Optical crosstalk between SiPMs in dual-ended readout

Abstract

Gamma-ray detection performance in scintillation crystals can be improved by coupling multiple photodetectors around the sensitive volume. Greater light collection leads to improved time and energy resolution. However, neighboring detector noise, both uncorrelated and correlated, can produce optical photons that may deteriorate their performance. We studied the impact of external optical crosstalk in a dual-ended readout setup using AFBR-S4N44P014M silicon photomultipliers (SiPMs) from Broadcom coupled to 20 mm thick bismuth germanate (BGO) crystal, with and without scintillation light. SiPMs were biased individually. The behavior of crosstalk was analytically modeled. The dark count rate increased up to 50 % when both SiPMs were biased with respect to only one active SiPM. At the same time, the crosstalk probability increased more than 2-fold at high overvoltages due to detection and re-emission of avalanche photons, limiting the operational range and causing an up to 5-fold increase in the total number of detected photons. Energy resolution in BGO was impacted, and the coincidence time resolution distribution shape between two BGO detectors was significantly altered- both in a non-intuitively manner. Experimental results validate the analytical model which rely on a single constant to predict dark count rate, crosstalk and light output enhancement. Crosstalk in multi-SiPM configurations, especially at high overvoltage, significantly impacts performance. Deep understanding of the effects of excess noise in SiPM is crucial for an optimal overall gamma detector operation. Findings extend to applications in high-energy physics, time-of-flight positron emission tomography using dual-ended or monolithic crystals, and cryogenic SiPM experiments in neutrino/dark matter studies.

Graphical Abstract

1. Introduction

Silicon photomultipliers (SiPMs) have largely substituted photomuliplier tubes (PMTs)) and are the photodetector of choice for applications dealing with light-based radiation detection, such as medical imaging instruments, high energy physics experiments, astronomy, homeland security, or light detection and ranging (LIDAR). They combine high detection efficiency for optical photons, excellent photon counting capability ranging from single photons up to several thousands of photons, low time jitter on the detection time, insensitivity to magnetic field, low operational voltage and high gain, and general robustness and small footprint [1–4]. SiPMs combined with scintillation materials such as inorganic crystals constitutes a competitive light-based radiation detector providing good timing and energy resolution [5, 6].

The disadvantages of SiPMs are the thermally generated counts in the absence of injected light (dark count rate (DCR)), hot-carrier-induced luminescence in the silicon creating secondary photons which can be detected by other single photon avalanche diodes (SPADs) (crosstalk, X-talk), carrier production outside the amplification region or in defect states which triggers a delayed signal (delayed crosstalk / after-pulsing), and pixel-to-pixel non-uniformity leading to gain variations [7, 8].

Recent R&D efforts have been made to reduce the probability of avalanche photons triggering neighboring SPADs (internal crosstalk) with deep trench isolation [9]. While this reduces the crosstalk rate and leads to better performance capabilities, avalanche photons can still escape the SPAD via the scintillator-interface before being back-reflected (external crosstalk). This optical crosstalk sets a limit to about 20 V excess bias voltage for this metal-trenched (MT) technology before the second breakdown appears [4, 10]. The emission spectrum of avalanche photons differs from typical scintillation light and convolved with the detection efficiency it peaks in the red to infrared regime between 600 nm to 1000 nm [11, 12].

Several concepts have been proposed to improve the light collection and reduce the light spread for better performance of light-based gamma-ray detectors, such as lateral side readout [13, 14] or dual-ended readout of high-aspect-ratio (HAR) crystals [15, 16]. The main disadvantage of coupling multiple SiPMs to the same gamma-ray sensitive volume is the potential increase of SiPM noise coming from avalanche photons produced in one SiPM and detected by the other one (and vice-versa), which can lead to a cascade of photons. While there exists work investigating the suppression of SiPM crosstalk with multiple SiPMs [17–20], there are no published reports of the measurements of the SiPM noise in a multi-SiPM configuration, nor a model that describes it.

In this study, we investigate the impact of SiPM crosstalk in a simple multi-SiPM configuration with two SiPMs coupled end-to-end to the same scintillating crystal (facing each other). Measurements in the dark and upon gamma-ray excitation for scintillation light were used to extract performance characteristics such as DCR, crosstalk probability, light output, and energy and timing resolution with and without the second SiPM being active. Analytic models were developed to describe the influence of biasing the additional SiPM and the framework can be extended to multi-SiPM configurations.

This work expands to the knowledge on how multiple SiPMs are impacting each other and provides accurate modeling to describe correlated noise and light output enhancement for a wide range of multi-SiPM configurations.

2. Materials and Methods

2.1. Detectors

Two pairs of NUV-MT SiPMs from Broadcom (AFBR-S4N44P014M) were used in this work. The SiPMs have a cell pitch of $40\mu \text{m}$ with an active area of about 3.6 × 3.7 mm² and have a epoxy protection layer. The NUV-MT SiPM technology developed by Fondazione Bruno Kessler (FBK) and Broadcom features an intrinsic single photon time resolution (SPTR) of about 55 ps full width at half maximum (FWHM), high photon detection efficiency (PDE) (63% at 420 nm) and good uniformity of breakdown voltage and gain among different devices [9, 21, 22]. SiPMs were soldered to printed circuit boards (PCB) to be used with custom dual-ended lownoise and high-frequency (LNHF) readout electronics [16] pictured on the top of figure 1. The readout electronics amplifies the voltage drop between SiPM anode and cathode to detect the signal of single photoelectrons [23]. The majority of the measurements were conducted with two pairs of polished BGO crystals (3×3×20 mm³, OST Photonics), wrapped in Teflon and coupled with Cargille Meltmount (n = 1.58) on both ends to the SiPMs. BGO was selected due to its recent interest in using Cherenkov light for fast timing in TOF-PET [24]. The relatively low light output of BGO simplifies the analysis, since there is no need to correct for SiPM saturation effects.

Figure 1:

Illustration of different types of measurements with SiPMs (blue) and BGO (black) coupled with Meltmount (light blue). Optical photons are illustrated as red arrow. (Top left:) Dark count and crosstalk measurement with the bare SiPM (top), second SiPM off (center) and both SiPMs on (bottom). (Top right:) Picture of crystal with SiPMs (top), scintillation measurement with lateral gamma-ray irradiation (center, bottom). The second SiPM is either off (center) or on (bottom). (Bottom:) Coincidence time resolution measurement with the radioactive source placed in between the two crystals. The outer (back) SiPMs are always on, while the inner (front) SiPMs are either off or on. The distances are not to scale.

2.2. Experimental setup and measurement configurations

The measurement setup was in a dark, temperature stabilized $(\approx {20}^{\circ }C)$ box. Three types of measurements were conducted with the different configurations illustrated on the bottom of figure 1. For all the measurements, the energy and timing signals of the back SiPMs from the LNHF readout electronics were connected to a fast oscilloscope (Tektronix MSO64B, 4 GHz, 25 Gs/s). The signals of the front SiPMs were not used in this study. Measurements were acquired once with the front SiPMs switched off and once with the front SiPM switched on and biased identical to the back SiPM. SiPMs were biased from close to the breakdown voltage up to 52 V (≈ 20 V excess bias) in steps of 1 V with external power supplies (ITECH, IT6863A).

2.3. Dark count and crosstalk measurement

The first set of measurements were performed without radioactive source, using solely dark counts as primary trigger. Three sets of measurements were taken by acquiring 50 traces with $200\mu \text{s}$ length each, namely with the bare SiPM; with the SiPMs glued to the crystal and the front SiPM unbiased; and with both SiPMs biased identically (top left configuration in figure 1). The LNHF readout electronics allows to identify the signal of a single triggered cell [23], with the SPAD signal amplitude plotted for different bias voltages on the left of figure 2. For each waveform, the number of crossing events passing a given threshold (corresponding to the triggering of at least one SiPM cell) of the signal was counted. The dark count rate as function of the threshold was plotted in a staircase plot [25] and the dark count rate $\left({\text{DCR}}_{0.5pe}\right)$ determined as counts at the threshold of half the single cell signal amplitude. The crosstalk probability $p$ was calculated as the ratio between at least two dark counts and at least one dark count $\left(p=DC{R}_{1.5pe}/DC{R}_{0.5pe}\right)$.

Figure 2:

(Left:) Breakdown voltage determination by measuring the single cell signal amplitude at different bias voltages. The inset shows the signal of a single dark count. For reference, the PDE at 500 nm is provided in blue on the right y-axis with the data taken from [9]. (Right:) Example SiPM waveform with a dark count event, dark count event with one/two crosstalk events and dark count with delayed crosstalk. A sweeping of the threshold was performed to count the number of signal crossings.

2.4. Light yield and energy resolution measurements

Light yield and energy resolution (ER) measurements were conducted by placing a ²²Na radioactive source (≈ 6 MBq) about 20 cm away from the BGO crystal, symmetrically centered with respect to the SiPMs to ensure uniform lateral radiation (illustration on the right of figure 1). The energybranch of the readout electronics (via operational amplifier Analog Devices AD8000) was used for triggering and the signal integral to measure the deposited energy (number of triggered cells) [26, 27]. Note, the readout circuit was not optimized for precise energy measurements but for 511 keV photopeak identification in the frame of accurate CTR measurements. No relevant SiPM saturation was observed, meaning the ratio between the photopeak positions corresponding to the 511 keV and 1.27 MeV peak stayed constant (≈ 2.5) among different measurements due to the relative low light yield of BGO. Therefore, no saturation correction was applied.

The selection of the region around the 511 keV photopeak to fit the Gaussian function had an impact on the calculated energy resolution. To ensure stability and reproducibility, we used a peak-finding algorithm (ROOT: TSpectrum, [28]) and fit a Gaussian function (without constant term) between 78 % and 125 % of the detected 511 keV photopeak position. While the statistical uncertainty of the calculated energy resolution amounts to less than 0.6% (absolute values), we estimate a systematic uncertainty of 1 %.

2.5. CTR measurement

The coincidence time resolution (CTR) was measured using the signal of the back SiPMs on each crystal. The signal of the front SiPM was not used in this nor other measurements. The ²²Na radioactive source was placed between the two crystals depicted on the bottom of figure 1. Energy signals were used to select on photopeak events (using an energy window of 440/511 and 700/511 times the 511 keV photopeak position) and the timing was determined via leading edge threshold with an optimum value at about 10 mV. In addition to the leading edge threshold, the signal rise time (between 20 mV and 100 mV) was stored to perform a time walk correction [29] to mitigate the impact coming from fluctuations on number of detected Cherenkov photons [30]. The time delay distribution was fitted with a sum of two Gaussian functions defined in equation 1.

$$f_{\left(t\right)}=\frac{N}{\sqrt{2\pi }}·\left[\frac{r_C}{{\sigma }_C}·\text{exp}\left(-\frac{(t-\mu {)}^2}{2{\sigma }_C^2}\right)+\frac{1-r_C}{{\sigma }_S}·\text{exp}\left(-\frac{(t-\mu {)}^2}{2{\sigma }_S^2}\right)\right]$$ (1)

Hereby $N$ expresses the total number of coincidence events, ${\sigma }_C\left({\sigma }_S\right)$ the standard deviation of the fast (slow) Gaussian, $r_C$ the abundance of the fast Gaussian, and $\mu$ the centroid of the distribution.

Since the shape of the time delay distribution is impacted by the presence of crosstalk, the Gaussian equivalent (GE) CTR is defined as a figure of merit (a single value instead of multiple fit parameters), following the work in [31] on the signal-to-noise ratio (SNR) of reconstructed TOF-PET images. For this, the sum of the value $\left(am{p}_{\mathit{\text{bin}}}\right)$ of each bin (with a bin width of $w$) in the coincidence time delay histogram is used according to equation 2, upon normalizing it with the total number of entries.

$$GE=0.664·\frac{w·\sum _{i}am{p}_{bin,i}}{w·\sum _{i}am{p}_{bin,i}^2}$$ (2)

The factor of 0.664 comes from the integration of a single Gaussian distribution $\left(\frac{1}{2\sqrt{\pi }}\right)$ and the conversion between standard deviation and FWHM $\left(2\sqrt{2\text{ln}\left(2\right)}\right)$ [16]. By using the raw histogram data, any influence of the fitting procedure or CTR modeling is omitted.

3. SiPM crosstalk model

Throughout this manuscript, we ignore delayed correlated pulses coming from trapped photons afterpulses or diffusion controlled delayed crosstalk pulses [32]. Since the probability of two dark counts happening at the same time is small (maximum one count every 200 ns, compared to a pulse width of less than 1 ns) we consider each signal crossing (in the dark) as an independent dark count, and each dark count with an amplitude greater than the single cell amplitude as a crosstalk event. To be accurate on the terminology, crosstalk-induced increase of the measured dark count rate is also labeled as dark count in this manuscript. Although these events come from an incident photon (from the other SiPM), they are initially triggered by intrinsic noise within the semiconductor structure.

3.1. Dark count enhancement

For both SiPMs, the dark count rate depends on the excess voltage, assuming a stable temperature. Upon a thermally-triggered dark count in one of the cells, the avalanche process in the SPAD produces hot-carrier-induced photons with an efficiency of around 3 · 10⁻⁵ photons with energy higher than 1.14 eV (below 1090 nm) per carrier crossing in the junction [33]. Since the gain (amount of charge flowing per avalanche) linearly scales with the excess voltage [11, 32], the amount of avalanche photons also increase with higher excess voltage $(N_{av}\propto V_{OV})$.

One or more avalanche photons may reach a neighboring cell from the same SiPM, either directly or upon reflection at the epoxy protection window and cause a crosstalk event (bare-configuration), which does not alter the count rate at the level of 1 SPAD. Though, with higher excess voltage, self-heating effects might occur which increases the dark count rate [34].

If the SiPM is coupled to the crystal (off-configuration), the crystal may act as temperature reservoir to mitigate SiPM self-heating effects. Additionally, avalanche photons may be reflected at the Meltmount-crystal interface, at the Teflon reflector from the crystal, or at the crystal-Meltmount-SiPM interface of the second SiPM and detected (external crosstalk). However, all the described effects will be at a short timescale and only alter the count rate above the level of 1 SPAD.

If the second SiPM is biased as well (on-configuration), two effects might occur: First, avalanche photons are detected by the second SiPM, which triggers an avalanche process that causes the generation of new photons, which may be detected by the first SiPM. The timescale for such effect is small, hence no change in the count rate below the level of 1 SPAD. Secondly, dark counts are generated in the second SiPM. These dark counts are not correlated and avalanche photons coming from this may reach the first SiPM. In this case an increase in the dark count rate in the first SiPM is expected. The amount of excess dark counts $\left({\text{DRC}}_{\mathit{\text{on}}}-{\text{DCR}}_{\mathit{\text{off}}}\right)$ depends on the initial dark count rate, the number of produced avalanche photons (which scales with the number of charge carriers, and thereby with the excess voltage), the weighted detection efficiency of avalanche photons $\left(PD{E}_{w,a}\right)$, and the probability of a photon being emitted toward the other SiPM and traversing through the crystal (avalanche photon transfer probability (APTP)), as summarized in equation 3.

$$DR{C}_{on}-DC{R}_{off}\propto DC{R}_{off}·N_{av}·PD{E}_{w,a}·\mathit{\text{APTP}}$$ (3)

Assuming a constant detection efficiency, equation 3 simplifies to equation 4, which merges the constant avalanche photon production (per excess voltage), the detection efficiency, and the APTP into a single constant $c$.

$$DR{C}_{on}-DC{R}_{off}=DC{R}_{off}·c·\left(V_{\mathit{\text{bias}}}-V_{\mathit{\text{brea}}k}\right)$$ (4)

3.2. Crosstalk enhancement

Similar considerations as for the dark count enhancement can be done to describe the SiPM crosstalk enhancement, though in this case we need to consider the count rate for at least one and two triggered cells.

Comparing configuration bare and off on the left of figure 1, the external crosstalk can be impacted by the presence of the crystal. While in [35] an increase of the crosstalk probability was reported for SiPMs without protection window when a crystal was coupled to it, we expect the opposite effect. The reason for this difference is the epoxy window, which has an optical interface with air (bare) and therefore higher probability for Fresnel reflections. By placing a refractive index-matching coupling agent between SiPM and crystal, these Fresnel reflections reduce, leading to a decrease of external crosstalk. In addition, the particular reflector configuration may impact the external crosstalk probability [23].

If the opposed SiPM is biased (on-configuration) avalanche photons from the first SiPM are detected, trigger a new avalanche process with a new generation of photons which may reach the first SiPM. Time-wise, it takes about twice the crystal propagation time (≈ 300 ps) plus the avalanche build up time, hence indistinguishable from a direct crosstalk event (within the same SiPM). Using the discussion from the subsection 3.1 and given the fact that photons need to traverse the crystal twice, and need to be detected and emitted twice, the excess crosstalk $\left({\text{P}}_{\mathit{\text{on}}}-{\text{P}}_{off}\right)$ probability is calculated according to equation 5.

$$P_{on}-P_{off}=c^2·{\left(V_{\mathit{\text{bias}}}-V_{\mathit{\text{break}}}\right)}^2$$ (5)

Again, we assume a constant detection efficiency as function of the excess voltage. Any enhancement in the dark count rate or change in temperature has no impact on the crosstalk probability, since the process is the same for any primary event seed and rate-dependent quantities cancel each other out by calculating the ratio.

3.3. Light output enhancement

A primary trigger may cause a chain of secondary events. To describe the probability distribution at a given overvoltage, a branching Poisson process stated in equation 6 without any SiPM saturation is considered [36].

$$\lambda =-\text{ln}(1-p)$$ (6)

For a non-random primary trigger (eg. dark count, N=1), the event distribution for multiple triggers $k=1,2,\dots$ is expressed according to equation 7, while for a random primary trigger (eg. light injection) with mean number $\mu$ of detected photons for multiple triggers $(k=0,1,2,..)$ based on equation 8 [36].

$$P_{(X=k)}=(\lambda ·k{)}^{k-1}·\frac{\text{exp}(-k·\lambda )}{k!}$$ (7)

$$P_{(X=k)}=(\mu +\lambda ·k{)}^{k-1}·\frac{\text{exp}(-\mu -k·\lambda )}{k!}$$ (8)

In both cases, having crosstalk $(\lambda >0)$ yields to an increase of the mean number of total triggers based on equation 9, with $\mu =1$ in the case of a single primary trigger.

$$N_{\mathit{\text{total}}}=\frac{\mu }{1-\lambda }$$ (9)

For a given number $\mu$ of impinging photons on the photodetector, the resulting signal response depends on the detection efficiency (for scintillation light), the SiPM gain $G$ (which linearly scales with the excess voltage, see figure 2), and the excess photon yield coming from SiPM crosstalk described in equation 9 and can be expressed according to equation 10.

$$\mathit{\text{Signal}}\approx PDE·G·\frac{\mu }{1-\lambda }$$ (10)

Combining equations 5, 6 with equation 10 while substituting the gain with the excess voltage, the signal response in the presence of SiPM crosstalk is expressed according to equation 11, where $S_0$ is a constant factor (which also includes $\mu$) derived from measurements at very low excess voltage where crosstalk is negligible.

$${\mathit{Signal}}_{\left(OV\right)}=S_0·PD{E}_{\left(OV\right)}·V_{OV}·\frac{1}{1+\text{ln}\left(1-P_{off,\left(OV\right)}-c^2·V_{OV}^2\right)}$$ (11)

For small excess voltage the decimal fraction can be approximated with a Taylor expansion to equation 12, which better represents the impact of the second SiPM using the previously introduced constant $c$.

$${\mathit{\text{Signal}}}_{\left(OV\right)}\approx S_0·PD{E}_{\left(OV\right)}·V_{OV}·\left(1+P_{off,\left(OV\right)}+c^2·V_{OV}^2\right)$$ (12)

4. Experimental results

4.1. SiPM noise enhancement

The measured staircase plot for three selected SiPM bias voltages for all three configurations is shown in figure 3. At the level of 1 SPAD, the dark count rate for bare and off is identical, while there is a noticeable difference above 1 SPAD (higher threshold values). At the bias voltage of 34 V, the DCR among the three measurement conditions fluctuates by less than 1 %, which in reverse [4] means the temperature fluctuation among different measurements was less than $0.2°C$. We observe a degradation in photon counting capability (the steps are more smeared) for the on-configuration already visible at 42 V from the transition from 2–3 triggered cells (between 150 mV to 200 mV).

Figure 3:

Measured dark count rate as function of the selected threshold to calculate the number of crossings within $200\mu \text{s}$. The steps corresponding to a dark count without crosstalk and a dark count with one or two additional crosstalk events are very well visible.

Figure 4 shows the extracted dark count rate and crosstalk probability as function of the SiPM bias voltage. In line with expectations from sections 3.1 and 3.2, switching the second SiPM on yields an increased dark count rate and additional crosstalk probability. The DCR and crosstalk probability for the bare SiPM as function of the bias voltage are in line with the specifications from the datasheet.

Figure 4:

Dark count rate (left) and crosstalk probability (right) as function of the SiPM bias voltage for the three measurement conditions.

These measurements allow extraction of the constant $c$ for different SiPM bias voltages by drawing the difference of the DCR or crosstalk probability between on and off (done in figure 5). Since the approximation on the constant weighted detection efficiency for infrared avalanche photons is not valid at low excess voltage, only measurements above 36 V bias voltage for the dark count rate, and above 38 V bias voltage for the crosstalk were considered. The latter is due to the fact that the PDE is squared in equation 5, hence deviations from the constant approximation have a stronger impact. In figure 5 the breakdown voltage was kept as an open fitting parameter to focus on the linear relationship and determination of the constant $c$ for both the dark count rate and crosstalk enhancement. For this particular detector configuration (20 mm thick BGO crystal, wrapped in Teflon and coupled on both crystal ends with Meltmount), the constant is determined to be $c=3.4\pm 0.1\%$.

Figure 5:

Normalized dark count rate and square root of the crosstalk probability as function of the applied SiPM bias voltage. For both quantities a fit function of the form $y=c·\left(V_{\mathit{\text{bias}}}-V_{\mathit{\text{break}}}\right)$ was used with $c$ and $V_{\mathit{\text{break}}}$ as free fit parameter.

It is subject of ongoing research to determine whether there is an underlying physics reason for the small discrepancy of resulting breakdown voltage of about 1 V (≈ 3 standard deviations from each other, both within the specifications of the sensor according to the datasheet) based on the dark count rate and crosstalk dataset. Different breakdown voltage determination methods exist in the literature which yield differences of up to 1 V as well [4, 37].

4.2. Light output and energy resolution

The left of figure 6 shows the measured light output measured at 44 V bias voltage. The histogram is in semi-logarithmic scale to also show the peak originating from the 1.27 MeV gamma-ray. As evident from the figure, the signal area is almost double when the second SiPM is on. The 511 keV photopeak position as function of the SiPM bias voltage is shown on the right of figure 6 in red (off) and blue (on), respectively. Additionally in black we show the expected signal from equation 11 using the PDE at 500 nm reported in [9] (figure 2) and assuming a breakdown voltage of 32.9 V and a value of $c=3.4\%$ (figure 5). Within the precision of the measurement, the model matches the experimental data very well and accurately explains the significant enhancement on the number of detected photons. The energy resolution is shown as dotted lines for different SiPM bias voltages on the right of figure 6. The interpretation for the dependency of the energy resolution on the overvoltage for both configurations is discussed in section 5.2

Figure 6:

(Left:) Measured signal area upon ²²Na excitation with the second SiPM off and on. A Gaussian fit function (black) around the 511 keV peak is applied to determine the position and the energy resolution. No relevant SiPM saturation is observed. (Right:) 511 keV photopeak position (solid lines) as function of the SiPM bias voltage with the second SiPM off (red), on (blue) and the model from equation 11 (black). The dotted lines (right y-axis) represent the measured energy resolution at 511 keV.

4.3. Coincidence time resolution

The coincidence time delay distribution was measured at three different SiPM bias voltages, with the front SiPM being off or on. The fit parameters and time resolution (FWHM / FWTM) are summarized for all the measurements in table 1. The distribution at the highest excess voltage is shown in figure 7 in a semi-logarithmic scale to better illustrate the different shape of the two distributions. The CTR in terms of FWHM improves $\left({\text{FWHM}}_{\mathit{\text{off}}}=282\pm 6\text{ps}\right)$ with the second SiPM not biased, compared to the on-configuration $\left({\text{FWHM}}_{\mathit{\text{off}}}=342\pm 7\text{ps}\right)$. Considering the tails, both distributions have a similar width of ${\text{FWTM}}_{\mathit{\text{off}}}=1372\pm 29\text{ps}$ and ${\text{FWTM}}_{\mathit{\text{on}}}=1344\pm 28\text{ps}$ indicating that the shape of the distributions is impacted by the second SiPM. The GE CTR is also comparable, but systematically (for different leading edge thresholds, different SiPM bias voltages, different energy cuts) always slightly better when the front SiPM is switched on.

Table 1.

Measured coincidence time resolution and abundance of the fast Gaussian component for different SiPM bias voltages with the second SiPM either off or on. The measurement error on the time resolution is about 3 %.

	38 V		42 V		47 V
	off	on	off	on	off	on
FWHM [ps]	499	551	305	352	282	342
FWTM [ps]	2493	2480	1538	1498	1372	1344
GE [ps]	1239	1214	763	750	688	677
fast [ps]	400	438	259	287	243	282
slow [ps]	2124	2074	1531	1308	1459	1204
${\text{r}}_{\text{fast}}[\%]$	35.0	36.4	40.1	40.0	42.4	43.0

Figure 7:

Coincidence time delay distribution of 20 mm thick BGO in dual-ended readout using only the back SiPMs and having the front SiPM switched off (left) or switched on (right). The dotted lines represent the individual Gaussian fit components and the blue solid line is the resulting fit function.

5. Discussion

The aim of this study was to evaluate the impact of crosstalk between two SiPMs coupled at opposed ends of a scintillation crystal. We have used the information of one SiPM to calculate performance metrics such as energy resolution or coincidence time resolution (in coincidence with another identical detector). Combining the information of both SiPMs for the on-configuration improves the performance metrics. For instance, with adequate correction methods, a CTR of 234 ps FWHM (386 ps GE) has been reported in [16]. However, this does not allow to quantify the role of enhanced SiPM noise, since then there is no off-configuration available to compare with.

Further, energy and timing resolution were determined with BGO, which is a scintillator with relatively low light yield and slow scintillation decay time. It is unclear to which extent crosstalk can impact detectors with a bright and fast scintillation emission. Accurate quantification and management of crosstalk photons is important when optimizing a specific detector configuration with multiple SiPMs. For the best performance, strategies that mitigate their impact should be employed.

5.1. Avalanche photon transfer probability

The SiPM noise enhancement can be characterized with the constant $c$, which is a single parameter which depends on the specific detector configuration. The models in equations 4, 5 and 11 describes very well the measured dark count rate enhancement, increase of crosstalk probability, as well as light output gain. In this study only one detector configuration (dual-ended readout with a 20 mm thick BGO crystal) was investigated. However, the models can be applied to other detector configurations where SiPMs may interact with each other, such as: (i) dual-ended readout with different scintillators or SiPMs, including high energy physics experiments such as the CMS barrel MIP timing detector [38]; (ii) single-ended readout with SiPMs and crystal arrays (the crosstalk of scintillation light has been studied in [39]); (iii) SiPM readout of (semi-)monolithic crystals, including photon counting applications [40]; (iv) or dark matter and neutrino experiments with a huge sensitive volume (liquid xenon, ice cube) surrounded by SiPMs [41, 42].

In the manuscript, we experimentally determined the constant, which also includes the detection detection probability for avalanche photons. A limitation of this work arises from the assumed constant detection efficiency, which is a accurate approximation for high overvoltage, but needs to be adjusted for low overvoltage (see figure 5). This requires an evaluation of the spectral distribution of SiPM crosstalk. Also, the avalanche photon transfer probability for both SiPMs was identical (or assumed to be identical). Especially edge- and corner channels in SiPM arrays have typically worse performance due to light leak, suggesting the transfer probability among different pixels is geometry dependent and not constant. For a full system, it might be sufficient to determine the performance characteristics of a single SiPM and simulate the geometric contributions of the particular detector configuration. Additionally, the different crosstalk probability between the bare-configuration and the others can be a calibration tool, for instance to evaluate the quality of the crystal coupling [43].

5.2. Impact of SiPM crosstalk on the energy resolution

Based on literature, we expect a worsening of the energy resolution for increased SiPM noise [4, 36, 44], as when the second SiPM is also biased and the crosstalk probability increases. In the case of dual-ended readout, however, the opposite is the case for most bias voltages as shown on the right of figure 6 for the dotted lines representing the obtained energy resolution. Despite the crosstalk probability increase having the second SiPM switched on, the measured energy resolution is better with the largest difference of $E{R}_{\mathit{\text{off}},44V}=32.4\pm 1.0\%$ and $E{R}_{\mathit{\text{on}},44V}=27.6\pm 1.0\%$. An explanation for this behavior is the fact that crosstalk photons coming from the second SiPM still carry information, since their primary seed (impinging on the second SiPM) is not used beforehand. This means, by switching the second SiPM on, it acts as a (poor) reflector by detecting scintillation light, triggering avalanche photons and re-emitting them toward the first SiPM. Therefore, at low overvoltage, no difference in the energy resolution is observed (because the crosstalk probability is too small) and with increased overvoltage the excess noise increases, but simultaneously the amount of “reflected” good photons also increases. Only when approaching a bias voltage above ≈ 46 V the energy resolution deteriorates and the high number of avalanche photons worsen the resolution. Hence, at very high overvoltage, the energy resolution worsens in line with literature for single-ended readout, while at low-to-medium excess voltage the better statistics counteracts SiPM noise.

The measured energy resolution by itself is not optimal and energy resolution values much better than ≈ 30% have been reported [45–47]. However, for our measurement the opposite crystal end was also coupled to a SiPM leading to severe light loss.

5.3. Impact of SiPM crosstalk on the CTR

Similar to the energy resolution, the measured time resolution should become worse for an increase of SiPM noise and crosstalk probability [48]. Though, the same reasoning on the restoration of information as in section 5.2 can also be made for the time delay distribution. About 18 Cherenkov photons are produced in BGO after a 511 keV gamma-ray photoelectric interaction [49, 50]. For the approximation of an isotropic emission for Cherenkov photons and considering the weighted SiPM detection efficiency over the Cherenkov emission, on average 1.4 Cherenkov photons are detected in a dual-ended configuration each on the front and back SiPM [15, 16]. This means there are large fluctuations and sometimes no Cherenkov photons are detected at all explaining the pronounced tail in the CTR distribution. If the front SiPM is active, Cherenkov photons may be detected at this SiPM, trigger an avalanche and produce avalanche photons. These photons, acting like the “ghost” of the primary Cherenkov photon, may traverse along the crystal and are detected by the back SiPM. In this case, the path length and thus the variance of the primary + “ghost” Cherenkov photon is larger as if the primary photon would be directly detected by the back SiPM, explaining the broadening of the narrow Gaussian from ≈ 280 ps to ≈ 340 ps. Additionally, the SiPM jitter is at least double since first the primary photon need to trigger an avalanche, to generate its “ghost” photon. Though, with all this delay, the Cherenkov “ghost” photon still arrives before scintillation light is produced, and therefore it reduces the tail in the distribution.

Similarly, if no Cherenkov photon is detected, the first produced scintillation photon might be detected by the front SiPM and the “ghost” of this photon is later detected in the back SiPM before any other real scintillation photon is produced/detected. While there is an additional variance on the photon detection time, this variance is smaller than the photostatistics term.

Another observation on the experimental CTR data is the impact of the rise time correction to improve the timing performance. We use the time difference between the signal crossing 20 mV and 100 mV to monitor fluctuations on the number of detected Cherenkov photons. While the correction improves the data in both cases, the improvement is more pronounced with the second SiPM off, with the GE CTR improving from 740 ps before correction to 688 ps after correction (off), compared to 696 ps to 677 ps (on). The quadratic difference on the improvement reduces from 272 ps (off) to only 162 ps (on). The most likely reason for this is that crosstalk photons from the other SiPM dilute the relationship between signal rise time and number of detected Cherenkov photons, as crosstalk photons have more time to arrive on the SiPM before the signal crosses the high leading edge threshold.

5.4. Potential SiPM crosstalk impact on the localization of the gamma-ray interaction

Depth-of-interaction determination in HAR crystals [45, 51], or 3D localization of the gamma-ray interaction point in monolithic crystals [52] rely on the distribution of the number of detected photons in different photodetectors. Positioning algorithms assume that optical photons are produced within a narrow $\left(𝒪100\mu \text{m}\right)$ region. With SiPM crosstalk, additional optical photons are produced for each photodetector and SiPMs detecting more scintillation emit proportionally more avalanche crosstalk photons. Those photons have a different origin, optical path, and light attenuation with respect to the primary scintillation photons, which will deteriorate the positioning estimation at high excess voltage. Validating the hypothesis is subject of ongoing investigation and beyond the scope of the current manuscript.

6. Conclusion

The quantification and importance of SiPM crosstalk in a dual-ended readout configuration was investigated. Already at low excess voltage, DCR and crosstalk probability increases due to the operation of the second SiPM. At high excess voltage, enhanced crosstalk probability is about 2-fold. The role of SiPM-induced crosstalk has been modeled using a single constant which combines the geometric configuration and the detection probability for avalanche photons and reproduces the experimental relationship. A non-intuitive energy resolution and time resolution improvement for certain bias voltages has been observed which can be described by crosstalk photons carrying information on the photostatistics and production time. The study introduces and validates a theoretical framework which can have implications in various applications where multiple SiPMs are used.

Highlights.

• Silicon photomultiplier dark count rate (DCR) and crosstalk probability $\left({\text{P}}_{XT}\right)$ is amplified in a dual-ended readout configuration compared to a single-ended one.

• DCR and ${\text{P}}_{XT}$ enhancement and number of triggered photoelectrons as function of the SiPM excess bias voltage can be analytically predicted.

• Crosstalk in multi-SiPM detector configurations significantly impacts the achievable performance.

Biographies

Dr. Nicolaus Kratochwil graduated in physics in 2018 and obtained a PhD in Natural Sciences in 2023, both at the University of Vienna, Austria. From 2018 to 2023 he worked as Doctoral Student and later as Senior Fellow Researcher at CERN, Switzerland. Since 2023 he works as Postdoctoral Scholar at the Department of Biomedical Engineering at UC Davis, USA. His research interests includes light-based radiation detectors for medical imaging instruments with a strong emphasis on fast timing with silicon photomultipliers and Cherenkov photons.

Dr. Emilie Roncali obtained her PhD in Biomedical Engineering at the Ecole Centrale de Paris, France in 2008. She has a joint Faculty appointment as Associate Professor with the Departments of Biomedical Engineering and Radiology at UC Davis, USA. Her research focuses on development on personalized dosimetry for radiopharmaceutical therapy and development of optical models for Monte Carlo simulation for light-based radiation instruments.

Dr. Joshua W. Cates completed his Ph.D in Nuclear Engineering at the University of Tennessee, USA in 2013. Currently he is a Staff Scientist at Lawrence Berkeley National Laboratory, where he works on projects involving radiation detection, molecular imaging and nuclear security applications. He is interested in improving time-of-flight PET systems with enhanced detector sensitivity and detector development for medical imaging.

Dr. Gerard Ariño-Estrada obtained his PhD from the Universitat Autonoma de Barcelona, Spain in 2015. He joined the department of Biomedical Engineering at UC Davis in 2016 where he worked as Postdoc, Assistant Project Scientist and now as Associate Research Professor. Since 2023 he also leads the CHLOE-PET project at IFAE, Spain to develop a new detector module for TOF-PET systems with Cherenkov light.