Anomalous Beam Transport through Gabor (Plasma) Lens Prototype

: 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 1.4MeV protons are reported here. Narrow pencil beams were imaged on a scintillator screen 67cm 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.


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 [2,[6][7][8].
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 laserdriven 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 [13,23,24]. 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 [16,20] it necessary to capture the beam using a strong-focusing element as close to the ion-production point as possible.
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.

The Gabor Lens
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 • 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 mm 2 . 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 volt-age 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 × 10 −7 mbar, with pressures of up to 3 × 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 (∼ 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, ∼ 5 × 10 −7 cm −3 , was produced.

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 mm 2 . 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.  Unstable plasma: plasma produced with higher magnetic field causing considerable sparking and therefore large variations in the output amplitude. 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.

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). 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. 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 µm as shown in figure 7. 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.

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. 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" were visible that corresponded to the beamlets produced by the holes in the aperture plate. The longer drift introduced in the Setup 2 configuration resulted in a larger number of beamlets being observed at lower magnification than in the Setup 1 configuration. The central axis of the lens passed 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 = 1.6 mrad and y = 0.5 mrad, respectively. 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 ringlike 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].  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 particlein-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 ns (corresponding to ≈ 6τ c , where τ c = 2π m/eB is the cyclotron period) to track the electron movement correctly. The grid had a transverse cell size ∆x = 0.14λ D and longitudinal cell size ∆z = 0.8λ D , where λ 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 macroparticles 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 [? ]. The diocotron modes observed in the simulations corresponded to higher order modes with an azimuthal mode number m > 1. Within the range of electron densities between 1 × 10 13 m −3 and 1 × 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 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 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.
Centroid r c = (x c , y c ): 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 ) = M 10 M 00 , M 01 M 00 , where M ij are the image moments M ij = ∑ x ∑ y x i y j I(x, y) of the pixel intensity I(x, y).

2.
Diameter, 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.
Eccentricity, E : The eccentricity is defined as the ratio D x 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 ±0.3 mm for D x , ±0.2 mm for D y , and ±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 was held at 15 kV, while the current through the magnetic coils was varied from 0 to 32 A. An approximately linear increase with magnetic field was observed. In addition to the circular motion of the plasma in the lens, there is a focusing force that increased 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 indicated that the driving mechanism associated with the observed plasma dynamics was a characteristic of the geometry and operation of the lens. 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 was observed, with the rate of increase in diameter getting larger at high magnetic fields. This indicated that the increase in plasma density with magnetic field was faster than linear and thus that the plasma trapping efficiency varied with the magnetic field strength. Since the points remained within the lines shown, there was 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 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 changed differently for each beam spot. The difference between the two focusing forces was more significant for those beam spots at greater distances from the beam axis. This indicated that the centre of the lens has a low plasma density, while further from the axis of the lens, the plasma density increased with radius. In addition, there was some perturbation that caused the motion of the plasma to evolve with time. Figures 14-17 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.

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 s trength, s howing a v ariable p lasma t rapping e fficiency. Th e x an d 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.