TlBr, TlCl, and lead fluoride (PbF₂) crystals with dimensions 3 × 3 × 3 mm³ are used for this study. The 3 × 3 × 3 mm³ geometry is of interest to evaluate the intrinsic performance of the crystal as a best-case scenario with low photon time spread and good light transfer efficiency. A 3 × 3 × 20 mm³ TlBr crystal is also used as it represents the conventional geometry used in PET. One of the 3 × 3 mm² faces is coupled to a SiPM using Cargille Meltmount with n = 1.58 and cutoff at 300 nm. The remaining surfaces are Teflon-wrapped or black-painted using a spray with a refractive index n = 1.5 [16], to emphasize the crystal properties. The transmission of the crystals is measured without wrapping, using a PerkinElmer LAMBDA spectrophotometer. Figure 1 displays the measured transmission as a function of the wavelength on the left. The cutoff for TlCl and TlBr is, respectively, at 400 and 440 nm, extrapolated from the curve at 50% of the slope. The relatively low transmission of TlBr and TlCl, with respect to PbF₂, is the result of the unpolished surface of these crystals. These and other relevant crystal properties are summarized in Table 1.

The measurements of the CTR are performed using the coincidence setup described in [2], where a ²²Na source with activity 2.7 MBq is placed between a small reference detector (2 × 2 × 3 mm³ LSO:Ce:Ca) and the crystal under test, both coupled to 4 × 4 mm² FBK NUV-HD SiPMs. The SiPMs are biased at 39 V, about 10 V above breakdown voltage. This setting ensures optimal conditions in terms of timing performance [2]. As described in Figure 2, high-frequency electronics is used to monitor the voltage drop between the SiPM anode and cathode with a very fast SiPM single-cell signal rise time, which also measures the number of triggered cells in the SiPM. The electronics of the high-frequency amplifier is described in depth in [18, 20]. The energy deposition in the reference detector is analyzed by integrating the output signal from a unity-gain amplification stage (voltage amplifier). This allows to select 511 keV photoelectric events and to minimize the presence of spurious coincidences in the dataset, due to the events that undergo a scattering process in the vicinity of the crystal. The signals are digitized using a LeCroy DDA735Zi oscilloscope, characterized by 3.5 GHz bandwidth, 20 Gs/s sampling rate (i.e. 50 ps binning), and a leading edge threshold that calculates the signal crossing time via sin(x)/x interpolation. The threshold for the reference detector is set at 20 mV (about 50% of the single-cell signal amplitude), while for the Cherenkov radiator at 10 mV, optimized to reach the best possible time resolution. Since the time information is extracted with a leading edge threshold, signals present a strong time-walk, which is observed in particular for low light intensities [21] or Cherenkov photons [4]. Therefore, a correction method is implemented using the SiPM signal slew rate.

The CTR of the described setup is defined as the full width at half maximum (FWHM) of the distribution of the time-stamps, and it represents the time resolution of the coincidence between the Cherenkov radiator-based detector and the reference detector. To reproduce the CTR assuming two identical detectors in coincidence, the value is multiplied by 2 , and the resolution of the reference crystal (CTR_reference = 61 ± 3 ps from [16]) is subtracted as: CTR = 2 ⋅ CTR measured 2 − CTR reference 2 . (1)

When selecting events with one-triggered avalanche diode (SPAD) or one Cherenkov photon detected, the single photon coincidence time resolution (SPCTR) is extracted as FWHM. SPCTR is useful to validate simulations, where SiPM cross talk is not included, and to better understand the fundamentals of light transport.

A Monte Carlo code based on the Geant4 tool kit is developed to model the experimental apparatus and reconstruct the time distributions. The geometry and composition of the crystal, optical coupling agent, and silicon photodetector are implemented. A particular focus is put on the model of the crystal surfaces, defined as optical surfaces using the unified model and setting a ground finish. Specular reflections are considered on a surface whose roughness is defined by the σ _α parameter [22, 23]. σ _α describes the Gaussian dispersion of the angle α [rad] between a micro-facet on which an optical photon interacts and the average surface (Figure 3).

The electron and Cherenkov light production and transport are tracked and saved for further analysis, together with the energy deposition. The simulation output is processed using the ROOT libraries to include the SiPM properties, hence a jitter in the arrival time due to SPTR and electronic noise, reproduced as Gaussian smearing with σ = 42 ps [2], and a weight to consider the photon detection efficiency as shown in Figure 1. The final time distribution is then convoluted with a Gaussian distribution to model the behavior of the reference detector and, finally, an uncorrelated dark count floor is added.

The σ _α parameters are tuned against experimental measurements, and the model is then used to analyze the contributions limiting the time resolution. Moreover, the setup performances are predicted when separately improving the crystal and SiPM properties. Given the complex structure of the time distribution, when studying the impact of the SiPM choice on the CTR, the figure of merit for the time resolution is extracted as the standard deviation within a fixed time window, instead of the FWHM. SiPM cross talk and after pulse effects are not considered in this model.

The 3 × 3 × 3 mm³ TlBr crystal is measured with the setup described in section 2.2. The time delay distribution of the crystal wrapped in Teflon and black-painted is displayed on the right of Figure 4 with the dashed line. The asymmetry in the distribution originates from the difference in performance between the reference crystal and TlBr. The distribution for the black painted crystal shows a more moderate tail. Indeed, the tail is attenuated when photons that undergo several reflections in the crystal are not detected. Similar behavior is presented in [13] when events with lower amplitude are removed from the dataset. To model the shape, a Crystal Ball fit function is used, which consists of a Gaussian convoluted with an exponential function [16]. The CTR values obtained after correction for the contribution of the reference detector according to Eq. (1) are CTR_{Black painting} = 204 ± 7 ps and CTR_Teflon = 269 ± 9 ps, assuming two identical TlBr crystals in coincidence. Despite the significant improvement in the time resolution achieved by painting the 3 × 3 × 3 mm³ TlBr black to suppress reflections in the crystal, the overall signal-to-noise ratio is not superior with respect to other reflectors for the low number of coincidence events detected. Aiming to obtain the best detector performance, a trade-off between sensitivity and timing must be considered [24]. The measurement is repeated with Teflon-wrapped TlCl and PbF₂ crystals of the same dimensions. The results are summarized in Table 2 and show agreement between the CTR values of TlCl and TlBr and an improvement using PbF₂.

The stated CTR values are affected by time-walk due to the leading edge discriminator of the oscilloscope and the low number of triggered SPADs in the SiPM. One strategy to mitigate this effect is to lower the leading edge threshold to just above the electronic noise floor. However, at such low thresholds, the electronic noise would deteriorate the time resolution more than the time-walk itself. Instead, the slew rate at 30 mV (about 75% of the single-cell amplitude) is measured for each event, and a correction method is implemented using this information.

The collected data are divided into 10 categories as shown on the left of Figure 4, each one containing the same number of events. For each category, the time delay distribution is separately analyzed, and the position of the maximum is evaluated by interpolation. The coincidence time resolution of the individual categories ranges from 324 to 180 ps, while a shift of the centroid of 110 ps can be observed between the first and last category. The time delay distributions are merged after subtracting for the position of the centroid, thus reducing the smearing due to time-walk. Further details on a similar correction are presented in [4, 25]. After time-walk correction, the values significantly improve. CTR_{Black painting} goes from 204 to 167 ps FWHM and CTR_Teflon from 269 to 214 ps FWHM (Figure 4 right).

The last measurement listed in Table 2 is performed with the Teflon wrapped 3 × 3 × 20 mm³ TlBr crystal. Locating the 3 × 3 mm² face in coincidence, the CTR value extracted after time-walk correction is 285 ± 9 ps FWHM. The value is significantly higher than the time resolution of the 3 mm long crystal, arguably due to the time spread introduced by the DOI of the incident gamma photons [26] and a lower LTE with respect to the shorter one [27, 28].

To determine the light propagation time in TlBr, the 3 × 3 × 20 mm³ crystal wrapped in Teflon is used to perform DOI-lateral-collimated measurements along the 20 mm long side of the crystal. The 3 × 20 mm² face is located in coincidence, and the reference detector is placed far from the source to have a collimated parallel beam of gamma photons. The source and the reference detector are moved in steps of 1 mm along the 20 mm side, to reproduce different depths of interaction from 0 mm DOI, at the far end of the TlBr crystal, to 20 mm DOI, near the SiPM. The resulting coincidence time delay histograms are displayed in Figure 5 on the left, selecting events in which only one photon is detected. There is an increasing delay (t _delay ) in the position of the maximum of the distributions, from 0 to 19 mm of DOI, due to the increasing distance that the produced optical photon must travel in the crystal before being detected, defined as: t delay = x ̄ c n a n d x ̄ = 20 mm − DOI mm ⋅ k , (2) where c n is the speed of the optical photons in the crystal and x ̄ is the average traveled distance. x ̄ depends on the DOI position, the position of the source, and on a coefficient k ≥ 1, which takes into account the multiple reflections of the photons on the long lateral sides (3 × 20 mm²) of the crystal.

A second peak, resolved for DOI positions between 11 and 19 mm, characterizes the time distributions [29, 30]. This peak is created by the photons that are produced toward the far end of the crystal, are back-reflected on the 3 × 3 mm² face not coupled to the SiPM, and finally detected. The resolution of the second peak is worse than that of the first one since it contains the time-stamp of photons traveling longer and thus more exposed to fluctuations in the number of internal reflections or lost photons. The time difference between the two peaks is defined as: t 2 nd peak − t 1 st peak = 2 x ′ ̄ c n a n d x ′ ̄ = DOI mm ⋅ k . (3)

Geant4 simulations are developed to extract complementary information to the measurements. In particular, the DOI-collimated measurements are used to tune the crystal surface parameters of TlBr in the Geant4 model. To simplify simulations, the correlated noise of the SiPM (cross talk, after pulsing) is not included and, to provide identical conditions between measurements and simulations, only events with one-triggered SPAD (one detected Cherenkov photon) are considered. The tuning of the parameters is required to correctly reproduce the coincidence time spectra since the surface state has a strong impact on the time resolution [31–33]. The lateral surfaces have a different roughness with respect to the front and back; therefore, two different parameters are considered: σ _{α 1} for the lateral faces and σ _{α 2} for the back and front. The two peaks characterizing the measured time distributions are used for the tuning since σ _{α 1} affects the resolution of both peaks while σ _{α 2} affects only the second one. Simulations are performed varying the values of σ _{α 1} and σ _{α 2}. The values that best fit the measurements are σ _{α 1} = 0.04 ± 0.01 and σ _{α 2} = 0.5 ± 0.1, compared to typical values of about 0.005–0.01 for a common well-polished surface. The resulting simulations are shown on the right of Figure 5, for different DOI positions. The left of Figure 6 shows the simulated normalized distributions in comparison with the measured ones at the extremes of the crystal, DOI = 1 and 19 mm. The agreement is maintained at all the DOI values: the position of the maximum of the time distribution (t _delay ), and the difference between the two peaks (t_{2nd peak} − t_{1st peak}) is analyzed as a function of the DOI (Figure 6 right). From the latter, the k coefficient is extracted using Eq. (3), which can be rewritten as: t 2 nd peak − t 1 st peak = 2 x ′ ̄ c n = 2 k c n DOI mm . (4)

The coefficients extracted from simulations and measurements are: k sim = 1.62 ± 0.05 and k meas = 1.69 ± 0.05 . (5)

Moreover, by integration of the baseline of the measured time distributions, the number of dark counts is extracted. Dark counts represent ∼15% of the total number of events when events with a single-triggered SPAD are selected. The dark count composition highly depends on the experimental condition and can be lowered by using a stronger radioactive source, a larger reference detector, or decreasing the distance between the two crystals.

The validated simulation model is used to study the contributions that limit the time resolution regarding the crystal and the photodetector properties, to assess possible improvements and the potential of the materials. The measurement performed with the 3 × 3 × 20 mm³ Teflon-wrapped TlBr crystal is considered. Placing the 3 × 3 mm² face in coincidence with the reference detector, the CTR extracted after time-walk correction, assuming two equal detectors in coincidence, is 285 ± 9 ps (Table 2). If events with one-triggered SPAD only are selected, the resulting time distribution, shown on the left of Figure 7 with the solid blue line, has a resolution (SPCTR) of 448 ± 15 ps.

This setup is simulated selecting events in which one photon only is detected and calculating the SPCTR according to Eq. (1). Simulations are performed considering, initially, an ideal photodetector having no electronic noise and SPTR equal to 0 ps (red dashed line in Figure 7). The SPCTR, in this case, is 337 ± 33 ps FWHM. When adding SPTR and electronic noise, SPCTR becomes 438 ± 20 ps FWHM (blue dashed line). Finally, dark counts are added as a constant component to match the experimental spectrum (green line). In this case, SPCTR does not change, since the time resolution is always calculated as the FWHM of the distribution, subtracting for the baseline. The value 438 ± 20 ps is in agreement with the SPCTR extracted from the experimental measurement. Indeed, a good agreement is observed in Figure 7. The SiPM SPTR has a significant impact on the overall detector performance. The impact of the PDE is analyzed in the following section, when all events, regardless of the number of detected photons, are considered. Moreover, the simulation is repeated considering a polished crystal by setting both σ _α parameters to 0.005. The extracted SPCTR = 306 ± 6 ps proves that, currently, the major improvement can be achieved by polishing the surface of the crystal.

The surface condition plays a fundamental role in time resolution [31, 32]. Unpolished surfaces have a greater tendency to promote internal reflection, as observed in the transmission plot in Figure 1, thus decreasing the number of photons lost (section 4.2) but severely increasing the PTS (Figure 7). Table 2 shows that the CTR values measured with TlCl and TlBr are compatible, while a significant difference is found with the resolution of PbF₂. One of the sources of this difference is the surface state. Indeed, while TlBr and TlCl present unpolished surfaces, PbF₂ presents polished.

The difference between the CTR values measured with TlCl/TlBr and PbF₂ is justified, on the one hand, by the difference in the surface state. On the other hand, this can be explained by the difference in the refractive index of the thallium-based crystals compared to PbF₂ (n = 2.32, 2.48 vs. n = 1.78), which promotes better extraction of photons in the latter, and by the different transmission cutoffs. TlBr and TlCl, unlike PbF₂, do not harvest Cherenkov photons between 300 and 400 nm. To study the contribution of the refractive index on the Cherenkov photon yield and the timing performance, the developed simulation tool kit is used to simulate a small crystal wrapped in Teflon, with dimensions 3 × 3 × 3 mm³. Regarding the constituent material, all the properties of TlBr are considered except for the refractive index, which is modified performing a scan in the range 1.7–2.6, and the transmission cutoff set to 300 nm wavelength. Polished and unpolished surfaces are investigated, considering the two cases previously described. The coincidence with the reference detector is implemented. The CTR of the corresponding output time distribution and the number of detected photons are extracted. Despite the increasing number of produced Cherenkov photons as a function of the refractive index n [34] as N p r o d u c e d ∼ 1 − 1 n 2 , (6) it is observed that the impact of the high refractive index on the CTR is almost negligible in the crystals with polished faces, while it shows a deterioration with the increasing refractive index for crystals with unpolished surfaces (Figure 8). At the same time, the mean number of detected photons is consistently greater in crystals with unpolished surfaces. These results suggest that, for crystals with unpolished surfaces, a greater fraction of photons undergoes more internal reflections before detection as the refractive index increases, while in crystals with polished surfaces these additional events are not detected.

From the photodetector side, relevant contributions to the CTR arise from the time response of the detection (SPTR), electronic noise, and PDE. To evaluate their impact on time resolution, simulations are performed using two different crystal geometries (3 × 3 × 3 mm³ and 3 × 3 × 20 mm³). The crystal properties are fixed to those of Teflon-wrapped TlBr and the σ _α parameters to 0.005, while the SPTR and PDE of the SiPM are varied. To simplify the analysis, the time delay distribution is considered in coincidence with a perfect reference detector (CTR = 0 ps), and the standard deviation within a fixed time window is saved as the figure of merit for the time resolution. The time delay distributions simulated for three specific (PDE [%], SPTR (FWHM) [ps]) values, with a short crystal and a long crystal, are displayed at the top of Figure 9. In particular, the case of a perfect PDE (100%) and a very good SPTR (50 ps FWHM) is compared to the case of a SiPM having a worse SPTR (100 ps FWHM) or a worse PDE (1%). While a high SPTR introduces only a spread in the time distribution, clearly visible on the peak, a low PDE increases the fraction of events in the tail, due to the low probability of detecting the fastest optical photon.

Varying the PDE in steps of 1% and the SPTR in steps of 2 ps FWHM, the resulting heat maps are shown at the bottom of Figure 9, where the isolines represent a change of 10 ps in standard deviation. The three black dots represent typical values of the parameters of different photodetectors, namely SiPMs from FBK (NUV-HD) and Hamamatsu (HPK S13360–3050 PE) [2] and an MCP-PMT with excellent time response but worse detection efficiency [14]. It is observed that, for the small crystal geometry, SPTR dominates the overall time performance, while PDE is less important. This is also the reason for the excellent measured CTR values in [14] for using small black-painted crystals. However, when approaching PET-sized geometry with longer crystals, it becomes clear that more attention needs to be drawn on the detection efficiency, possibly making SiPMs a more favorable choice of photodetector compared to fast MCP-PMTs.

Assuming two equal detectors in coincidence, CTR values of about 210 ps FWHM are measured with small Teflon-wrapped TlBr and TlCl crystals. These values are significantly better compared to those around 400 ps reported in the literature [11, 13], despite the acceptance of all events, without any selection on the number of triggered SPADs. The coincidence time resolution setup used differs from that described in the literature in using a SiPM with a better SPTR, higher SiPM overvoltage, and high bandwidth readout. In particular, FBK NUV-HD with an intrinsic SPTR  ≈ 70 ps is used compared to HPK S14160 with an intrinsic SPTR  ≈  120 ps [2]. Good SPTR and high bandwidth readout are crucial to optimize the timing performance in the case of low light intensities [20, 25]. In addition, time-walk correction significantly improves the timing performance. This correction method could have a practical implementation in an ASIC with two trigger levels, where the first is used as the primary time-stamp and the second to calculate the slew rate around the leading edge threshold [35]. However, the measured CTR values can be further optimized, by polishing the crystal and improving light transfer and light collection.

In the simulations, a digital-like approach is used for the SiPM, and the impact of optical cross talk is ignored, focusing on one-triggered SPAD events. Both simplifications provide valid results. However, the study of the analog SiPM response and the model of SiPM cross talk in the case of a few detected photons would provide more accurate results and will be subjected to future work.