3D Photoelectron Track Polarimeter

• Three-Dimensional Photoelectron Track Polarimeter is a photoelectric X-ray polarimeter that reconstructs full 3D electron tracks to minimize projection ambiguities and improve polarization measurements.
• It integrates advanced algorithms—such as voxelisation, ridge finding, and machine-learning methods—with diverse detector architectures like TPC, GridPix, and hybrid optical systems to boost modulation factors.
• Demonstrated improvements include a 5–17% relative gain in modulation in the soft X-ray band and high rate capability, supporting future missions with enhanced sensitivity and energy reach.

A three-dimensional photoelectron track polarimeter is a photoelectric X-ray polarimeter that reconstructs the full trajectory of the photoelectron in three spatial dimensions rather than only its two-dimensional projection. In this class of detector, the polarization information is carried by the initial photoelectron emission direction, while the third coordinate is typically recovered through time projection in a gas volume or by combining complementary imaging and timing observables. Relative to two-dimensional photoelectron imaging, the three-dimensional approach is pursued to reduce projection ambiguities, improve reconstruction of the initial track segment, and extend performance to more demanding regimes of energy, rate, and event topology (Kim et al., 2023, Manikantan et al., 2 Sep 2025).

1. Physical basis of three-dimensional photoelectron polarimetry

Photoelectric X-ray polarimetry relies on the anisotropy of the photoelectron emission direction with respect to the electric-field vector of the incident photon. In the formulation used for soft X-ray polarimetry, the differential cross section is written as

$$\frac{{\rm d}\sigma_{\rm ph}}{{\rm d}\Omega} \propto \frac{\cos^2 \varphi \,\sin^2 \theta}{\left(1 - \beta \cos\theta\right)^4},$$

where $\theta$ is the polar angle of the photoelectron with respect to the photon propagation direction, $\varphi$ is the azimuthal angle measured from the polarization vector, and $\beta$ is the photoelectron speed divided by the speed of light (Kim et al., 2023). In practical polarimetry the measured modulation curve is commonly parameterized as

$$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$

with $A$ a normalization constant, $P$ the degree of linear polarization, $\phi_0$ the polarization angle, and $\mu$ the modulation factor for a 100% polarized beam (Fiorina et al., 30 Oct 2025).

The distinction between two-dimensional and three-dimensional systems is not a change in the underlying photoelectric physics but in the fidelity with which the track is represented. Traditional gas pixel polarimeters reconstruct the photoelectron from a single projected charge image, so information along the drift direction is integrated away. Three-dimensional approaches instead seek a volumetric charge cloud or a three-dimensional ridgeline, allowing the initial segment of the track to be localized more accurately when the track is short, strongly scattered, or unfavorably oriented with respect to the readout plane (Kim et al., 2023, Tilly et al., 2023).

The gain from adding the third coordinate is energy dependent rather than uniform. A simulation study using a three-dimensional reconstruction algorithm reported an improvement of the modulation factor with three-dimensional track reconstruction as $\sim 5\%$ (relative) in the $\theta$0 keV range and $\theta$1 (relative) in the $\theta$2 keV range compared to a current two-dimensional polarimetry system (Kim et al., 2023). This indicates that three-dimensional reconstruction is most consequential where track projection losses are most severe, rather than automatically transforming performance at all energies.

2. Detector architectures and readout topologies

Several detector realizations embody the three-dimensional photoelectron track polarimeter concept. One route is the time projection chamber with electronic or optical readout. Another is the hybrid micromesh–pixel architecture exemplified by GridPix. A third, at higher electron energies, combines optical imaging with photomultiplier timing in large-volume TPCs (Tilly et al., 2023, Fiorina et al., 30 Oct 2025, Manikantan et al., 2 Sep 2025).

The MIGDAL optical TPC illustrates the hybrid optical-electronic strategy at few-keV electron energies. It operates at 50 Torr in CF$\theta$3 and CF$\theta$4/Ar mixtures, with a 3 cm drift region, a double glass GEM amplification stage, an indium tin oxide strip readout, and an sCMOS camera imaging the GEM scintillation. The camera provides the $\theta$5 projection with $\theta$6m sampling, while the strip readout provides an $\theta$7 projection through timing with 833 $\theta$8m pitch. Combining the shared $\theta$9 coordinate yields full three-dimensional track reconstruction of low-energy electrons (Tilly et al., 2023). Although developed for Migdal-effect studies, the detector addresses the same technical regime as soft-X-ray photoelectron polarimetry: few-keV electrons, millimeter-scale tracks, diffusion, and the need to recover initial direction.

A different implementation is the Hype-X concept based on TIMEPIX3. In the simulation study devoted explicitly to three-dimensional photoelectron track reconstruction, each pixel is assumed to provide $\varphi$0 and $\varphi$1, time of arrival for the $\varphi$2 coordinate, and time over threshold as a proxy for charge, producing a 3D charge cloud $\varphi$3 in pure dimethyl ether at 800 mbar (Kim et al., 2023). This architecture is conceived as an evolution of gas pixel polarimetry toward a true three-dimensional image of the photoelectron track.

At higher energies, a triple-GEM optical TPC has been demonstrated for wide-field hard X-ray polarimetry. That prototype uses a cylindrical active volume of radius $\varphi$4 cm and height $\varphi$5 cm, a triple-GEM amplification stage, an sCMOS camera, and a PMT. In the reported first results, the PMT timing was not yet used event by event, so the analysis remained two-dimensional, but the hardware is already a time-projection system in which the third coordinate is accessible via the PMT signal (Fiorina et al., 30 Oct 2025).

GridPix represents a more radical detector-level redesign. It couples an InGrid structure to a Timepix3 ASIC with $\varphi$6 pitch and $\varphi$7 sensitive area. Each pixel records ToA with 1.56 ns resolution and ToT as a digital amplitude, enabling three-dimensional track imaging through drift-time reconstruction (Manikantan et al., 2 Sep 2025). In the comparison given for a representative GPD/XPOL versus GridPix/Timepix3, the active area is similar, but GridPix adds 3D track imaging, much smaller diffusion in the transfer gap, and a maximum usable rate of $\varphi$8 cts s$\varphi$9 at 10% dead time under the assumption of $\beta$0 pixels per track (Manikantan et al., 2 Sep 2025).

A mission-oriented variant is the transmissive TPC polarimeter designed for hard-X-ray focusing telescopes. It uses pure DME at $\beta$1 atmosphere, a 24 cm sensitive absorption depth, 128 strips with 140 $\beta$2m pitch, and a rear Be window so that harder photons not absorbed in the gas can continue to a focal-plane detector. The transmission at 6.4 keV is stated to be as high as 80%, making the design relevant when a polarimeter must coexist with downstream spectroscopy or imaging hardware (Li et al., 2015).

3. Three-dimensional reconstruction methodologies

Three-dimensional photoelectron track polarimetry is as much an algorithmic problem as a detector problem. The central tasks are to reconstruct the interaction point, identify the initial low-$\beta$3 segment, suppress diffusion and readout artifacts, and infer the emission direction with a calibrated uncertainty.

In the MIGDAL optical TPC, two complementary reconstruction chains are described. The first is voxelisation: once the $\beta$4 camera image and the $\beta$5 strip-time map are brought onto a common grid, the three-dimensional voxel intensity is formed as

$\beta$6

where $\beta$7 indexes the shared $\beta$8 coordinate, $\beta$9 is $$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$0, and $$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$1 is $$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$2 (Tilly et al., 2023). The second is the RidgeFinder method, which applies dark subtraction, Fourier Gaussian filtering, Lucy–Richardson deconvolution with a diffusion-matched Gaussian point spread function, and then Steger’s ridgeline algorithm to extract a two-dimensional track spine in the camera image. The $$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$3 coordinate is then assigned from the strip timing through

$$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$4

producing a three-dimensional curve that follows the particle trajectory (Tilly et al., 2023).

The Hype-X three-dimensional reconstruction algorithm uses a moments-based procedure adapted to a 3D charge cloud. A regression plane

$$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$5

is fitted by minimizing

$$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$6

after which the track is rotated into that plane, reduced to a two-dimensional moments analysis in the rotated frame, and refined around the inferred interaction point by selecting the initial segment while excluding the Bragg-peak region (Kim et al., 2023). This construction is explicitly motivated by the need to preserve three-dimensional asymmetries that would be washed out in a planar projection.

Another important line of development comes from graph-based track finding. A shortest-path algorithm was introduced for two-dimensional gas-pixel images in which the measured charge distribution is converted into a graph, the longest shortest path is used as the primary path, and the interaction point is identified at the low-charge end of the reconstructed path (Li et al., 2016). The same work emphasizes that the useful information needed for polarimetry is stored mostly in the initial part of the track where less energy is deposited, and it explicitly notes that the method should also work for other polarimetric techniques such as a time projection chamber (Li et al., 2016). A plausible implication is that its generalization from pixels to voxels is natural for three-dimensional track polarimeters.

Machine-learning reconstruction has also become part of the field. Deep neural network methods have been used to determine photoelectron emission directions, photon absorption points, and photon energies from two-dimensional track images, while also estimating predictive uncertainties. These uncertainties can be folded into a weighted maximum likelihood for polarization analysis, and a classifier can be trained to reject events with little polarization sensitivity (Peirson, 2022). In the formulation adopted there, the optimal event weights are

$$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$7

where $$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$8 is the event-level concentration parameter of the angular posterior (Peirson, 2022). This framework is detector-agnostic in principle and therefore directly relevant to three-dimensional implementations.

4. Demonstrated performance and energy reach

The performance landscape of three-dimensional photoelectron track polarimetry is defined by two regimes that have both seen concrete demonstrations: few-keV electrons in low-pressure or finely instrumented gas volumes, and tens-of-keV electrons in larger-volume TPCs for hard-X-ray polarimetry.

At low energies, the MIGDAL optical TPC demonstrated three-dimensional reconstruction of electrons at 50 Torr in CF$$M(\phi) = A \left[1 + \mu P \cos\left(2(\phi - \phi_0)\right)\right],$$9 and CF$A$0/Ar mixtures, including 5.9 keV $A$1Fe photoelectron tracks, 2.9 keV electrons from the Ar escape peak, and faint low-$A$2 beta tracks (Tilly et al., 2023). The assessment is explicitly qualitative rather than a full efficiency study, but the work establishes that full 3D tracks are reconstructable down to approximately the few-keV range relevant for soft-X-ray polarimetry.

The Hype-X simulation study quantified the specific benefit of three-dimensional reconstruction in the IXPE-like soft-X-ray band. It reported a relative improvement of $A$3 in modulation factor over $A$4 keV and $A$5 over $A$6 keV compared to the current two-dimensional system (Kim et al., 2023). The same study further stated that this is equivalent to add a further telescope to the three-telescope systems now employed in space on board the IXPE mission (Kim et al., 2023). This comparison underscores that even moderate gains in $A$7 can be operationally significant.

In the hard-X-ray regime, the triple-GEM time projection chamber reported first results on fully reconstructing electrons in the 10–60 keV range, obtaining angular resolutions as good as $A$8, and inferring modulation factors up to 0.9 (Fiorina et al., 30 Oct 2025). In the more detailed breakdown, angular resolutions are better than $A$9 for energies above 10 keV and better than $P$0 for electrons between 20–60 keV, with inferred $P$1 above 10 keV and $P$2 in the 20–60 keV range (Fiorina et al., 30 Oct 2025). This establishes that photoelectric-effect polarimetry can be extended well beyond the classical $P$3 keV band when long tracks and large gas volumes are available.

The reconstruction algorithm itself can also be a limiting factor. In a two-dimensional GPD study that is directly relevant to later three-dimensional generalizations, a graph-based reconstruction method improved the modulation factor from $P$4 to $P$5 at 13 keV and from $P$6 to $P$7 at 15 keV relative to a moment-analysis baseline, while showing limited improvement below about 7 keV (Li et al., 2016). This is a useful corrective to the common assumption that all performance gains must come from detector hardware: algorithmic extraction of the initial segment remains a major lever.

Rate capability is another defining performance axis. The current GridPix prototype was run linearly up to at least 7000 cts s$P$8 with negligible dead time, and the comparison table gives $P$9 cts s$\phi_0$0 as the maximum usable rate at 10% dead time for GridPix/Timepix3 under the assumption of $\phi_0$1 pixels per track (Manikantan et al., 2 Sep 2025). For bright X-ray sources and future large-area optics, this is as consequential as raw modulation.

5. Calibration, systematics, and design constraints

Three-dimensional photoelectron track polarimetry is constrained by gas transport, track statistics, and event-class systematics. These do not disappear when the third coordinate is added; they become more explicitly modelable.

Track length statistics provide one design anchor. In a negative-ion TPC, photoelectron track length distributions between 3 and 8 keV were measured in gas and found to be best fit by a lognormal distribution (Prieskorn et al., 2014). The mean track length followed a power law

$\phi_0$2

with $\phi_0$3 for Ne+CO$\phi_0$4+CH$\phi_0$5NO$\phi_0$6 and $\phi_0$7 for CO$\phi_0$8+CH$\phi_0$9NO$\mu$0 (Prieskorn et al., 2014). Because pixel size and drift geometry must be chosen relative to the shortest usable tracks rather than only to the mean, these measurements are directly relevant to three-dimensional detector design.

Diffusion and drift calibration are the second anchor. In the optical-TPC and Hype-X work, the $\mu$1 coordinate is written in the time-projection form $\mu$2, with the drift velocity $\mu$3 requiring calibration for each gas mixture and field (Tilly et al., 2023, Kim et al., 2023). The diffusion model is correspondingly expressed through

$\mu$4

or

$\mu$5

which determines how much deconvolution or uncertainty inflation is needed in reconstruction (Tilly et al., 2023). In a true three-dimensional polarimeter, this calibration affects the quality of the volumetric track and hence the inference of the initial direction, even if the polarization-sensitive azimuth still lies in the transverse plane.

Event selection is the third anchor. Neural-network analysis for IXPE-type data showed that conversions outside the fiducial gas volume have little polarization sensitivity and complicate both energy reconstruction and polarization estimation. The same study used a classifier to reject such events and incorporated event-level angular uncertainties into a weighted maximum likelihood for the Stokes parameters (Peirson, 2022). In that framework the effective number of events becomes

$\mu$6

and the weighted MDP is

$\mu$7

Applied to IXPE-specific simulations, the deep-learning-based method achieved a $\mu$8 improvement in minimum detectable polarization relative to standard analysis (Peirson, 2022). A plausible implication is that three-dimensional detectors, which can localize the interaction point and boundary proximity more directly, should make this style of quality-weighted analysis more effective rather than less.

A common concern is that more elaborate reconstruction may inject spurious modulation. In the graph-based reconstruction study, unpolarized simulations yielded measured modulations of $\mu$9 at 6 keV and $\sim 5\%$0 at 15 keV, consistent with zero within uncertainties (Li et al., 2016). This does not eliminate systematic-risk questions, but it shows that algorithmic sophistication is not inherently synonymous with artificial polarization signal.

6. Scientific roles and future development

Three-dimensional photoelectron track polarimeters are being developed for both focal-plane and wide-field use. In the hard-X-ray TPC prototype, the explicitly stated target classes include gamma-ray bursts, solar flares, magnetar giant flares, tidal disruption events, and binary neutron star mergers, all of which benefit from wide field of view and sensitivity at tens of keV (Fiorina et al., 30 Oct 2025). The transmissive TPC concept addresses a different mission architecture: a polarimeter placed above the focal plane with a rear window so that hard X-rays can penetrate through it and reach a downstream detector. In the reported design, the transmission is nearly 80% at 6 keV and the expected sensitivity is 3% MDP for a 1 mCrab source in $\sim 5\%$1 s (Li et al., 2015).

The post-IXPE development trajectory is represented most clearly by GridPix. The current program is aimed at a next generation three-dimensional photoelectron track polarimeter based on an InGrid–Timepix3 detector, motivated by the need to overcome XPOL dead time and GEM gain instabilities in future missions with 10–20× larger effective area than IXPE (Manikantan et al., 2 Sep 2025). Preliminary proton beam irradiation runs verified both ASIC tolerance to high radiation doses and the capability of the Bonn cyclotron facility to operate at sufficiently low rates for controlled tests, while a dose exceeding the expected 20-year LEO exposure by about two orders of magnitude produced no significant degradation in threshold maps, noise scans, or $\sim 5\%$2Fe response (Manikantan et al., 2 Sep 2025). Heavy-ion tests with $\sim 5\%$3N$\sim 5\%$4 are planned to probe spark behavior and single-event robustness more aggressively (Manikantan et al., 2 Sep 2025).

The field is therefore defined by convergence rather than by a single canonical instrument. Optical TPCs have shown qualitative three-dimensional reconstruction down to the few-keV regime (Tilly et al., 2023). TIMEPIX3-based concepts have shown that the modulation gain from 3D reconstruction is concentrated at the energies where projection losses are most damaging (Kim et al., 2023). Triple-GEM optical TPCs have already demonstrated hard-X-ray electron tracking with strong inferred modulation (Fiorina et al., 30 Oct 2025). GridPix has shown the readout-rate and radiation-hardness properties required for space deployment (Manikantan et al., 2 Sep 2025). What unifies these efforts is the same objective: to reconstruct the initial photoelectron direction with sufficient geometric fidelity, event-by-event quality control, and instrumental robustness that the full statistical power of future X-ray polarimetry missions can be realized.