture the low-energy ($\sim 15$ MeV) ion flux produced in the laser-target
interaction is to exploit the strong focusing forces that can be provided by a
cloud of electrons trapped within a cylindrical volume by crossed electric and
magnetic fields. Such an electron-plasma lens was initially proposed by Gabor
in 1947 [27]. The use of electron-plasma lenses of the Gabor type to capture
and focus proton and ion beams has been studied by a number of authors [28,
29, 30, 31, 32, 33, 34]. Such a lens has the potential to decrease the
magnetic field required in the first focusing element by a factor of more than
40 compared with that required for a conventional beam-capture solenoid of the
same focusing strength [35]. Consider, for example, a 25 MeV proton beam. A
magnetic field strength of 0.06 T is required to achieve a focal length of 1 m
using a Gabor lens with an anode length of 0.3 m. To achieve the same focal
length using a solenoid requires a field of 2.6 T. Such strong focusing is
particularly important when beams are produced with a large divergence angle
[14]. The Gabor lens is therefore the ideal focusing element by which to
capture a laser-accelerated proton or ion beam. Its compactness and relatively
low price are key if it is to be exploited in particle-beam therapy
facilities. Furthermore, it has been shown in simulation that a Gabor-lens-
based system is capable of capturing laser-generated proton beams at energies
as high as 250 MeV, the energy required to serve a proton-beam therapy
facility [13].

Following the initial proposal by Gabor, several groups have reported stable
operation of a space-charge lens under a variety of electrode and magnetic
field configurations [36, 28, 29, 37, 30]. Experiments with ion beams
confirmed the focusing capability of the Gabor lens and observed emittance
growth [30, 38, 34]. The mechanism for electron production and the
inhomogeneity in the electron density within the lens were believed to cause
the observed growth in emittance [38].

The focusing strength of a Gabor lens is determined by the electron density.
The theoretical maximum electron density is related to the electric and
magnetic field strength [13]. Careful design of the field configuration
allowed certain lenses to operate at electron densities of 61% of the
theoretical maximum [35]. At high pressure, the electrons are lost due to the
radial expansion of the plasma driven by collisions with neutral atoms. At low
pressure, this radial transport can be caused by small azimuthal asymmetries
in the applied electric or magnetic fields [37]. Further work was directed
towards the design of an electrostatic lens for the space-charge-compensated
transport of a high intensity heavy ion beam [39]. In this case, the absence
of emittance growth due to the lens was reported [39].

Advances in simulation and finite-element analysis have been exploited to
calculate the expected focusing strength of a space-charge lens and to study
the resulting phase-space transformation on a beam passing through the lens.
Good agreement between experimental results and beam-transport simulations
[35, 34] suggest that plasma instabilities are a likely cause of beam
aberrations. Experimental observations and numerical results [40] have
confirmed that the confined plasma is vulnerable to the diocotron instability
[41]. Further studies are required to characterise the configurations under
which a Gabor space-charge lens operates in a stable regime.

Plasma-lens focusing for electron beams is being developed by the CERN Linear
Electron Accelerator for Research (CLEAR) collaboration [42]. Evidence of
aberrations in the CLEAR lens due to radial non-uniformity of the plasma
temperature has been observed [43, 44]. Previous work at the Stanford Linear
Accelerator Center (SLAC) demonstrated the plasma-lens focusing of 28.5 GeV
electron [45] and positron [46] beams. Numerical simulations were able to
describe the observed non-linear focusing force in this experiment [47].
Discharge-capillary active plasma lenses were also investigated as compact
devices for focusing 100 MeV-level electron beams produced by a gas-jet-based
laser-plasma accelerator. Both weak and strong chromatic effects [48] were
observed with the potential to cause emittance degradation [49]. Research at
SLAC seeks to demonstrate the use of such a plasma lens for staging in plasma
wakefield acceleration or in radially-symmetric final focusing for linear
colliders [50]. As compact and tunable devices, active plasma lenses [51, 52]
are a promising solution for the extraction and transport of the witness bunch
while removing the driver without loss of beam quality [53, 54, 55]. Finally,
electron plasmas were studied in multicell traps [56, 57] to develop methods
by which the number of accumulated positrons could be increased compared to
the present limits coming from the requirement for long-term confinement.

In this paper we report the performance of a Gabor lens prototype constructed
at Imperial College London. The lens was exposed to 1.4 MeV protons at the
Surrey Proton Beam Centre [58]. The effect on the beam is presented and
compared to the results of a simulation of the impact of plasma instabilities
on the focusing forces produced by the lens.

##  2 The Gabor Lens

Figure 1:  Internal structure of the IC Gabor lens viewed in longitudinal
cross-section. The main components are: 1-central anode, 2-end electrodes,
3-end flanges, 4-vacuum tube, 5-pancake coils, 6-outer tube, 7-high-voltage
feed-through.

A schematic of the prototype Gabor lens is shown in figure 1. The total length
of the lens, from end flange to end flange, was 540 mm. The central anode was
formed of a copper cylinder with an inner diameter of 85.7 mm and a length of
444 mm. The copper cylinder was 1.6 mm thick and had four rows of four 10 mm
diameter holes forming lines along the axis of the cylinder spaced by
$90^{\circ}$ so that the volume inside the anode would be evacuated
efficiently. Two ceramic isolating spacers were used to maintain the position
of the central high-voltage electrode and to electrically isolate it from the
vacuum tube. A 15 mm copper high-voltage (HV) connector was soldered to the
central electrode to provide a socket for the high-voltage feed-through
designed for voltages up to 60–65 kV.

The two end electrodes were formed of two copper cylinders with a length of 34
mm, an inner diameter of 66.7 mm, and a wall thickness of 1.6 mm. The ends of
the cylinders were manufactured with rounded edges to reduce the likelihood of
sparking, with a minimum gap of 16.8 mm to the high-voltage anode. The end
electrodes were press fitted into the mild steed end flanges. The outer tube
and end flanges were connected to ground.

The configuration of the pancake coils is shown in figure 1. The input and
output tails of the coils exited through the 50 mm gap in the outer tube of
the lens. A water cooled copper conductor was used with a square cross-
sectional area of 10.87 mm2. The base configuration of the coil included four
windings. The number of windings was locally increased to seven at specific
positions to generate a more uniform magnetic field. A maximum magnetic field
of approximately 55 mT was achieved at 45 A.

A power supply (Glassman LP 60–46) was used to regulate the current that flows
through the coils. Typical values for the current were in the range of 14 A to
30 A. The voltage for the central electrode was provided by a high-voltage
supply of the Glassman Series FR type with typical values of between 8 kV and
20 kV.

The pumping system was comprised of a roughing pump (Edwards 5) and a turbo
molecular pump (Leybold Turbovac 151) with a pressure gauge (Leybold Penninvac
PTR 90 N) and a pressure-gauge monitor (Leybold Graphix One). The lowest
pressure achieved was $3\times 10^{-7}$ mbar, with pressures of up to $3\times
10^{-5}$ mbar when a non-neutral plasma was established inside the lens.
Settings of HV and current that produced a stable plasma could be
distinguished clearly from those which gave rise to an unstable plasma at low
pressure ($\sim 10^{-7}$ mbar). For higher pressures, this distinction was
more difficult to observe.

Plasma in the lens was produced by increasing the high voltage applied to the
anode and the current in the magnetic coils. A significant increase in
pressure was observed when a stable plasma was first established in the lens.
Simulation of the plasma discharge within the lens indicated that a high
electron density, $\sim 5\times 10^{-7}$ Cm-3, was produced.

##  3 Plasma Characterisation

The operation of the lens was tested over the range of available anode
voltages and coil currents to identify the regime for which a stable plasma
could be produced. Measurements of the plasma in the lens were made using the
Medusa voltage sensor shown in figure 2. The sensor detects the current of
ions and electrons discharged by the lens and was composed of 16 equal
segments with a total area of 122.5 mm2. The detector segments were connected
either in concentric circles or in a sector arrangement, with four segments
combined and fed into one channel of an oscilloscope. The Medusa detector was
used to characterise the range of high voltage and current settings which
would produce a stable plasma within the lens. A schematic diagram of the
experimental setup for these primary studies is given in figure 3. For a
constant current through the coils of the lens, producing a constant magnetic
field, the high voltage was increased from zero until plasma was produced in
the lens. The presence of a stable plasma in the lens was indicated by a
steady voltage read from the Medusa detector. The high voltage was then
increased further, until instability in the plasma, characterised by sparking,
was observed as an extreme variation in the output voltage reading.

Figure 2:  The segmented detector used for measuring the current of electrons
and ions exiting the Gabor Lens. The detector is divided in to 16 sections of
equal area which were combined in sector (middle) or concentric circle (right)
arrangements.

Figure 4 shows the amplitude responses observed in the Medusa detector that
are typical of three modes of operation:

  * •

Plasma off: high voltage and current through coils below the threshold for
plasma to be produced;

  * •

Stable plasma: plasma produced with high voltage below $25$ kV and current
below $27$ A; and

  * •

Unstable plasma: plasma produced with higher magnetic field causing
considerable sparking and therefore large variations in the output amplitude.

Figure 3:  Schematic of the Gabor lens, current detector, and oscilloscope.
The high voltage supply maintains the drop across the electrodes. Expelled
ions hit the detector, and the current signal is converted to a voltage output
signal in the oscilloscope.

The mean and standard deviation of the Medusa voltage measurement of the
different plasma regions are shown in table 1. The mean increases slightly
when the plasma is switched on, while the standard deviation remains largely
unchanged. As the current applied through the coils becomes large, and the
unstable region is reached, the mean and standard deviation rose and a large
amount of noise was observed. This is shown in figure 4, where the level of
noise is similar with the plasma off and the plasma on. The noise increases
appreciably only when the unstable region is reached. The frequency spectrum
of the voltage signal was studied in each of the three modes of operation. In
the unstable region, the low-frequency noise is increased significantly while
the high-frequency noise remains largely unchanged. The characterisation of
the operating regimes described in this section was used to verify that the
lens produced a stable plasma during the beam test.

Figure 4:  Amplitude of signal from the Medusa detector in three regions: plasma off, plasma on, and plasma on in unstable region. The time range is given 0.0004 s, and the time resolution of the measurement is $4\times 10^{-7}$ s. The voltage range is 0.5 V with a resolution of 0.01 V. Table 1:  Mean and Standard Deviation values for Plasma Off, On, and Unstable regions shown in figure 4.  | Mean (V) |  | Standard
---
Deviation (V)
Plasma Off | 0.008 | 0.019
Plasma On | 0.114 | 0.020
Plasma Unstable | 0.797 | 0.133

##  4 Beam Test Setup

The prototype Gabor lens was exposed to proton beams with a kinetic energy of
1.4 MeV at the Ion Beam Facility at the University of Surrey [58] in October
2017. Schematic diagrams of the two setups used in the beam tests are shown in
figure 5. The proton beam entered the lens through a section of evacuated beam
pipe. The length of the drift on the first day of data taking was
approximately 380 mm (Setup 1 in figure 5). On the second day the length of
the drift was extended to approximately 680 mm to exploit the divergence of
the beam to illuminate a larger area at the front face of the lens (Setup 2 in
figure 5).

Figure 5:  Schematics of the day 1 beam test setup, Setup 1 (top) and the day
2 beam test setup, Setup 2 (bottom). The setup includes the Gabor lens,
aperture, and beam pipes.

Narrow “beamlets” were created using an aperture plate placed at the entrance
to the lens (see figure 6). The holes in the aperture plate were 2 mm in
diameter and arranged in a pattern designed to minimise the overlap of the
outgoing beamlets under a focusing force that is rotationally symmetric about
the beam axis.

Figure 6:  Photograph of the aperture placed in the beamline upstream of the
Gabor Lens. 30 holes of 2 mm width are drilled in a symmetrical pattern around
one further hole on the axis. The surrounding holes are pitched at an angle of
20∘.

A further section of evacuated beam pipe of length 670 mm was attached to the
downstream flange of the prototype lens. A phosphor screen was installed on
the downstream flange as indicated in figure 5. The phosphor screen used was a
P43 phosphor surface on an aluminised pyrex substrate with an effective area
of diameter 44.9 mm and a thickness of 10-15 $\mu$m. Photographs of the image
of the beam on the phosphor screen were acquired with a DSLR camera using an
exposure long compared to the beam spill.

Figure 7:  Photograph and schematic of the phosphor screen used for imaging
the beam. The screen was composed of a P43 phosphor surface on a substrate of
aluminized pyrex, and the scale and resolution of the screen are shown on the
schematic.

##  5 Characterisation of lens performance

The lens was set up on the beam line and stable operation of the lens was
established as described in section 3. The voltage-current characteristics of
the lens measured using the Medusa detector with the lens on the beam line are
compared with those measured at Imperial in figure 8. The two sets of
measurements show similar features indicating that the lens was operating in a
similar manner to the operation in the lab at Imperial. Figure 8 was used to
determine that the lens was operating in the stable regime.

Figure 8:  Voltage-current characteristic of the Gabor lens measured with the
lens on the beam line (blue crosses) compared to measurements in the
laboratory at Imperial (orange crosses). During the beam test the gas pressure
in the lens was 10% higher than the pressure at which the lens operated in the
laboratory.

Images of beam impinging on the phosphor screen were taken with the lens
turned off in both the Setup 1 and Setup 2 configurations (see figure 9).
Distinct “spots” are visible that correspond to the beamlets produced by the
holes in the aperture plate. The longer drift introduced in the Setup 2
configuration results in a larger number of beamlets being observed at lower
magnification than in the Setup 1 configuration. The central axis of the lens
passes through the centre of the second beamlet from the right in both Setup 1
and Setup 2. Measurement of the diameter of the beam spots and the centre-to-
centre distances allowed the divergence of the beam to be determined. The
divergence in the $x$ and $y$ directions was determined to be $x^{\prime}=1.6$
mrad and $y^{\prime}=0.5$ mrad, respectively.

Figure 9:  Observed camera image of the 3 beam spots beyond the aperture in
the Setup 1 configuration, left, and with 6 beam spots beyond the aperture in
the Setup 2 configuration, right. Both images were taken with the lens off.
The dashed lines indicate the beam axis and the central beam spot.

Images of the beam with the lens operating at a voltage of 20 kV are shown for
currents of 28 A and 33 A in figure 10. The figure shows that the effect of
the lens is to produce ring-like structures on the phosphor screen. The
diameter and eccentricity of the rings increases with radial distance from the
beam axis. The brightness of the image is observed to vary around the ring.
This effect is seen more clearly in figure 11 which shows the intensity
distribution plotted as a function of position on the phosphor screen. The
ring-like structure of the beam spots is clearly visible against the low
background and the non-uniformity of the intensity distribution is also
observed. The integrated intensity as well as the intensity distribution
around the ring differs from ring to ring. Similar behaviour has previously
been reported in [59, 60].

Figure 10:  Observed camera image of the 6 beam spots beyond the aperture in
the Setup 2 configuration with the lens on at a current through the coils of
28 A, left and 33 A, right. Both images were taken with a lens voltage of 20
kV. An additional spot is visible on the left hand side, as the lens focusing
is increased. The dashed lines indicate the beam axis.  Figure 11:  3D plot of
the scintillator measurement of the 6 beam spots in the Setup 2 configuration,
with the lens on. The image is shown looking down along the beam axis.  Figure
12: PIC [61] simulation of a plasma instability that was observed to focus the
proton pencil beams into rings. Left: The averaged density of plasma in a
transverse cross-section through the lens at four different time steps during
the evolution of the instability. The green spots mark the entry position of
the pencil beams. Right: Number of macroparticles hitting a screen 67 cm
downstream of the lens.

To understand better the distribution of the space charge inside the lens and
the plasma dynamics during the beam test, a particle-in-cell (PIC) code VSIM
[61] was used to study the main characteristics of a plasma instability that
converts pencil beams into rings. The geometry of the central anode, the two
end electrodes, and the vacuum tube was reproduced in VSim. A 3D magnetic
field map obtained using the actual configuration of the coil was imported
from a separate finite-element-analysis package. Thus, the field map described
the radially confining magnetic field. The voltage on the central anode was
set such that the longitudinal-confinement condition imposed a limit on the
maximum electron density which was equal to that imposed by the radial-
confinement theory.

The electron cloud was modelled [62] as a collision-less plasma using the
particle-in-cell (PIC) method in VSim [61]. The electrostatic potential was
calculated from the charge density by solving Poisson’s equation on a 3D grid.
Then, the macro-particles were advanced in time according to the Lorentz
equation. The time step was set to $0.2\,\mathrm{ns}$ (corresponding to
$\approx 6\tau_{c}$, where $\tau_{c}=2\pi m/eB$ is the cyclotron period) to
track the electron movement correctly. The grid had a transverse cell size
$\Delta x=0.14\lambda_{D}$ and longitudinal cell size $\Delta
z=0.8\lambda_{D}$, where $\lambda_{D}$ is the Debye length. Each macro-
particle represented approximately 2000 real electrons.

To drive an instability the electrons were loaded at the beginning of the
simulation as a plasma column displaced from the central axis or with a large
positive radial gradient in the electron density. The six, 1.4 MeV, proton
pencil beams were modelled as macro-particles entering the lens and left to
propagate through the electron plasma. The simulation registered the
distribution of the proton macro-particles that hit the exit plane of the
lens. These macro-particles were then tracked separately through an additional
drift space of 67 cm.

The protons were propagated through the electron plasma using VSIM to simulate
the impact of a number of plasma instabilities that have been observed
experimentally [63, 64]: a hollow electron ring and the diocotron instability
[41]. The diocotron modes observed in the simulations corresponded to higher
order modes with an azimuthal mode number $\mathrm{m}>1$. Within the range of
electron densities between $1\times 10^{13}$ m-3 and $1\times 10^{15}$ m-3, no
ring formation was observed in the simulations. The instabilities named above
show good azimuthal symmetry during their evolution and, hence, would focus
the pencil beams to the same position at all times. A displacement of the bulk
of the plasma from the central axis and the rotation of the focusing centre
are necessary for the formation of rings. An example of such an instability is
shown in figure 12 and consists of a region of high electron density and a
region of low electron density that rotate around the beam axis. Figure 12
shows the result of tracking six proton pencil beams through the instability.
Rings are formed on a screen downstream of the lens. In the PIC simulation,
the instability was gradually damped due to the absence of a driving
mechanism. As the bulk of the plasma approaches the central axis, each pencil
beam is focused on a ring with a radius that decreases and a centre that
shifts with time. Thus, in the simulation, each pencil beam is transformed
into a set of overlapped rings. By contrast, in the experiment, each pencil
beam produced a single ring. This experimental outcome was observed
consistently throughout the two days of the beam test.

A Monte Carlo particle-tracking code, BDSIM [65], was used to simulate the
formation of the rings on a screen downstream of the lens for a simplified
plasma distribution. The electron cloud was modelled according to the main
features of the $\mathrm{m}=1$ diocotron mode, as a longitudinal column of
electrons with azimuthal symmetry. The central axis of the column was
displaced from the beam axis and rotated around the beam axis with a constant
period that was larger than the transit time of the protons through the lens.
The electron cloud had a region of constant density in the centre and a
negative radial gradient up to the walls of the anode [62]. A 3D time-
dependent electric field map was calculated from the rotating plasma column.
To study the focusing effect of the plasma and the lens, the six proton pencil
beams were tracked through the electric and magnetic field maps. The initial
phase-space of the protons upstream of the lens was tuned such that the
intensity profile of the pencil beams on the screen obtained from the
simulation matches the images taken during the beam test with the lens off.

Figure 13: The effect of a plasma column rotating around the beam axis on six
proton pencil beams as simulated with BDSIM [65] for electron densities of $0$
m-3, $1.8\times 10^{14}$ m-3, and $2.8\times 10^{14}$ m-3. Increasing the
density of the plasma modifies the separation and the width of the rings. The
plasma column has a radius $\mathrm{r_{c}}=14$ mm and an offset $\mathrm{D}=7$
mm from the central axis of the lens.

Figure 13 shows the focusing effect of a rotating plasma column on the pencil
beams as a result of the particle tracking. A variation of the separation
between the rings and the width of each ring is seen as a function of the
density of the plasma. The shape and thickness of the rings are influenced by
changes in the electron density and in the radius of rotation of the plasma
column. The eccentricity of the rings increases for the pencil beams that are
further away from the beam axis as a result of the different focusing
strengths in the $x$ and $y$ directions. This geometrical effect depends on
the relative position of the pencil beam with respect to the rotation axis of
the plasma column. As in the experimental observations, the brightness of each
ring is seen to vary along the circumference. The simulations indicated that
the position and extent of the intensity peak is dependent on the ratio
between the period of rotation of the plasma column and the transit time of
the protons through the lens.

A systematic study of the characteristics of the rings was carried out as a
function of the voltage and current settings at which the lens was operated
during the beam test. Three parameters were used to characterise the rings:

  1. 1.

Centroid $\mathbf{r_{c}}=\left(x_{c},y_{c}\right)$: The centroid was taken to
be the weighted average of all the pixels constituting a ring above a fixed
intensity threshold
$(x_{c},y_{c})=\left(\frac{M_{10}}{M_{00}},\frac{M_{01}}{M_{00}}\right)$,
where $M_{ij}$ are the image moments $M_{ij}=\sum_{x}\sum_{y}x^{i}y^{j}I(x,y)$
of the pixel intensity $I(x,y)$.

  2. 2.

Diameter, $\mathcal{D}_{x,y}$: The diameter of the ring (or of the beam spot
in images taken with the lens off) was determined along the $x$ and $y$
directions separately. The diameter is defined as the width of a beam spot or
ring along the $x$ or $y$ direction after an intensity cutoff was applied to a
camera image.

  3. 3.

Eccentricity, $\mathcal{E}$: The eccentricity is defined as the ratio
$\frac{\mathcal{D}_{x}}{\mathcal{D}_{y}}$.

In order to extract the diameter and the position of the centroid of each
ring, an intensity cutoff was applied to the camera images. This procedure
introduces an associated uncertainty in the calculations. For the parameters
given above, the uncertainties are $\pm 0.3$ mm for $\mathcal{D}_{x}$, $\pm
0.2$ mm for $\mathcal{D}_{y}$, and $\pm 0.05$ mm for $x_{c}$ and $y_{c}$.

A comparison of the effects of electric field only and magnetic field only in
the lens is shown in figure 14. This plot presents data from Setup 1, in which
the 3 spots are those shown in figure 9, with the rightmost point
corresponding to the beam axis. As expected, applying only an electric
potential does not influence the particle transport, while a variation in the
magnetic field leads to a more significant change in the focusing strength.
Comparison between magnetic field only data and the results of particle
transport in the magnetic field shows good agreement in direction as well as
in magnitude. This is true for all three of the observed pencil beamlets. The
small variation around the pencil beam on the beam axis indicates that the
beam axis, aperture axis, and lens axis were not identical.

Figure 14:  Position of the centroid of the 3 beam spots for varying magnetic
fields and high voltages. Squares and Circles represent variation in current
through the magnetic coils only, in simulation and data respectively, with no
applied high voltage. Crosses represent variation in high voltage (and
therefore electric field) with no current through the magnetic coils.

Figure 15 shows the variation in $x$ and $y$ position of the three beam spot
centroids in Setup 1, under the effects of both applied magnetic field and
high voltage. In this case, the high voltage is held at 15 kV, while the
current through the magnetic coils is varied from 0 to 32 A. An approximately
linear increase with magnetic field is observed. In addition to the circular
motion of the plasma in the lens, there is a focusing force that increases
with the external magnetic field. A possible cause for this effect is a
misalignment between the beam axis and the symmetry axis of the coil. The two
sets of data, for 15 kV and 20 kV, were taken on consecutive days which
indicates that the driving mechanism associated with the observed plasma
dynamics is a characteristic of the geometry and operation of the lens.

Figure 15:  The $x$ and $y$ position of the centroids of the beam spots imaged
with the Setup 1 (blue) and Setup 2 (red) with increasing magnetic field
strength.

The variation in $x$ and $y$ diameter of the six beam spots from Setup 2 is
given in Figure 16: with a constant applied voltage of 15 kV, the variation in
spot diameter with magnetic field is shown. A non-linear increase in spot size
with change in magnetic fields is observed, with the rate of increase in
diameter getting larger at high magnetic fields. This indicates that the
increase in plasma density with magnetic field is faster than linear and thus
that the plasma trapping efficiency varies with the magnetic field strength.
Since the points remain within the lines shown, there is an indication of a
trend for the change in diameter for a given spot. The ratio variation being
solely dependent on initial position of the spot indicates that this effect is
caused by the density distribution of the plasma in the transverse plane.

Figure 16:  The variation in the $x$ and $y$ diameter of the 6 spot data with
increasing magnetic field strength.

Figure 17 shows the change in $xy$ ratio of the six beam spots from Setup 2
with variation in magnetic field. The focusing force in the $x$ direction
compared to the focusing force in the $y$ direction changes differently for
each beam spot. The difference between the two focusing forces is more
significant for those beam spots at greater distances from the beam axis. This
indicates that the centre of the lens has a low plasma density, while further
from the axis of the lens, the plasma density increases with radius. In
addition, there is some perturbation that causes the motion of the plasma to
evolve with time. Figures 14 to 17 may be understood in terms of a position
dependent plasma density, with the bulk of the plasma shifted radially from
the lens centre and rotating around the central axis.

Figure 17:  The ratio of the $x$ diameter to the $y$ diameter for beam spots
in the 6 spot data with increasing magnetic field strength.

##  6 Conclusions

We characterised the operation of an electron plasma (Gabor) lens based on a
test at the Surrey Proton Beam Centre with a 1.4 MeV proton beam. Prior
measurements at Imperial College indicated that the lens had a stable regime
of operation over a range of applied voltage and current. During the beam
test, the lens was observed to transform pencil beams into rings. The presence
of a plasma was confirmed by matching the measured focusing effect with
particle transport calculations. An evaluation of the focusing strength showed
that the density of the trapped electrons depends on the strength of the
radially confining magnetic field, effects that were well described by
simulation. Since the same focusing effects and ring patterns were observed on
consecutive days, the plasma instability was associated with the geometry and
operation of the lens.

The formation of rings indicates that the plasma column is excited into a
coherent off-axis rotation. The size of the rings increases with increasing
current through the coil. A reproducible modulation of the intensity was
observed around the circumference of each ring. The position of the centroids
of the rings varied non-linearly with the external magnetic field strength,
showing a variable plasma trapping efficiency. The $x$ and $y$ diameters, and
the eccentricity of the rings were seen to depend on their position with
respect to the beam axis, as a result of the different $x$ and $y$ focusing
forces experienced by each pencil beam.

Both particle-in-cell and particle-tracking simulations showed that a rotation
of the bulk of the plasma transforms pencil beams into rings. The size and
width of the rings were shown to be determined by the density of the plasma.
Rings with size, eccentricity, and intensity modulation similar to the
experimental images were reproduced with a simulation of particle transport
through a plasma characterised by the $m=1$ diocotron instability.

The results described here indicate the presence of a mechanism that drives
the rotation of the plasma column. Further investigations are required to
identify and describe the exact mechanism that needs to be avoided for the
lens to be operated as a reliable focusing device.

## Acknowledgements

We wish to acknowledge the work of Piero Posocco at the 2017 beam test. We
would also like to the thank all the staff at the University of Surrey Ion
Beam Facility, and in particular Vladimir Palitsin, for his support throughout
both days of the beam test. We gratefully acknowledge the Plasma Physics Group
at Imperial College London who were generous in sharing their scientific
insight and provided technical support at various stages of the project. The
work was financially supported by the STFC through the Imperial Impact
Acceleration Account. One of us (TSD) gratefully acknowledges the support from
the Class of 1964 Scholarship.

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