*3.1. Measurement Principles*

The polarization of X-ray or *γ*-ray photons can be measured by studying the properties of the particles created during their interaction within the detector. For all the three possible interaction mechanisms, namely, the photo-electric effect, Compton scattering, and pair production, a dependency exists of the orientation of the outgoing products on the polarization vector of the incoming photon. This is illustrated in Figure 3 for the three processes. For the photo-electric effect, it is the azimuthal direction of the outgoing electron that shows a dependency on the polarization vector of the incoming photon; for Compton scattering, it is the azimuthal scattering angle of the photon; and for pair-production, it is the plane defined by the electron-positron pair.

The differential cross section for photo-absorption (via the photo-electric effect) has a dependency on *φ*, which is defined as the azimuthal angle between the polarization vector of the incoming photon ~*p*, as shown in Figure 3, and the projection of the velocity vector of the final state electron ~*β* = ~*v*/*c* (where *β*ˆ = ˆ*k*2) on to the plane normal to the momentum vector~*k*<sup>1</sup> of the photon,

$$\frac{d\sigma}{d\Omega} \propto \cos^2 \phi \quad , \quad \phi = \cos^{-1} \left( \frac{\vec{\mathcal{J}} \cdot \vec{p}}{\beta p \sin \theta} \right) = \cos^{-1} \left( \frac{\vec{k}\_2 \cdot \vec{p}}{k\_2 p \sin \theta} \right) \, , \tag{2}$$

where *d*Ω = sin *θdθdφ* is the unit solid angle, and the polar angle *θ* is given by cos *θ* = *β*ˆ · <sup>ˆ</sup>*k*<sup>1</sup> <sup>=</sup> <sup>ˆ</sup>*k*<sup>2</sup> · ˆ*k*1. Similarly, for the differential cross section of Compton scattering the dependence on *φ*, here the angle between the polarization vector of the incoming photon ~*p* and the projection of the momentum vector of the outgoing photon~*k*<sup>2</sup> on to the plane normal to the momentum vector~*k*<sup>1</sup> of the incoming photon, where *φ* = cos−<sup>1</sup> <sup>~</sup>*k*2·~*<sup>p</sup> k*<sup>2</sup> *p* sin *θ* as in Equation (2), is

$$\frac{d\sigma}{d\Omega} = \frac{r\_o^2}{2} \frac{E'^2}{E^2} \left(\frac{E'}{E} + \frac{E}{E'} - 2\sin^2\theta\cos^2\phi\right). \tag{3}$$

Here, *r*<sup>0</sup> = *e* <sup>2</sup>/*mec* 2 is the classical electron radius with *e* being the elementary charge, *E* is the initial photon energy, *E* 0 the final photon energy, and *θ* = cos−<sup>1</sup> ( ˆ*k*2 · ˆ*k*1) is the polar scattering angle.

**Figure 3.** Illustration of the angular dependency of the interaction product on the polarization vector of the incoming photon for the three interaction mechanisms: photo-electric effect (**left**), Compton scattering (**middle**), and pair production (**right**). The incoming photon is shown in blue, its polarization vector in green, and the secondary product(s) in red. The *θ* angle (as used in Equations (2) and (3)) is defined as the angle between the incoming photon direction and its secondary product. The *φ* angle (again as used in Equations (2) and (3)) is defined as the angle between the projections of the polarization vector and the momentum vector of the secondary product(s) onto the *x*-*y* plane. The *η* angle is the azimuthal angle between the *x*-axis and the projection of the momentum vector of the secondary particle onto the *x*–*y* plane. The *θ* and *η* angles can be directly measured in a detector, while *φ* is measured indirectly.

> Finally, for pair production the differential cross section is *dσ*/*d*Ω ∝ 1 + *A*(cos 2*φ*), where *A* is the polarization asymmetry of the conversion process (which has dependencies on the photon energy and properties of the target), and *φ* is the angle between the polarization vector of the incoming photon ~*p* and the plane defined by the momentum vectors of the electron–positron pair,~*k*±.

> The general concept for polarimetry in the three energy regimes where these cross sections dominate is therefore similar: one needs to detect the interaction itself and subsequently track the secondary particle, be it an electron, photon, or electron–positron pair. This requirement indicates the first difficulty in polarimetry: simply absorbing the incoming photon flux, as is the case in, for example, standard spectrometry, is not sufficient. The requirement to track the secondary product significantly reduces the efficiency of the detector.

> After measuring the properties of the secondary particles, a histogram of *φ* can be made, which shows a sinusoidal variation with a period of 180◦ referred to as a modulation curve. The amplitude of this is proportional to the polarization degree (PD) and the phase related to the polarization angle (PA). As can be derived from, for example, the Compton scattering cross section, the amplitude for a 100% polarized beam will depend on the energy of the incoming photons as well as on the polar scattering angle. Whereas the energy depends on the source, the polar scattering angle is indirectly influenced by the instrument design. For example, using a detector with a thin large surface perpendicular to the incoming flux, it is more likely to detect photons scattering with a polar angle of 90◦ , which have a larger sensitivity to polarization than those scattering forward or backward. The relative amplitude, meaning the ratio of the amplitude of the sinusoidal over its mean,

is directly proportional to the PD. The relative amplitude one detects for a 100% polarized beam is known as the *M*100, and it depends on the source spectrum, source location in the sky, the instrument design, and the analysis. Although for specific circumstances the *M*<sup>100</sup> can be measured on the ground using, for example, mono-energetic beams with a specific incoming angle w.r.t the detector, its dependency on the energy, incoming angle, and instrument conditions, such as its temperature, implies that the *M*<sup>100</sup> required in the analysis of real sources can only be achieved using simulations. The large dependency on simulations provides a source for potential systematic errors in the analysis, which can easily dominate the statistical error in the measurements.

Additionally, it should be noted that in practice retrieving the polarization is significantly more complicated as both instrumental and geometrical effects (such as the incoming angle of the photons w.r.t. the detector and the presence and orientation of materials around the detector) are added to the polarization-induced signal in the modulation curve. In order to retrieve the polarization signal one can, for example, divide it by a simulated modulation curve for an unpolarized signal as illustrated in the first column of Figure 4. This method is often used, for example in [188]. A second option is to model these effects together with the signal and fit the uncorrected curve with this simulated response, as was for example done in [189]. In either case, it requires a highly detailed understanding of the instrument.

In polarization analysis, any imperfections in modelling the instrument will likely result in an overestimation of the polarization. As illustrated in Figure 4, for a modulation curve resulting from an unpolarized flux, removing any instrumental effects from the modulation curve should result in a perfectly flat distribution. This is illustrated in the middle column of this figure. Any error in the model of the instrumental or geometrical effects will, however, result in a non-flat distribution, which, when fitted with a harmonic function, will result in some level of polarization to be detected. It is therefore in practice impossible to measure a PD of 0% as it would require both an infinite amount of statistics, and more importantly, a perfect modelling of all the instrumental effects. On the other extreme, for a PD of 100%, imperfections in the modelling can result in a lower amplitude, but can still also increase it further resulting in measuring a nonphysical PD. Overall, due to the nature of the measurement, both statistical and systematic errors tend to inflate the PD rather than decrease it. Since it is not possible to test the modelling of the instruments when in orbit, as there are no polarization calibration sources, this issue exists for all measurements and can only be minimized by extensive testing of the instrument both on the ground and in-orbit.

A final figure of merit often used in polarimetry is the minimal detectable polarization (MDP) [190]. For GRBs the MDP is best expressed as

$$\text{MDP} = \frac{2\sqrt{-\ln(1-\text{C.L.})}}{M\_{100}\text{C}\_s}\sqrt{\text{C}\_s + \text{C}\_b} \,. \tag{4}$$

Here, *C*.*L*. is the confidence level, *C<sup>s</sup>* is the number of signal events, and *C<sup>b</sup>* the number of background events. The MDP expresses the minimum level of polarization of the source that can be distinguished from being unpolarized for a given confidence level. It can therefore be seen as a sensitivity of a given polarimeter for a given observation. Whereas this is highly useful for polarimeters observing point sources, for GRB polarimeters, there is an issue related to the *M*100. For wide-field-of-view instruments, such as polarimeters designed for GRB observations, the value of *M*<sup>100</sup> can start to depend on the PA of the source. For example, in POLAR, the *M*<sup>100</sup> was found to depend on the PA for GRBs with a large off-axis incoming angle [187]. This is a result of only being able to resolve two dimensions of the scattering interactions in the detector, making it insensitive (so *M*<sup>100</sup> = 0) to certain values of PA when the *γ*-ray photons enter the detector perpendicular to the readout plane [187]. As in such cases the MDP becomes dependent on PA, it loses its use as a figure of merit. However, as the MDP remains highly used in the community and remains the best measure of sensitvity for polarimeters, we used it here in this work

as well, although with a small adaptation. In order to remove the PA dependence, we used the mean MDP where the *M*<sup>100</sup> is averaged over all possible values of the a priori unknown PA.

**Figure 4.** Illustration of recovering the polarization signal from a raw modulation curve. The left column illustrates the ideal case with a high PD value, with a raw measured modulation curve (**top**), abd the perfectly simulated instrumental and geometrical effects (**middle**), which pollute the raw modulation curve. The (**bottom**) panel shows the modulation curve after correction from the instrumental and geometrical effects, which results in a perfect harmonic function. The middle column illustrates the same but for an unpolarized signal resulting in a flat distribution. The right column shows the same for an unpolarized signal; however, random small errors were added to the instrumental and geometrical effects, thereby simulating a non-perfect understanding of the instrument. The result is a non-flat distribution, which, when fitted, shows a low level of polarization.
