The Crilin calorimeter prototype used for the beam test (Proto-0) was developed in the ambit of the Muon Collider experiment [3]. Proto-0 houses two 10 × 10 × 40 mm³ crystals. The mechanical structure was realised via fused-deposition modelling in acrylonitrile styrene acrylate (ASA) with an overall size of 61 × 40 × 44 mm³ (Figure 1).

The baseline choice of crystal for the Crilin calorimeter is lead fluoride, PbF₂. PbF₂ is a Cherenkov crystal [4] offering intrinsically fast emission, in line with the aforementioned stringent timing requirements. Alternative crystal choices are also under investigation, such as a recent formulation of lead tungstate with ultra-fast emission [5], now commercially available from Crytur [6] as PWO-UF. This material features high density, good light yield, high radiation resistance and fast response speed by combining the prompt Cherenkov emission with a fast scintillation component, yielding a dominant emission with a decay time τ < 0.7 ns.

Both PbF₂ and PWO-UF crystals were employed during the beam test. Table 1 summarises the properties of these crystals. For the beam test, the crystals were wrapped in 100-μm-thick aluminized Mylar foil and tested one at a time in dedicated runs. Each crystal was readout by a 2 × 2 matrix of 3 × 3 mm² Hamamatsu S14160-3010 PS SMD silicon photomultipliers [7], with 10-μm pixel size, mounted on a dedicated PCB (SiPM board). The crystals were optically coupled to the SiPMs by direct contact without the use of optical grease. The left and right sides of the crystal were each read out by a pair of SiPMs connected in series, providing two independent readout channels for each crystal (see inset in Figure 3). The signals were transmitted from the SiPMs to the FEE board via micro-coaxial transmission lines. For the test, a two-channel prototype version of the Crilin front-end electronics (FEE) was used. On the FEE board, after proper termination, the pulses were processed first by a high-speed, non-inverting amplification stage with gain 4. The first stage output drove a pole-zero cancellation circuit, followed by a second, non-inverting stage (with gain 2) to drive the digitisation section. The FEE circuit has a dynamic range of 2 V and an overall gain of 8. External HV supplies were used for biasing.

All measurements were carried out in the H2 beamline at the CERN SPS with a 120-GeV electron beam and the setup illustrated in Figure 2. The trigger was obtained from the coincidence of the two scintillator counters, S1 and S2. The beam was tracked with a beam telescope consisting of two stations of two 9.5 × 9.5 cm² planes of silicon-microstrip tracking detectors, C1 and C2, spaced 15.4 m apart. Each single-sided tracking plane was 410 μm thick and had a spatial resolution of 47 μm, yielding an angular resolution for beam particles of about 3 μrad. Single particle event selection was performed by rejecting multi-cluster hits in the tracking detectors.

The prototype was placed on a 4-axis motorised stage, with 2 axes of rotation and 2 axes of translation, for alignment (Figure 2, bottom). Two different data-taking configurations were employed.

• forward orientation, with the beam incident on the upstream face of the crystal and the SiPMs downstream facing upstream (front incidence);

SiPM signals from the two readout channels of Proto-0 were sampled at 5 GS/s using a CAEN V1742 switched-capacitor digitiser. For PWO-UF runs, as a consequence of the higher light yield, a 6 dB attenuator was placed before the digitiser inputs to halve the signal amplitude, due to the maximum 1 V dynamic range of the V1742 (in contrast to the 2 V output dynamic range of the FEE). The charge and amplitude values for PWO-UF reported in the text already account for the presence of the attenuator and are scaled to represent the true values output by the FEE system. The SiPM pairs in series were biased at 83.5 V, which corresponds to a 3.75 V overvoltage for each photosensor (V _br = 38 V).

Pulse charges were evaluated by integrating each waveform over the range [T _peak-20 ns, T _peak+140 ns], where T_peak represents the waveform peak time, and dividing by the 50 Ω input impedance of the digitiser. Offline equalisation was carried out on the residual minor imbalances in charge between the two readout channels, less than 5% for PbF₂ and less than 2% for PWO-UF, due to small non-uniformities in SiPM gains and optical couplings. For the PbF₂ runs, events with signal of at least 50 pC on both readout channels (80 pC for PWO-UF) were selected, and fiducial cuts were made on the extrapolated position of the beam particle at the crystal face.

The pulse timing was evaluated using a waveform template fit procedure. SiPM pulse templates are sets of nodes with polynomial interpolation and fixed proportions, which can be fit to each waveform using a three-parameter optimisation. For each channel, waveform templates were generated by aligning and averaging signals from a large dataset of hits: for each sampled waveform, a pseudo-timing was extracted by applying a polynomial spline interpolation to the rising edge and peak, using a constant fraction technique (CF) applied to the spline function (Figure 4, left). The CF value employed for reconstruction was 12% of the peak amplitude, optimised by minimising the timing resolution, as shown in Figure 5 (top left). Finally, using the pseudo-timing information, all processed waveforms were aligned and, after proper normalisation, averaged into wave templates. A comparison between the waveform templates for PbF₂ and PWO-UF crystals is shown in the right panel of Figure 4: a sharper rising edge and narrower pulse shape is observed for PbF₂ due to different light generation and transport dynamics, as discussed later.

To reconstruct the pulse timing, templates were fitted to the rising edge using a three-parameter minimisation (scale, baseline and time offset) over the range [T _peak − 20 ns; T _peak − 2 ns]. An example of the application of the template fit is shown in Figure 4, left. The fit range bounds were optimised by minimising the timing resolution, as before. It should be noted that this reconstruction procedure does not introduce any significant time-amplitude slewing, so that no correction was necessary in data. The fitting procedure resulted in the normalised χ ² distribution for the fitted waveforms shown in Figure 5, bottom right. Pulses used for the analysis were required to have χ ² < 30. It was verified that the cut on χ ² and choice of fit range did not introduce any significant bias with respect to particle hit position, as seen in Figure 5, bottom left; such a bias might be expected due to the variation of the waveform shape as a function of particle hit position, which will be discussed in Section 4. Similarly, it was verified that the aforementioned selection cuts did not result in any bias with respect to pulse amplitude and timing. To ensure that the reconstructed timing information was free of any significant bias with respect to the digitiser sampling frequency, the plot in Figure 5, top right, was produced, which shows that the distribution of the reconstructed time modulo the digitiser sampling period is flat.

For PbF₂ crystals, which represent the baseline choice for the Crilin design, a detailed simulation was also implemented to study the optical transport of Cherenkov photons. The relevant optical properties and surfaces were simulated. In particular, a dielectric-dielectric optical boundary between the PbF₂ crystal and Mylar wrapping was implemented, based on the LUT model [9]. The interface between the crystal and the four SiPM packages, made of silicone resin, was simulated using a polished dielectric-dielectric boundary (UNIFIED model). As shown in Figure 6, four 3 × 3 mm² active regions made of silicon were used to reproduce the active areas of the SiPMs. Sensitive detectors attached to the four silicon regions were used to score the energy, position, and timing of optical photon hits.

Optical photons arriving on the sensitive detector volumes representing the SiPMs for each readout channel were counted and used to simulate the corresponding signal waveform (Figure 6). To evaluate the number of detected photoelectrons, optical photon hits were weighted offline based on the spectral response of the photodetector, which has a peak PDE (photon detection efficiency) of 18% at 450 nm [7]).

For each simulated event and each readout channel, a SiPM pulse template representing the contribution of each individual pixel (single photoelectron response) was convoluted with the arrival times of optical photons over an interval of [−1, +100] ns with respect to the particle hit time on the crystal surface. The resulting pseudo-waveforms were fitted using the template method discussed above (Section 2.2) to extract timing information.

To evaluate the prototype response in terms of output charge per unit of deposited energy, mean-charge distributions for events with the track incident within a square 5 × 5 mm² fiducial region centred on the front face of the crystal were compared to the analogous distributions from Geant4 simulations carried out using a planar, 120-GeV electron source of the same dimensions. For both types of crystals, a total of 10⁵ events were generated by resampling x-y beam positions from the ones actually tracked during the test beam. The MC energy histogram showed a most probable energy deposit of about 4.9 GeV for PbF₂ and 5.8 GeV for PWO-UF. For all runs, the histogram of the deposited energy distribution from the MC was fitted to that for data using normalisation and scale parameters. From the fit procedure, the scale factors 29, 36, 67 and 77 pC/GeV were obtained for the cases of PbF₂ back, PbF₂ front, PWO-UF back, and PWO-UF front, respectively. After fitting, the data-MC consistency in shape was checked using a Kolmorogov-Smirnov test, resulting in a p-value > 0.5 for all PbF₂ runs and > 0.3 for all PWO-UF runs. An example of the data-MC overlay for PbF₂ is shown in Figure 7, where the range 50–300 pC was used for the MC shape fit. A comparison between the energy scale factors for PbF₂ and PWO-UF is shown in Figure 7, along with the most probable values (MPV) and sigma values for the respective deposited energy distributions, as obtained from fits with a Landau distribution convoluted with a Gaussian resolution function.

Once the scale factors have been determined, an estimate of the light yield can be derived from the knowledge of the SiPM gain. In particular, the SiPMs used have a nominal gain of 1.8 × 10⁵ at V _op, as previously characterised [3]. Accounting for the charge gain of the amplifier in the FEE, the light yield values 0.26, 0.30, 0.58, 0.67 p.e./MeV were obtained for the cases of PbF₂ back, PbF₂ front, PWO-UF back, and PWO-UF front, respectively. For comparison, the simulation gives a light yield of 0.38 p. e./MeV for the case of PbF₂ in front configuration, after weighting according to the PDE of the SiPMs.

For runs carried out in the front configuration, the waveform shape, along with the charge and timing distributions, presented some variation as a function of the particle hit position on the crystal. This behaviour is assumed to be associated with light transport effects inside the crystal that give rise to asymmetries in the light collected by the SiPMs for each of the two readout channels. These asymmetries are ultimately reflected in the apparent light yield and signal timing for each channel, as discussed in the following.

For all experimental configurations, the distribution of the time difference between the two readout channels ΔT = T ₁ − T ₀ was used to study the time resolution of the system as a function of deposited energy. A 0.8 × 0.8 cm² fiducial cut centered on the crystal face was applied for all runs. The distribution of ΔT as a function of deposited energy E _dep for PbF₂ runs is shown in Figure 11. The value of E _dep is obtained from the mean of the charge values from both SiPMs, using the scaling factors discussed in Section 4.1. As shown in Figure 11, top left, for front-configuration runs, this distribution is split into two populations due to the position-dependent light transport effects described in Section 4.2. In order to evaluate the timing resolution in the front configuration, a correction was developed based on the dependence of ΔT on the charge asymmetry variable A = (Q ₁ − Q ₀)/(Q ₁ + Q ₀), without relying directly on any information derived from the tracking system, as shown in Figure 11, top-right. Approximating the ΔT-A relationship with a straight line, a linear fit yields a slope of about −8 ns for PbF₂ (about −4 ns for PWO-UF).

In order to correct the position dependence of the ΔT distribution, a spline function was fitted to the profile of the ΔT-A distribution and used to obtain an event-by-event correction for the timing offset due to positional effects. The corrected ΔT vs. E _dep histograms were then filled, as shown for PbF₂ in Figure 11, bottom left. The σ T 1 − T 0 time resolution could then be evaluated from the sigma of a Gaussian distribution fit to the corrected ΔT distributions for slices of E _dep. No correction was applied for runs carried out in back configuration. The results are summarised in Figure 12 for all runs, where the time resolution as a function of deposited energy was fitted using the function σ MT = σ T 1 − T 0 / 2 = a / E dep ⊕ b , where the subscript MT refers to the resolution obtained for the mean time for the two readout channels of the single calorimeter cell. Even after the correction, the residual position-dependent light-transport effects spoil the timing performance for the front configuration, despite the generally higher light yields. Due to the purely Cherenkov nature of the light emission from PbF₂  for both configurations, the time resolution for PbF₂ is better than that for PWO-UF, despite the fact that the light yield for PbF₂ is only about half of that for PWO-UF.

For runs in the front configuration, it should be noted that, to correctly account for the charge imbalance between the two readout channels (in the worst-case, ±8% for PbF₂), the time resolution should ideally be modelled as 2σ _MT = σ _T (E ₀) ⊕ σ _T (E ₁) instead of 2 σ MT = σ T 1 − T 0 ( E ) , where the subscript MT refers to the mean-time resolution, E ≡ (E ₀ + E ₁)/2, and σ T = σ T 0 = σ T 1 represents the resolution as a function of energy of a single readout channel (assumed to be identical). By generating trial time distributions in a toy-MC simulation assuming σ T ( E ) = σ T 1 − T 0 ( E ) / 2 , it was verified that the worst-case discrepancy in the fit parameters obtained with the σ _T (E) fit model is O(1%) for either PbF₂ or PWO-UF in the energy range of interest.