A roadmap for sole Cherenkov radiators with SiPMs in TOF-PET

The crystals used for this study were produced by Epic-Crystals and were fully polished. The tested crystals had a cross section of 2 × 2 mm² and a length ranging from 3 to 30 mm. Depending on the measurement conditions, the crystals were either kept as they are (Bare), wrapped with more than 5 layers of Teflon (Teflon), black painted using black spray (LUXENS) with refractive index of n = 1.5 (Black) or a combination using enhanced specular reflector (ESR) opposite to the readout, as summarized in table 1, before optically coupling to the SiPMs with Cargille Meltmount (n=1.582). The Cherenkov emission probability follows a 1/λ² distribution, so that more photons are produced at shorter wavelength. PbF₂ is in principle transparent up to around 250 nm, but the coupling agent and the used SiPMs only allowed to use the emission up to 300 nm.

For one measurement a quartz (SiO₂) glass (2 × 2 × 4 mm³, black painted) was tested. This pixel was cut from a larger piece and kept unpolished.

One advantage of PbF₂ compared to other inorganic scintillating crystals is the low cost of the row materials and moderate melting point, yielding to a cost effective crystal solution. In addition, PbF₂ features high effective atomic number and high density, yielding to short gamma attenuation length or high stopping power. Key properties of PbF₂ and other well known materials are summarized in table 2.

For all measurements NUV-HD SiPMs from fondazione bruno kessler (FBK) were used, having a cross section of 4 × 4 mm² and 40 μm² SPAD pitch and no protective resin window. The performance of these devices was extensively studied in the past (Gundacker et al 2016b, Gola et al 2019) and holds the current record in terms of excellent intrinsic SPTR of 69 ± 6 ps FWHM and CTR of 58 ± 3 ps FWHM for small LSO:Ce:Ca crystals (Gundacker et al 2019). The same configuration (LSO + FBK) was used as reference detector, but with a slightly different photopeak region for better statistics.

The CTR was measured with the setup described in Gundacker et al (2019) in a temperature stabilized environment (≈ 18 ^◦C). A ²²Na source with an activity of 3 MBq emits two gammas back to back which were detected in coincidence by the reference detector (2 × 2 × 3 mm³ LSO:CeCa, CTR = 61.2 ± 3.0 ps FWHM) and Cherenkov radiator under test. For the reference detector the signal was split, where one part was amplified using HF electronics for the time signal and the other part via an analog operational amplifier (AD8000) for the energy signal. The signals were digitized with a LeCroy DDA735Zi oscilloscope, having a bandwidth of 3.5 GHz and sampling rate of 20 Gs s⁻¹. For the other detector side (PbF₂) only the HF signal was used, and the amplitude and charge of the HF signal was measured within a time window of 5 ns around the signal. Moreover the slew rate (dV/dt) at different threshold positions (5, 20, 30 mV) and the signal rise time (5–20 mV) were recorded. The single SPAD signal amplitude at operational condition (39 V bias voltage) was 44 mV, while the time difference was calculated via leading edge discrimination at 10 mV. All time resolution values were corrected for the reference detector contribution, assuming two identical Cherenkov radiators in coincidence. The duration of one measurement condition (SiPM overvoltage, leading edge threshold, DOI position) changed between one hour and two days, but it was ensured that at least 10k coincidence events were acquired after photopeak selection of the reference detector.

An illustration of the setup is sketched on the left of figure 1, while on the right a screen-shot of the oscilloscope with typical detector pulses is depicted.

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Measurements presented in section 4.4 were performed in a DOI collimated configuration where the reference crystal was placed at a distance of 60 mm and the source of 10 mm from the PbF₂ crystal. The crystal was irradiated from the lateral side and moved vertically for different DOI measurements.

The used custom made electronics is fully described in Cates et al (2018), Gundacker et al (2019) to have low influence of electronic noise, which is of utmost importance to achieve excellent time resolution, particular for low light intensities (Cates and Levin 2019, Kratochwil et al 2020a, Gundacker et al 2020a). A 3 GHz balun transformer and two BGA616 bipolar monolithic microwave integrated circuit amplifiers with 570 mW total power consumption were used to obtain the time response of a single channel.

The number of detected Cherenkov photons is very low in the order of few triggered SPADs only and not distinguishable from random triggered dark count events (≤5 counts per μs (Gola et al 2019)). Crosstalk happens when secondary photons are produced during the SPAD avalanche process, which subsequently are detected by another SPAD. The crosstalk probability was deduced with the same setup as for the CTR measurements, but without the radioactive ²²Na source in the test bench and by triggering only on the HF signal of the SiPM coupled to the crystal. A histogram of the measured charge was drawn and the number of triggered SPADs (N₁, N₂, ...N_k ) were counted. Because of a very small probability of two dark count events appearing in the same time window, it can be assumed with certainty that all events above one triggered SPAD are due to crosstalk. The timescale of delayed crosstalk and after-pulsing was well above the used 5 ns time window (Acerbi and Gundacker 2019). To model the distribution of triggered SPADs a branching Poisson process (Borel λ) was used (Vinogradov 2012), because a simple geometric chain is not accurate in the case of high crosstalk probabilities. Equation (1) denotes the probability P_k,DCR of measuring the signal of k triggered SPADs after the initial dark count event, as function of the crosstalk parameter λ.

$$\begin{eqnarray}&&{P}_{k,{DCR}}=\displaystyle \frac{{(\lambda \cdot k)}^{k-1}\cdot {\exp }(-k\cdot \lambda )}{k!}\,\,\,\,\,\,{\rm{for}}\,\,\,k\geqslant 1.\end{eqnarray} \tag{ 1 }$$

The case k = 1 describes the probability of not having crosstalk while 1 − P_k=1,DCR is the probability of having one or more SPADs triggered due to crosstalk. λ is extracted via chi-square minimization (Berkson 1980) using the measured values N_k by ${P}_{k,{DCR}}=\tfrac{{N}_{k}}{{\sum }_{i\geqslant 1}{N}_{i}}$ and the conversion to the geometric model can be done via ${p}_{{geom}}=1-{\rm{\exp }}(-\lambda )$.

To estimate the mean μ number of detected Cherenkov photons a generalized Poisson (μ, λ) (Vinogradov 2012) was used, where the probability P_k of having k triggered SPADs was calculated as in equation (2). This distribution allows to have zero triggered SPADs (no photons detected) which are not measured. Therefore a new distribution shown in equation (3) was constructed by dividing over the probability of one-SAPD events to enable the extraction of μ using the measured crosstalk parameter λ, again via chi-square minimization.

$$\begin{eqnarray}&&{P}_{k}=\displaystyle \frac{\mu \cdot {(\mu +\lambda \cdot k)}^{k-1}\cdot {\rm{\exp }}(-\mu -k\cdot \lambda )}{k!}\,\,\,\,\,\,{\rm{for}}\,\,\,k\geqslant 0\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\tilde{{P}_{k}}=\displaystyle \frac{{P}_{k}}{{P}_{1}}=\displaystyle \frac{{N}_{k}}{{N}_{1}}=\displaystyle \frac{{(\mu +\lambda \cdot k)}^{k-1}\cdot {\rm{\exp }}(\lambda -k\cdot \lambda )}{k!}\,\,\,\,\,\,{\rm{for}}\,\,\,k\geqslant 2.\end{eqnarray} \tag{ 3 }$$

From the estimated mean Cherenkov photon number μ the probability of detecting one or more photons (k ≥ 1) in coincidence can be calculated according to equation (4), since the probability for P₀ is the simplified expression ${\rm{\exp }}(-\mu )$

$$\begin{eqnarray}&&{P}_{k\geqslant 1,{coinc}}={(1-{\rm{\exp }}(-\mu ))}^{2}.\end{eqnarray} \tag{ 4 }$$

The square in equation (4) is required when assuming two identical detectors, as for both detector arms more than zero photons need to be detected in coincidence.

The measured time delay distribution shows, in dependency of the condition more or less, a pronounced tail coming from photons which are bouncing long time in the crystal before being detected. For standard scintillation these tails are not that pronounced, as the trigger is on the first few detected photons and very late photons do not contribute. In the case of pure Cherenkov radiators, in several cases the first detected photon is also the last one. To account for this tail a Crystal Ball function, which consists of a Gaussian core portion and a power-law low-end tail, was used. The FWHM and full width at tenth maximum (FWTM) of the fit function was calculated and corrected for the reference contribution. When the distance between the Cherenkov radiator and the ²²Na source was large, a constant floor of events in the time delay distribution was observed coming from DCR. For reasons of consistency for all the measurements a constant term was added to the fit function to account for those events.

A strong time walk was observed depending on the number of detected photons, similar as reported in Dolenec et al (2016b). Time walk correction based on the amplitude or charge were tested with certain success, but clearly better performance was achieved when corrections based on the measured slew rate (SR) were made (see sections 4.1, 5.1 and table 4). A similar correction based on the signal rise time is described in the case of BGO in Kratochwil et al (2020b).

Table 4. SPCTR, CTR before/after time walk correction (FWHM and FWTM) and detection efficiency corrected CTR [ps] accounting for not detected photons upon gamma interaction for various PbF₂ crystal configurations. Energy ranges for LSO:CeCa (Teflon) and BGO (Teflon) are used without SiPM saturation considerations. Values for BGO are taken from (Kratochwil et al 2020a). Errors are ≈4% of the values (up to 10% for det. $\overline{{CTR}}$ due to error on μ). The leading edge threshold was set to 10 mV for all the measurements (LSO 20mV) which is close to the optimal settings.

Surface condition	Geometry	SPCTR	CTR	CTR cor	CTR cor	det. $\overline{{CTR}}$
or Material	(mm3)	FWHM	FWHM	FWHM	FWTM	FWHM
Teflon	2 × 2 × 3	166	190	142	293	184
Bare	2 × 2 × 3	132	214	132	277	186
Black+Teflon	2 × 2 × 3	173	230	159	348	234
Black	2 × 2 × 3	113	235	134	269	514
Teflon	2 × 2 × 20	263	261	215	582	290
Black+Teflon	2 × 2 × 20	200	269	203	527	344
Black	2 × 2 × 20	145	186	159	303	2252
LSO [100–665 keV]	2 × 2 × 3	/	142	123	224	123
BGO [100–665 keV]	2 × 2 × 3	/	238	193	567	193
BGO [440–665 keV]	2 × 2 × 3	/	169	151	331	151
BGO [100–665 keV]	2 × 2 × 20	/	334	298	1469	298
BGO [440–665 keV]	2 × 2 × 20	/	288	258	891	258

The left of figure 2 shows the measured probability of triggering one, two and more SPADs due to the initial dark count event and a following cascade of crosstalk for different bias voltages and with Teflon wrapping or black painted crystals. On the right of figure 2 the calculated crosstalk parameter λ as function of the bias voltage is drawn for all tested surface conditions for 2 × 2 × 3 mm³ sized crystals. We observed a higher crosstalk probability when the crystal is wrapped in Teflon, as it behaves like a reflector for the secondary photons generated by each avalanche. The lowest crosstalk parameter was measured for a black painted crystals and for black paint opposite to the readout side, since in this case most of the created avalanche photons are absorbed and do not contribute. The change of the correlated noise when coupling a crystal to the SiPM is in agreement with previous measurements using different crystal cross sections (Gundacker et al 2016b).

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A histogram of the measured SiPM signal charge for 20 mm long crystals is presented on the left of figure 3. While the black painted crystal had mostly one or two SPAD events and the occurrence droped very fast for higher SPAD number, it was less steep when the crystal was wrapped in Teflon. When counting the events having k SPADs triggered and using the calculated crosstalk parameter and equation (3) the mean number of detected Cherenkov photons was evaluated, as shown on the right of figure 3. The calculated mean Cherenkov photon number μ denotes all events including cases where the 511 keV gamma only deposited a fraction of its energy in the crystal. For larger crystal volumes (e.g. 4 × 4 × 30 mm³ instead of 2 × 2 × 3 mm³) on average more energy is deposited in the crystal (due to a higher fraction of Compton events being contained in the crystal) and therefore also more Cherenkov photons are produced and detected (see section 5.2). Similarly higher Cherenkov photon numbers are expected when selecting on photopeak events only (Arino-Estrada et al 2018).

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Results in terms of crosstalk, Cherenkov photon yield and probability P_coinc of detecting an event in coincidence are summarized in table 3.

Table 3. Crosstalk parameter and mean number of detected Cherenkov photons for various crystal configurations. The error of λ is ± 0.02 and for μ ± 10%, while P_coinc is derived from μ.

Surface condition	Geometry (mm3)	λ @ 35 V	μ	Pcoinc (%)
Teflon	2 × 2 × 3	0.36	2.08	${77}_{-5}^{+3}$
Bare	2 × 2 × 3	0.33	1.86	${71}_{-5}^{+5}$
Black+Teflon	2 × 2 × 3	0.31	1.74	${68}_{-5}^{+5}$
Black	2 × 2 × 3	0.28	0.71	${26}_{-4}^{+3}$
Teflon	2 × 2 × 20	0.32	1.95	${74}_{-6}^{+4}$
Black+Teflon	2 × 2 × 20	0.3	1.46	${59}_{-6}^{+5}$
Black	2 × 2 × 20	0.28	0.32	${7}_{-1}^{+2}$
Quartz black	2 × 2 × 4	0.29	0.49	${15}_{-2}^{+2}$

Based on simulations with Geant4, a weak directionality of the Cherenkov emission was expected (Brunner et al 2014, Roncali et al 2019). In particular when a Compton interaction occurs, the momentum of the recoil electron depends on the direction of the incident γ-ray and therefore also the Cherenkov emission cone. To validate this assumption the 2 × 2 × 3 mm³ sized crystal was black painted opposite to the readout side and wrapped in Teflon on the lateral faces to enhance the expected effect. The PbF₂ crystal was placed far away (≈15 cm) from the reference crystal and the ²²Na source to have an almost parallel beam of gammas without precise alignment. The system of crystal, SiPM and amplifier was rotated so that irradiation from three different orientations with respect to the readout side were possible (head on, side, back), as illustrated in figure 4. A minor drop to 94 ± 6 % of μ with respect to classical head-on condition for the side irradiation was calculated, but for back irradiation this value drops to 54 ± 6 % indicating a strong directionality of the emission.

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When selecting on coincidence events, having a photopeak on the reference crystal, the time delay distribution shows a non-Gaussian structure and a tail coming from the fluctuating number of triggered SPADs. The excellent single photon counting capability of the SiPM allows to select on events with different number of triggered SPADs, as illustrated on the top of figure 5. The peak of the distribution using all events is moved to the left compared to one, two or three triggered SPADs and more pronounced for the Teflon condition. The reason for this time walk is coming from a steeper signal which passes the leading edge discrimination faster than for a low number of triggered SPADs. A similar observation is reported by Liu et al (2016) for low light intensity detection with a Phillips digital SiPM.

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The ratio of all events to one to three triggered SPADs is greater for the Teflon case than for the Black condition, since the average number of detected Cherenkov photons is larger in the first case leading to a different distribution of activated SPADs.

On the bottom left a scattered plot of the coincidence time delay t against the slew rate SR is shown. A polynomial function f(SR) was used to fit the data and to correct the individual time stamps t_cor,i = t_i − f(SR). When drawing a histogram of the slew rate, peaks coming from one, two, and more triggered SPADs are visible (see section 5.1). The time delay distribution after correction is shown on the bottom right of figure 5. A large improvement of the time resolution after correction, in particular with respect to the tails is observed.

The measured time delay distribution for 20 mm long crystal length is shown in figure 6 for all three tested surface conditions.

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The measured time resolutions for all configurations are summarized in table 4.

Single photon coincidence time resolution (SPCTR) denotes events where only one SPAD was triggered. This way the photon time spread (PTS) contribution can be estimated for different surface contributions and lengths. Comparing Teflon with Black, an additional CTR contribution of ≈86 ± 6 ps FWHM (single side, 122 ps in coincidence, $86=\sqrt{{166}^{2}-{113}^{2}}$/$\sqrt{2}$) for small and ≈ 155 ± 9 ps for long crystals is observed in Teflon. This is due to a larger PTS in the crystal when wrapped in Teflon as compared to the black painted case, because of multiple reflections and scattering at the surface of the crystal.

Moreover, comparing Teflon wrapped PbF₂ and BGO, having a similar refractive index and a similar Cherenkov photon yield, the CTR is better for PbF₂ (2 × 2 × 3 mm³: 142 ± 5 ps FWHM PbF₂, 151 ± 3 ps BGO/2 × 2 × 20 mm³: 215 ± 8 ps PbF₂, 259 ± 3 ps BGO). This confirms, that slow scintillation photons lead to a deterioration of the time resolution, since they are not promptly produced and lead to a broadening of the time delay distribution when the initial Cherenkov photons are not detected. This is the price paid for a higher detection efficiency, since in the case of sole Cherenkov radiators these events are not detected.

It is interesting to notice that SPCTR is better than CTR after correction for crystals having low μ with Black configuration, while for Teflon high number of detected Cherenkov photons lead to a better time resolution compared to only one detected photon without crosstalk. The trend that SPCTR is better than the CTR was also observed in Dolenec et al (2016a) in the case of low photon number.

The detection efficiency corrected CTR is normalized based on the detection probability P_coinc from equation (4) and table 3 (${\det }.\overline{{CTR}}$ = CTR_cor /P_coinc ) in order to compare the results with each other in terms of effective sensitivity. For standard scintillators having high light output both quantities are the same, since there are always photons detected upon gamma interaction in the crystal. In the case of sole Cherenkov radiators reducing the PTS by loosing photons inevitably lead to an overall drop of PET detector performance by much lower effective sensitivity (Schaart et al 2021).

While black painted crystals are most likely not suited for PET given the low sensitivity and therefore high ${\det }.\,\overline{{CTR}}$, they provide interesting features to study the contributions of measured time resolution. The case of one triggered SPAD is a crosstalk-free environment where light transfer efficiency (LTE) and PDE do not matter and the PTS is only coming from different DOIs and a small smearing due to photon propagation in the crystal (see section 4.4).

The measured SPCTR for 2 × 2 × 3 mm³ black painted crystals is solely determined by the SiPM properties: the intrinsic SPTR of the used SIPM is 69 ± 6 ps FWHM (Gundacker et al 2020a) with an already subtracted electronic noise contribution of 45 ± 5 ps. Adding these two contributions in quadrature and multiplying with $\sqrt{2}$ for a coincidence condition, the expected SPCTR is 116.5 ± 11.5 ps FWHM, which is in good agreement with the measured value of 112.7 ± 4.5 ps. Small differences can come from illuminating only a 2 × 2 mm² SiPM cross section compared to all the SiPM area which improves the SPTR (Nemallapudi et al 2016) and that the photon emission is not only at a fixed wavelength but continuous (Piull et al 2012).

SPCTR measurements with different crystal lengths were performed to study the impact of DOI on the timing resolution. Analytic models on DOI-induced time bias exist for standard scintillators (Toussaint et al 2019, Loignon-Houle et al 2020a) where the light transport and rise and decay time lead to additional uncertainty and in Cates et al (2015) DOI contribution has been shown of not being Gaussian like. In figure 7 the DOI bias was modeled by taking the time difference of different velocities of the gamma and optical photons at the extreme cases (small and large DOIs). This is certainly a very simple approximation, as no weighting for the DOIs was used. Due to crystal attenuation certain DOIs are more probable to interact, while photons produced at short DOIs far away from the SiPM need to travel longer and are more likely lost. Also the contribution of the PTS on top of the SPTR was assumed to be Gaussian like, which might only be valid for small DOI contribution with respect to the SPTR. For black painted crystals the approximation of the speed of the optical photons (v = c/n) matches the data, since mainly direct photons are detected. Even when having very high number of prompt photons this intrinsic limit cannot be surpassed without accessing the DOI information.

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To confirm that the SPCTR is not affected by the used material in case of Black surface, time resolution of a 2 × 2 × 4 mm³ Quartz glass was measured. While having a high floor of dark count events since the stopping power of Quartz is poor, a comparable time resolution of 123 ± 7 ps FWHM as for PbF₂ was measured. The small deterioration with respect to PbF₂ might be due to enhanced reflections, as the Quartz was depolished, while the additional millimeter crystal length should only contribute to around 0.5 ps and is negligible.

To further evaluate the photon time propagation in the crystal, DOI collimated measurements were performed. The DOI setup is shown in the inset of figure 8, left. Depending on the experimental conditions, the photon can directly travel toward the SiPM or travel away from it being later reflected at the opposite crystal face leading eventually to its detection. Figure 8 shows the coincidence time delay histogram for two surface conditions: on the left for Teflon and on the right for Black. In the second case no reflections are visible, and only photons propagating directly to the SiPM contribute to the SPCTR. The center of the distribution is moving, since by changing the DOI position the distance of the two SiPMs is changing as well. The measured SPCTR slightly deteriorates from 132 ± 4 ps to 150 ± 4 ps when decreasing the DOI due to a larger dispersion of the photon paths (shown as inset in figure 8, right). When the crystal is fully wrapped in Teflon photons can bounce between the surfaces and are kept longer in the crystal, largely increasing the variation of the travel time and therefore the SPCTR. In the case of interaction close to the SiPM one can identify the wave of photons which are back-reflected at the end of the crystal and reach the SiPM at later times. This propagation of photons in high aspect ratio crystals is well understood and simulations are shown for example in Gundacker et al (2013), but to the authors knowledge have never been experimentally visualized. In the case of BGO (Kratochwil et al 2020c) and also LSO (Loignon-Houle et al 2020b) the CTR is better at short DOI positions, as the two waves of photons reach the SiPM at the same time, effectively increasing the initial photon time density. Work is ongoing to use this effect for extracting DOI information (Loignon-Houle et al 2020b). The distributions were normalized based on the total number of valid coincidences, since, in particular for the black painted configuration, a strong attenuation was observed.

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In order to better visualize the time structure caused by the reflections opposite to the SiPM while keeping the time spread small, another DOI measurement for a 30 mm long PbF₂ crystal was performed. Here the small 2 × 2 mm² face opposite to the SiPM was covered by ESR for mirror-like reflections. The four lateral sides were black painted to suppress reflections and time dispersion of the photon wave. Five different DOI positions were measured and to better discuss the results they are split in figure 9. For shallow DOI positions the back reflected wave is not or only barely visible, as the difference of travel distance of the two photon waves is too little to be resolved. Only faintly visible is a second peak being about 350 ps delayed with respect to the main peak coming from Fresnel reflections at the interface between crystal, Meltmount and SiPM. The reflections coming from the ESR are well visible on the right of figure 9 for DOI positions at 20 and 28.5 mm close to the SiPM. The abundance of measured reflected photons largely decreases with longer DOI positions, as the black paint leads to a short effective attenuation length. The measured time differences for figure 9 are all, within the experimental resolution, in excellent agreement with DOI positions and the speed of the photons inside the crystal (n/c ≈6.5 ps/mm at 300 nm).

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In section 4 and figure 5, time resolution for one and more triggered SPADs are shown. Naively it is expected that more triggered SPADs lead to better time resolution, as more photons are detected. In this high crosstalk environment however, most events populating two triggered SPADs are caused by one detected photon and one crosstalk event. Those events do not carry better time information than the case of having only one triggered SPAD.

On top of this, the crosstalk introduces an additional time jitter. Depending when crosstalk happens the signal reaches the leading edge threshold earlier or later. Triggering on even smaller thresholds (e.g. 5 mV instead of 10 mV) leads to better time resolution for most events since crosstalk has less time to propagate, but worse timing for one SPAD events due to a higher electronic noise contribution, since the slew rate (dV/dt) at lower threshold generally is smaller. This is also the reason why time walk correction is so crucial, as to some extent it can correct the time walk caused by crosstalk.

For 20 mm long Black crystals using all events a CTR of 186 ± 7 ps FWHM without correction was calculated, compared to 174 ± 7 ps for a correction based on the measured amplitude, 177 ± 7 ps based on the charge, 180 ± 7 ps based on the signal rise time between 5 and 20 mV and 159 ± 6 ps based on the slew rate at 30 mV (see table 4). A correction based on the signal rise time at higher values (e.g. 20–60 mV) lead to similar time resolutions as the slew rate correction at 30 mV, but does not work for one-SPAD events since the signal amplitude is too low to pass the second threshold. Classical correction methods (e.g. based on the amplitude or integrated charge) are not that effective, as these quantities consider all the triggered SPADs even after the leading edge threshold. As a result the measured amplitude is only weakly correlated with the time walk, while the slew rate has a stronger dependency.

In the case of HF electronics, we have shown that corrections based on the slope of the signal (slew rate, rise time) will outperform corrections based on the amplitude or charge, demonstrated in figure 10 for two-SPAD events. Only when selecting on a high number of triggered SPADs it is more likely that two or more "real" Cherenkov photons are responsible for those events. In this case they carry improved time information, which allows to overcome the SPCTR limit. For example, sub-100 ps (99 ± 4 ps FWHM, 191 ± 6 ps FWTM) CTR was measured for 2 × 2 × 3 mm³ long Teflon wrapped PbF₂ crystals for 21% of all events when selecting on the highest slew rate (4.3% in coincidence).

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The most simple case to be considered is a two SPAD event, as in this case either two real photons are detected or only one photon with one crosstalk. When looking at the distribution of the slew rate for those events (filtering two-SPAD events based on the charge), a structure is visible, shown on the left of figure 10. Selecting only on certain slew rates of the two-SPAD events the center of the time delay distribution is moving illustrated on the right of figure 10 and the measured CTR is improving compared to all two-SPAD events. These results confirm the importance of the time walk correction based on the slew rate and might indicate towards an identification of crosstalk events compared to two real photon events. More detailed measurements with different light intensities (e.g. with laser) are required for validation.

The intrinsic Cherenkov photon yield for an electron with a given speed can directly be calculated using the Frank-Tamm-formular (Leo 1994, Brunner et al 2014). Uncertainties arise, since recoil electrons not always have the same energy. For photo ionization on average 20 optical Cherenkov photons were simulated for PbF₂ (Canot et al 2019) (above 250 nm), while for BGO which has higher refractive index, 17 ± 3 Cherenkov photons were experimentally measured between 300 and 800 nm (Gundacker et al 2020a).

The measured number of Cherenkov photons stated in table 3 for small Teflon wrapped crystals is μ = 2.08 ± 0.21 including crosstalk and μ = 2.28 without crosstalk considerations and a worse robust fit function. To estimate the intrinsic Cherenkov photon yield several correction factors need to be included: LTE in the crystal (c_LTE ≈ 0.6) (Gundacker et al 2020a), weighted PDE starting at 300 nm (c_PDE ≈ 39%) (Gundacker et al 2020a), and correction factors regarding energy c_energy = 0.7 and Cherenkov emission in the deep UV c_VUV = 0.77. The correction on the deposited energy c_energy is needed, since in the experiment both photo-absorption and Compton events were considered, where for the latter the Cherenkov photon yield is lower. The value was derived from Geant4 simulations on a 2 × 2 × 3 mm³ BGO crystal comparing the intrinsic Cherenkov photon yield with 511 keV γ-irradiation for deposited energies above 100 keV and for only 511 keV deposition (Terragni 2020). PbF₂ is transparent down to 250 nm, while in the measurement Cherenkov photons between 250 and 300 nm were not detected due to the non-transparent optical glue and non-VUV sensitive SiPMs. The correction factor c_VUV was calculated by comparing the integral of the 1/λ² Cherenkov distribution between 250–900 nm and 300–900 nm.

Including all the correction factors the intrinsic Cherenkov photon yield upon 511 keV gamma interaction for PbF₂ between 250 and 900 nm was calculated according to equation (5).

$$\begin{eqnarray}&&{{ILY}}_{{{PbF}}_{2}}=\mu \cdot \displaystyle \frac{1}{{c}_{{LTE}}}\cdot \displaystyle \frac{1}{{c}_{{PDE}}}\cdot \displaystyle \frac{1}{{c}_{{energy}}}\cdot \displaystyle \frac{1}{{c}_{{VUV}}}=16.5\,\pm \,3.3.\end{eqnarray} \tag{ 5 }$$

In the case of Teflon only around 20% of the data was used for the Cherenkov photon yield calculations, since for many events the number of triggered SPADs was exceeding the used range of the oscilloscope giving an additional uncertainty. Repeating the calculation for different SiPM overvoltage does not change the calculated value significantly, e.g. μ = 1.81 (38 V), μ = 1.75 (37 V), μ = 1.88 (36 V), μ = 1.86 (35 V), μ = 1.82 (34 V) was calculated for the bare crystal. Thorough measurements with different crystal volumes and better dynamic acquisition ranges together with simulations are required for a more precise estimation.

The measured CTR values scale close to the analytic calculation considering DOI and SPTR impact. Even for 20 mm long crystals wrapped in Teflon, the contribution of the PTS is below 200 ps FWHM. As a result literature values on measured SiPM SPTR values with different acquisition systems can be reviewed to estimate the timing performance for Cherenkov radiators.

Using a NINO (Anghinolfi et al 2004) chip, SPTR values of 94 ± 5 ps FWHM for 1 × 1 mm² (175 ± 7 ps for 3 × 3 mm²) SiPM size have been measured in Nemallapudi et al (2016) with FBK NUV SiPMs, which translates to 133 ps (247ps) CTR for two detectors in coincidence. Using the same SiPMs as in this work (4 × 4 mm² FBK NUV-HD) and NINO chip, about 130 ps FWHM have been measured with this ASIC (Gundacker et al 2019). For SiPMs from Hamamatsu (S13360-3050PE), SPTR values of 214 ps with FlexTOT v2 (Sarasola et al 2017), and 176 ps FWHM with FASTIC ASIC (Sanchez and Ballabriga 2021) have been reported, compared to the estimated intrinsic SPTR value of 135 ± 8 ps FWHM (Gundacker et al 2019).

Another solution might be the use of digital SiPMs (Liu et al 2016), and to use the timestamp from the first triggered SPAD. For instance using 3 × 3 × 20 mm³ BGO crystals and utilizing the Cherenkov emission, CTR values down to 330 ps FWHM have been reported (Brunner and Schaart 2017). Based on values presented in table 4, the time resolution should improve when replacing BGO with PbF₂.

One major drawback of SiPMs are dark count events, which lead to a indistinguishable signal from the detection of few Cherenkov photons. DCR events are not correlated, so the probability of triggering two SPADs simultaneously is negligible. However, crosstalk can cause two or more SPADs to be triggered simultaneously, as described in equation (1). The probability distribution of detecting a noise event for a given crosstalk probability as function of the trigger threshold is shown on the left of figure 11. When increasing the trigger threshold, it becomes at the same time less likely to detect a real event coming from Cherenkov photons after γ-interaction in the crystal. This drop of detection probability is illustrated on the right of figure 11. This analytic calculations are derived from equations (1) and (2). When on average a high number of Cherenkov photons are detected and for low crosstalk, DCR events can successfully be suppressed by increasing the trigger threshold to 2 or more triggered SPADs without large loss of detection efficiency. Thanks to the excellent stopping power and high sensitivity of PbF₂, such a selection will not harm the overall sensitivity too much, compared to less dense materials.

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Sufficiently good energy resolution in PET is required to reject detected gammas which scatter in the body and loose some energy, as such events would otherwise lead to misplaced line of responses. In the case of PbF₂ or other pure Cherenkov radiators (ignoring novel concepts like Cherenkov charge-induction (Arino-Estrada et al 2019)) it is not possible to extract the energy of the gamma at all. However, the Cherenkov production process intrinsically provides a lower energy threshold. Based on simulations (Canot et al 2019) most of the 511 keV γ interactions lead to an electron with kinetic energy of 423 keV with side peaks at 495 and 507 keV coming from the K, L and M-shell photo ionization of lead. If the interacting γ has lower energy it results in further lower energy of the recoil electron. An electron with kinetic energy of 200 keV will produce on average about 4 times lower number of Cherenkov photons compared to a kinetic energy of 400 keV (Canot et al 2019). Hence gammas which are scattered in the body having less energy, produce fewer Cherenkov photons and are therefore less likely detected compared to non scattered gammas. This can be seen as different value of μ depending on the deposited energy.

Using equation (2) the detection probability for different values of μ can be analytically calculated. In figure 12 λ was fixed, while μ was modified depending on the deposited energy. The conversion of energy to mean number of detected Cherenkov photons depends on the crystal configuration and the SiPM, while the nonlinear Cherenkov photon yield was approximated based on simulations. Lastly the cutoff energy, where no Cherenkov photons are produced (μ = 0) is around 100 keV, giving an additional filter for highly scattered gammas.

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Both not-wanted types of events (DCR, scattered gammas) have on average lower number of triggered SPADs compared to true 511 keV gamma interaction. To overcome the tradeoff between a "clean" signal (high threshold) and high statistics (low threshold), a weighted reconstruction could be an option where events with high number of triggered SPADs are given more priority or serve as a seed in the reconstruction process. A similar concept with a weighting based on different timing kernels for BGO was recently demonstrated (Efhimiou et al 2020). Overall, such a detector design is a completely new way of thinking (Efthimiou 2020), focusing mostly on cost and sensitivity, while energy resolution and scatter correction in the reconstruction are the weak points. Thorough simulation and image reconstruction for the case of sole Cherenkov radiators, including DCR background evaluating the feasibility of this approach are subject of future studies.

In this work only Cherenkov photons starting from 300 nm were used. With transparent optical coupling, as recently studied for BaF₂ crystals (Pots et al 2020), and using SiPMs with extended detection efficiency in the VUV below 200 nm developed for liquid xenon (e.g. HPK-VUV S13370-3050CN or FBK-VUV (Capasso et al 2020, Gundacker et al 2020b)), more Cherenkov photons can be harvested giving a better time resolution and higher detection probability.

Further, there is a deterioration on time resolution due to crosstalk when having multiple SPADs triggered. Here the requirement is to reduce the crosstalk probability while keeping the good SPTR and PDE. Promising candidates for this are currently under development (Gola et al 2019) and subject for future work. A critical aspect in improving the time resolution significantly below 100 ps is the SPTR of the photodetector. Here 3D integration (Nolet et al 2018) could be a viable solution with already reported SPTR values of 17.5 ps FWHM for a single SPAD including the time to digital converter. Also nano-structured light concentrators and nanophotonics (Enoch et al 2021) is an encouraging research line to improve both the SPTR and detection efficiency, while reducing crosstalk probability.

In the case of high aspect ratio crystals Cherenkov photons can travel towards or away from the SiPM, before being reflected and later detected by the photodetector. Unless the arrival time of the photons can be resolved in time, the second photon does not carry substantial time information, as it arrives too late to contribute to the development of the signal and time estimation. If the first photon is not detected, there is an increased time delay with respect to the time of interaction, responsible for the observed tail. Those problems can be solved when placing a second SiPM on the other side of the crystal. The measurement with the face black painted opposite to the SiPM imitates a double sided readout configuration having only one SiPM active. Here the SPCTR and also the time resolution after correction is better compared to having the crystal fully wrapped in Teflon, with lower number of detected Cherenkov photons. Depending on the requirement, a trigger in coincidence could also substantially decrease the DCR contribution without a large loss in sensitivity. Using double sided readout with twice as long (e.g. 30 mm) crystals might be more favorable than having a normal sized crystal (e.g. 15 mm) with only single sided readout, despite the same ratio of sensitive volume to readout channels.

Also light sharing (Pizzichemi et al 2019) might be an option, but special attention need to be given to false dark count events.

Preserving some momentum of the incoming gamma photon with respect to the produced Cherenkov photons was expected due to Compton kinematics and from simulations. To the authors knowledge it is the first time that an experimental confirmation was given. For monolithic crystals and when covering all the crystal surface with SiPMs, which have an excellent detection efficiency close to 100% (Somlai-Schweiger and Ziegler 2015), the recovery of the direction of the incoming gamma might be feasible, like for a Compton camera. This directionality might be also of interest for high energy particle detectors, as it gives another difference between scintillation and Cherenkov radiation besides the emission wavelength and scintillation kinetics for dual readout calorimetry (Lucchini et al 2013, Ferrari et al 2019).

From an instrumental point of view the SPCTR can provide a new method for measuring the SPTR of a SiPM. This kind of measurements does not require a picosecond laser and can be also done with two SiPMs and two black painted PbF₂ crystals in coincidence or one SiPM against a reference detector. The drawback is a lower count rate, hence longer measurement time and a large emission spectrum. However, the crystal does not necessarily have to be glued to the SiPM and an optical filter can be placed between the crystal and SiPM. Such a scheme would allow SPTR measurements at different wavelengths. Black painted PbF₂ crystals can also be used for calibration purposes to extract the impulse response function for time correlated single photon counting measurements (Gundacker et al 2018, Martinazzoli et al 2020).