First experimental time-of-flight-based proton radiography using low gain avalanche diodes

As ions travel through matter, they lose energy and slow down, which leads to a material and energy-dependent TOF, calculated as

$$\begin{eqnarray}&&\mathrm{TOF}={\int }_{0}^{L}\displaystyle \frac{{\rm{d}}s}{v(\vec{{\bf{x}}}(s))}={\int }_{0}^{L}\displaystyle \frac{{\rm{d}}s}{c\cdot \tfrac{{E}_{\mathrm{kin}}(\vec{{\bf{x}}}(s))}{{E}_{\mathrm{kin}}(\vec{{\bf{x}}}(s))+{m}_{0}{c}^{2}}\sqrt{1+2\cdot \tfrac{{m}_{0}{c}^{2}}{{E}_{\mathrm{kin}}(\vec{{\bf{x}}}(s))}}}\end{eqnarray} \tag{ 1 }$$

with m₀ as the rest mass of the ion, c as the speed of light and ${E}_{\mathrm{kin}}(\vec{{\bf{x}}}(s))$ and $v(\vec{{\bf{x}}}(s))$ as the kinetic energy and corresponding velocity at position $\vec{{\bf{x}}}(s)$.

While the TOF in equation (1) strongly depends on the tissue composition, beam energy and ion's path, it was shown that, for a small pathlength increment Δx inside the traversed tissue, the difference in TOF per unit pathlength with respect to the TOF in vacuum (TOF_vac), i.e. without any energy loss, is directly related to the stopping power (SP) of the medium according to

$$\begin{eqnarray}&&\displaystyle \frac{\mathrm{TOF}-{\mathrm{TOF}}_{\mathrm{vac}}}{{\rm{\Delta }}x}=\displaystyle \frac{{\rm{\Delta }}\mathrm{TOF}}{{\rm{\Delta }}x}:= \mathrm{SDP}(E)=f(E)\cdot \mathrm{SP}(E)\end{eqnarray} \tag{ 2 }$$

with f(E) as a solely energy-dependent term, ΔTOF = TOF − TOF_vac and SDP(E) as the so-called slowing-down power of the traversed medium at the beam energy E.

Using equation (2), one can easily see that the relative slowing down power (RSDP), i.e. the SDP in matter (SDP_mat) relative to the SDP in water (${\mathrm{SDP}}_{{{\rm{H}}}_{2}{\rm{O}}}$) is equal to the RSP since f(E) cancels out and

$$\begin{eqnarray}&&\mathrm{RSDP}:= \displaystyle \frac{{\mathrm{SDP}}_{\mathrm{mat}}(E)}{{\mathrm{SDP}}_{{{\rm{H}}}_{2}{\rm{O}}}(E)}=\displaystyle \frac{{\mathrm{SP}}_{\mathrm{mat}}(E)}{{\mathrm{SP}}_{{{\rm{H}}}_{2}{\rm{O}}}(E)}=\mathrm{RSP}\end{eqnarray} \tag{ 3 }$$

with SP_mat and ${\mathrm{SP}}_{{{\rm{H}}}_{2}{\rm{O}}}$ denoting the SP in the material and water, respectively.

In standard iCT, the RSP distribution inside the patient can be reconstructed by measuring the particle path of multiple ions travelling through the patient and the corresponding WETs, which are a measure for the total energy loss along the path (ΔE) for a given primary energy E₀. The respective inverse problem reads as follows

$$\begin{eqnarray}&&\mathrm{WET}({E}_{0},{\rm{\Delta }}E)={\int }_{0}^{L}\mathrm{RSP}(\vec{{\bf{x}}}(s)){\rm{d}}s\approx {\int }_{0}^{L}\mathrm{RSDP}(\vec{{\bf{x}}}(s)){\rm{d}}s,\end{eqnarray} \tag{ 4 }$$

with $\mathrm{RSP}(\vec{{\bf{x}}}(s))$ and $\mathrm{RSDP}(\vec{{\bf{x}}}(s))$ denoting the RSP and RSDP at positon $\vec{{\bf{x}}}(s)$ and L the total pathlength of the ion's trajectory inside the patient. Inserting equations (2) and (3) into the right-hand-side of equation (4) yields the following image reconstruction problem for Sandwich TOF-iCT

$$\begin{eqnarray}&&{\int }_{0}^{\mathrm{TOF}-{\mathrm{TOF}}_{\mathrm{vac}}}\displaystyle \frac{{\rm{d}}{\rm{\Delta }}\mathrm{TOF}}{{\mathrm{SDP}}_{{{\rm{H}}}_{2}{\rm{O}}}({\rm{\Delta }}\mathrm{TOF}(E(\vec{{\bf{x}}}(s))))}={\int }_{0}^{L}\mathrm{RSP}(\vec{{\bf{x}}}(s)){\rm{d}}s,\end{eqnarray} \tag{ 5 }$$

where ${\mathrm{SDP}}_{{{\rm{H}}}_{2}{\rm{O}}}$ is given as a function of ${\rm{\Delta }}\mathrm{TOF}(E(\vec{{\bf{x}}}(s)))$. To obtain the relation between ΔTOF and E, and therefore ${\mathrm{SDP}}_{{{\rm{H}}}_{2}{\rm{O}}}({\rm{\Delta }}\mathrm{TOF}(E(\vec{{\bf{x}}}(s))))$, one can correlate the cumulative TOF increase in water for different water thicknesses and a given primary beam energy E₀ to the corresponding residual energy $E(\vec{{\bf{x}}}(s))$ using the initial condition ${\rm{\Delta }}\mathrm{TOF}(E(\vec{{\bf{x}}}(s=0)))=0$. The correlation ${\rm{\Delta }}\mathrm{TOF}(E(\vec{{\bf{x}}}(s)))$ for water can be either obtained via MC simulations or analytically as described in Ulrich-Pur et al (2023). After determining ${\mathrm{SDP}}_{{{\rm{H}}}_{2}{\rm{O}}}({\rm{\Delta }}\mathrm{TOF}(E(\vec{{\bf{x}}}(s))))$ and measuring the total increase in TOF (TOF − TOF_vac) and the total pathlength L, one can reconstruct the RSP using standard iCT reconstruction algorithms, e.g. as defined in Rit et al (2013).

However, actually measuring the total increase in TOF requires prior knowledge of the velocity distribution along the particle path, which can only be estimated, e.g. via MC simulation. Therefore, as a first step, a much simpler approach was introduced in Ulrich-Pur et al (2023). Instead of estimating the true increase in TOF, the increase in TOF w.r.t the TOF in air (TOF_air), i.e. the TOF through the scanner without the phantom, is determined. For each particle, the resulting TOF increase is then mapped to the corresponding WET of the traversed medium using a fifth-order polynomial

$$\begin{eqnarray}&&\mathrm{TOF}-{\mathrm{TOF}}_{\mathrm{air}}(\mathrm{WET},{E}_{0})=\displaystyle \sum _{i=0}^{5}{a}_{i}({E}_{0})\cdot {\mathrm{WET}}^{i}\end{eqnarray} \tag{ 6 }$$

with a_i (E₀) as the fit parameters for an ion with primary beam energy E₀. To obtain the fit parameters a_i (E₀), a calibration run has to be performed prior to the actual imaging experiment, where TOF − TOF_air(WET, E₀) has to be measured for a fixed beam energy E₀ and different absorbers with known WET.

While this simplified calibration model is easy to implement, it also introduces a systematic dependence of the WET estimation on the system parameters of the used TOF-iCT system (Ulrich-Pur et al 2023). Thus, an optimized model should be developed to further advance this imaging modality. However, since the study presented here focuses mainly on the first experimental realisation of Sandwich TOF-iCT, improving the WET estimation algorithms for this modality was out of the scope. Therefore, the simplified WET calibration method described in equation (6) was also chosen for the pRad experiment conducted at MedAustron.

Four single-sided LGAD strip sensors were used for the TOF-iCT demonstrator, each measuring ≈1 × 1 cm² and containing 86 strips with a strip length of 8.6 mm and a pitch of 100 μm (figure 1(a)). Similar to the HADES T₀ detector, which stems from the same R&D production run as the LGADs used within this work, all sensors underwent a thinning procedure to reduce the overall material budget of the detector, resulting in a total sensor thickness of 200 μm.

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In order to bias and read out each sensor, four detector modules were developed. As shown in figure 1(c), each module was designed with a centrally located rectangular cut-out in which a 3D-printed plastic grid was placed (dark green grid). The sensor was then glued onto this grid and connected to the discrete FEE consisting of a two-stage pre-amplifier indicated as dark blue strips and the corresponding low-voltage (LV) power lines to drive the pre-amplifiers as well as high-voltage (HV) power lines for biasing the LGAD. The pre-amplified signals were then forwarded to custom FPGA-based time-to-digital converter (TDC) boards (DiRICH5s1 boards), which also feature another amplification stage and a leading edge discriminator (indicated as pink strips) as well as an optical link for the readout (highlighted in yellow) and low voltage differential signal (LVDS) lines for triggering (grey connectors). Using the DiRICH5s1, the leading and falling edge of the amplified LGAD signal could be determined, which allowed measuring the time-of-arrival (ToA) of the particle as well as the time-over-threshold (ToT), i.e. a measure of the signal duration and therefore also signal height. While each DiRICH5s1 has 32 readout channels, the channel configuration inside the discrete FEE required four of those FPGA-based TDC boards to read out all 86 channels of a single LGAD sensor. However, constraints related to the number of available DiRICH5s1 boards, which was 14 in total, meant that only every second channel in the final LGAD was effectively connected. This still allowed full particle tracking inside the last sensor as the cluster size, i.e. the number of fired strips per particle, was mostly greater than 3 for the used proton beam energies, and therefore, this sensor required a minor adaptation of the 4D-cluster finder algorithm, which will be briefly outlined in section 2.4.1.

Given that an individual LGAD module can measure just one spatial coordinate, we paired two of the aforementioned modules orthogonally to each other to create a single 4D-tracking layer. Since mechanical constraints prohibited mounting the LGADs in close proximity, they were installed at a distance of 1.64 cm between the individual LGAD sensors. For each sensor, an additional light-tight enclosure was constructed using a 3D-printed plastic holder featuring a circular aperture with an area greater than the LGAD size. This cutout was subsequently sealed with light-tight tape to ensure total light isolation. An image of a completely assembled 4D-tracking layer can be seen in figure 1(b).

Figure 2 illustrates the structural design of the TOF-iCT demonstrator, which consists of two 4D-tracking layers separated by a distance of d_{4D−stations} = 27 cm. While d_LGAD denotes the previously mentioned spacing between the individual LGAD sensors per 4D-tracking layer, d_tape describes the gap between the sensor and the light-tight tape. The latter was determined by the dimensions of the light-tight enclosure, measuring 0.5 cm.

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The DAQ system of the TOF-iCT demonstrator (figure 3) is a modified version of the HADES DAQ system (Michel et al 2011). Central to this system is the TrbNet (Michel et al 2013), a custom FPGA-based network protocol used for triggering, data transfer and slow control of the FEE of the individual detectors, e.g. the DiRICH5s1 boards on the LGAD modules. The main hardware component of the DAQ system is the FPGA-based multi-purpose trigger and readout board version 3 (TRB3), which houses five FPGAs (Neiser et al 2013). One of these FPGAs acts as the central controller and the other four establish the optical connections to the LGAD modules using supplementary adapter boards equipped with small form-factor pluggable (SFP) modules. While the synchronisation, data transfer and slow control operations are carried out through these optical links, additional LVDS connections provide the physical trigger to the DiRICH5s1 boards. Both the synchronisation and triggering are managed by the so-called central triggering system (CTS), which is also integrated into the central controller FPGA. Leveraging the capability of the CTS to accept external trigger inputs, an auxiliary FPGA-based logic board (i.e. the Trb3sc) was utilized to set the trigger condition for the readout. A logical OR across all channels of the third LGAD was chosen for this purpose, meaning that a trigger was initiated and forwarded to the CTS inside the TRB3 via LVDS whenever the signal in any channel of the third LGAD exceeded the threshold set in the corresponding leading-edge discriminators. The CTS then generated an output trigger, which was synchronously dispatched to all the LGAD modules through a fan-out board linked to the TRB3. Once an event was triggered, all incoming signals that fell within a predefined time window were sent from the DiRICH5s1 boards to the main FPGA and then to the readout PC via ethernet. For this experiment, the time window was set to 250 ns prior to and 50 ns after the trigger's arrival to account for the different delays in the readout chain and to guarantee the simultaneous measurement of a single particle in all detector modules. On the readout PC, the C++-based data acquisition backbone core framework (DABC) and Go4 analysis framework (Go4) were running to combine the data streams and build complete events, perform online monitoring and store the pre-processed data to disk. A more detailed description of the HADES DAQ and all used components and protocols can be found on the TRB Homepage (trb.gsi.de).

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First, each TDC had to be calibrated using internal calibration pulses issued from the CTS. Details about the FPGA-based TDC calibration can be found on the TRB Homepage (trb.gsi.de). Then, the thresholds in each leading-edge discriminator were set to a fixed value above the noise level (≈20 mV), which was low enough to capture the smaller, capacitively-coupled signals (Pietraszko et al 2020) in the strips adjacent to the actual strip hit by the particle. Since the noise level was assumed to be constant for the entire experiment, the thresholds were only set at the beginning of the experiment.

However, slight variations in the effective threshold levels, attributed to different noise levels, and differences in signal amplification across individual LGAD channels could still be expected. Given that these factors can significantly impact the signal response, especially the obtained ToT, normalizing the ToT measurement for each channel was essential to maintain a uniform response across all sensors. Therefore, each 4D-tracking module was irradiated with protons of a fixed beam energy E₀ and the corresponding ToT distributions were recorded in both sensors of every 4D-tracking module. First, to estimate the position of the main signal peak, the most probable ToT value (${\mathrm{ToT}}_{\mathrm{MPV}}{| }_{{E}_{0}}$) was determined for each LGAD channel. This was done by fitting a Gaussian to the maximum of the obtained ToT distributions using only the ToT values in the vicinity of the ToT peak. Then, the ToT spectra were normalized by setting the ${\mathrm{ToT}}_{\mathrm{MPV}}{| }_{{E}_{0}}$ to 20 ns in every LGAD channel using the following re-scaling function

$$\begin{eqnarray}&&\mathrm{ToT}(i,j)=\displaystyle \frac{20\,\mathrm{ns}}{{\mathrm{ToT}}_{\mathrm{MPV}}{| }_{{E}_{0}}(i,j)}* {\mathrm{ToT}}_{\mathrm{raw}}(i,j).\end{eqnarray} \tag{ 7 }$$

Here, ToT_raw(i, j) and ToT(i, j) denote the raw and the corresponding normalized ToT value measured in channel i of LGAD j.

Since both the ToA and the ToT are determined using a predefined, fixed threshold, their measured values strongly depend on the actual amplitude of the physical signal. This amplitude-dependence of the ToA, also known as the time-walk effect, can lead to ToA variations in the order of several nanoseconds, depending on the used readout electronics and resulting signal shape. As those variations can significantly deteriorate the precision of the time measurement, correcting for the time-walk effect is essential. The time-walk correction in each 4D-tracking layer was done in two steps using the data sets recorded for the ToT normalisation (section 2.3.2). First, to obtain the time-walk trend, the time difference spectra between every channel on one LGAD and a reference channel on the other LGAD of a single 4D-tracking layer were determined with respect to the corresponding signal amplitudes on the first LGAD, given via the related ToT of those signals. A 4D-clustering procedure (section 2.4.1) in addition to a ToT cut to the signals in the reference channels was applied to avoid redundancy by selecting only hits, which stem from the actual particle and not from noise events or capacitive-coupling between the neighbouring strips. Then, to estimate the time walk trend in each LGAD strip, the corresponding time difference vs ToT spectrum was divided into 50 ps × 50 ps bins and the most probable time difference value (T_Diff,MPV(ToT)) was obtained for each ToT bin using a Gaussian fit. The obtained T_Diff,MPV(ToT) values were then stored in a look-up-table (LUT) and used to remove the signal amplitude dependence of the measured ToA in this channel. This was done by determining the T_Diff,MPV(ToT) for the measured ToT and subtracting it from the corresponding ToA (ToA_meas(ToT)) according to

$$\begin{eqnarray}&&{\mathrm{ToA}}_{\mathrm{corr}}={\mathrm{ToA}}_{\mathrm{meas}}(\mathrm{ToT})-{{\rm{T}}}_{\mathrm{Diff},\mathrm{MPV}}(\mathrm{ToT}),\end{eqnarray} \tag{ 8 }$$

with ToA_corr as the time-walk corrected ToA. After calibrating every channel inside the first LGAD, the same procedure was applied to the second LGAD of the same 4D-tracking layer by choosing one channel in the first LGAD as the reference channel.

While the previously mentioned time-walk calibration procedure removes any ToT dependence of the measured ToA, it also has the advantageous effect of synchronising the time measurement across the entire 4D-tracking layer. By subtracting T_Diff,MPV(ToT) from the measured ToA, the mean time difference between the calibrated channel on the one LGAD and corresponding reference channel on the second LGAD of the same 4D-tracking layer will always be zero for the same beam energy. Since the same reference channel on one LGAD is used to calculate T_Diff,MPV(ToT) for calibrating all channels on its partner LGAD, the mean time difference between all those channels and the corresponding reference channel is also zero. This also means that no offset in the measured ToA between the individual LGAD channels caused e.g. different signal propagation times inside the individual channels or the non-zero propagation time of the particles travelling between the sensors should remain. Consequently, for the same beam energy, all channels inside the 4D-tracking layer should measure precisely the same mean ToA with the ToA of the reference channel defining the global reference time. However, since the time-walk calibration procedure has to be done once for one LGAD and once for the partnering LGAD of the same 4D-tracking layer, it is important to carefully choose the order of the calibration as this will define the final reference ToA for the entire 4D-tracking layer. For instance, if LGAD₁ (figure 2) is calibrated after LGAD₂, the reference channel in LGAD₁ defines the final reference ToA of the corresponding 4D-tracking layer. For all following experiments, one central strip in LGAD₂ and one central strip LGAD₃ were used to define the final reference ToAs.

As discussed in Pietraszko et al (2020), a particle hit inside the used LGADs results in a signal above the discriminator threshold across a cluster of strips mainly due to capacitive coupling between the individual detector channels. While the strip closest to the actual particle hit position yields the highest signal (highest ToT), the neighbouring strips detect a reduced signal, depending on the capacitive properties of the LGAD strip detector. To estimate the actual hit position for each particle, those clusters have to be identified and analyzed. For that purpose, a 4D cluster finder algorithm was implemented, which will be summarized briefly in the following. For every hit inside the LGAD, coincident signals within a given time window (±15 ns before and ±1 ns after the time-walk calibration) and strip distance to the strip position of the corresponding hit (±12 strips) were determined. For LGAD₁, LGAD₂ and LGAD₃ (figure 2), where all channels could be read out, a cluster was found if the strips of those coincident hits were consecutive in space. For the last LGAD (LGAD₄), where only every second channel was connected, a cluster was found if the coincident hits where within the previously described time window and strip range. Once a 4D cluster was found, the true ToA and hit position of the particle hit inside the LGAD were estimated using the strip signal inside the cluster with the largest ToT, i.e. closest to the actual particle position.

After applying the 4D cluster finder procedure to the calibrated data and estimating the corresponding particle hit positions and ToAs for each cluster, the 4D-particle tracks inside the TOF-iCT demonstrator could be reconstructed using a custom 4D-tracking algorithm. However, given that the TOF-iCT demonstrator consists of two 4D-tracking layers, only a straight-line track model could be utilised. First, a track candidate was identified if every LGAD had at least one cluster within a triggered event. Then, to simplify the 4D-tracking finding procedure, only events with exactly one cluster per layer were selected for the subsequent analysis to guarantee that only one particle passed through the TOF-iCT demonstrator.

In order to align the 4D-tracking layers of the TOF-iCT demonstrator, an available laser-positioning system at MedAustron was utilised. Following this, a more accurate, software-based detector alignment technique similar to the pre-alignment procedure defined in Dannheim et al (2021) was applied. For that purpose, the TOF-iCT demonstrator was irradiated with 800 MeV protons and the hit positions were recorded on both 4D-tracking layers using the 4D-track candidates described in section 2.4.2. Then, the lateral displacement r_x and r_y between the first and second 4D-tracking layer were calculated for each particle according to

$$\begin{eqnarray}&&{r}_{x}={x}_{\mathrm{back}}-{x}_{\mathrm{front}}\quad \mathrm{and}\quad {r}_{y}={y}_{\mathrm{back}}-{y}_{\mathrm{front}},\end{eqnarray} \tag{ 9 }$$

with x_front, y_front, x_back, y_back denoting the x and y hit position in the first and second 4D-tracking layer, respectively. Any lateral misalignment of the layers would then manifest as a non-zero mean in the distributions of r_x and r_y since the average trajectory of all particles should theoretically be a straight line. To mitigate this misalignment, the first 4D-tracking layer was designated as the reference layer and the second layer's hit positions were adjusted by subtracting the mean offsets ${\hat{r}}_{x}$ and ${\hat{r}}_{y}$, which were obtained by fitting a 1D Gaussian to both the r_x and r_y distributions.

Using the calibrated and aligned TOF-iCT demonstrator, the TOF through the scanner could be determined. For all of the following experiments, the TOF was defined as the difference between the mean ToA per 4D-tracking layer using

$$\begin{eqnarray}&&\mathrm{TOF}=\displaystyle \frac{{\mathrm{ToA}}_{\mathrm{corr}}(3)+{\mathrm{ToA}}_{\mathrm{corr}}(4)}{2}-\displaystyle \frac{{\mathrm{ToA}}_{\mathrm{corr}}(1)+{\mathrm{ToA}}_{\mathrm{corr}}(2)}{2}\end{eqnarray} \tag{ 10 }$$

with ToA_corr(j) as the calibrated ToA measured in LGAD j. As detailed in section 2.3, the reference ToA channels for the offsets correction between the individual LGAD layers were chosen such, that the measured TOF in equation (10) reflects the TOF between LGAD₂ and LGAD₃ (figure 2).

In order to assess the homogeneity of the TOF measurement for the entire active area of the TOF-iCT demonstrator, the position dependence of the TOF was investigated using different proton beams with energies ranging from 83 MeV to 800 MeV. This was done by projecting the TOF of every particle (equation (10)) onto the last 4D-tracking layer by combining all 86 LGAD channels per sensor to 86 × 86 pixels. For each pixel, the TOF distribution was recorded and the corresponding most probable value and standard deviation were estimated using a Gaussian fit. The resulting TOF values were then compared to the theoretical TOF through vacuum for the same flight path length and primary beam energy to assess the accuracy of the TOF measurement.

Using the same data set as obtained in section 2.5.1, the intrinsic time resolution per LGAD channel was estimated for all of the employed beam energies. This was done by calculating the time difference

$$\begin{eqnarray}&&{{\rm{T}}}_{\mathrm{Diff}}(i,j)={\mathrm{ToA}}_{\mathrm{corr}}({i}_{\mathrm{ref}},k\ne j)-{\mathrm{ToA}}_{\mathrm{corr}}(i,j)\end{eqnarray} \tag{ 11 }$$

between every channel i in LGAD_j and a reference channel i_ref on LGAD_{k≠ j} of the same 4D-tracking layer (figure 2). A Gaussian was then fitted to the resulting T_Diff(i, j) distributions to estimate the corresponding standard deviations ${\sigma }_{{{\rm{T}}}_{\mathrm{Diff}}}(i,j)$. Finally, to approximate the intrinsic time resolution for every LGAD channel (σ_ToA(i, j)), the obtained ${\sigma }_{{{\rm{T}}}_{\mathrm{Diff}}}(i,j)$ was divided by $\sqrt{2}$ since ${\sigma }_{{{\rm{T}}}_{\mathrm{Diff}}(i,j)}\approx \sqrt{2}\cdot {\sigma }_{\mathrm{ToA}}(i,j)$, which can be derived from a Gaussian error propagation of equation (11) and assuming a similar time-resolution in each LGAD (σ_ToA(i, j) ≈ σ_ToA) for a fixed beam energy.

First, a calibration curve according to section 2.1.5 was recorded for each beam energy. This involved irradiating 1.65 mm thick PMMA slabs (figure 4(b)) with an RSP of 1.012 and measuring the corresponding TOF per pixel inside the TOF-iCT demonstrator as detailed in section 2.5.1. Then, the median over those TOF per pixel values was calculated and used to determine the calibration parameters a_i (E₀) (see equation (6)).

As shown in figure 5(a), the calibration phantom was mounted in front of the third LGAD. It was placed adjacent to the light-tight enclosure covering the LGAD to minimize the energy loss in air between the phantom and the last 4D-tracking station.

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After determining the WET calibration curves, the pRads of the aluminium stair phantom were recorded. For that purpose, the phantom was mounted on a rotating table and placed 1.9 cm in front of the third LGAD (figure 5(b)). However, given that each LGAD consisted of 86 strips with a strip pitch of 100 μm, only a partial section of the phantom could be imaged. The specific area imaged is delineated in figure 4(a) by the red region of interest (ROI).

In order to acquire the pRad image for each beam energy, the TOF through the scanner was measured and projected onto the last 4D-tracking layer as outlined in section 2.5.1. A pixel size of 2 × 2 strips was selected for the pRad images, which is equivalent to an area of 200 μm × 200 μm per pixel. For each pixel, the MPV of the TOF, i.e. TOF_MPV(x, y), was obtained by applying a Gaussian fit to the resulting TOF distributions. Using the WET calibration curves as determined in section 2.6.1, the collected TOF_MPV(x, y) values were converted into the corresponding WET.

To assess the pRad image quality with respect to the WET accuracy, the WET values, which were obtained according to section 2.6.2, were collected inside the centre of the aluminium stair phantom using a square-shaped ROI with a size of 24 × 24 pixels. Based on the resulting WET distribution, the median WET (WET_median) was calculated for that ROI. Then, to evaluate the WET accuracy, the relative WET error (_WET) was determined as follows

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{WET}}=\displaystyle \frac{{\mathrm{WET}}_{\mathrm{median}}-{\mathrm{WET}}_{\mathrm{theo}}}{{\mathrm{WET}}_{\mathrm{theo}}},\end{eqnarray} \tag{ 12 }$$

using WET_theo as the theoretical WET for a 1 cm thick aluminium layer, which, according to the NIST PSTAR database (Berger et al 2017), is 20.968 mm.

In addition to the WET accuracy, the WET resolution was estimated. For that purpose, the quartile coefficient of dispersion (QCOD_WET) was calculated inside the previously specified ROI using

$$\begin{eqnarray}&&{\mathrm{QCOD}}_{\mathrm{WET}}=\displaystyle \frac{{{\rm{Q}}}_{3,\mathrm{WET}}-{{\rm{Q}}}_{1,\mathrm{WET}}}{{{\rm{Q}}}_{3,\mathrm{WET}}+{{\rm{Q}}}_{1,\mathrm{WET}}},\end{eqnarray} \tag{ 13 }$$

with Q_1,WET and Q_3,WET as the first and third quartiles and Q_3,WET − Q_1,WET as the interquartile range (IQR_WET) of the corresponding WET distribution.

Figure 6(a) shows a normalized ToT distribution within a single LGAD at 800 MeV. As outlined in section 2.3.2, the primary signal peak (marked in red) was shifted to 20 ns for each beam energy, ensuring a consistent response across all LGAD channels. Besides this main signal peak, also secondary peaks are present at lower ToT values (highlighted in blue), which originate from capacitively-coupled hits in the strips adjacent to the main hit position. In order to remove those capacitive-coupled signals, the 4D-cluster finder algorithm (section 2.4.1) was applied to the recorded data set. Depending on the used beam energy, the mean cluster size, i.e. the number of fired strips per 4D cluster, ranged between three (800 MeV) and five (83 MeV) strips. The main reason is that, for lower beam energies, the signal increases, leading to signals above threshold even at strips further away from the actual hit strip.

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Once the clusters were identified, the strips with the highest ToT values were used as an estimate for the true particle hit position. The corresponding ToT distribution is depicted in figure 6(b). Now, only the main signal peak can be recognized. However, a low-ToT background with much lower statistics than the main signal is still visible. As described in Pietraszko et al (2020), this low-ToT background stems from border effects at certain regions inside the sensor, which show a lower gain than the nominal one, e.g. between two strips or at the border of the sensor.

Without any time-walk and offset correction, the time difference between the individual LGAD sensors per 4D-tracking layer shows a strong channel dependence due to different propagation times and time-walk properties in each channel (figure 7(a)). However, after applying the time-walk and offset corrections as outlined in section 2.3.3, a uniform response across the whole sensor for the given beam energy could be achieved. The resulting mean time difference spectrum was then centred at 0 ns and showed a reduced variability in each channel (figure 7(b)). The latter is caused by the time-walk calibration, as the time-walk effect widens the time difference distribution due to its signal amplitude dependence of the ToA. To demonstrate this time-walk effect, figure 7(c) depicts an example time difference vs ToT distribution for a single LGAD channel with respect to a reference channel on the second LGAD of the same 4D-tracking layer. As mentioned in section 2.3.3, a ToT cut was then applied on the reference channel to guarantee that the hits on the first LGAD were only correlated once with the true particle hit on the reference channel of the second LGAD. Although those ToT cuts for the reference channel were applied on the normalized ToT values, the time-walk calibration was done using the raw, non-normalized ToT values (ToT_raw), as this provided a more accurate correction of the time-walk effect. Thus, the ToT values depicted in figure 7(c) also represent the ToT_raw. In this graph, the signal amplitude-dependence of the ToA, which is in the order of ≈ ns, is clearly visible. To mitigate this time-walk effect, we conducted a time-walk calibration as outlined in section 2.3.3. The resulting time difference distribution for the calibrated data is depicted in figure 7(d). No further ToT-dependence on the ToA can be recognized and the main TDiff signal peak is now centred around zero.

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For each of the used beam energies, the median TOF per pixel and interquartile range were calculated. Then, the median TOF per pixel was compared to the calculated TOF in vacuum after subtracting the corresponding TOF values at 800 MeV (figure 8(a)).

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Setting the TOF at 800 MeV to zero enabled the correction of any extra offset between the measured and theoretical TOF values, which allowed for a more accurate comparison. While the relative difference between the measured median TOF and TOF in vacuum was 1.55% for 83 MeV, it was lower than 0.74% for all other beam energies and reached its lowest value of 0.004% at 252.7 MeV. This decrease at higher beam energies can be explained by the reduced energy loss along the particle path at increased beam energies.

Using the same data sets, the intrinsic time resolution per LGAD was calculated according to section 2.5.1. The resulting time resolution per pixel and corresponding interquartile ranges are depicted in figure 8(b). While the blue data points represent the intrinsic time resolution with respect to the primary beam energy, the red diamond-shaped data points represent the corresponding energy loss in MIPs, i.e. the ratio between the expected energy loss for the used beam energy and the expected energy loss of a MIP. Both of those values were calculated using the NIST PSTAR database (Berger et al 2017). As shown in figure 8(b), the intrinsic time resolution strongly depends on the beam energy and improves with lower beam energies, i.e. higher energy loss inside the detector. While 800 MeV yielded a median time resolution of 69.3 ps, 83 MeV showed an intrinsic time resolution of 41.8 ps.

For each beam energy, the median WET was then calculated, represented by the dotted (83 MeV) and dashed (100.4 MeV) lines in figure 11. The determined WET_median values were 28.75 mm and 31.72 mm for 83 MeV and 100.4 MeV, respectively. In order to compare the obtained median WET values to the theoretical WET from the NIST database (20.968 mm), the relative WET errors were estimated according to equation (12). The results are summarised in table 1 showing a relative WET error of _WET = 37.09% for 83 MeV and _WET = 51.12% for 100.4 MeV.

Table 1. Summary of the pRad image quality parameters obtained in the ROI.

Parameter	83 MeV	100 MeV
WETtheo	20.968 mm
WETmedian	28.75 mm	31.72 mm
WET	37.09%	51.12%
IQRWET	2.98 mm	3.78 mm
QCODWET	5.19%	6.00%

As described in section 2.6.3, the WET resolution was estimated by calculating QCOD_WET according to equation (13). The corresponding IQR is highlighted as the darker-shaded area in figures 11(a) and (b). While the pRad at 83 MeV yielded a QCOD_WET of 5.19%, the pRad at 100.4 MeV showed a slightly higher QCOD_WET of 6% (table 1).