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\begin{document}
\title{Performance of the Hall D Tagger Microscope as a Function of Rate}
\author{R.T. Jones\\
University of Connecticut, Storrs, CT}
\date{December 30, 2010}
\maketitle
\begin{abstract}
The Hall D tagger microscope is responsible for determining the energy
and indirectly, the polarization, of the photon beam for the GlueX experiment.
A detailed design for the tagger microscope has been completed and prototyped
by the University of Connecticut group, based on scintillating fiber detectors
and a silicon photomultiplier (SiPM) readout.  An earlier Monte Carlo study
indicated that this design would be capable of running at the maximum design
luminosity of $10^8$ $\gamma/s$ on target in the primary polarized peak, with
a time resolution better than 200~ps, a detection efficiency of 95\% and a
tagging ratio close to 70\%.  This conclusion has recently come into question,
as more information has come to light about the long-term performance of
SiPM devices, in particular the increase in dark rate
that has been observed as a result of their exposure to neutron radiation.
This paper presents results from a detailed Monte Carlo study of the
performance of a single tagger microscope channel, in terms of detection
efficiency and time resolution, as a function of both beam intensity and
and detector dark rate.  This study improves upon the earlier work in
that it takes into explicit account published performance parameters of
specific SiPM devices, whose values in the prior study were in some cases
only guessed.  The conclusion of the study is that the CPTA 2~mm SiPM
from Photonique that was used in the prototype is not capable of reaching
the stated performance goals, but that the 3~mm Hamamatsu device does
satisfy these requirements at the full operating beam intensity for Gluex,
in the presence of a dark rate as high as $10^{10}$~Hz, which is a factor
2500 higher than the dark rate of the pristine device.  Extrapolating 
existing measurements of dark rate as a function of neutron dose, this
corresponds to a lifetime limit of 550,000 hours (63 years) of continuous
running in the unshielded Hall D tagger microscope at the nominal beam
intensity of $10^8$~$\gamma$/s.  This remarkable robustness of the readout,
in the presence of elevated dark rate, is a consequence of the combination
of large photon yield in the microscope, and the fast pixel recovery time
of the Hamamatsu SiPM.
\end{abstract}

\newpage

\section{Introduction}

The Hall D tagging spectrometer detector package consists of two 
subsystems: a coarse-grained array of 190 counters spanning the range
3.0~-~11.7 GeV photon energy with 30~MeV bins, and a fine-grained ``microscope''
covering the range 8.3~-~9.1~GeV photon energy with 8~MeV bins.
The microscope is focused on this specific photon energy range because
significant beam polarization exists only in a restricted energy
region around the primary coherent peak in the window 8.4~-~9.0 GeV.
GlueX has been designed to be able to run at collimated
beam intensities up to $10^8$/s within this energy window, which requires
that the microscope be capable of running at a total tagged rate of 
$2.4\times 10^8$/s integrated over the same energy range.  To handle this
rate, the microscope is segmented into 100 non-overlapping energy channels
and covers the somewhat larger energy region 8.3~-~9.1 GeV to allow for
small variations in the coherent peak position while running.  This
leads to a modest requirement on the microscope readout electronics that
it run efficiently and with good time resolution at rates up to 4~MHz
per channel.  Table~\ref{specs} lists the design goals for the tagger
microscope, which enable near-optimum performance of the tagger for GlueX
physics, subject to the constraints of a coherent bremsstrahlung source.

\begin{table}[b]
\caption{\label{specs}
Design specifications for the Hall D tagger microscope.  These goals
enable near-optimum performance of the tagger for GlueX physics, within
the constraints of a coherent bremsstrahlung source designed for 40\%
polarization at 9~GeV produced using 12 GeV electrons.}
\begin{center}
\begin{tabular}{lrl}
\hline \hline
\multicolumn{1}{c}{parameter} & value & unit \\
\hline
tagging rate per energy bin (peak) & 4.0 & MHz \\
tagged electron detection efficiency & 95 & \% \\
tagged electron time resolution & 200 & ps \\
post-collimator tagging ratio & 70 & \% \\
\hline \hline
\end{tabular}
\end{center}
\end{table}

\subsection{optimization of microscope design}

One property in Table~\ref{specs} that requires explanation is the 
{\em post-collimator tagging ratio}.  Also sometimes known as ``tagging
efficiency'', the {\em tagging ratio} is the rate of tagged photons in
a particular energy bin that reach the target, divided by the total
counting rate in the corresponding tagger channel.  Ordinarily one would
like this ratio to be close to unity, but there is a trade-off between
polarization and tagging efficiency in the Hall D photon beam design that
comes from the choice of the collimation angle; photons that strike the
collimator do not contribute to the numerator in the tagging ratio, but
their corresponding electrons do get counted in the denominator.  The
particular choice made for Gluex results in a near-optimum polarization
of 40\% at 9~GeV, but at a tagging ratio that is around 50\%.  This poor
tagging ratio can be significantly improved by collimating the
post-bremsstrahlung electrons to within a thin band around the horizontal
scattering plane, so that those with large vertical projections of their
momentum vector are not detected in the tagging counters.  The photons
produced by these electrons have a similarly large vertical momentum
component, and so are blocked by the collimator.  Making the tagger blind
to these electrons improves the tagging ratio without impacting the rate
of tagged photons that reach the target, hence improving the performance
of the tagger by decreasing the rate of accidental tags.  

The only reliable way to achieve this within the existing spectrometer
geometry, without introducing more background from scattering than is
eliminated by collimation, is to vertically segment the tagging detector
array, and count only the vertical segment that subtends the momentum
region corresponding to the photon collimator.  Monte Carlo studies have
shown that a 2~mm stripe across the focal plane contains nearly all of
the photons that pass through the collimator, while eliminating about half
of those that do not pass.  The improved value of 70\% in Table~\ref{specs}
reflects this capability.  The microscope focal plane is constructed as a
two-dimensional array of $2\times 2$~mm$^2$ scintillator cells, whose 100
columns constitute the tagged energy bins and whose 5 rows are used both
for tagging (central row, 70\% of count rate) and active monitoring of
the out-of-plane scattering rates (2 rows above and 2 below), for a total
of 500 readout channels.

\subsection{overview of readout design}

The readout of light pulses from the individual square scintillating fibers
that make up the detector array is done using SiPM's.  Clear acrylic fibers
glued to the exit end of the 2~cm scintillating fibers carry the light from
the spectrometer focal plane to a coupler which aligns the ends of the light
guides with the positions of the SiPM's mounted on a circuit board. 
Also integrated together in the same detector-mounted electronics package
are the preamplifiers and summing circuits for the readout signals, and
the programmable bias voltage control for the SiPM's.  The decoupling of
the readout from the detectors with clear-fiber light guides allows the
electronics to be located some distance from the detectors in a shielded area.
In the present design, the electronics are located within the same housing
less than 50~cm from the detectors.

\subsection{lifetime from radiation damage}

A critical concern with the design outlined above has arisen within the
past year, as new data have come to light regarding the sensitivity of SiPM
devices to ionizing radiation, particularly protons and
neutrons~\cite{musienko2009}.  In a comprehensive study
by Y. Qiang~\cite{yi2010}, SiPM devices from CPTA (Photonique), Sensl, and
Hamamatsu were exposed to varying calibrated doses of neutron radiation
up to 150 rem.  The results of the study are more or less uniform, that
gain and quantum efficiency are not affected after exposure to neutrons,
but that dark current increases in a linear fashion with neutron dose,
with a slope that is large enough to produce measurable effects after
only a few hours of operation in the Hall D tagger area.

A simulation study recently published by A. Somov~\cite{somov2010} reports
neutron dose estimates in the position of the tagger microscope of 180~mrem/h
at nominal beam intensity of $10^8$~$\gamma$/s.  This dose rate can be
combined with the results from Ref.~\cite{yi2010} to estimate what the
dark rate would be after 10 years of operation of the Hall D beam.  The
CPTA $2\times 2$~mm$^2$ device from Photonique starts out with a dark rate
of 10~MHz.  After 40~rem of neutrons at an operating temperature of 24~C,
the persistent dark rate has increased by a factor 5 to 50~MHz, corresponding
to 200 hours of tagger operation at full intensity.  Extrapolating forward
with a constant slope, the dark rate is $7\times 10^9$~Hz after 30,000
hours of operation.  Periodic annealing of the electronics boards at
60~C might reduce that rate estimate by 30\%.  The Hamamatsu S10931 device
starts off at a somewhat lower dark rate of 4~MHz and scales up to about
$3\times 10^9$Hz after 10 years.  The following sections present the
results of a detailed study investigating the effects of increased dark
rate on the detector performance, specifically in terms of time resolution
and detection efficiency.

\section{Simulation Model}

The model presented here is comparable to that reported
in Ref.~\cite{retiere2009}, and a simplified version of the more
complex model reported in Ref.~\cite{seifert2009}.
A SiPM consists of many identical photodiodes arranged in a square grid
array on the surface of a silicon wafer.  The cells of the grid are
separated from each other by v-shaped trenches that cut down through the
active layer of the device, and serve to electrically and optically isolate
the pixels from their neighbors.  Each pixel acts as an independent
avalanche photodiode (APD) operating in Geiger mode.  Quenching of the
Geiger pulse is provided by a resistor $R_q$ which operates in conjunction
with the internal capacitance $C_1$ of the diode structure to determine the
charge $q_1$ released by an individual avalanche event and the time constant
$\tau_1$ for recharging the junction to its full reverse-bias operating
potential $V_b$ which is set a small amount $\Delta V$ above the breakdown
threshold potential $V_0$.
\begin{equation}
q_1=C_1\;\Delta V
\label{eq:q1}
\end{equation}
\begin{equation}
\tau_1=R_q\,C_1
\label{eq:tau1}
\end{equation}
These quantities are closely related to the device operating parameters.
The value $q_1$ is also known as the gain of the device.  The pixel (or
microcell) recovery time, which measures the time from the onset of an
avalanche to the moment when the diode is recharged to 95\% of its full
change, is equal to $3\tau_1$. 
The value of $C_1$ is roughly proportional to the area of the pixel, and
varies by more than an order of magnitude between different devices, 
although for very small pixels a significant part of $C_1$ comes from the
interconnects on the chip.
Eq.~\ref{eq:q1} explains why, roughly speaking, SiPM's with high gain
have fewer pixels per unit area than those with lower gain.  Typical
values for $C_1$ are in the range 15-150~fC.
The value of $R_q$ is also very important to understanding the behavior
of the SiPM at high rates.  Values vary from 100~k$\Omega$ to 100~M$\Omega$,
depending on the manufacturer.

The photon detection efficiency (PDE) is the product of the ratio of 
the active area to the total area of the chip, and the quantum efficiency
$\epsilon$ of the APD's.  The quantum efficiency is itself a product of several
device and intrinsic properties, but tends to be in the range 50\% - 70\%
around 500~nm for most devices, when operating at the optimum bias voltage.
For the purpose of modeling the microscope readout, only the PDE is of any
practical interest.  It falls in the range 15\%-25\% at 490~nm, the peak
wavelength for BCF-20 scintillator, for the devices considered in this
study.

\subsection{signal light yield}

A Geant simulation of energy loss by a high-energy electron traveling
through a 2~cm plastic scintillator fiber is shown by the yellow shaded
region in the left panel of Fig.~\ref{fig:dE}.  The most-probable
energy loss of 3.6~MeV agrees with expectations of roughly 1.8~MeV/cm in
plastic.  A fit to this spectrum with the Landau distribution was used
to generate the signal pulse heights $y_i$ for this study.
\begin{equation}
y_i = y_{mp}(1+p\ell_i)
\label{eq:phdist}
\end{equation}
where the pseudo-random sequence $\ell_i$ is distributed according to the
Landau distribution given in Eq.~\ref{eq:landau}, and free parameters
$y_{mp}$, $p$ are displayed in Table~\ref{phdistpars}.
\begin{equation}
f(\ell) = 12\pi i \int_{c-i\infty}^{c+i\infty}\exp(\ell s + s\ln{s}) ds
\label{eq:landau}
\end{equation}

\begin{table}
\caption{\label{phdistpars}
Parameter values in the model pulse height density function
Fig.~\ref{eq:phdist}.  Values are dimensionless. }
\begin{center}
\begin{tabular}{lrl}
\hline \hline
\multicolumn{1}{c}{parameter} & value \\
\hline
$y_{mp}$ & 200 \\
$p$ & 0.17 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}

The pulse height distribution of signal pulses produced in the readout
model simulation is shown for comparison in the right panel of
Fig.~\ref{fig:dE}.  The horizontal axis is plotted in units of $q_1$
so that the effects of photon statistics are easy to estimate.  The
smearing out of the low-energy edge of the Landau peak in the right
panel, as compared to the one in the left panel, is the result of
the Poisson statistics in the discrete amplitude response of the SiPM.

\begin{figure}[t]
\begin{center}
\includegraphics[width=3.0in]{MCpulseheight.eps}
\includegraphics[width=3.0in]{modelpulseheight.eps}
\caption{\label{fig:dE} 
Energy deposition spectrum (left panel) and modeled photoelectron
yield in a SiPM (right panel) for a 3~GeV post-bremsstrahlung electron
in the Hall D tagger Geant simulation.  In the left panel, the yellow
shaded area corresponds to post-bremsstrahlung electron tracks, the blue
peak near zero comes from gamma ray conversions in the plastic, and the
black histogram is the complete spectrum, including the mostly low-energy
hits from random room background.  The spectrum in the right panel includes
only the post-bremsstrahlung electron component, excluding the few percent
below the Landau edge in the left plot that come from tracks entering or
exiting the fiber from the side, that do not travel the full length inside
the fiber.  Cerenkov photons produced in the clear light guides fall outside
the capture cone of the fiber, and are not included in the statistical model.
}
\end{center}
\end{figure}

The conversion factor from energy deposition to
average photoelectron count (pixels firing) was obtained in the following
way.  Saint Gobain advertises that about 8000 photons are produced in the
scintillation peak around 490~nm, per MeV of energy deposited by a
minimum-ionizing particle.  The capture fraction for the double-clad
optical fiber is about 4.3\% in each direction.  With a 90\% reflective
coating on the upstream end of the fiber, a total of 650 photons per MeV
are captured within the transmission aperture of the fiber.  Assuming
33\% losses the glue joint and along the length of the clear light guide,
one expects a most-probable yield of 1600 green photons incident on the
surface of the SiPM.  A conservative PDE estimate of 15\% leads to a
most-probable pulse height of 230 pixels.  This value has been rounded
down to 200 pixels to produce the results reported below.

\subsection{amplifier response}

\begin{figure}[t]
\begin{center}
\includegraphics[width=3.0in]{wiki_pulse.eps}
\includegraphics[width=3.0in]{wiki_pulse_tail.eps}
\caption{\label{fig:impres}
Detailed analytical model for a pulse from the UConn SiPM preamplifier
circuit.  The data points are computed from the Fourier transform
coefficients of the circuit model, and the curve is a piece-wise analytic
interpolation of the points.  The second panel shows an expanded view
of the tail of the pulse plotted in the first panel.  The charge integral
of the pulse is zero.  The amplitude of the pulse is normalized to a
maximum value of unity.
}
\end{center}
\end{figure}

The response of the preamplifier and summing circuit to a single pixel
firing has been modeled by a detailed impulse-response analysis of the
circuit.  The source is treated as a fast current spike with a total charge
$q_1$ landing on input capacitance $C_n$ and being drained off through the
input impedance $Z_i$ of the the amplifier. The parameter $Z_i$ is a complex
number that is frequency-dependent, but its value is on the order of 
10~$\Omega$.  The preamplifier circuit was designed with a special input
stage with low impedance in order to prevent the large capacitance $C_n$
from producing long signal fall times $C_n\,|Z_i|$.  The device capacitance
$C_n$ that appears in the expression for the single-pixel pulse width is
{\em not} the $C_1$ that determines the gain factor according to
Eq.~\ref{eq:q1}, but includes the capacitance of all pixels in the device,
plus the extra capacitance contained in the interconnection network on the
chip.  The value of $C_n$ is in the range 150~-~300~pF for SiPM's that are
large enough to read out a $2\times 2$~mm fiber.  The shape of the output
voltage pulse driven into 50~$\Omega$ by the preamp in response to a single
photoelectron is shown in Fig.~\ref{fig:impres}.  This shape is consistent
with multi-photon pulses observed in bench tests with a fast-pulse laser
diode, after unfolding with the diode pulse shape which was about 4~ns FWHM.

Note the long tail shown in the right panel of Fig.~\ref{fig:impres}.  The
vertical scale has been expanded, relative to that in the left panel, to make
the variation in the signal visible.  It is extremely important to model the
long tail correctly in the simulation because, at high rate, the tails from
many pulses stack up on top of one another and produce a wandering baseline
that underlies the fast component of following pulses.  These baseline
variations are equivalent to a meandering threshold in the discriminator,
and limit the time resolution of the device at high rates.  These effects
are carefully modeled in this study.

\subsection{pixel recovery}

The primary purpose for the quench resistor is to isolate the diode from
the bias voltage source, so that the junction potential drops below
breakdown threshold as soon as a charge of $q_1$ has accumulated across
the diode junction capacitance.  To function in this way, the value of $R_1$
must be large enough to delay the recharging of the capacitance until all
of the charges from the avalanche have been swept out of the depletion
zone.  For bulk of the charges, 100~ps would be sufficient delay to 
accomplish this~\cite{vandam2009}, but a few of the charges inevitably
get trapped in weakly bound localized states around defects.
As soon as the junction potential is restored above breakdown threshold,
a single one of these trapped charges can thermally excite out of its
trap and induce a second avalanche.

This process known as ``after-pulsing'' is a property of all
SiPM devices, although their effects are most noticeable in devices with
small values of $R_q$ like the ones from Hamamatsu.  When these after-pulses
were first observed in the Hamamatsu SiPM tested by our group, they were
thought to be a defect of the device, but since then we have learned that
they are the signature of a fast pixel recovery time, without which the
device cannot function well at high rates.  Provided that their probability
remains significantly less than unity, after-pulses are a feature of fast
SiPM devices, and simply add a long-tail component to the pulse shape for
multi-photon pulses.  As shown below, these after-pulses set the effective
lower bound on the discriminator threshold that can be used in the microscope
readout.

Several groups have published detailed measurements of the pixel recovery
and after-pulsing behavior of SiPM's from a variety of manufacturers
\cite{musienko2009,retiere2009,britvitch2006,para2007}.
Ref.~\cite{britvitch2006} lists measured pixel recovery times for 14
different SiPM devices from 5 manufacturers, including CPTA and Hamamatsu.
The value of 2.8~$\mu$s that they report for the CPTA $40\times 40$~$\mu$m$^2$
556 pixel device is consistent with the large value of $R_q$ of 7~M$\Omega$
reported in Ref.~\cite{musienko2009} for this device.  Ref.~\cite{britvitch2006}
reports a wide range of recovery times for the 10 different Hamamatsu SiPM's
they examined.  Ref.~\cite{musienko2009} reports a $R_q$ of 110~k$\Omega$
for the MPPC S10362-11-050C (400 50-micron pixels).  Ref~\cite{S10362} reports
a gain of $1\times 10^6$ for this device at $\Delta V=1$~V. Using
Eqs.~\ref{eq:q1}-\ref{eq:tau1}, these numbers give $\tau_1=16$~ns, and a
95\% pixel recovery time of 48~ns for this device.  Analyzing the data 
shown in Fig.~15 in Ref.~\cite{S10362} gives a more precise estimate of
$\tau_1=10.9\pm 1.4$~ns, which is more or less consistent with the value
of 50~ns that is reported in Ref.~\cite{britvitch2006} for the last two
Hamamatsu devices listed in their Table~1.

Ref.~\cite{musienko2009}
reports a much longer recovery time close to 200~ns for the S10362-11-050C.
This may be due to the peculiar way that those authors used to measure the
recovery time, by saturating the entire device with a 200,000 photon pulse
and then following it after a variable delay with a smaller pulse whose
amplitude is measured as a function of the delay.  This method would induce
additional recovery delays from whole-device effects, in addition to the
single-pixel recovery times, so its disagreement with the manufacturer's
reported value of 50~ns for the S10362 single-pixel recovery time is not
surprising.  Whole-device effects are probably less important for devices
like the CPTA SiPM whose large value of $R_q$ leads to single-pixel recovery
times on the order of microseconds.

Hamamatsu markets its MPPC devices in three pixel sizes, for each choice
of device size: 25, 50, and 100 microns.  The gains of these devices vary
by an order of magnitude, from $2\times 10^5$ to $2\times 10^6$.
Eq.~\ref{eq:tau1} implies that recovery times should scale with the gain,
for fixed $R_q$.  However slide 18 of Ref.~\cite{para2007} reports that
Hamamatsu increases the value of $R_q$ for devices with smaller pixels,
which would offset this effect.  This makes sense if the value of $R_q$ 
was adapted by the manufacturer for each device, to keep the value of
$\tau_1$ constant, at the level needed to adequately suppress after-pulsing,
while allowing for a maximum performance at high rates.  Therefore a fixed
value of $\tau_1=15$~ns was taken for the Hamamatsu SiPM simulated in this
study, which is considered to be appropriate regardless of the actual pixel
size chosen.  Pixel recovery is modeled by assigning to each pixel that
fires a time-dependent gain $g_1(t)$
\begin{equation}
g_1(t)=q_1(1-e^{-t/\tau_1})
\label{eq:g_1(t)}
\end{equation}
where $t$ is the time since that pixel last fired.

\subsection{after-pulsing}

The physical origin of after-pulse avalanches predicts that all pixels have
independently random after-pulse events with exponential time distributions
whose time constants are the lifetimes of the traps.  Although in principle
there may be many different time constants, one for each kind of traps,
Ref.~\cite{retiere2009} reports that the observed after-pulse time
distribution for MPPC devices can be adequately fit with just two time
constants, one at 18~ns and the other around 70~ns.  The probability
integral for each of these after-pulse components depends steeply on the
operating voltage $\Delta V$.  For this study, I assumed a value of
$\Delta V=1$~V, for which the two after-pulsing probabilities shown
in Fig.~4 of Ref.~\cite{retiere2009} are 6\% and 5\%, respectively.

\subsection{cross talk}

The primary source of cross talk in a SiPM is an infrared photon produced
in an avalanche in one pixel that is absorbed in a neighboring pixel and
causes it to avalanche.  Unlike after-pulses, cross-talk pulses occur 
essentially simultaneously with the avalanche that triggers them, and
amounts to little more than a small increase in the overall gain of the
device.  Because they increase the observed pulse-height in multiples of
the single-pixel response, they lead to larger widths than would be expected
based on Poisson statistics.  This effect does not impact the performance
of the microscope in a significant way, but it is included in the simulation
for completeness.

\subsection{dark current}

Dark current is caused by the random firing of pixels due to thermal
excitations of charges creating spontaneous avalanches.  Dark current
$I_d$ is related to dark rate per pixel $r_1$ through the following relation.
\begin{equation}
I_d=Nq_1r_1
\label{eq:darkrate}
\end{equation}
where $N$ is the number of pixels in the device.  The pixel dark rate $r_1$
generally scales with the volume of the depleted zone in the diode, varying
from 300~Hz in the pristine MPPC S10931-025P to 9~kHz in the MPPC S10931-100P
at room temperature and nominal operating bias~\cite{S10931}.

The dark rates reported above have a negligible effect on the operation of
the SiPM in the microscope readout.  However, these numbers do not remain
constant but increase over time as a result of exposure to elevated levels
of ionizing radiation.  Ref.~\cite{musienko2009} reports results from
exposing SiPM's from a variety of manufacturers to 83~MeV protons.  The
principal effect seen in the SiPM's as a result of the exposure was an
increase in the dark rate.  They report the total radiation dose in their
measurement as $2\times 10^10$~/cm$^2$ of ``1 MeV neutron equivalent flux''.
Using the conversion factor 1~rem $=2.4\times 10^7$~$n_{eq}/$cm$^2$, this
dose is equivalent to 830~rem.  After irradiation, they observed a 30-fold
increase in the dark rate for the Hamamatsu MPPC, and a similar factor for
the CPTA/Photonique device.  Ref.~\cite{yi2010} reports a factor of 12
increase in dark current for the Hamamatsu MPPC after only 153~rem of
neutron exposure.  Comparison of these two results indicates that the
slope of dark current vs.\ neutron dose is decreasing with increasing dose.
This is also suggested by the slight negative curvature of the dark current
vs.\ dose seen in the fourth plot in Ref.~\cite{yi2010}.

Operating the Hall D photon beam over a 10-year period (30,000 operating
hours) at full beam intensity, an unshielded microscope readout would
see a neutron dose of 5500~rem~\cite{somov2010}.  Surrounding the readout
electronics with plastic shielding could easily reduce this number to
800~rem~\cite{somov2010}, which brings the dose into the range covered
by Ref.~\cite{musienko2009}, eliminating uncertainties from a large
extrapolation.  These estimates for dark rate increase vary from a factor
30, for a shielded readout solution, to a factor 400, for an unshielded
readout and linear extrapolation based on Ref.~\cite{yi2010}.  In this
study a number of scenarios are explored, with dark rates varying from
a factor of 2 to 25,000.

\section{Simulation Results}

The parameter values used by the simulation model to produce the results
reported below are given in Table~\ref{modelpars}.  Note that the gain of
the device is not included in the list.  This is because all pulse heights
are expressed in units of the single-pixel charge $q_1$.  It is assumed that
the preamplifier is capable of rescaling the output level to a range that
is appropriate for the discriminator and ADC, and that the additional
electronic noise from the preamplifier is negligible compared to the noise
associated with after-pulsing and dark current when operating at rates
appropriate for tagging.  For reference, the prototype preamplifier built
by the UConn group is designed to produce 2~mV peak output per photoelectron,
for a most-probable expected output of 450~mV and an optimum discriminator
threshold close to 100~mV.  Judging by the ability to easily resolve 
individual photoelectron peaks in the SiPM pulse-height spectrum of the
preamplifier output, the RMS electronic noise added by the preamp is smaller
than 1~mV.

\begin{table}[t]
\caption{\label{modelpars}
Parameters used by the simulation model to produce the results of this
study.  The CPTA device under study is the SSPM-0606BG4-PCB marketed by
Photonique.  The Hamamatsu device is the S10631-025P.  Many of these
parameters are not device-dependent and so are the same in both simulations.}
\begin{center}
\begin{tabular}{lrr}
\hline \hline
\multicolumn{1}{c}{parameter} &
\multicolumn{1}{c}{CPTA SSPM} &
\multicolumn{1}{c}{Hamamatsu MPPC} \\
\hline
most-probable amplitude (photoelectrons) 	& 200 	& 200 \\
scintillator time constant (ns) 		& 2.7 	& 2.7 \\
discriminator threshold (photoelectrons) 	& 50 	& 50 \\
discriminator minimum pulse separation (ns)	& 10	& 10 \\
preamplifier impulse rise time (ns)		& 2.4	& 2.4 \\
preamplifier impulse fall time (ns) 		& 5.3	& 5.3 \\
preamplifier peak delay time (ns)		& 9.5	& 9.5 \\
preamplifier 50\% delay time (ns)		& 7.8	& 7.8 \\
mean tagged electron rate (MHz)			& 0.1-5 & 0.1-5 \\
length of trace simulated ($\mu$s)		& 3	& 3  \\
length of trace analyzed ($\mu$s)		& 1	& 1  \\
size of time step simulated (ps)		& 100	& 100 \\
dark rate for whole device (MHz)		& 10-$10^5$ & 10-$10^5$ \\
total number of pixels in device		& 1700	& 14400 \\
pixel recovery time constant (ns)		& 1000	& 15 \\
cross-talk probability				& 0.13	& 0.02 \\
short after-pulse time constant (ns)		& 	& 18 \\
short after-pulse probability			&	& 0.06 \\
long after-pulse time constant (ns)		& 	& 70 \\
long after-pulse probability			&	& 0.05 \\
number of traces generated per setting		& 20000 & 20000 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}

An important aspect of the simulation is its ability to take into account
pile-up effects and the effects from AC couplings between various stages 
in the preamplifier, which create long oscillating tails on the back ends
of the output pulses.  The simulation started at t=0 with the state of
the readout in quiescent condition and then ran forward for 3~$\mu$s, but
only the last 1~$\mu$s was actually analyzed.  This is what is meant by
the parameters ``length of trace simulated'' and ``length of trace analyzed''
in Table~\ref{modelpars}.

The simulation consists of building up a trace of the output voltage on
the preamplifier output in the following way.  The elementary event in the
simulation is the firing of a single pixel.  Each time a pixel fires, a
copy of the single-pixel output pulse shown in Fig.~\ref{fig:impres} is
added with an appropriate time offset to the output trace.  First dark-rate
pulses are distributed randomly, one at a time, with the total number of
them equal to the dark rate times the trace length.  After that, signal
pulses are generated with pulse heights distributed as illustrated in
Fig.~\ref{fig:dE}.  Based on the generated pulse height, a Poisson random
number of pixels are added to the trace, with their times starting at the
$t_0$ of the pulse and distributed randomly according to the exponential
decay with the time constant of the scintillator.  After-pulses and
cross-talk pulses are generated automatically each time a pixel event is
produced.  Each pixel generated modifies a full 2~$\mu$s of the output
trace, because of the long oscillating tail that is present in the
preamplifier response function.  A sample trace is shown in
Fig.~\ref{fig:traces}.  This trace was produced for the Hamamatsu MPPC
with a signal rate of 4~MHz and a dark rate of 10~GHz.  Both dark pulses
and after-pulsing effects are visible, on top of the normal output pulse
shape.

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{trace0.eps}
\includegraphics[width=3.0in]{trace1.eps}
\caption{
\label{fig:traces}
Sample trace taken from the simulation of the Hamamatsu MPPC at a signal
rate of 4~MHz and a dark rate of 10~GHz.  The y axis is output voltage
expressed in units where a single pixel has a maximum amplitude of 1,
as shown in Fig.~\ref{fig:dE}.  The second panel is a zoomed view of
the first, showing the largest pulse in greater detail.
}
\end{center}
\end{figure}

Once the trace is generated, it is analyzed to simulate the behavior of
a leading-edge discriminator acting on the simulated signal.  The analyzer
steps through the trace watching for it to transition from below the
threshold level to above.  When this happens, the trace values below and
above the threshold are interpolated to find the crossing time $t_0$.  The
value of the trace amplitude a fixed time interval after $t_0$ is recorded
as the pulse height for that pulse.  For comparison with the generated 
value, the analyzer goes through the list of generated signal pulses and
finds the one whose generated time plus a predefined offset is closest to
the $t_0$ and matches up the generated time and amplitude with that found
by the analyzer.  The amplitude spectrum of the discriminated trace pulses
analyzed in this way, divided by the generated spectrum, gives the detection
efficiency of the readout as a function of pulse height.  The ratio of the
integrals of the two spectra gives the detection efficiency.  It might be
feared that measuring the efficiency this way might produce a bias on the
high side, because spurious (noise) triggers are also counted, but the
threshold used in this study was too high for this to occur at an
appreciable rate.  Its presence would be indicated by an extended flat
background outside the peak in the time resolution plot, and this is not
seen except at the highest dark rate $10^{11}$~Hz considered in this study.

\subsection{results for the CPTA 2~mm device}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{ampSSPM0.eps}
\includegraphics[width=3.0in]{effSSPM0.eps}
\caption{
\label{fig:ampSSPM0}
Amplitude spectrum (left panel) and detection efficiency as a function
of amplitude (right panel) for the CPTA device simulation at a signal
rate of 100~kHz and a dark rate of 10~MHz. The solid line in the left
panel is the generated spectrum, and the dashed line is the spectrum
of the detected pulses.  The spectrum-weighted efficiency in this
simulation is 99.8\%.
}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{twalkSSPM0.eps}
\includegraphics[width=3.0in]{tresSSPM0.eps}
\caption{
\label{fig:tresSSPM0}
Time walk correction (left panel) and time resolution (right panel) as
a function of pulse amplitude, for the CPTA device simulation at a signal
rate of 100~kHz and a dark rate of 10~MHz.  The spectrum-weighted RMS time
resolution from this simulation is 122~ps.
}
\end{center}
\end{figure}

To obtain baseline performance figures, the simulation was first carried
out at a low signal rate of 100~kHz and a pristine dark rate for this
device of $10^7$~Hz.  The generated and observed pulse height spectra
are shown in the first panel of Fig.~\ref{fig:ampSSPM0}.  The second panel
shows the discriminator detection efficiency as a function of generated
pulse amplitude.  Fig.~\ref{fig:tresSSPM0} shows the time-walk correction
and RMS time resolution as a function of pulse amplitude under the same
conditions.  At 100~kHZ the CPTA device easily meets all of the performance
requirements for the microscope readout.

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{ampSSPM1.eps}
\includegraphics[width=3.0in]{ampSSPM4.eps}
\caption{
\label{fig:ampSSPM}
Pulse height amplitude spectrum for the CPTA simulation at a signal
rate of 1~MHz (left panel) and 4~MHz (right panel).  The solid lines
are the generated amplitude spectra and the dashed lines are the detected
pulse heights.  The spectrum-weighted detection efficiencies are 98.5\%
and 92.5\%, respectively.
}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{tresSSPM0.eps}
\includegraphics[width=3.0in]{tresSSPM2.eps}
\caption{
\label{fig:tresSSPM}
RMS time resolution for the CPTA device at a signal rate of 4~MHz
and dark rate of 10~MHz (left panel) and 1~GHz (right panel).  The
corresponding spectrum-averaged resolutions are 176~ps and 228~ps,
respectively.  The discriminator detection efficiencies at 50~pe
threshold are 92.5\% and 55.3\%, respectively.
}
\end{center}
\end{figure}

Fig.~\ref{fig:ampSSPM} shows the generated and measured pulse amplitude
spectra at signal rates of 1~MHz and 4~MHz.  At the lower signal
rate, there is a noticeable compression of the dynamic range coming from
the limited number of 1700 pixels.  This tends not to affect the time
resolution very much because the missed photons tend to come from the
tail of the pulse, whereas the leading edge is not affected.  However
at the operating rate of 4~MHz one starts to see a significant depression
of the gain.  This comes about in the simulation because of the high rate
compared to the long pixel recovery time of 2.8~$\mu$s.
The product of the rate and the average of 350
photoelectrons per pulse means that the average pixel firing rate is
800~kHz, with the result that the pixels rarely recover their full 
operating potential between firings.  This might not be a problem if
it were consistent, but the random nature of the events means that the
effective gain is fluctuating from one signal pulse to the next by as
much as a factor 2.  Because of this, the pristine device detection
efficiency drops from 98.5\% to 92.5\% between 1~MHz and 4~MHz counting rate.

These observations have nothing to do with elevated dark rate.  The
the simulations reported so far were run under pre-radiated conditions
for the CPTA SSPM device.
Things eventually get even worse when the dark rate is increased.  The
pristine dark rate is 6~KHz per pixel, small compared to the 800~kHz
per pixel under operating conditions at 4~MHz.  Dark rate begins to
be noticeable as soon it becomes reaches the same magnitude as the signal
rate per pixel, which occurs at about a factor of 100 times the pristine
rate, or $10^9$~Hz.  Fig.~\ref{fig:tresSSPM} compares the time resolutions
at 4~MHz signal rate between device dark rates of $10^7$~Hz and
$10^9$~Hz.  The time resolution has degraded somewhat with increasing dark
rate, but the most important effect is on the detection efficiency.
At 4~MHz signal rate and $10^9$~Hz dark rate, the detection efficiency
at the discriminator threshold of 50~pe has dropped to only 56\%.

In summary, the pre-radiated CPTA device marginally satisfies the efficiency
specification of 95\% at the full operating rate of a microscope channel,
which is somewhat under 4~MHz, but it is already operating at a current
higher than is optimum for this device, because of its long pixel recovery
time.  If radiation damage causes the dark
rate to increase by a factor 100 above that of the pristine device, the gain
fluctuations caused by the long pixel recovery time seriously impact the
efficiency of the detector, to the point where it cannot function adequately
for the purposes of tagging.  One could recover most of the lost efficiency
by reducing the discriminator threshold to 20~pe, or perhaps even to 10~pe,
but at some level it begins to see a substantial rate of false hits from
dark current pile-up.  Further optimization of the simulation based on the
CPTA device was not attempted because running this device above 1~MHz seems
already to be stretching its capabilities, even in the absence of elevated
dark current.  Beside that, it would seem to be a waste of time to squeeze
the last ounce of performance out of the Photonique SSPM, when one compares
it to the Hamamatsu MPPC device treated in the next section.

\subsection{results for the Hamamatsu 3~mm device}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{ampMPPC4.eps}
\includegraphics[width=3.0in]{effMPPC4.eps}
\caption{
\label{fig:ampMPPC4}
Amplitude spectrum (left panel) and detection efficiency as a function
of amplitude (right panel) for the Hamamatsu device simulation at a signal
rate of 4~MHz and a dark rate of 10~MHz. The solid line in the left
panel is the generated spectrum, and the dashed line is the spectrum
of the detected pulses.  The spectrum-weighted efficiency in this
simulation is 94.8\%.
}
\end{center}
\end{figure}

Fig.~\ref{fig:ampMPPC4} shows the amplitude spectrum, and the detection
efficiency as a function of generated amplitude, for the pristine MPPC
device running at 4~MHz.  The Hamamatsu MPPC considered in this simulation
has a considerably smaller geometric fill factor than the CPTA SSPM device
(25 micron vs.\ 50 micron pixels), but its intrinsic quantum efficiency peaks
in the green-blue region, whereas the CPTA device QE peaks in the red.  The
net effect is that the two devices have similar PDE values for light from
BCF-20 scintillator, here taken to be 15\%.  This is why the generated
pulse height distribution comes out to be the same in the two simulations,
as can be seen by comparing the solid histgrams in Figs~\ref{fig:ampSSPM}
and \ref{fig:ampMPPC4}, left panel.  The measured pulse height distributions
are different, however, because the non-linearity and pixel-recovery effects
are different in the two cases.  Note that the large number of pixels and
the fast pixel recovery time make the measured spectrum much closer to the
generated one in the case of the MPPC device than was seen for the SSPM,
but in both cases some amount of dynamic range compression is visible in
the suppression of the Landau tail.

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{twalkMPPC4.eps}
\includegraphics[width=3.0in]{tresMPPC4.eps}
\caption{
\label{fig:tresMPPC4}
Time walk correction (left panel) and time resolution (right panel) as
a function of pulse amplitude, for the MPPC device simulation at a signal
rate of 4~MHz and a dark rate of 10~MHz.  The spectrum-weighted RMS time
resolution from this simulation is 181~ps.
}
\end{center}
\end{figure}

The time walk correction and RMS time resolution functions are shown in
Fig.~\ref{fig:tresMPPC4} for the pristine MPPC running at the maximum 
expected operating signal rate of 4~MHz.  Comparison with
Fig.~\ref{fig:ampMPPC4} shows that the time resolution in the region of
the peak amplitude is better than 200~ps, satisfying the design goal for
RMS time resolution under running conditions.

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{ampMPPC4a.eps}
\includegraphics[width=3.0in]{ampMPPC4b.eps}
\caption{
\label{fig:ampMPPC4ab}
Amplitude spectra for the MPPC device operating at a signal rate of 4~MHz
with a whole-device dark rate of 10~GHz (left panel) and 100~GHz (right panel).
The solid line is the generated spectrum, and the dashed line is the spectrum
of the detected pulses.
}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{tresMPPC4a.eps}
\includegraphics[width=3.0in]{tresMPPC4b.eps}
\caption{
\label{fig:tresMPPC4ab}
RMS time resolution as a function of pulse amplitude for the MPPC device
operating at a signal rate of 4~MHz with a whole-device dark rate of 10~GHz
(left panel) and 100~GHz (right panel).  The discriminator threshold was
left fixed at 50 pe.
}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{effMPPC4a.eps}
\includegraphics[width=3.0in]{effMPPC4b.eps}
\caption{
\label{fig:effMPPC4ab}
Detection efficiency as a function of generated amplitude for the MPPC device
operating at a signal rate of 4~MHz and a dark rate of 10~GHz (left panel)
and 100~GHz (right panel). The discrimator threshold was left fixed at
50 pe.
}
\end{center}
\end{figure}

To track how these performance parameters degrade with increasing dark rate,
the plots in Figs.~\ref{fig:ampMPPC4}-\ref{fig:tresMPPC4} were repeated at
device dark rates of 100~MHz, 1~GHz, 10~GHz, and 100~GHz.  Dark rate effects
were not noticeable at 100~MHz and 1~GHz, but they become significant at
10~GHz, and at 100~GHz the performance of the device is badly impaired.
Fig.~\ref{fig:ampMPPC4ab} shows the generated and measured amplitude
spectra for the highest two dark rates simulated.   In all of these dark
rate studies, the signal rate was kept constant at 4~MHz so that all
differences from what was shown in Figs.~\ref{fig:ampMPPC4}-\ref{fig:tresMPPC4}
would be due to dark rate effects alone.  Visual comparison of these spectra
suggest that the readout is still functioning at 10~GHz dark rate essentially
as it did when it was new at 4~MHz, but at 100~GHz it is showing substantial
pe yield fluctuations.  The corresponding RMS time resolutions and detection
efficiencies are shown in Figs.~\ref{fig:tresMPPC4ab}-\ref{fig:effMPPC4ab}.

\begin{table}
\caption{\label{tres_eff}
Time resolution and detection efficiency for different levels of dark
rate.  Dark rates stated are whole-device counts per second.}
\begin{center}
\begin{tabular}{ccrl@{\hspace{12mm}}rlrl}
\hline \hline
device & signal & 
\multicolumn{2}{c}{\hspace*{-8mm}dark rate} &
\multicolumn{2}{c}{\hspace*{-9mm}time resolution} &
\multicolumn{2}{c}{efficiency} \\
\hline
SSPM   &    4 MHz  &   10&MHz  &     166&ps   &        92.5&\% \\
       &           &  100&MHz  &     167&ps   &        91.7&\% \\
       &           &    1&GHz  &     195&ps   &        55.3&\% \\
       &           &   10&GHz  &      - &     &         0.0&\% \\
MPPC   &    4 MHz  &   10&MHz  &     181&ps   &        94.8&\% \\
       &           &  100&MHz  &     183&ps   &        94.9&\% \\
       &           &    1&GHz  &     188&ps   &        94.9&\% \\
       &           &   10&GHz  &     225&ps   &        94.7&\% \\
       &           &  100&GHz  &     381&ps   &        73.8&\% \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

These results are summarized in Table~\ref{tres_eff} for the full range of
dark rates studied.  Considered together, they show that the readout is
able to function at an acceptable level at a dark rate up to as high as 
10~GHz, but not much higher.

\section{Conclusions}

Operating the Hall D photon beam over a 10-year period (30,000 operating
hours) at full beam intensity, an unshielded microscope readout would
see a neutron dose of 5500~rem~\cite{somov2010}.  Surrounding the readout
electronics with plastic shielding could easily reduce this number to
800~rem~\cite{somov2010}, which brings the dose into the range covered
by Ref.~\cite{musienko2009}, eliminating uncertainties from a large
extrapolation.  Linear extrapolation of dark rate increase from 150~rem
to 5500~rem, based on Ref.~\cite{yi2010}, predicts a growth in dark rate
by the factor 440 for the Hamamatsu MPPC device.  Linear extrapolation of
the dark rate increase from 830~rem to 5500~rem, based on
Ref.~\cite{musienko2009}, predicts a growth in dark rate by the factor
200.  The smaller value is consistent with the slower-than-linear growth
of dark rate with radiation dose shown in Ref.~\cite{yi2010}.  For the
Hamamatsu S10931-025P MPPC, these predictions lead to dark rates of
$1.8\times 10^9$~Hz and $8\times 10^8$~Hz, respectively, at the end of
the 10-year period.  Both of these rates are far less than the upper
bound of 10~GHz for the dark rate, based on this simulation study.

These estimates do reply on extrapolations well beyond the range of
existing radiation damage measurements.  Evidence up to 800~rem points
to less-than-linear dark rate growth with radiation dose, but there is
some risk entailed in assuming that this trend continues up to
5500~rem.  Somov has shown~\cite{somov2010} that a relatively simple
shielding scenario, employing sheets of plastic as neutron moderators
around the readout, can reduce the neutron dose seen by the MPPC devices
by a factor of 6.  Doing this would bring the projected dose over a 10-year
period down to the same level as reported in Ref.~\cite{musienko2009},
thus avoiding extrapolation uncertainties.

Based on this study, the impact of radiation damage on the performance
of the tagger microscope readout based on Hamamatsu MPPC devices is
expected to be negligible over the projected lifetime of the detector.
Shielding the readout with sheets of plastic to moderate fast neutrons
is recommended to control risks associated with extrapolation of
existing radiation damage measurements.  It would also be useful to
extend the range of existing radiation damage measurements on these
devices up to the level of 5000~rem, to test the validity of the
extrapolation model for the dark rate, and also to check if other
effects on the operating characteristics begin to emerge at these
higher levels of radiation damage.


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\end{document}
