Noninvasive 3D Field Mapping of Complex Static Electric Fields

Many upcoming experiments in antimatter research require low-energy antiproton beams. With a kinetic energy in the order of 100 keV, the standard magnetic components to control and focus the beams become less effective. Therefore, electrostatic components are being developed and installed in transfer lines and storage rings. However, there is no equipment available to precisely map and check the electric field generated by these elements. Instead, one has to trust in simulations and, therefore, depend on tight fabrication tolerances. Here we present, for the first time, a noninvasive way to experimentally probe the electrostatic field in a 3D volume with a microsensor. Using the example of an electrostatic quadrupole focusing component, we find excellent agreement between a simulated and real field. Furthermore, it is shown that the spatial resolution of the probe is limited by the electric field curvature which is almost zero for the quadrupole. With a sensor resolution of $61\mathrm{V}/\mathrm{m}/\sqrt{\mathrm{Hz}}$, the field deviation due to a noncompliance with the tolerances can be resolved. We anticipate that this compact and practical field strength probe will be relevant also for other scientific and technological disciplines such as atmospheric electricity or safeguarding near power infrastructure.

Figure 1

Sketch of the FODO assembly. Elements of the simplified model are highlighted. Different colors represent the applied voltage at the poles: red ($+$) and blue ($-$). Frontal and rearward shielding plates for mitigating of the fringe field influence are highlighted in grey.

Figure 2

(a) Cross section of the $E$ field recalculated from $F_{\mathrm{y}}$ and $F_{\mathrm{z}}$ obtained on the $yz$ plane in the EQP center. The black dots denote the positions of the MEMS geometry. A linear interpolation has been applied between them. (b) Potential at the cross section. For investigating the spatial resolution, the MEMS was moved along the horizontal line in 1 mm steps. (c) Field components obtained along the line. (d) Force obtained in the simulations (red circles) compared to the force calculated from the data in (c) by $F_{\mathrm{y}}=\alpha E_{\mathrm{y}}^2$.

Figure 3

(a) Setup for the EQP field characterization. (b) The zoom highlights the light path through the probe tip. (c) Schematic of the measurement and map of the field strength in the front half of the EQP. The data points were interpolated and represented as isosurfaces. (d) Different views on the visualized measurement data.

Figure 4

(a) Output voltage $U_{\text{tot}}=U_{\mathrm{y}}+U_{\mathrm{z}}$ from the EQP center cross section. (b) Corresponding field strength $|E|$ obtained with the respective responsivities $R_{\mathrm{y},\mathrm{z}}$. The black dots denote the positions of the probe. The data were linearly interpolated between them. (c) Spatial resolution of the $E$ field probe in terms of recorded voltage. The MEMS was moved along a line as in Fig. 2 with a step size of $100\mu \mathrm{m}$. (d) $|E_{\mathrm{y}}|$ recalculated from the data in (c) compared to the simulations.

Figure 5

Measurement precision as function of the input $E$ field. The horizontal lines denote the precision necessary to resolve the perturbation due to a transverse offset of one electrode by the fabrication tolerances. Below roughly $100\mathrm{V}/\mathrm{m}$ the curves are flat due to the noise floor of the sensor. Then the relative standard deviation decreases in the same way as the applied field increases. Above $2\mathrm{kV}/\mathrm{m}$, the curves become flatter again, pointing out the precision limit of the current sensor and setup.