# Anomalous Beam Transport through Gabor (Plasma) Lens Prototype
## 5 Characterisation of Lens Performance
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]:  
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. a hollow electron ring and the diocotron instability [65]. The diocotron modes observed in the simulations corresponded to higher order modes with an azimuthal mode number \(\mathbf{m}>1\). 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 \(\mathbf{m}=1\) 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  
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. 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.