EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-LHCb-DP-2021-005 November 23, 2021

Characterisation of signal-induced noise in Hamamatsu R11265 Multianode
Photomultiplier Tubes

M. Andreotti1, S. Capelli2,a, G. Cavallero3,∗, S. Chiozzi1, A. Cotta
Ramusino1, C. D’Ambrosio3, M. Fiorini1,b, E. Franzoso1,b, C. Frei3, S.
Gallorini†, S. Gambetta4,3, C. Giugliano1,b, C. Gotti2, T. Gys3, F. Keizer3,
M. Maino††, B. Malecki3, L. Minzoni†††, S. Mitchell4, I. Neri1, A.
Petrolini5,c, D. Piedigrossi3, G. Robertson4, A. Sergi5,c, G. Simi6, I.
Slazyk1,b, M. Smith7, J. Webster4, S.A. Wotton8.

1INFN Sezione di Ferrara, Ferrara, Italy
2INFN Sezione di Milano-Bicocca, Milano, Italy
3European Organization for Nuclear Research (CERN), Geneva, Switzerland
4School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
5INFN Sezione di Genova, Genova, Italy
6Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
7Imperial College London, London, United Kingdom
8Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
aUniversità di Milano Bicocca, Milano, Italy
bUniversità di Ferrara, Ferrara, Italy
cUniversità di Genova, Genova, Italy
† Formerly at 6
†† Formerly at 2,a
††† Formerly at 1,b
∗Corresponding author

Signal-induced noise is observed in Hamamatsu R11265 Multianode
Photomultiplier Tubes, manifesting up to several microseconds after the single
photoelectron response signal and localised in specific anodes. The mean
number of noise pulses varies between devices, and shows significant
dependence on the applied high-voltage. The characterisation of this noise and
the mitigation strategies to perform optimal single-photon counting at 40 MHz,
as required by the LHCb Ring-Imaging Cherenkov detectors, are reported.

Published in JINST 16 (2021) P11030

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

The LHCb experiment [1] at the Large Hadron Collider (LHC) is designed to
search for New Physics beyond the Standard Model in the study of $C\\!P$
violating processes and rare $b$\- and $c$-hadron decays. The excellent
performance of the LHCb detector during LHC Run 1 and 2 allowed to extend the
core physics programme to include additional flavour-physics measurements,
such as lepton flavour universality tests and conventional and exotic hadron
spectroscopy [2].

Two Ring-Imaging Cherenkov (RICH) detectors [3] provide long-lived charged
hadron discrimination in the momentum range 2-100
$\text{\,Ge\kern-1.00006ptV\\!/}c$. This particle identification (PID)
capability allows to reduce the combinatorial background and to discriminate
between decay modes of otherwise identical topology: the RICH detectors are
therefore crucial to fulfil the flavour-physics programme of LHCb [3].

The LHCb experiment is undergoing a major upgrade [4] to enable operation at
an instantaneous luminosity of $2\times
10^{33}\text{\,cm}^{-2}\text{\,s}^{-1}$ during the LHC Run 3 which will start
in 2022. To fully exploit the flavour-physics potential with this five-fold
increase in the instantaneous luminosity, the whole LHCb detector will be read
out at the full 40 MHz LHC bunch crossing rate and a flexible full software
trigger [5] will be implemented. The Hybrid Photon Detectors (HPDs) used in
the RICH system during LHC Run 1 and 2, with embedded readout electronics
working at 1 MHz trigger rate [1], will be replaced with Multianode
Photomultiplier Tubes (MaPMTs) and new external Front-End (FE) electronics
[6], able to provide single photon counting capability at the 40 MHz bunch
crossing rate. The high-rate and high-occupancy environment imposes strict
requirements on the properties of the photon detectors, including the demand
for a low count rate caused by dark noise and other sources of internal
instrumental noise. The opto-electronics chain that will be used in the
upgraded LHCb RICH detectors is described in Sec. 2.

During the last year of data-taking before the Long Shutdown 2, an upgrade
photon detector module prototype was installed behind the RICH2 HPD plane, in
order to evaluate the behaviour in the LHC environment (see Fig. 1). The
module was integrated and operated together within the LHCb detector
framework, including services, data acquisition (DAQ) and control systems.
During these tests, out-of-time hits, i.e. delayed with respect to the
expected arrival time of Cherenkov photons relative to a $pp$ collision, were
detected, indicating the presence of an unexpected source of noise. The
observation of this noise has been made possible by acquiring data in an LHC
collision scheme with isolated bunches, and with a 3$\,\upmu\text{s}$-wide
acquisition time window, with the aim to synchronise the module with the LHC
clock.

Figure 1: 1: picture of a photon detector module prototype installed in the
LHCb cavern during the 2018 operations. 1: schematic view of the setup. The
photon detector module, integrating sixteen MaPMTs, is installed behind the
HPD plane. Given the HPD staggered configuration, Cherenkov photons reach
MaPMTs with the relative intensity reported in the figure.

These noise pulses have been further investigated in the laboratory by means
of different experimental setups and light sources, as described in Sec. 3.
The characterisation of these signal-correlated events is described in Sec. 4,
and a model to estimate the amount of pile-up effects at high rates is
developed and validated in Sec. 5. Mitigation strategies have been extensively
studied, and the most effective solutions are reported in Sec. 6. Finally the
impact of this noise on the PID performance of the RICH detectors is assessed
using the LHCb simulation framework as described in Sec. 7.

##  2 The LHCb RICH opto-electronics chain for LHC Run 3

The detection of Cherenkov light requires single-photon counting capability in
the UV and visible wavelength ranges and at the LHC bunch crossing rate of 40
MHz. In addition, given the large event multiplicities, position-sensitive
photon detectors are necessary to achieve the required Cherenkov angle
resolution, distinguish Cherenkov rings close in space and to assign each
reconstructed ring pattern to the corresponding charged particle track.

A custom version of Hamamatsu R11265 MaPMTs (namely R13742) were selected as
photon detectors for RICH1 and the central region of RICH2, where the expected
occupancy is up to the 30%.111A custom version of Hamamatsu R12699 (namely
R13743) with a coarser granularity are used in the outer region of RICH2 where
the occupancy is about one order of magnitude lower. The noise characterised
in this paper is largely absent in such MaPMTs. These tubes are
26.2$\times$26.2$\text{\,mm}^{2}$ devices composed by $8\times 8$ pixels, each
of size 2.9$\times$2.9$\text{\,mm}^{2}$, and are displayed in Fig. 2. About
3000 tubes have been qualified to fulfil several requirements such as an
average gain (across all pixels) larger than ${10^{6}}$, a maximum gain
variation of 1:4 among pixels, and a dark count rate lower than 2.5
kHz/$\text{\,cm}^{2}$. The difference between R13742 MaPMTs and R11265 MaPMTs
stands on such technical specifications, while other relevant changes are not
present in the internal structure of the photon detectors.

(a)(b)

Figure 2:  (a) The MaPMTs selected for the Run 3 RICH detectors with the
2-inches R13743 on the left and the 1-inch R11265 on the right. (b) Scheme of
the electrode structure for the metal channel dynodes and the electron
trajectories [7].

The current pulses from the MaPMT anodes are read out by a fast, radiation
hard, low power consumption, 8-channel ASIC named CLARO [8]. Each CLARO
channel contains an amplifier and a discriminator with programmable threshold.

Four MaPMTs and the associated FE electronics are grouped into a unit called
Elementary Cell (EC), whose schematic view is reproduced in Fig. 3. The MaPMTs
are housed by a baseboard, providing the bias to the photon detectors through
four custom voltage dividers, and routing the signals from the anodes to the
CLARO inputs. Four FE Boards (FEBs) are plugged into the rear side of the
baseboard, each FEB hosting eight CLARO ASICs that convert the anode pulses
into binary signals. The latter are routed by a backboard to a digital board,
namely the Photon Detector Module Digital Board (PDMDB), that captures and
formats the binary outputs from the CLARO channels through three Kintex-7
Field-Programmable Gate Arrays (FPGAs). Four ECs and two PDMDBs form a Photon
Detector Module (PDM) as shown in Fig. 1.

Figure 3: Exploded view of the elementary cell. The magnetic shield protects
the MaPMTs from the effects of the residual magnetic field in the RICH1
region.

##  3 Experimental setups

Several experimental setups have been used to study and characterise the
observed noise, and to ensure that these effects are genuine and do not arise
from artefacts of the light source or DAQ system used.

As mentioned in Sec. 1 the noise events were first observed while operating a
photon detector module prototype in the LHCb environment. The light source was
the Cherenkov photons produced by charged particle tracks in the
$\mathrm{CF_{4}}$ gas, i.e. the RICH2 radiator, while the DAQ system was a
prototype of the pcie40-based back-end system that will be used in the
upgraded LHCb experiment [9]. The same readout setup is used to perform
measurements in the RICH laboratory at CERN. In the latter case, the
illumination system is based on a pulsed laser light source synchronised with
the DAQ clock and producing single photons at a rate up to 10 MHz. Analogue
measurements by probing the anode output of one MaPMT with an oscilloscope are
performed as well to exclude that the observed noise was related to
digitisation features.

The characterisation of the noise is done on the full MaPMT production by
means of a dedicated quality assurance setup allowing to acquire data sets
with large statistics. Since most of the studies described in this paper are
based on the output of such measurements, the setup is described in detail in
the following section.

###  3.1 The elementary cell quality assurance setup

Given the complexity and the large number of ECs (1056 units) needed to equip
the RICH focal planes, a Quality Assurance (QA) test setup and protocol have
been developed in order to characterise the complete EC production. The ECQA
was performed in two sites (Ferrara, Italy and Edinburgh, United Kingdom). Two
test stations were deployed in each location.

The test setup consists of a light-tight box that is internally divided in two
sections. The back section houses four Digital Boards (DBs), comprising two
ALTERA MAX10 FPGAs each. A DB hosts two 12-bit, 8-channel Analogue-to-Digital
Converters (ADC) to monitor the CLARO power consumption, together with the
current and temperature of the FEBs, baseboards and backboards. The front
section contains the ECs and is separated from the back section by a black,
light-tight, panel. The illumination system consists of an external blue LED
driver. The pulsed LED light is injected in multimode optical fibers that are
fed through in the box. Light is reflected off a flat mirror placed inside the
box, providing uniform illumination on the MaPMTs. The setup is shown in Fig.
4.

Figure 4: Schematic 4 and picture 4 of the ECQA setup. The numerical aperture
of each fibre is used to achieve a uniform illumination on the photon
detectors.

Environmental temperature and humidity that may affect the MaPMT performance
are monitored using a Raspberry Pi and are stabilised by an air-cooling fan
system. A system controller, based on the Cyclone V GT FPGA development board,
is placed outside the box. It is connected to the DBs through a Universal
Asynchronous Receiver/Transmitter (UART) interconnection module and to a PC
via a Gbit Ethernet interface. The High Voltage (HV) for the MaPMTs is
provided by an ISEG ECH 242 high voltage crate equipped with an ISEG crate
controller model CC24 Master6 and an ISEG 8-channels HV power supply board EHS
8020n.

Data acquisition is managed by control software based on Python for the low-
level communication with the hardware. A Graphical User Interface (GUI) is
implemented in National Instruments LabVIEW. The software controls the HV
crate, configures the DBs together with the CLARO ASICs and triggers the LED
driver to generate light pulses synchronous to the DAQ clock. The measurements
required to validate ECs are carried out automatically by the software and are
monitored without operator’s intervention. The procedure performs the
validation of up to 4 ECs simultaneously for each test station, which
corresponds to 16 Hamamatsu R11265 MaPMTs. Raw data are stored to disk and the
parameters characterising the behaviour of each single EC are extracted and
saved to a database.

##  4 Characterisation and results

The setup described in Sec. 3.1 is used to record the time spectrum of the
noise pulses, hereafter referred to as Signal Induced Noise (SIN) pulses. The
SIN acronym has been first introduced in the literature to identify an
analogous phenomenon observed in photomultipliers used in LIDAR experiments
[10]. The recorded time spectra are binned in 25 ns time-slots, and the total
DAQ time-window used for data analysis is 6.325$\,\upmu\text{s}$. The
repetition rate of the laser source, unless stated otherwise, is 100 kHz with
$10^{6}$ laser pulses per run. The illumination is uniform across the whole
photocathode surface. All but those measurements used to study the SIN
dependence on the HV, are performed biasing the MaPMTs at 1000 V.

###  4.1 SIN pulses time distribution and probability

A typical spectrum for a pixel affected by SIN is shown in Fig. 5. The
distribution is dominated by the primary laser signal, seen in the leftmost
part of the plot as a peak of events populating the 100–125 ns time slots
(signal slots). The events after the signal slots are due to SIN pulses.

Figure 5: Time distribution of detected events in 5 linear and 5 logarithmic
vertical-axis scale for a pixel (number 61) affected by SIN. The MaPMT under
study has serial number FB4500. The corresponding mean number of SIN pulses,
treated as a Poisson-distributed variable, is $R_{61}=1.034\pm 0.010$.

The probability of generating $n_{\text{sin}}$ noise pulses is treated as
Poisson distributed, i.e. the processes creating multiple SIN pulses from a
single photon converted at the photocathode are considered, in first
approximation, as independent [11]. In vacuum photomultipliers, these
processes are typically due to the ionisation of the residual gas inside the
tube caused by the interaction with electrons, generating ion feed-back to the
photocathode. But other sources of signal-correlated noise such as light
emission are also reported in the literature [12]. The mean number of
processes generating SIN pulses is defined as $\mu_{\text{sin}}$. The time
profile shown in Fig. 5 indicates that the number of detected SIN pulses is a
function of the time elapsed from the detected primary signal hit.

The probability $P_{m}$ to detect at least one SIN count induced by the
primary signal $m$ time slots after the laser pulse is given by the number of
counts $N_{m}$ in bin $m$ divided by the number of counts $S$ in the signal
slot. Indicating with $\bar{P}_{k}$ the probability to detect zero counts in
time slot $k$, the following identity can be written:

$\displaystyle
P_{m}=\frac{N_{m}}{S}=\left(\prod_{k=1}^{m-1}\bar{P}_{k}\right)(1-\bar{P}_{m})+\left(1-\prod_{k=1}^{m-1}\bar{P}_{k}\right)(1-\bar{P}_{m})=1-\bar{P}_{m}.$
(1)

In a first approximation, i.e. neglecting the contribution arising from dark
counts, the first term in the sum gives the probability to detect a primary
SIN pulse in slot m, while the second term describes the probability to detect
in the same slot higher-order SIN pulses originating from cascade effects.

Contributions to $P_{k}$ arise from both dark counts and SIN pulses. Dark
counts, obeying Poisson statistics and occurring with a mean number of counts
$\mu_{d}$ in a $25\text{\,ns}$ bin, contribute to $P_{m}$ independently of the
presence of signal. $\bar{P}_{\text{d}}=e^{-\mu_{\text{d}}}$ and
$\bar{P}_{\text{sin},m}=e^{-\mu_{\text{sin}}(m)}$ are the probabilities to
have zero dark counts and SIN pulses, respectively. The probability to detect
zero counts in time slot $m$ is therefore given by

$\displaystyle\bar{P}_{m}=\bar{P}_{\text{d}}\cdot\bar{P}_{\text{sin},m}.$ (2)

The time profile of the SIN pulse probability is introduced through a
dependence of the mean number of processes on the slot under consideration.
Considering explicitly and separately dark counts and SIN pulse contributions,
and using Eqs. 1 and 2, the probability to detect at least one SIN pulse in
slot $m$ after the signal can be written as

$\displaystyle P_{\text{sin},m}=\left(\frac{N_{m}}{S}-1\right)e^{\mu_{d}}+1.$
(3)

Knowing the dark count rate from the MaPMT QA characterisation, it is possible
to determine the SIN pulse probability in each slot $m$.

The probability $P_{\text{sin}}$ to detect at least one SIN pulse in the full
time range where SIN pulses occur is given by

$\displaystyle P_{\text{sin}}=1-\prod_{m=1}^{M}(1-P_{\text{sin},m})$ (4)

where the upper limit of the time range is defined as $25\text{\,ns}\times M$.
The typical value of maximum dark count rate per pixel is
$\nu_{d}=1\text{\,kHz}$, resulting in a maximum mean number of dark counts per
$25\text{\,ns}$ bin of $\mu_{d}\sim 10^{-5}$. The dark counts contribution can
therefore be safely neglected, since it is at least one order of magnitude
lower than the intensity of SIN pulses. With this approximation, and given
that $N_{m}\ll S$, Eq. 4 can be written as

$\displaystyle P_{\text{sin}}=1-\prod_{m=1}^{M}(1-P_{\text{sin},m})\approx
1-\prod_{m=1}^{M}\left(1-\frac{N_{m}}{S}\right)\approx
1-\prod_{m=1}^{M}e^{-\frac{N_{m}}{S}}=1-e^{-\frac{B}{S}},$ (5)

where $B=\sum_{m=1}^{M}N_{m}$ is the total number of hits after the signal.
The mean number of SIN pulses $\mu_{\text{sin}}$ is therefore given to a good
approximation by the ratio $B/S$, and can be inferred from the time
distribution like the one shown in Fig. 5. For this reason the figure of merit
used to assess the effect of SIN on a pixel-by-pixel basis is the ratio
$\mu_{\text{SIN}}^{p}=B_{p}/S_{p}$, representing the mean number of SIN pulses
for the pixel $p$ under study.

The mean number of SIN pulses is different for each pixel within the MaPMT
under study, and for the same pixel between different MaPMTs, as shown in Fig.
6, where the distributions only show background events. However, a common
pattern of pixels affected by SIN is found, and is described in Sec. 4.2. The
same figure of merit is used to search for correlations with parameters such
as gain and dark count rate, and to study the dependence on the supplied high
voltage, the latter being described in Sec. 4.3.

Figure 6: SIN pulse probability up to $m=100$ time slots after the signal,
corresponding to $2.5\,\upmu\text{s}$. The same pixel (number 61) is shown for
two different MaPMTs, to highlight the qualitative similarity between
different devices. The distribution in Fig. 6 is extracted from the spectrum
shown in Fig. 5. Slot number 1 in the SIN probability distributions
corresponds to slot number 6 in the whole distribution of Fig. 5, i.e. the
signal slot is not shown here.

###  4.2 Localisation of SIN and correlation with MaPMT parameters

Time spectra such as those shown in Fig. 5 were acquired for all pixels of
several MaPMTs. The time spectra for four pixels and the map of the
$\mu_{\text{SIN}}^{p}$ values for the MaPMT with serial number FB4439,
representative of the average quality of R11265 MaPMTs, are reported in Fig.
7.

Figure 7: 7: time-spectra in logarithmic vertical scale for four pixels of
tube FB4439. 7: map of the mean number of SIN pulses ($\mu_{\text{SIN}}$) for
all the pixels of tube FB4439 (pixel-0 is top-left, pixel-7 is top right and
pixel-63 is bottom-right).

The time spectrum for pixel 28 shows that this pixel is essentially unaffected
by SIN, since the events after the signal, beside a negligible component of
fast after-pulses, are consistent with dark counts. The time spectra for the
other three pixels clearly show the effect of SIN, with different intensities
that depend on the pixel position within the MaPMT.

Table 1: Mean number of SIN pulses estimated for four MaPMTs. For comparison, the first line in the table corresponds to FB4439, showing the numerical ratios computed from the time distributions and map displayed in Fig. 7. MaPMT | $\mu_{\text{sin}}^{0}$ | $\mu_{\text{sin}}^{5}$ | $\mu_{\text{sin}}^{28}$ | $\mu_{\text{sin}}^{61}$
---|---|---|---|---
FB4439 | $0.2457\hskip 2.84526pt\pm\hskip 2.84526pt0.0035$ | $1.695\hskip 2.84526pt\pm\hskip 2.84526pt0.013$ | $0.0109\hskip 2.84526pt\pm\hskip 2.84526pt0.0007$ | $2.139\hskip 2.84526pt\pm\hskip 2.84526pt0.018$
FB2294 | $0.0745\hskip 2.84526pt\pm\hskip 2.84526pt0.0012$ | $0.3715\hskip 2.84526pt\pm\hskip 2.84526pt0.0029$ | $0.0076\hskip 2.84526pt\pm\hskip 2.84526pt0.0004$ | $0.491\hskip 2.84526pt\pm\hskip 2.84526pt0.003$
FB2312 | $0.132\hskip 2.84526pt\pm\hskip 2.84526pt0.0017$ | $0.747\hskip 2.84526pt\pm\hskip 2.84526pt0.004$ | $0.0081\hskip 2.84526pt\pm\hskip 2.84526pt0.0004$ | $2.231\hskip 2.84526pt\pm\hskip 2.84526pt0.011$
FB4500 | $0.266\hskip 2.84526pt\pm\hskip 2.84526pt0.004$ | $1.398\hskip 2.84526pt\pm\hskip 2.84526pt0.012$ | $0.0103\hskip 2.84526pt\pm\hskip 2.84526pt0.0007$ | $1.034\hskip 2.84526pt\pm\hskip 2.84526pt0.010$

The localisation of SIN can be visualised by computing the
$\mu_{\text{SIN}}^{p}$ map, reproduced in Fig. 7. The SIN effect is
essentially concentrated in the top and bottom rows of the MaPMT. After
computing the same map for different MaPMTs, it is observed that the
localisation described above is a general feature of SIN, but the absolute
values of $\mu_{\text{SIN}}^{p}$ differ between different units, as reported
in Table 1.

The correlation between the mean number of SIN pulses and the average gain,
blue sensitivity index, average dark current and pixel dark counts for pixel
61 of 1400 MaPMTs of series FB was studied. No clear correlation is observed
with respect to these variables. A weak correlation against the pixel gain,
increasing at lower high-voltages, is instead visible in Fig. 8. This
correlation coefficient is determined for each pixel of these 1400 MaPMTs, in
order to produce the corresponding correlation maps. The increasing
$\mu_{\text{SIN}}$-to-gain correlation coefficient with lower high-voltages
matches the MaPMT regions affected by SIN.

Figure 8: Correlation between the pixel gain at 8 1000 V, 8 950 V, 8 900 V and
8 850 V and the mean number of SIN events at the corresponding HV.

Two MaPMTs with a large SIN rate have been subjected to a high illumination
rate for a period approximately equivalent to the total integrated charge
expected at the end of Run 3. The results of such measurements show that there
is no degradation in the MaPMT properties such as the quantum efficiency,
while the gain decrease shows no connection between ageing effects and SIN.

Other tests have been performed by using different voltage dividers, allowing
to change the potential difference either between the photocathode and the
first dynode or between the last dynode and the anode. No significant changes
in the mean number of SIN events and in the time distributions were observed.

###  4.3 High-voltage dependence and peaking structures

The SIN probability as a function of the time elapsed from the signal slot and
for different applied high voltages is shown in Fig. 9 for pixel 61 of one
MaPMT. As explained in Sec. 4.1, each bin content corresponds to $N_{m}/S$,
i.e. histogram entries are normalised to the signal counts. To highlight the
common time characteristics, the per-bin SIN probability distribution averaged
over a sample of randomly selected 20 MaPMTs is reported in Fig. 9, for the
same pixel 61.

Figure 9: 9: comparison between the SIN time distributions at different HV
values for anode 61 of one MaPMT. 9: Comparison between the SIN time
distributions at different HV values for anode 61 averaged over twenty MaPMTs.
The characteristics time structure can be seen.

The mean number of SIN pulses has an exponential dependence on the high
voltage, as shown in Fig. 10. The comparison between the gain and SIN curves
versus the high voltage is reported in Fig. 10, where the measurements at 1000
V are used as reference.

Figure 10: 10: mean number of SIN pulses as a function of the applied HV for
the same anode 61 of Fig. 9. 10: relative value of gain (black) and of the
mean number of SIN pulses (red) as a function of the applied HV for the same
anode and normalised to the measurement at 1000 V.

The distributions shown in Fig. 9 present similar peaking structures: the
first, second and third peaks appear at $175\pm 25\text{\,ns}$, $300\pm
25\text{\,ns}$ and $400\pm 25\text{\,ns}$ after the signal, respectively,
where the 25 ns uncertainty reflects the DAQ time binning. Even if there is no
observable shift in time as the high voltage changes, such peaking structures
are usually due to ion feedback: in order to identify possible ions
contributing to SIN pulses, the transit times can be determined using the
classical relation

$t=L\sqrt{\frac{2m}{qV}},$ (6)

where $q$ is the ion charge, $m$ is the ion mass, $V$ and $L$ are the
potential difference and the distance, respectively, between the start and end
point of the ion trajectory, and a uniform electric field is assumed. The
voltage divider in the baseboard provides a potential difference between the
photocathode and the first dynode of $V_{1}=153.3$ V and 130.3 V when biasing
the MaPMT at 1000 V and 850 V, respectively. The distances between the
photocathode and the anode, and between the photocathode and the first dynode,
are estimated to be 13 mm and 2 mm, respectively. The assumption of a uniform
electric field between the photocathode and the first dynode is a very good
approximation, besides the corner pixels where small edge effects due to the
focusing optics are expected. More significant edge effects affect the
configuration of the electric field between the photocathode and the anode, in
particular on the peripheral pixels, since the walls of the MaPMT are at the
same potential as that of the photocathode. For this reason, Eq. 6 only gives
an approximate value of the transit times between the photocathode and the
anode. The transit times calculated from the relation above for various ion
candidates [11, 13, 14], for 1000 V and 850 V biases, and between photocathode
to first dynode and photocathode to anode are reported in Tab. 2.

Table 2: Transit times, expressed in ns, for various ion candidates reported in the first column. The corresponding atomic masses A are reported in the second column and expressed in units of atomic mass (1 u = $1.66\times 10^{-27}$ kg). The other columns represent the transit times ($\Delta t$) for, from left to right, photocathode (PK) to first dynode (Dy) at 1000 V, photocathode to first dynode at 850 V, photocathode to anode (An) at 1000 V, photocathode to anode at 850 V, respectively. Ion | A [u] |  $\Delta t_{\text{PK-Dy}}$ [ns] |  $\Delta t_{\text{PK-Dy}}$ [ns] |  $\Delta t_{\text{PK-An}}$ [ns] |  $\Delta t_{\text{PK-An}}$ [ns]
---|---|---|---|---|---
|  | V=153.3 V | V=130.3 V | V=1000 V | V=850 V
$\text{H}_{2}^{+}$ | 2.0 | 33.0 | 35.8 | 84.0 | 91.1
$\text{He}^{+}$ | 4.0 | 46.5 | 50.5 | 118.4 | 128.4
$\text{CH}_{4}^{+}$ | 16.0 | 93.1 | 101.0 | 237.0 | 257.1
$\text{K}^{+}$ | 39.1 | 145.4 | 157.7 | 370.1 | 401.4
$\text{Sb}^{+}$ | 121.8 | 256.6 | 278.3 | 653.0 | 708.3
$\text{Cs}^{+}$ | 132.9 | 268.1 | 290.7 | 682.3 | 740.0

In order to gather information on the nature of SIN, the same per-bin SIN
probability, averaged over 300 MaPMTs, is studied for other pixels, as shown
in Fig. 11. Different components are present depending on the pixel under
consideration. Fig. 11 refers to a central row of pixels, for which the SIN
probability is at least a factor ten lower, on average, with respect to the
most affected top and bottom rows. A narrow peak centred at $\Delta
t=100\text{\,ns}$ after the signal is visible in all the pixels, while two
other peaks, centred at $\Delta t=275\text{\,ns}$ and a wider one at $\Delta
t=850\text{\,ns}$ are only visible for the peripheral pixels 32 and 39, but
also present with at least a 20 times lower probability for the neighbouring
pixels, with decreasing rate while moving towards the centre of the tube.

Figure 11: 11: SIN pulses per-bin probability for the least affected pixels.
11: SIN pulses per-bin probability for the most affected pixels. Note that the
maximum probability is approximately ten times more with respect to 11.

Such peaking structures, and the localisation of SIN events described in Sec.
4.2, can be interpreted considering the internal structure of the photon
detectors. Mechanical apertures are present at the tube periphery in the top
and bottom regions most affected by SIN effects and, to a lesser extent, in
the left- and right-most columns. Photons and ions produced at the anode can
therefore propagate freely through these apertures towards the photocathode.
In the central region of the tube, the dynode chains are screening these
effects. The narrow peak centred at $\Delta t=100\text{\,ns}$ can be
associated to $\text{CH}_{4}^{+}$ feed-back between the photocathode and the
first dynode. The other structures at $\Delta t=275\text{\,ns}$ and $\Delta
t=850\text{\,ns}$ are compatible with $\text{CH}_{4}^{+}$ and $\text{Cs}^{+}$
feed-back, respectively, between the photocathode and the anode. A component
corresponding to $\text{Cs}^{+}$ feed-back between the photocathode and the
first dynode could be present but shadowed by the tail of the first peak. Fig.
11 displays the SIN time distribution for four pixels belonging to the most
affected rows. The dominant feature of such distributions is the exponential
tail, indicating the presence of a mechanism consistent with internal light
emission and fluorescence decay. The positions of the peaking structures are
shifted towards later times due to the convolution with such effects and to
the actual electric field configuration, and are compatible with both ion
feed-back and cascade processes. The time gap of approximately
$200\text{\,ns}$ between the signal and the first peak suggests that the
internal light emission is mainly amplified by the larger number of
photoelectrons produced by the interaction between ions and the photocathode.
A shoulder around 800 ns is also visible and can be associated to the same
effects observed in the central row of pixels due to $\text{Cs}^{+}$ feedback
between the photocathode and the anode. Details on electron-impact ionisation
cross-sections for $\text{CH}_{4}$ and Cs can be found in Refs. [15] and [16],
respectively.

###  4.4 Analogue measurements

Further investigations have been performed with an oscilloscope by probing the
output of one MaPMT channel, in order to validate the interpretation of the
time distribution of SIN events. An example of recorded waveform showing the
presence of out-of-time events is reproduced in Fig. 12.

Figure 12: Example of a waveform recorded with the oscilloscope. The MaPMT
response to the laser signal is in correspondence of t=0 ns, while two SIN
pulses can be observed at later times.

The data collected with the oscilloscope, though statistically limited, are
used to draw qualitative conclusions on the nature of SIN pulses. In
particular, the SIN amplitude versus the elapsed time is determined as
reported in Fig. 13. The main contribution to SIN events, corresponding to the
region around 400 ns is due to pulses having single-photon amplitude. In this
region, the dominant mechanism responsible for this source of noise is
attributed to light emission generated from the anode side of the MaPMTs.

Figure 13: SIN amplitudes expressed in single-photoelectron (PE) units as a
function of the elapsed time from the signal.

The events in the band centred at $\Delta t=200\text{\,ns}$ have larger
amplitudes, equivalent to up to four photoelectrons. This cluster of events
confirms the interpretation of the first peak as due to ion feed-back.
However, the rate of ion feed-back events is approximately one order of
magnitude lower with respect to the SIN pulses produced by the dominant light
emission mechanism.

The waveforms acquired with the oscilloscope can also be used to determine the
amount of higher-order events, i.e. noise signals generated by photoelectrons
arising from SIN pulses rather than from primary signal photons, for the
corresponding pixel of this particular MaPMT. Such higher-order components are
shown in Fig. 14 in which the majority of events are seen to be of the first
order, i.e. the peaking structures are mainly due to independent mechanisms
and not to cascade effects. Approximately 16% of the total is due to second-
order SIN pulses that accumulate around $\Delta t=400\text{\,ns}$, while
around 4% are due to the third order. However, the amount of higher order SIN
pulses may be underestimated since the primary may be present in adjacent
pixels and not detected in this particular anode.

Figure 14: SIN time distribution with order components as described in the
legend.

##  5 Pile-up studies

As described in Sec. 4, the SIN time distribution extends over several
microseconds after the primary signal. The probability to detect noise events
is conditioned to the presence of signal in previous time slots. As a
consequence, in case of a high illumination rate, the detected photon
occupancy is increased in the regions affected by SIN effects. As determined
from simulations, the spatial distribution of Cherenkov photons across the
photon detector planes of the LHCb RICH detectors is expected to vary
significantly, with a peak detection rate of $10\text{\,MHz}$ in the central
region of RICH1, corresponding to an occupancy of approximately 30%.

The pile-up studies were performed in the laboratory at CERN, employing an
upgrade photon detector module, a pcie40-based DAQ and slow control and a
laser-based illumination system. The acquisition is synchronous with the laser
pulses, the trigger signal being sent to both the DAQ boards and the laser
driver. The DAQ time window is one 25 ns slot wide and a trigger rate
$\nu_{t}$ is equivalent to introducing a dead time
$\tau_{d}=1/\nu_{t}-25\text{\,ns}$ in the DAQ system. Measurements for
$\nu_{t}$ up to $10\text{\,MHz}$ are performed in order to evaluate the
increase in detected counts due to SIN pulses. The laser intensity is tuned to
get an occupancy of approximately $1\%$ when running at
$\nu_{t}=100\text{\,kHz}$, corresponding to a dead time
$\tau_{d}=10\,\upmu\text{s}$, i.e. larger than the typical duration of SIN
pulses.

The probability $\bar{P}_{m}$ to have no hits detected by a pixel in a 25 ns
time slot $m$ is given by the probability $\bar{P}=e^{-(\mu_{s}+\mu_{d})}$ to
detect no signal or dark counts in this time slot multiplied by the
probability to have no SIN pulses in time slot $m$ due to signal or dark
counts in previous time slots. $P_{\text{ck}}=1-\bar{P_{\text{ck}}}$ indicates
the detection probability without SIN, with the subscript ck to indicate
Cherenkov signal events (or any signal other than SIN background sources),
that can be determined from simulation and takes the same mean value for each
25 ns time slot. The SIN pulse probability can be determined from the
measurement of the SIN time distribution described in Sec. 4.1. Since this
distribution includes the contribution from higher-order SIN pulses, the
probability to have hits in slot $m$ is given by

$\displaystyle
P_{m}=1-\bar{P}_{\text{ck}}\prod_{i=1}^{m}(1-P_{\text{sin},i}P_{\text{ck}}),$
(7)

where $P_{\text{sin},i}$ denotes the probability to detect a SIN pulse $i$
slots after the corresponding detected photon or dark count. The duration of
SIN pulses given a signal is limited in time and their probability is null $M$
time slots after the signal, where $M$ corresponds to the maximum duration of
SIN pulses (e.g. $M=160$ for a duration of 4$\,\upmu\text{s}$). Consequently,
the probability to detect hits in the presence of SIN for $m>M$ converges to

$\displaystyle
P=1-\bar{P}_{\text{ck}}\prod_{i=1}^{M}(1-P_{\text{sin},i}P_{\text{ck}})\approx
1-\bar{P}_{\text{ck}}e^{-\sum_{i=1}^{M}P_{\text{sin},i}P_{\text{ck}}}=1-\bar{P}_{\text{ck}}e^{-\mu_{\text{sin}}P_{\text{ck}}},$
(8)

where the approximation $P_{\text{sin},i}P_{\text{ck}}\ll 1$ is used and
$\mu_{\text{sin}}$ is the mean number of SIN pulses. The probability is now
independent from the slot $m$ under study.

The introduction of a dead time, resulting from the DAQ strategy in the high-
rate measurements described in this section, requires the rewriting of Eq. 7
as

$\displaystyle P_{m}=1-\sum_{n=0}^{N}\delta(m-n\Delta
m)\left[\bar{P}_{\text{ck}}\prod_{i=1}^{m}\left(1-\sum_{k=1}^{m/\Delta
m}\delta(i-k\Delta m)P_{\text{sin},i}P_{\text{ck}}\right)\right],$ (9)

where $N=10^{6}$ is the total number of acquired time slots, i.e. the number
of trigger pulses, and $\Delta m$ is the number of $25\text{\,ns}$ time slots
between each trigger pulses, given by $\Delta m=40\text{\,MHz}/\nu_{t}$. When
$\nu_{t}=100\text{\,kHz}$, the probability to detect a hit in the acquired
slot $m$ is equal to the counting probability without SIN,
$P_{m}=1-e^{-(\mu_{s}+\mu_{d})}$, since $\Delta m=400$ and
$P_{\text{sin},k\Delta m}=0$. No dead time, i.e. when the acquisition runs at
$40\text{\,MHz}$ as expected during the data taking corresponding to the
collisions mode at the LHC, means $\Delta m=1$ and Eq. 9 reduces to Eq. 7.

To check the above relations, the ratio between the hit maps acquired at
different illumination rates is determined. The ratio between the occupancy at
10 MHz and 100 kHz for one MaPMT is shown in Fig. 15. The expected values
based on the model described above agree with the measurements, as reported in
Fig. 15 for one pixel. In particular, the validity of Eq. 8 allows to assess
the impact of SIN on the PID performance, described in Sec. 7, with the
knowledge of the mean number of SIN pulses and the occupancy values across the
LHCb RICH photon detectors planes. Other measurements at larger occupancies,
e.g. at 10%, have been performed and give consistent results.

Figure 15: 15: ratio of detected counts at $10\text{\,MHz}$ and
$100\text{\,kHz}$ for MaPMT FB3036. 15: ratio of the counting probabilities
with and without SIN for pixel 61 of MaPMT FB3036, at different illumination
rates, for the experimental (red) and expected (black) values. The mean number
of SIN pulses is $\mu_{\text{SIN}}^{61}=4.329\pm 0.031$.

##  6 Mitigation strategies

Different strategies to mitigate the SIN effects are described in the present
section. They will be adopted to operate the R11265 MaPMTs in the 40 MHz
single-photon counting mode.

###  6.1 Optimisation of the high-voltage operating point

Given the strong dependence of the mean number of SIN pulses on the applied
HV, the MaPMTs will be operated at the lowest possible HV, as a compromise
between a high single-photon detection efficiency and a low SIN background
rate. While operations at low HV is sufficient to recover the expected
performance for a large region of the RICH detectors acceptance, additional
strategies are required in the central region of RICH1 where an average
illumination rate of 10 MHz is expected.

###  6.2 New MaPMTs with reduced SIN effects

A new series of R11265 MaPMTs, referred to as the FD series, has been produced
by Hamamatsu to reduce the contributions from SIN pulses. The comparison of
the per-bin SIN pulse probability of pixel 61 between the FB and FD series,
averaged over 200 MaPMTs, is reported in Fig. 16. A change in the internal
mechanical design of the tube results in a large reduction of SIN effects. The
FD-series MaPMTs will be used in the high-occupancy region of RICH1.

Figure 16: 16: Comparison of SIN distributions between the original FB-series
and the new FD-series MaPMTs. The maximum per-bin probability is more than
halved, with a faster recovery of tail events. 16: per-bin SIN pulse
probability for 4 pixels of the FD series.

###  6.3 Nanosecond time gate at the front-end readout

The prompt emission of Cherenkov radiation and the focusing mirror geometry
result in an excellent intrinsic time resolution of the RICH detectors. Per-
track Cherenkov photons from a single proton-proton interaction reach the
MaPMT plane with a spread of $\mathcal{O}(10\,$ps) for RICH1 and
$\mathcal{O}(100\,$ps) for RICH2. The larger spread for RICH2 is due to the
size and position of the detector. Within the LHC environment the time spread
of the signal is dominated by the space and time distributions of the proton
collisions at the interaction point. Fig. 17 shows the photon arrival time at
the RICH1 MaPMT plane following a bunch crossing at time zero, simulated in
the LHCb framework. A time gate of a few nanoseconds can be used to collect
all Cherenkov photon signals (populating the peak labelled S) whilst rejecting
background from beam interactions and MaPMT noise. The time gate covers a
small fraction of the 25 ns between LHC bunch crossings and therefore allows
up to an order of magnitude suppression of out-of-time background hits in the
front-end output data.

Figure 17: RICH1 MaPMT photon detector hit time distribution for LHC beam
interactions, showing the signal peak (S) within a 3.125 ns time gate. As
indicated by the shaded regions, the time gate excludes MaPMT hits during most
of the 25 ns period between bunch crossings.

During LHC Run 3, the time gate will be implemented in the FPGAs of the PDMDB.
The programmable logic in the FPGA is adapted to sample the digital signals
from the CLARO front-end chips at 320 MHz using the de-serialiser embedded in
every input-output logic block. The FPGA logic can be configured to detect
specific input patterns and to apply a time gate with a minimal width of 3.125
ns around the signals from the CLAROs. The gate can be synchronised with the
signal peak by shifting the sampling clock phase with respect to the LHC
bunch-crossing clock. Given the overall time resolution of the electronic
readout chain, a 6.25 ns time gate may be required to achieve the best PID
performance. Since the SIN probability is approximately constant over the
relatively short 25 ns time between bunch crossings, the signal-to-SIN ratio
is expected to improve by a factor of four or eight using a time gate of 6.25
ns or 3.125 ns, respectively.

##  7 Effects on the PID performance

The LHCb simulation framework is used to evaluate the effects of SIN on the
PID performance. The SIN modelling is introduced during the digitisation of
the detector response, which is conducted within the Boole application.

###  7.1 Simulation input

In the Run 3 simulation, various pixel properties, such as the average pixel
gain and the mean number of SIN pulses, will be read from a dedicated database
populated with data from the commissioning phase and the quality assurance
measurements. Although the detector installation is ongoing at the time of
writing this paper, the MaPMTs in the simulation are placed at their expected
positions, which allows to reproduce realistic detector properties and assess
the effect of SIN on the PID performance. Due to the presence of the high-
occupancy areas in the RICH1 detector, a dedicated procedure is applied to
group the MaPMTs into ECs based on their gain and SIN characteristics and then
populate them optimally in different occupancy regions.

###  7.2 SIN model implementation

The implementation of the SIN model in the simulation is based on the
considerations described in Sec. 5, which take into account the pile-up
effects due to the expected high detection-rates. As mentioned there,
modelling SIN requires knowledge of $\mu_{\text{sin}}$ in the given pixel as
well as pixel occupancy values in the absence of SIN (corresponding to
$P_{ck}$). The latter are based on the simulation and have the same average
values throughout all $25\text{\,ns}$ time slots.

Additional contributions due to SIN are added in each pixel with a probability
$x_{SIN}$ that satisfies the relation $P=P_{ck}+x_{SIN}$ (both contributions
to the occupancy are treated independently here), where $P$ is the detection
probability in a $25\text{\,ns}$ time-slot including the presence of SIN, as
defined in Sec. 5. Following Eq. 8 the SIN deposits are added to each event at
the digitisation stage with the probability:

$\displaystyle
x_{SIN}=\left(1-P_{ck}\right)\left(1-e^{-\mu_{\text{sin}}{}P_{ck}}\right),$
(10)

which depends on the pixel properties and its position on the photon detector
plane, as indicated before.

Since a full time-simulation of the RICH detector response is performed, SIN
contributions are also added in the two $25\text{\,ns}$ time windows preceding
the signal one, and a random time-of-arrival is assigned to them within each
of the slots. It allows to emulate a situation in which a SIN pulse arrives
before the signal, which can result in the failure to record an actual
Cherenkov hit. Contributions of all types (meaning the signal ones, the SIN
ones, etc.) are then processed in the same way to provide the final CLARO time
response. In the current simulation, this response is sampled into eight
$3.125\text{\,ns}$ slots and the patterns accepted for further processing can
be configured (which aims to reproduce the behaviour of the RICH digital
boards). In a nominal case, a pattern configuration corresponding to a level
detection in a specified $3.125\text{\,ns}$ slot after the signal peak is
applied, but more refined scenarios with a time gate around CLARO signals, as
described in Sec. 6.3, are explored as well.

###  7.3 Simulation results

The effects of the presence of SIN, as evaluated using the simulation, are
presented in Fig. 18. Two scenarios of the detector configuration are
considered. The nominal case corresponds to the pixel properties while
operating the MaPMTs at $1000\text{\,V}$ with the standard level detection
described in Sec. 7.2. An optimised one implements some of the mitigation
strategies described in Sec. 6. It refers to using the operating point of
$900\text{\,V}$ with the MaPMTs in RICH1 populated according to the procedure
mentioned in Sec. 7.1 and a preliminary configuration with an edge detection
and a $9.375\text{\,ns}$ time gate around the signal peak (defined separately
for RICH1 and RICH2). Both scenarios account for SIN, as well as spillover
effects from adjacent bunch crossings.

A substantial increase in the average pixel occupancy per MaPMT due to the
additional SIN deposits is observed in the nominal scenario (see Fig. 18). The
points at values close to unity correspond to the R12699 MaPMTs, where no
significant SIN effect is found. Introducing the SIN-mitigation strategies in
the optimised configuration allows to considerably reduce the increase in the
occupancy related to SIN to a level of $<10\%$ in all MaPMTs, which is crucial
in terms of an optimal operation of the RICH detectors. It is worth noting
that despite the essential contributions from SIN deposits in the nominal
scenario, the PID algorithms are relatively robust with respect to the sources
of random noise like SIN. Nevertheless, a sizeable improvement in the PID
performance is observed after implementing the SIN-reduction measures, as
illustrated in Fig. 18, and, in this context, the effect of SIN appears to be
nearly fully alleviated (see the reference curve for the nominal scenario
without simulating SIN). Preliminary studies suggest that, after introducing
the mitigation strategies described in Sec. 6, SIN should have no significant
impact on physics performance during the Run 3 operations.

Figure 18:  18: Ratio of the average channel (pixel) occupancy per MaPMT in
RICH2 from the LHCb simulation with and without SIN using the nominal detector
configuration described in the text. 18: Comparison of the PID performance
obtained from the LHCb simulation including SIN with the nominal (red) and
optimised (blue) scenarios as stated in the text. A case corresponding to the
nominal configuration without simulating the SIN effect (black) is also shown
for reference. The curves illustrate the probability of misidentifying a pion
as a kaon, as a function of the efficiency of identifying a kaon.

##  8 Conclusions

Signal-induced noise is observed in Hamamatsu R11265 MaPMTs. The noise is
localised in specific regions of the tube periphery and extends over 3 to 4
$\,\upmu\text{s}$ after the primary signal. It originates from combined
effects of internal light emission and ion feed-back. A formalism is developed
to assess the impact of this noise on the PID capabilities of the LHCb RICH
detectors during LHC Run 3. Based on these studies, mitigation strategies are
being adopted and involve the following solutions: use of an appropriate high-
voltage operating point as a compromise between optimal single-photon
detection efficiency and minimal number of SIN pulses; installation of new
MaPMTs with reduced SIN effects in the RICH1 regions of highest occupancy;
implementation of a nanosecond time gate in the digital readout board firmware
to improve the signal-to-noise ratio. The expected RICH performance is
evaluated by means of full simulation studies carried out with the LHCb
software stack. It is shown that excellent PID can be achieved after these
mitigation strategies are applied.

## Acknowledgements

We would like to thank the LHCb RICH team for supporting this publication, and
in particular Antonis Papanestis for the review. We express our gratitude to
our colleagues in the CERN accelerator departments for the excellent
performance of the LHC. We thank the technical and administrative staff at the
LHCb institutes. We acknowledge support from CERN and from the national
agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China);
CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO
(Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE (Russia);
MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United
Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC
(Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and NERSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from ARC and ARDC (Australia); AvH Foundation
(Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union);
A*MIDEX, ANR, IPhU and Labex P2IO, and Région Auvergne-Rhône-Alpes (France);
Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP,
Fundamental Research Funds for the Central Universities, and Sci. & Tech.
Program of Guangzhou (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal
and GENCAT (Spain); the Leverhulme Trust, the Royal Society and UKRI (United
Kingdom).

## References

  * [1] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [2] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [3] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [4] LHCb collaboration, _Framework TDR for the LHCb Upgrade: Technical Design Report_ , CERN-LHCC-2012-007, 2012
  * [5] LHCb collaboration, _LHCb Trigger and Online Technical Design Report_ , CERN-LHCC-2014-016, 2014
  * [6] LHCb collaboration, _LHCb PID Upgrade Technical Design Report_ , CERN-LHCC-2013-022, 2013
  * [7] Hamamatsu Photonics, _Photomultiplier tubes, Basics and Applications, Fourth Edition_ , https://www.hamamatsu.com/resources/pdf/etd/PMT_handbook_v4E.pdf, 2017
  * [8] M. Baszczyk et al., _CLARO: an ASIC for high rate single photon counting with multi-anode photomultipliers_ , JINST 12 (2017) P08019
  * [9] T. Colombo et al., _The LHCb Online system in 2020: trigger-free read-out with (almost exclusively) off-the-shelf hardware_ , J. Phys. Conf. Ser. 1085 (2018) 032041
  * [10] F. Cairo et al., _A survey of the signal‐induced noise in photomultiplier detection of wide dynamics luminous signals_ , Rev. Sci. Instrum. 67 (1996) 3274
  * [11] J. Haser et al., _Afterpulse Measurements of R7081 Photomultipliers for the Double Chooz Experiment_ , JINST 8 (2013) P04029, arXiv:1301.2508
  * [12] V. C. Antochi et al., _Improved quality tests of R11410-21 photomultiplier tubes for the XENONnT experiment_ , JINST 16 (2021) P08033, arXiv:2104.15051
  * [13] KM3NeT, V. Giordano, S. Aiello, and E. Leonora, _Noise and spurious pulses for Cherenkov light detection with 10-inch and 3-inch photomultipliers_ , AIP Conf. Proc. 1630 (2015) 122
  * [14] KM3NeT, S. Aiello et al., _Characterisation of the Hamamatsu photomultipliers for the KM3NeT Neutrino Telescope_ , JINST 13 (2018) P05035
  * [15] H. C. Straub et al., _Absolute partial cross sections for electron-impact ionization of $\text{CH}_{4}$ from threshold to 1000 eV_, The Journal of Chemical Physics 106 (1997) 4430, arXiv:https://doi.org/10.1063/1.473468
  * [16] M. Łukomski et al., _Electron-impact ionization cross sections out of the ground and $6\phantom{\rule{1.99997pt}{0.0pt}}^{2}p$ excited states of cesium_, Phys. Rev. A 74 (2006) 032708