# Parametric Study of the Performance of an Electrostatic Analyzer With an Hemispheric Field-of-View Based on the Donut Topology

Gwendal Hénaff, Matthieu Berthomier

► To cite this version:

Gwendal Hénaff, Matthieu Berthomier. Parametric Study of the Performance of an Electrostatic Analyzer With an Hemispheric Field-of-View Based on the Donut Topology. Journal of Geophysical Research Space Physics, 2025, 130 (3), 10.1029/2024JA033367. [https://doi.org/10.1002/2024JA033373]

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# JGR Space Physics

## METHOD

10.1029/2024JA033367

**Key Points:**

* A parametric study of electrostatic analyzers based on the donut topology with a 0–20 keV energy range was conducted

* A set of 3 construction parameters can be used to carefully tune the geometric factor and energy resolution at a specific angular resolution

* It suggests to be a versatile concept for a variety of applications from space weather to performance intensive science missions

**Correspondence to:**

G. Hénaff, gwendal.henaff@lpp.polytechnique.fr

**Citation:**

Hénaff, G., & Berthomier, M. (2025). Parametric study of the performance of an electrostatic analyzer with an hemispheric field-of-view based on the donut topology. *Journal of Geophysical Research: Space Physics*, 130, e2024JA033367. https://doi.org/10.1029/2024JA033367

Received 23 SEP 2024

Accepted 4 MAR 2025

©2025. The Author(s). This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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## Parametric Study of the Performance of an Electrostatic Analyzer With an Hemispheric Field-of-View Based on the Donut Topology

Gwendal Hénaff¹ and Matthieu Berthomier¹

¹Laboratoire de Physique des Plasmas (LPP), CNRS/UMR7648, École polytechnique, Institut Polytechnique de Paris, Sorbonne Université, Université Paris-Saclay, Observatoire de Paris, Paris, France

**Abstract** We carried out a parametric study of the optical performance of an electrostatic analyzer based on the donut topology. The instantaneous hemispheric field-of-view of the optics eliminates the need of electrostatic deflectors, which are usually added to the energy analyzer in other designs to cover such a wide field-of-view. Parametrization of the donut topology and the use of parallel computing have enabled a wide parametric study of the instrument's performance as a function of the angle resolution of the instrument. We have identified a limited number of geometric parameters, including the outer radius of the detection system, which determine the geometric factor and energy resolution of the instrument for a given angular resolution. The average geometric factor per pixel of this 3D plasma camera varies in the range $4 \cdot 10^{-4} - 3 \cdot 10^{-3} \text{cm}^2 \cdot \text{sr.eV/eV}$ with an energy resolution between 9% and 14% for an energy limit of 20 keV. Our results suggest that a wide range of space missions could benefit from this new instrument concept. A low-angular-resolution version of the instrument could be installed on a nano-satellite platform, for example, for space weather monitoring on low-Earth orbit. For space missions requiring high-angular-resolution measurements, a high-temporal-resolution plasma camera would be able to provide the detailed distribution function of charged particles on larger platforms.

## 1. Introduction

Electrostatic analyzers measure thermal and supra-thermal charged particles distribution functions in the eV to several 10s keV energy range. Therefore, they are key instruments of any space plasmas mission. In the 80s, the development of the iconic top-hat electrostatic analyzer by Carlson et al. (1982) had been a major advance due to its uniform Field of View (FoV) and near-Gaussian energy response. Benefiting from a well-established mathematical description, top-hat sensors were since widely used (Young et al. (1988) and references therein). However, the relatively small FoV of the top-hat optics—typically 5° around the 360° viewing plane—had a strong influence on the design of many space missions. For resources constrained multi-satellite missions such as Cluster (Johnstone et al., 1997; Rème et al., 1997) and THEMIS (McFadden et al., 2008), the top-hat design implied that the time resolution of plasma measurements was limited by the satellite spin period, which is at best a few seconds. Many other missions, and particularly planetary missions using three-axis stabilized platforms, had to use either actuators to rotate the whole instrument (Young et al., 2004) or electrostatic deflection systems to enlarge the instrument field of view (Burch et al., 2007; Halekas et al., 2015; McComas et al., 2017; Saito et al., 2010; Sauvaud et al., 2008; Yokota et al., 2004, 2021). However, the need to scan all viewing directions complicates the design of these instruments while limiting their temporal resolution to seconds typically. Furthermore, the hemispherical FoV of these sensors can only be achieved within a limited energy range, typically below 15 keV/q due to the low K-factor of the deflection system, K being the ratio between the energy of deflected particles and the voltage applied to the deflectors. For example, the Codice-Lo ion composition experiment for the IMAP mission includes a deflection system working in the range of $\pm 45^\circ$ but it requires 0.6 V/ eV to reach a $40^\circ$ deflection (Desai et al., 2016).

Considering that the deflection voltage to apply evolves proportionally to the desired deflection angle, if a 0.6 V/ eV is needed to reach $40^\circ$ of deflection, 0.675 V/eV are needed to reach $45^\circ$. Therefore deflecting a 15 keV/q particle at such angles implies using a 10 kV sweeping high-voltage supply to reach 15 keV/q. Given the limitation of high-voltage opto-couplers, this typically sets the energy limit of the hemispherical FoV of such a design. The limited temporal resolution of such systems can be addressed by distributing multiple sensors around the azimuthal plane of the spacecraft. This approach has been implemented for the FPI instrument suite onboard MMS (Pollock et al., 2016), enabling very high temporal resolutions: 30 ms for electrons and 150 ms for ions.
<figure><img src="image_3.png" /><figcaption>**Figure 1.** 3D sketch of the donut topology pointing upwards, with 16 azimuths sectors ($\Phi = 22.5^{\circ}$) and 4 polar channels.</figcaption></figure>

However, this method comes with significant mass and power requirements, with 16 individual top-hats per spacecraft.

Alternatively, there has been a few attempts to develop plasma analyzers that are not based on the “top-hat” concept in order to maximize their instantaneous FoV. The DYMIO and FONEMA instruments on board the MARS 96 satellite were plasma spectrometers with 3D FoV capabilities on a three-axis stabilized platform (Berthelier et al., 1998; James et al., 1998). However, DYMIO was designed for thermal ions below 700 eV and it only provided a coarse coverage of phase space with 32 independent entrance windows. FONEMA, which was coupling an electrostatic mirror with a Thomson parabola analyzer, had a similar ion phase space coverage with 36 entrance windows between 20 eV and 8 keV in 125 ms. Another attempt was the HYDRA instrument on the Polar spacecraft, which was a collection of cylindrical electron and ion analyzers for particles in the 2 eV–35 keV energy range (Scudder et al., 1995) It provided an even sparser sampling of phase space with only 12 entrance windows and 0.5 s time cadence. More recently, the PICAM optical concept was proposed (Vaisberg et al., 2016). PICAM has a truly 3-D instantaneous FOV, but the voltages applied to its electrostatic mirrors have to be equal to the particle's energy with a limit of 3 keV and 8 s time resolution (Orsini et al., 2010).

In this report, we present the parametric study of the optical performance of an electrostatic analyzer that has a 3D instantaneous FoV. This 3D plasma camera is based on the optical concept of the donut analyzer first presented by Morel et al. (2017). Our wide parametric study provides a deep understanding of the various trade-off that can be made to optimize the optical design of this instrument for a specific application. After the presentation in Section 2 of the instrument concept, we define the instrument parameters in Section 3 and we describe the numerical tools that were developed to perform the study in Section 4. Then, we illustrate our approach by analyzing the performances of a specific instrument model in Section 5. Sections 6 and 7 present the results of our parametric study of low and high angular resolution 3D plasma cameras, respectively. Section 8 provides a summary of our numerical results and discusses the potential use of the 3D plasma camera concept in various space missions.

## 2. Instrument Concept

The 3D plasma camera has a donut-like topology where individual toroidal electrostatic analyzers are stacked on each other. Each channel of the analyzer is pointing toward a specific direction providing an instantaneous hemispheric Field-of-View (FoV) of the surrounding plasma. As depicted in Figure 1, the toroids of the analyzer have a common axis of symmetry around which individual sectors cover all azimuth directions. The number of channels in the other (polar) dimension and the number of sectors in the azimuth dimension set the angular resolution of the instrument. For instance, with 4 polar channels and 16 azimuth sectors, the 3D plasma camera has 64 individual look-directions with a $2\pi sr$ FoV and a $22.5^{\circ}$ angular resolution. The native angular resolution of the instrument is therefore embedded into its topology and it cannot be adjusted in-flight as it is the case for standard instruments using deflection systems. Therefore, it is of high importance to define the required angular
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Cut view of the 3D plasma camera topology with its construction parameters. The Micro-Channel Plates based detection system takes the form of a disk of outer radius $R_d$ with a center hole of radius $R_h$.</figcaption></figure>

resolution of the measurements performed by the 3D plasma camera at the start of its development. The purpose of this research was to study how the choice of a particular angular resolution would impact the performance of the 3D plasma camera.

Such a topology implies that each toroidal electrode of the electrostatic optics can be biased at a given voltage in an independent way. This creates a radial electric field between the toroids that deflects charged particles whose energy falls in the energy bandwidth of the analyzer toward the detection plane of the instrument. In this plane, a Micro-Channel Plates (MCP) based detection system can be used to image the particles phase space at the selected energy. This typically requires using a pixelated anode underneath the MCP that will collect MCP charge clouds, each pixel corresponding to a specific look-direction. Charged sensitive amplifiers running in parallel can be coupled to the anode and appropriate digital electronics can be used to accumulate number counts at a given energy level. A classic high-voltage sweep of the electrodes bias allows reconstructing the full particle distribution function with the energy resolution of the analyzer. While the effective time resolution of such a 3D plasma camera depends on many parameters, including the geometric factor (i.e., the sensitivity) of the electrostatic optics, the efficiency of the detection system, the expected charged particles flux, and the available power budget for its implementation, it should be emphasized that such an instrument topology removes the need of scanning different look directions with a charged particle deflection system ahead of the electrostatic analyzer. It means that for a given geometric factor and a given accumulation time per energy level, the 3D plasma camera will provide 3D measurements of the particle distribution function at a higher-time resolution than standard designs with deflectors. Also, the 3D plasma camera concept can be used to measure either electrons or ions depending on the polarity of the electrode bias.

### 3. Instrument Parameters

The key challenge in the parametric study of the optical properties of the 3D plasma camera consists in identifying, among the many parameters that geometrically define this donut analyzer, the parameters that will have the strongest impact on the optics performance. This is needed to understand the trends in the physical behavior of the optics that will guide the development of a specific version of this sensor for a given space mission.

We first define the set of parameters that we used to describe the donut geometry, as shown in Figure 2

* $R_d$: outer radius of the MCP detector

* $R_h$: center hole radius of the MCP detector

* $R_a$: main radius of the toroidal electrodes

* n: number of polar channels

* $R_i$: mean radius of polar channel i

* $D_i$: gap between inner and outer electrodes of channel i
* $\theta_i$: cut angle of polar channel i

* ep: thickness of the toroidal electrodes

In addition, we define $\Phi$ as the angular width in azimuth of the sectors around the main axis of symmetry of the instrument (see Figure 1). Together with the number of polar channels $n$, this defines the target angular resolution of the instrument. To simplify the study, we also assume that the cut angles of the polar channels can be uniformly spaced between 0° and 90° to cover all polar angles evenly, which means that $\theta_i = (i - \frac{1}{2}) \cdot 90^{\circ}/n$, with $n = 4$ and $i \in [1:4]$. The instrument parameters are linked together according to

$$ 
\left. 
\begin{aligned}
R_a &= R_d + R_1 - D_1/2 - ep \\
R_h &= R_a - R_n - D_n/2 - ep \\
R_i &= R_{i-1} + (D_{i-1} + D_i)/2 + ep
\end{aligned}
\right\} \qquad (1) $$

Equation 1 illustrates the fact that the donut analyzer is built by successively adding nested toroids, which means that the properties of the outer channels highly depend on the choice of the parameters that define the innermost channels, including the gap and mean radius of the first channel. Another key parameter of any electrostatic analyzer is the analyzer constant $K$. Here $K_i$ is the coefficient of proportionality between the selected energy per charge $E/q$ and the voltage difference $\Delta V_i(E)$ between the inner and outer electrodes of polar channel i. We can write $K_i$ as

$$ K_i = \frac{E}{q\Delta V_i(E)} \qquad (2) $$

Similar to spherical and toroidal analyzers, the donut analyzer does not exhibit an ortho-radial electric field between its plates, as illustrated in Figure 4. Consequently, a charged particle entering the instrument is subjected solely to a radial electrostatic force generated by the two plates. At the mean Radius $R_i$, this force can be expressed as $F_e = q \cdot \Delta V_i / D_i$. Assuming that the particle follows the curvature of the analyzer, this electrostatic force must equate the centripetal force $F_c$. From the expression of the radial acceleration and the law of motion, we can write $F_c = m \cdot \frac{v_i^2}{R_i}$. Equating $F_c$ and $F_e$, we obtain $m \cdot \frac{v_i^2}{R_i} = q \cdot \Delta V_i / D_i$. This can be rearranged knowing the expression of the kinetic energy $E = \frac{1}{2} m \cdot v_i^2$. It gives: $2 \cdot E / R_i = q \cdot V / D_i$.

From the definition of $K$ at Equation 2, we can write

$$ K_i = \frac{R_i}{2D_i} \qquad (3) $$

This relation is similar for spherical and toroidal analyzers where the electric field doesn't exhibit a radial component (Stude, 2016; Wuest & Evans, 2007).

A key parameter that defines the capability of an electrostatic analyzer is the maximum energy per charge $E_{max}$ of the particles it can detect. In this study, we set $E_{max}$ to 20 keV, which is a standard value for electrostatic analyzers. This defines, for a given analyzer constant, the maximum bias of the electrodes $\Delta V_i(E_{max})$. We also set the maximum acceptable electric field $EF_{max}$ in the instrument to 3 kV/mm. These quantities are of importance because they will impact the size, volume, and mass of the instrument. However, in such a complex system as the 3D plasma camera is, as will be shown in the following sections, we can still adjust other instrument parameters to avoid a linear increase of its size as a function of the maximum energy per charge it can detect. This will practically be achieved by adjusting the size of other key instrument parameters such as the outer radius of the MCP $R_d$ to keep the size of the instrument within a specified range. The maximum detectable energy per charge will also impact the design of the high-voltage power supply needed to bias the electrodes. In addition, having several analyzers with different $K_i$ factors would increase the complexity, volume, mass, and cost of the instrument. To simplify the design, we restricted our study to the case of a common analyzer constant among all channels. Hereafter, we will have $K_i = K_{const}$.

Finally, as shown by Morel et al. (2017), the topology of the 3D plasma camera requires introducing separators between azimuth sectors to prevent particles from different azimuth directions from mixing in the detection plane.
<figure><img src="image_5.png" /><figcaption>Figure 3. Shape of the separators between two azimuthal separators.</figcaption></figure>

However, since the inner and outer toroidal electrodes of a given polar channel cannot be connected to each other, the separation between azimuth sectors can only be made of partial separators to keep a minimum stand-off distance $d_{min}$ between the inner and outer toroidal electrodes which is defined as

$$d_{min} = \frac{\Delta V_{max}}{EF_{max}} \qquad (4)$$

The stand-off distance $d_{min}$ plays a role in the shape of the inter-sectors separators, as illustrated in Figure 3, where $d_{min}$ is the minimal distance acceptable between two differently polarized elements. Particles flying in the analyzer see a gap in between separators named j. When this gap becomes large with respect to $H_{clois}$, particles will fly from one sector to another.

Given Equation 2, this sets $d_{min}$ as

$$d_{min} = \frac{E_{max}}{qK_{const} \cdot EF_{max}} \qquad (5)$$

As can be seen from Equation 3, imposing the same K factor to all channels implies that outer channels of increasing mean radius $R_i$ will have wider gap between their inner and outer electrode. By taking the same stand-off distance $d = d_{min}$ for all channels, the ratio $d/D_i$ will decrease for outer channels which reduces the possibility of cross-talk in azimuth for those channels. Equation 5 also means that the stand-off distance determines the K factor of the analyzer.

The combination of Equations 1, 3, and 5 indicates how the instrument model can be built from only three parameters [$R_d, d, D_1$] by using the following relationship between these free parameters and the other instrument parameters.

$$ \left. \begin{aligned} K_{\text{const}} &= E_{\text{max}}/(q \cdot EF_{\text{max}} \cdot d) \\ R_i &= (R_{i-1} + D_{i-1}/2 + ep)/[1 - 1/(4K_{\text{const}})] \\ D_i &= R_i/(2K_{\text{const}}) \\ R_a &= R_d + R_1 - D_1/2 - ep \\ R_h &= R_a - R_n - D_n/2 - ep \end{aligned} \right\} \qquad (6) $$

Indeed, with $d, D_1$ and $R_d$ chosen, and knowing that parameters $ep, EF_{max}$ and $E_{max}$ have been already fixed, with see in Equation 6 that $K_{const}, R_i, D_i, R_a$ and $R_h$ are fully defined for all values of i. We used the MCP outer radius $R_d$ as a free parameter instead of the main radius of the toroids $R_a$ because of the limited MCP size manufacturers can produce.
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Cut-view of a donut analyzer model showing electrostatic potential iso-contours, the start surface in yellow, particles flying directions in green and red, and the detection plane in blue.</figcaption></figure>

4. Numerical Tools

The classic method to optimize the performance of an electrostatic analyzer through numerical means involves building first a geometrical model of its electrodes, either through a standard mechanical CAD software or through an application specific software. Performing a parametric study of our electrostatic optics also requires the capability to parametrize its complex geometry, which practically excludes the use of standard CAD software. Instead, we used the SIMION-8 software (Dahl, 2000) that compiles the information on the shape of the electrodes from a geometry text file to produce a 3D model of the instrument on a cartesian grid with a specified grid resolution. We typically used a 0.1 mm grid resolution. Our geometry file is produced by a home-made code written in C language that takes as an input the set of instrument parameters that was described in Section 3. The 3D model includes all relevant grids and appropriate boundary conditions.

The next step consists in solving the Laplace equation with the SIMION Laplace solver to obtain the 3D map of the electrostatic potential within the instrument. This step is followed by injecting particles with a wide range of energy and directions at the optics entrance, using a particle pusher and counting which particles reach the MCP detector. We did not "fly" particles with the Monte-Carlo based SIMION pusher because it does not take advantage of multi-core architectures. Instead, we used a particle-pusher algorithm based on the Runge-Kutta fourth-order method that was written in Fortran. Particle trajectories are handled in parallel by a computer cluster with 64 cores and 256 GB of RAM using the Open MPI library (Open MPI v5.0.x, 2023).

This tool is key to investigate the wide space of instrument parameters and to optimize the 3D plasma camera optical design.

Particles trajectories of varying velocity direction, energy, and initial position are launched in the instrument model. The algorithm aims to count which of these particles reach the analyzer's MCP detector. Saving their initial velocity direction, start position and start energy, we can compute the angular and energy acceptance of the instrument and the geometric factor of each polar channel. For each particle, this involves the following steps

* The particle is flown with a specific energy and velocity direction from the entrance of the polar channel in direction Dr1 toward the entrance grid.

* If the particle reaches the grid, it is flown back with the same energy toward the analyzer in direction Dr2.

* If the particle reaches the detector plane Pz, its initial energy and angle direction is saved.
Output files generated by the simulation include the number of particles of a given initial azimuth angle, polar angle, and energy that have reached the MCP detector. They also include a 2D image of the particles in the detection plane, which is useful to optimize the definition of the pixels. Using these files, we can calculate the azimuth and polar acceptance of each channel, its energy acceptance, and its numerical K factor. We can also estimate the cross-talk level between channels, either in the polar or in the azimuth direction.

## 5. Optical Performance of a Specific Model

The donut analyzer displays a rotational symmetry around its main toroidal axis, therefore allowing to only simulate the n polar channels of a single sector in azimuth to fully characterize its performances. To illustrate our approach with an example, we first consider an instrument described by the following parameters:

* $E_{max} = 20$ keV

* $n = 4$

* $\Phi = 22.5^{\circ}$

* $R_d = 45$ mm

* $d = 1.1$ mm

* $D_1 = 1.8$ mm

* $ep = 2.5$ mm

Once the instrument model has been automatically built with these parameters, charged particles are flown in the simulation box with a specific set of energies $E_{start}$, polar angles $\theta_{start}$, and azimuth angles $\phi_{start}$. Start energies range from 750 eV to 1250 eV with a 6.25 eV step. Polar angles range from -10° to 10° with a 0.25° step in the instrument's aperture reference frame. Azimuth angles range from -25° to 25° with a 1° step in the instrument's aperture reference frame. Each triplet [$E_{start}, \theta_{start}, \phi_{start}$] is used multiple times to scan the instrument's aperture in a homogeneous way. Typically, we use 64 different initial positions and we fly 21 · 10⁶ particles per channel, which is 85 · 10⁶ particles for a four channels instrument, allowing good statistics. It takes about 3 hr to run the complete simulation.

After completing the simulation, the azimuth, polar and energy response of each channel can be plotted, as shown in Figure 5. We note that the peaks of the energy response of the 4 channels are not perfectly aligned, which is due to the approximation made in Equation 3. As expected, the four channels have different optical properties. However, this is also a known feature of electrostatic analyzers that include electrostatic deflectors (Collinson & Kataria, 2010). For example, the angular resolution of the FPI-DES electron sensor of the MMS NASA mission varies between 4.2° and 8.7° as a function of the deflection angle while it varies between 2.8° and 5.7° for the FPI-DIS ion sensor (Pollock et al. (2016), Figures 32 and 34).

The sensitivity of an electrostatic analyzer is given by its geometric factor $GF$ expressed in cm²·str.eV/eV. Different authors may have different approaches when defining the geometric factor (see e.g., Collinson et al., 2012). In this paper, we neither include the MCP efficiency (Fraser, 2002) nor the grids transparency in the definition of the geometric factor. These effects reduce the sensitivity of the instrument by an energy dependent factor $Q(E)$. Each pixel of the 2D anode corresponds to a specific polar/azimuth direction ($\Theta_i, \Phi_j$) and the number of counts $C_{ij}$ on this pixel linearly varies with the charged particles differential energy flux $J(E, \Theta_i, \Phi_j)$ and with the integration time period $\Delta t$ such as

$$C_{ij} = GF_i \cdot J(E, \Theta_i, \Phi_j) \cdot Q(E) \cdot \Delta t \quad (7)$$

where the geometric factor is given by

$$GF_i = A_i \iiint R_i(E, \theta, \phi) \cdot dE/E \cdot \cos^2 \theta \cdot \cos \phi \cdot d\theta d\phi \quad (8)$$

with $A_i$ the analyzer's entrance area in cm², and $R_i(E, \theta, \phi)$ the analyzer's response function at a given energy E and along the direction ($\theta, \phi$) as defined by Collinson et al. (2012). $R_i(E, \theta, \phi)$ corresponds to the fraction of the incoming particles that will eventually reach the detector and $A_i \cdot R_i(E, \theta, \phi)$ is the effective entrance surface of the instrument. The response function $R_i(E, \theta, \phi)$ does not explicitly include entrance position dependence because
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Polar, azimuth and energy response of an instrument with $E_{max}$ = 20 keV, $n$ = 4, $\Phi$ = 22.5°, $R_d$ = 45 mm, $d$ = 1.1 mm, and $D_1$ = 1.8 mm. In this configuration 512 particles were flown per triplet [$E_{start}$, $\theta_{start}$, $\phi_{start}$], totaling a simulation run with 685.10⁶ particles flown.</figcaption></figure>
<caption>**Table 1**</caption>

Optical Performances of the 3D Plasma Camera Model With $E_{max}$ = 20 keV, n = 4, $\Phi$ = 22.5°, $R_d$ = 45 mm, d = 1.1 mm, and $D_1$ = 1.8 mm

<table><thead><tr><th></th><th>Δθ</th><th>Δφ</th><th>ΔE/E</th><th>GF (cm².str.eV/eV)</th><th>K</th><th>A (mm²)</th></tr></thead><tbody><tr><td># 1</td><td>5.25°</td><td>15°</td><td>9.9%</td><td>8.3 · 10<sup>-4</sup></td><td>5.8</td><td>48.2</td></tr><tr><td># 2</td><td>5.5°</td><td>19°</td><td>8.4%</td><td>7.7 · 10<sup>-4</sup></td><td>5.8</td><td>58.9</td></tr><tr><td># 3</td><td>5.5°</td><td>22°</td><td>9.4%</td><td>9.0 · 10<sup>-4</sup></td><td>6.0</td><td>69.5</td></tr><tr><td># 4</td><td>8.5°</td><td>24°</td><td>13.2%</td><td>13.3 · 10<sup>-4</sup></td><td>6.0</td><td>80.1</td></tr></tbody></table>

particle trajectories are simulated for random entrance positions 64 times on each set of $[E, θ, φ]$, uniformly distributed across the analyzer's entrance. This randomization ensures that any dependence on entrance position is averaged out, leading to a representative calculation of the geometric factor.

This 3D quantity is directly estimated from our simulations over the whole range of energy and velocity directions where it is non-zero. Instead of using an approximation involving the full-widths at half-maximum of the energy and angular responses, we perform the exact integration of Equation 8. The overall performance of this specific instrument model is given in Table 1. It provides the angular acceptances, the energy resolution, the geometric factor and the calculated analyzer's constant of each polar channel.

It is worth noticing that the estimated K factor is not too far from its approximation of 6.06 given by Equation 6. The optical properties of the fourth polar channel significantly differ from those of the first three channels with larger angular and energy acceptances. This is due to the combined effect of a large inter-electrode distance $D_4$ = 3.1 mm and of a small $R_h/R_d$ ratio. In the following sections, we will investigate these effects in detail.

# 6. Low-Angular-Resolution Instruments

In this paper, we aim to give an overview of the behavior of the donut analyzer. Since we anticipate that its angular resolution will strongly impact the resources that will be needed to implement it, we organized our study according to the expected angular resolution of the instrument. We first present our parametric study of low-angular-resolution instruments with $\Phi$ = 22.5° and n = 4 which implies a 64 pixels anode board. Using the same 20 keV energy limit as before, we are left with three free parameters [$R_d, d, D_1$] to define the instrument geometry.

## 6.1. The Torus Shape Factor

The torus topology implies that the size of the innermost pixels in the detector plane will be impacted by the ratio between the major-radius $R_a$ and the minor-radius $R_4$ of the largest torus. The surface area of these pixels will shrink to unacceptable values when the torus factor $f_t = R_4/R_a$ is too large. To illustrate this phenomenon, we studied the effect of increasing the spacing between the electrodes of the first polar channel $D_1$ on the energy response of an instrument with $R_d$ = 36 mm and d = 1.1 mm. Equation 6 shows that, at constant d (hence K) value, $R_4$ will strongly increase with $D_1$ while $R_a$ will only slightly increase, resulting in larger $f_t$ factors. We show the energy response of the models corresponding to $D_1$ = [1.9, 2.2, 2.6, 3.2] mm in Figure 6. This corresponds to $f_t$ = [0.68, 0.71, 0.76, 0.81] respectively. We observe a severe distortion of the energy response of the 2 outermost channels when $f_t$ increases. This effect is likely caused by the narrow space left between two separators in azimuth when $f_t$ is too large, which severely constrains the iso-potentials and distorts the energy response. To prevent this effect, we found that an empirical maximum value to consider for the torus shape factor is $f_{t,max}$ = 2/3.

## 6.2. Reduction of the Parameters Space

Equation 6 shows that $f_{t,max}$ also determines the minimum detector radius $R_{d,min}$ that can be used for a couple of [$D_1, d$] parameters. It allows us to draw the iso-contours of $R_{d,min}$ in Figure 7 in the [$D_1, d$] parameters space. In order to reduce the parameters space to be investigated, we also included in this Figure the following constraints:

1. *Manufacturing constraints:* The minimum stand-off distance between the inner and the outer toroidal electrodes d cannot be too small. This minimum may vary depending on the fabrication process of the electrodes. If we assume that a suitable process should provide an accuracy of the order of 0.1 mm, we can use 1 mm as a good order of magnitude for the minimum d value we need to consider.

2. *Cross-talk in azimuth:* The stand-off distance d also needs to be small enough to limit the cross-talk between adjacent sectors in azimuth since d defines the opening between the partial separators that isolate these sectors from each other. We established empirically that d should be at most 60% of the regular spacing $D_1$ between the electrodes of the first channel.

Finally, the available surface to accommodate a sensor on a space platform can be a strict design element. Since low-angular resolution models target micro and nano-satellites, we constrained the total diameter of the optics to
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Effect of increasing *f*ᵢ on the energy response of the donut analyzer. From top-left to bottom-right: *f*ᵢ = [0.68, 0.71, 0.76, 0.81].</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Iso-contours of the minimum detector radius *R*<sub>d</sub> (mm) in parameters space [*D*<sub>1</sub>, *d*] for which instruments satisfying relevant physical constraints can be evaluated.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Regions of parameters space [$D_1, d$] for 3 values of minimum $R_d$ = [40, 45, 50] mm shown in iso-contours, for which instrument models with a total diameter less than 19 cm are possible.</figcaption></figure>

19 cm to be able to fit the instrument on a 20 × 20 cm² platform with a 5% margin. In Figure 8, we show the region of parameters space where, for a given value of $R_d$, we have such instrument models whose total diameter would not exceed the 19 cm limit. We considered three specific values of $R_d$: 40, 45, and 50 mm. It can be seen that this size constraint excludes instrument models with large values of $D_1$ which would translate, for a given $d$ and $K$ value, into a large value of $R_1$ that would increase all toroidal radii.

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Average geometric factor and energy resolution of 3D plasma camera models of less than 19 cm diameter with 22.5° angle resolution and a 20 keV energy limit. Instruments with an MCP outer radii $R_d$ = [40, 45, 50] correspond to orange, green, and blue dots, respectively. Each dot is associated with a specific set of $D_1, d$ parameters within the regions displayed in Figure 8. Dashed gray lines separate set of parameters [$D_1, R_d$] with the same K factors.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Iso-contours of the minimum detector radius $R_d$ in parameters space [$D_1, d$] for which instruments with a high-angular-resolution of 11.25° satisfying relevant physical constraints can be evaluated.</figcaption></figure>

## 6.3. Optical Performances

We used Figure 8 to identify the set of [$R_d, D_1, d$] parameters for which detailed numerical simulations were needed. By sampling the [$D_1, d$] parameters space at 0.1 mm resolution, we found 32 sets of instrument parameters that satisfy all the above-mentioned constraints. Each instrument model was studied following the approach described in Section 5. Several metrics can be considered to compare the performance of these models. We used the average geometric factor of the four channels GF and their average energy resolution $\Delta E/E$ to present their optical performance in Figure 9. The knowledge of GF allows to quickly compare with reference instruments and to anticipate the differential energy flux that will be detectable at a given time resolution (see Equation 7). The energy resolution also provides the number of energy steps needed to cover a specific energy range.

Models with different values of $R_d$ correspond to a specific color in Figure 8. For a given set of [$D_1, d$] parameters, increasing $R_d$ increases the geometric factor, in some cases by more than 50% for a modest increase of the detector's radius of only 1 cm. We also observed that for a fixed $R_d$, increasing $d$ increases the geometric factor but decreases the energy resolution. This is consistent with Equation 2 since electrostatic analyzers with lower K factors usually accept more particles and have higher sensitivity at the expense of their energy resolving power. This parametric study gives us a broad range of instrument models to choose from, depending on the application. A highly capable instrument in terms of GF may correspond to $R_d = 50$ mm, $d = 1.2$ mm, $D_1 = 2.3$ mm with $GF = 1.5 \cdot 10^{-3} \text{ cm}^2 \cdot \text{sr} \cdot \text{eV}/\text{eV}$ per pixel. Considering a 35% overall detection efficiency, we have an effective $GF_{eff} = 5 \cdot 10^{-4} \text{ cm}^2 \cdot \text{sr} \cdot \text{eV}/\text{eV}$ per pixel which is similar or higher than the effective geometric factor of electrostatic analyzers that have flown on many space science missions (see e.g., Pollock et al., 2016).

## 7. High-Angular-Resolution Instruments

A similar method of optimization is used to study potential solutions for geometries with a high 11.25° angle resolution corresponding to 8 polar channels and 32 sectors in azimuth for a total of 256 look directions and as many pixels for the anode board. The same maximum energy of 20 keV is considered.

Such high-angular-resolution instruments target applications where a fine analysis of the plasma environment is needed and will be best fit for legacy scientific missions where the volume and mass budgets are less constrained than for small-satellite missions. We therefore considered a larger maximum instrument diameter of 30 cm. The same constraints as in Section 6 are imposed such that $d < 1$ mm, $f_i < 2/3$, and $d < 0.6 \cdot D_1$. We chose to study instrument models with MCP outer radii $R_d = [85, 90, 95]$ mm. Figure 10 provides the iso-contours of
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Comparative simulated performances of instrument models with a high 11.25° resolution.</figcaption></figure>

instrument models satisfying all these constraints for the three values of $R_d$ in the [$D_1, d$] parameters space. Again, by sampling this parameters space at 0.1 mm resolution, we found 25 sets of instrument parameters that met our requirements. Numerical simulations to study the 25 instrument models took a total of 136 hr, each model being described by its 8 polar channels.

We show the comparative performances of these instruments based on their average GF and energy resolution $\Delta E/E$ in Figure 11. The general optical behavior of these high-angular resolution 3D plasma cameras is similar to the low-angular-resolution models: the GF strongly depends on the MCP outer radius $R_d$. Larger MCP outer radii offer the possibility of instruments with GF in excess of $10^{-3} \text{ cm}^2 \cdot \text{sr} \cdot \text{eV/eV}$ per pixel. Varying the stand-off distance $d$, which is inversely proportional to the K factor of the analyzer, also allows adjusting the GF and the energy resolution of the instrument for a given value of $R_d$. Comparatively, varying the inter-electrode spacing of the first polar channel $D_1$ has a weaker impact on the optical performances. However, this parameter is already severely constrained by the chosen set of [$R_d, D_1$] parameters. While these instrument models reach relatively lower GF than the low-angular-resolution models, it is important to note that it reaches the level of GF in the $0.4 - 1.1 \cdot 10^{-3} \text{ cm}^2 \cdot \text{sr} \cdot \text{eV/eV}$ per pixel range that is appropriate to many space science missions.

In order to illustrate the performances of high-angular-resolution instrument models, we display the azimuth versus polar response of a specific instrument model with $R_d = 90$ mm, $D_1 = 2$ mm, and $d = 1$ mm in Figure 12. This Figure shows the iso-contours of the instrument response function $R_i(\theta, \phi)$ that has been integrated over all energies and normalized to its maximum value for each polar channel $i$. Underneath is the combined polar response of the 8 instrument channels after integration over all energies and azimuth look directions. We note the relatively dense coverage of phase space provided by this high-angular-resolution version of the 3D plasma camera.

The detailed performances of this model is given by Table 2. We see that the 3D plasma camera provides a dense cover of all azimuth look directions with $\Delta\phi = 12° - 14°$, and almost a complete cover of polar look directions. The average energy resolution $\Delta E/E = 9.7\%$ is also appropriate for the detailed investigation of many plasma processes.

## 8. Summary and Discussion

This research work allowed a detailed investigation of the capability of the 3D plasma camera whose design is based on the innovative donut analyzer topology. We had to develop appropriate numerical tools to perform our
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Top: iso-contours of the azimuth versus polar normalized response function $R_d(\theta, \phi)$ of a 3D plasma camera with $R_d = 90$ mm, $D_1 = 2$ mm, and $d = 1$ mm. The normalized response function has been integrated over all energies. Bottom: combined polar response of the 8 polar channels of the instrument after integration over all energies and azimuth look directions. Dotted lines represent the theoretical viewing direction in the polar dimension.</figcaption></figure>

parametric study. The building of the optics geometry as well as the evaluation of the instrument optical performance through the launch of charged particles was entirely parametrized. Using parallel computing, we were able to let vary the energy and the particles velocity direction, and to scan the whole entrance surface of the instrument with 10s of millions of particles in a matter of hours. We have obtained the exact response function, energy resolution, and angular acceptance of the donut analyzer. We have calculated the exact geometric factor of the instrument, without using the full-width at half-maximum of its angle or energy response function, which is often the case for top-hat analyzers with deflectors, as in Collinson et al. (2012).

Given the important number of geometrical parameters that define the donut topology, we had to impose a number of physical constraints to reduce the study to the effect of 3 driving parameters, namely the MCP outer radius $R_d$,

the inter-electrode distance of the first channel $D_1$, and the stand-off distance $d$, which is related to the $K$ factor of the analyzer. We also imposed a common $K$ factor for the $n$ polar channels of the optics, which would simplify the development of the high-voltage power supply of the 3D plasma camera. Our analysis showed that the variation of the 3 driving parameters had to be limited by mechanical constraints and by the need to avoid the cross-talk in azimuth that is inherent in the donut topology.

The parametric study was carried out at a given maximum energy of 20 keV and was organized according to the expected angular resolution of the optics, which translates directly into a number of polar channels and azimuth sectors and, finally, into a number of pixels in the detection plane. The main outcome of the study is that, at any given angular resolution, a wide range of geometric factors can be obtained depending on the size of the detector and on the $K$ factor of the analyzer. Overall, the averaged geometric factor per pixel of the 3D plasma camera varies between $4 \cdot 10^{-4} \text{cm}^2 \cdot \text{sr} \cdot \text{eV}/\text{eV}$ and almost

<table><caption>**Table 2**
Optical Performances of the 3D Plasma Camera Model With $E_{max} = 20$ keV, $n = 8$, $\Phi = 11.25^{\circ}$, $R_d = 90$ mm, $d = 1$ mm, and $D_1 = 2$ mm</caption><thead><tr><th></th><th>Δθ</th><th>Δφ</th><th>ΔE/E</th><th>GF(cm<sup>2</sup>.str.eV/eV)</th><th>K</th><th>A (mm<sup>2</sup>)</th></tr></thead><tbody><tr><td># 1</td><td>5.0°</td><td>13°</td><td>8.6%</td><td>5.0 · 10<sup>-4</sup></td><td>6.3</td><td>47.4</td></tr><tr><td># 2</td><td>4.8°</td><td>12°</td><td>7.9%</td><td>4.9 · 10<sup>-4</sup></td><td>6.3</td><td>56.5</td></tr><tr><td># 3</td><td>4.8°</td><td>12°</td><td>7.8%</td><td>4.9 · 10<sup>-4</sup></td><td>6.4</td><td>66.0</td></tr><tr><td># 4</td><td>5.0°</td><td>12°</td><td>8.3%</td><td>5.0 · 10<sup>-4</sup></td><td>6.5</td><td>76.1</td></tr><tr><td># 5</td><td>5.3°</td><td>12°</td><td>9.5%</td><td>5.2 · 10<sup>-4</sup></td><td>6.6</td><td>86.6</td></tr><tr><td># 6</td><td>6.3°</td><td>12°</td><td>10.5%</td><td>5.6 · 10<sup>-4</sup></td><td>6.8</td><td>97.3</td></tr><tr><td># 7</td><td>6.3°</td><td>13°</td><td>11.0%</td><td>6.2 · 10<sup>-4</sup></td><td>6.8</td><td>108.3</td></tr><tr><td># 8</td><td>5.8°</td><td>14°</td><td>14.0%</td><td>6.9 · 10<sup>-4</sup></td><td>6.8</td><td>119.1</td></tr></tbody></table>
3 · 10⁻³ cm²·sr·eV/eV with an energy resolution between 9% and 14%. This suggests the high-time resolution capability of the 3D plasma camera that takes advantage of a large entrance surface and of the suppression of the angular scan needed by standard optical designs. We had to assume a maximum allowable volume for the donut optics to identify these trends, which should help in the design of the 3D plasma camera for any specific application. We noticed there was some channel to channel variation in the optical performance of the instrument, as it is known to be the case for top-hat analyzers with electrostatic deflectors. Homogenizing the geometric factors among polar channels may be a good way to choose between different sets of instrument parameters with similar averaged performance.

A wide range of space missions can benefit from this new instrument concept. In the field of space weather, a low-
angular-resolution instrument that can be installed on a 20 × 20 cm nano-satellite platform seems feasible. It
would allow the monitoring of charged particle precipitations on low-Earth orbits with enhanced accuracy due to
the much wider FoV of the plasma camera than one of standard top-hat analyzers. This is the topic of our current
research, with a prototype instrument in the final stages of development. The enabling technology and experi-
mental performance of this prototype will be presented in a forthcoming publication. At the other end of the
spectrum, performance-intensive science missions would benefit from a larger-scale 3D plasma camera: the ESA
Plasma Observatory M7 mission candidate includes an electron plasma camera whose optical design is in its early
stages of development, and which is based on the high-angle resolution version of the instrument described in
Section 7. Compared with NASA MMS mission, which uses 4 dual-head sensors, the Plasma Observatory
mission plans to use just 2 plasma cameras, which should reduce the complexity of data calibration and the cost of
payload development, while keeping sensor mass and volume compatible with the use of a medium-size platform.
Finally, we would like to point out that the development of such a 3D plasma camera requires the design of a
complex detection system, which will necessarily be based on multi-channel charge sensitive amplifiers due to the
high number of anode pixels. A first version of such an ASIC has been developed for and used by the Solar Orbiter
SWA-EAS instrument (Owen et al., 2020), and an improved version of the component is currently being
developed for the 3D plasma camera.

Data Availability Statement

Raw simulation results of all the models presented in this paper are available in their original binary format at
https://zenodo.org/records/14609013 (Hénaff, 2025).

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