# Anomalous Beam Transport through Gabor (Plasma) Lens Prototype
## 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. 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 aluminised pyrex, and the scale and resolution of the screen are shown on the schematic. 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 \(x^{\prime}=1.6\,\mathrm{mrad}\) and \(y^{\prime}=0.5\,\mathrm{mrad}\), 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  
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 [PERSON]'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 \(\approx 6\tau_{c}\), 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.