# Anomalous Beam Transport through Gabor (Plasma) Lens Prototype
## 5 Characterisation of Lens Performance
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 \(\mathbf{r_{c}}=(x_{c},y_{c})\): The centroid was taken to be the weighted average of all the pixels constituting a ring above a fixed intensity threshold \((x_{c},y_{c})=\left(\frac{M_{01}}{M_{00}},\frac{M_{01}}{M_{00}}\right)\), 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. 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. 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 \(0\,\mathrm{m}^{-3}\), \(1.8\times 10^{14}\,\mathrm{m}^{-3}\), and \(2.8\times 10^{14}\,\mathrm{m}^{-3}\). Increasing the density of the plasma modifies the separation and the width of the rings. The plasma column has a radius \(\mathrm{r_{c}}=14\,\mathrm{mm}\) and an offset \(\mathrm{D}=7\,\mathrm{mm}\) from the central axis of the lens. Ecccentricity, \(\mathcal{E}\): 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 \(\mathcal{x}_{c}\) 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 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  
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.