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\begin{document}
\title{Influence of Readout Non-uniformity on Resolution in a Calorimeter}
\author{R.T. Jones \\
University of Connecticut, Storrs, CT}
\date{March 27, 2008}
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\vspace{2cm}
\begin{abstract}
The intrinsic energy resolution of a calorimeter is limited by
sampling fluctuations and photon detection statistics.  The reconstructed
photon energy resolution actually achieved in an experiment is generally
somewhat worse than the intrinsic resolution because of a number of factors,
which include external energy loss through early conversions and energy
leakage from the calorimeter, as well as various imperfections in the
calorimeter response.  Many of the latter effects can be mitigated through a
careful offline calibration procedure, but those arising from
variations in the readout sensitivity that occur between different segments
within a single digitization channel cannot.  This report examines the general
problem of readout non-uniformities and their effect on calorimeter resolution,
and shows that they contribute both to the statistical and floor terms
in the usual energy resolution formula, with the dominant effects seen
in the floor term.  General formulas are derived that can be used to calculate
the final calorimeter resolution resulting from a given
distribution of sensitivities at different scales of segmentation.
When formulas are applied to the specific case of the
GlueX barrel calorimeter, the results show that the final resolution is
surprisingly insensitive to large variations in the gain between
individual SiPM tiles or Planacon anode segments, once the offline
inter-channel gain calibration is carried out, provided that light
guides with suitable properties are used.
\end{abstract}

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The GlueX barrel calorimeter energy measurement is based on a sum of
signals from many independent sensors.  These sensors are summed in
subgroups to form electronic signals for digitization.  Each of these
digitized signals is multiplied by its own gain constant and
then summed over contiguous regions to generate the calorimeter energy
measurement for a shower.  The channel gain constants are not known
a-priori, so they must be determined from the data using a calibration
procedure.  Offline calibration procedures are used by virtually all calorimeter
experiments to determine these gain constants in a time-dependent fashion
based on the raw data collected by the experiment.  One such bootstrap
procedure that has proved effective for high-energy photoproduction
experiments is presented in Ref.~\cite{bootstrap-04}.  Discussion of 
the methods involved is outside the scope of this note, except to point
out that the best one can do using such a calibration procedure is to
asymptotically approach the calorimeter resolution that is permitted by
the intrinsic resolution plus other limiting factors.  One of these
other limiting factors is non-uniform gain that may occur within a segment
of the calorimeter subtended by a single
electronics channel.  For example, a light guide may have a non-uniform
probability of delivering a photon that enters its collection end to the
surface of the phototube at the other end, and the phototube may have a 
non-uniform photoelectron efficiency across the face of its photocathode.  

Such non-uniformities are present in every calorimeter, but they are of
particular concern in the region of the GlueX barrel calorimeter being read out
using silicon photomultiplier (SiPM) arrays.  The SiPM arrays are composed
of 3mm x 3mm tiles arranged in 4 x 4 groups and summed electronically to form a
single digitization channel.  Early prototypes of the tiled SiPM modules
indicate that variations in the product of the photon detection efficiency
(PDE) times the gain of individual tiles within a single module may be as
large as a factor 2 or even larger, unless special measures are taken to
match them.  {\em Special measures always entail costs.}
Therefore it is important to obtain quantitative information on the
impact that fine-grained variations in the calorimeter sensitivity have
on its energy resolution.  This is the issue addressed in this report.

\section{The Basic Model}

Consider a calorimeter as a large array composed of many identical sampling
elements.  The sampling element functions as the calorimetric unit cell
in the following analysis.  In the GlueX barrel calorimeter it may be
considered to be a single fiber together with the
surrounding lead and epoxy that constitutes its unit cell.  In a homogeneous
calorimeter such as the lead-glass array, a sampling element may be taken to
be a 1 mm$^3$ cube within a lead glass block.  The details of how the unit
cell is conceived is not important, provided that the following assumptions
are satisfied.
\begin{enumerate}
\item There are many elements per readout channel.
\item There are many readout channels in the full calorimeter.
\item A single gain constant $g_i$ describes the proportionality between
the energy deposition $E_i$ in cell $i$ and its mean contribution
$\left< y_i\right>$ to the response measured in the readout channel to
which the element belongs.
\item Apart from the different constants $g_i$, the response of all cells
in the calorimeter to a given deposited energy $E_i$ within that cell is the same.
\end{enumerate}

Consider an ensemble of many showers, each of which deposits exactly the same total 
contained\footnote{The final calorimeter energy uncertainty also contains
a contribution from external energy loss (backsplash, leakage out the
ends, early conversions) but these are independent from the gain
non-uniformity contribution and may be considered separately.}
energy $E$ in the calorimeter.  The showers are randomly distributed within
the acceptance of the calorimeter, however that is defined.  For each event
in the sample,
\begin{equation}
E = \sum_{i=0}^N E_i
\end{equation}
where $N$ represents the total number of cells in the entire calorimeter,
summed across all readout channels.  Note that this is an exact relation,
because $E$ is the total contained energy and $E_i$ is the local deposition in cell
$i$.  The $\{E_i\}$ vary from event to event in the sample, but $E$ is the
same for all events and is henceforth treated as a constant.  To visualize
this, the reader should note that most of the $E_i$ are zero for a given
event, and the non-zero values are mostly grouped in a limited region within
the vicinity of the shower axis.

The intrinsic response of the calorimeter in the model is encapsulated in
the distribution of the random variable $\hat{s_i} \sim R(E_i)$
where $R$ depends in a non-trivial way on the energy deposition $E_i$. 
The details of this distribution are not important, however, but only
its mean and its variance.  These are defined as follows.
\begin{equation}
\left<\hat{s_i}\right> = E_i
\label{eq:smean}
\end{equation}
\begin{equation}
\left<\hat{s_i}^2\right> - \left<\hat{s_i}\right>^2 = \alpha^2E_i
\label{eq:svar}
\end{equation}
where the constant $\alpha$ is the same for all cells.
Eq.~\ref{eq:smean} merely defines the scale for random
variable $\hat{s_i}$\footnote{The hat notation in the
random variable $\hat{s_i}$ is used to denote the abstract random variable
that represents a population for any given event with depositions $\{E_i\}$.  
Henceforth one is concerned with the concrete values $s_{ik}$ that the random
variable takes on for various events $k$ in the sample ensemble.  In what
follows, the event index $k$ is dropped and expectation values 
$\left< ..\right>$ of variables without hats are used to denote ensemble
averages of the concrete values, evaluated in the large-N limit of ensemble size.},
whereas Eq.~\ref{eq:svar} has 
important physical content.  The meaning of the constant $\alpha$ is clarified
by evaluating the variance of the random variable $s = \sum_{i=1}^{N}s_i$
which represents the total energy measured by all readout channels across
the calorimeter.  Before it is possible to compute the
variance of $s$, an expression is needed for the variance of the sample
values $s_i$ for cell $i$.  The distribution of these values is a convolution
of two fluctuations: shower deposition variations that cause the $\{E_i\}$ to be
different for every event, and sampling errors that produce a different $s_i$
for event even when $E_i$ is the same.  These two fluctuations are independent,
so their covariances are zero.
\begin{equation}
\left< s_i^2\right> - \left< s_i\right>^2 = \alpha^2\left<E_i\right> + 
\left(\left< E_i^2\right> - \left< E_i\right>^2\right)
\label{eq:vars}
\end{equation}
whereas
\begin{equation}
\left< s_i s_j\right> - \left< s_i\right>\left< s_j\right> = 
\left< E_i E_j\right> - \left< E_i\right>\left< E_j\right>
\label{eq:covars}
\end{equation}
because the sampling fluctuations in different cells are uncorrelated.
Note that the same is not true of the energy depositions, because the
constraint of energy conservation requires them to be correlated.
With these results in hand, one can now easily compute the mean and
variance of the total measured energy $s$.
\begin{equation}
\left< s\right> = E
\label{eq:newmeans}
\end{equation}
\begin{eqnarray}
V(s) &=& \sum_{i=1}^{N}\left(\left< s_i^2\right> - \left< s_i\right>^2\right) +
         \sum_{i\ne j}^{N}\left(\left< s_is_j\right> -
                                \left< s_i\right>\left< s_j\right>\right)
\nonumber\\
     &=& \alpha^2 \sum_{i=1}^N \left< E_i\right> + 
             \sum_{i=1}^N \left(\left< E_i^2\right> - \left< E_i\right>^2\right) +
             \sum_{i\ne j}^N \left(\left< E_i E_j\right>
                                 - \left< E_i\right>\left< E_j\right>\right)
\nonumber\\
     &=& \alpha^2 E
\label{eq:newvars}
\end{eqnarray}
where use has been made of the fact that
\begin{eqnarray}
V_E &=&
\sum_{i=1}^N \left(\left< E_i^2\right> - \left< E_i\right>^2\right) +
\sum_{i\ne j}^N \left(\left< E_i E_j\right> -
 \left< E_i\right>\left< E_j\right>\right)
\nonumber\\
&=& 0
\label{eq:varE}
\end{eqnarray}
by energy conservation. Eqs.~\ref{eq:newmeans}-\ref{eq:newvars} lead
to the familiar result for the intrinsic calorimeter energy resolution.
\begin{equation}
\frac{\sigma_s}{\left< s\right>} = \frac{\alpha}{\sqrt{E}}
\label{eq:sigmas}
\end{equation}
Eq.~\ref{eq:sigmas} shows that $\alpha$ is nothing other than the
coefficient of the statistical error term in the familiar expression
for the intrinsic energy resolution of the calorimeter, and that in
this idealized model there is no floor term\footnote{In is shown below
how gain non-uniformities lead naturally to the appearance of a floor term.}.
For the GlueX barrel calorimeter, it has a numerical value close to
$\alpha=0.05$.

The variable $s$ represents the total shower energy measured by the
calorimeter for the ideal case where all elements contribute to the
observed response with the same
weight factor.  Gain non-uniformities are introduced to the model by supposing
that the readout produces a distorted sum $y$ of the individual cell responses,
which is similar to $s$ except that each cell gets weighted by a gain factor
$g_i$.
\begin{equation}
y = \sum_{i=1}^{N}y_i = \sum_{i=1}^N g_i s_i
\label{eq:g_i}
\end{equation} 
Similar to what was done above for $s$, the moments of the $y$ distribution are
computed as follows.
\begin{eqnarray}
\left< y\right> &=& \sum_{i=1}^N g_i \left< s_i\right>
\nonumber\\
                &=& g E
\end{eqnarray}
where the new constant $g$ represents a global gain constant that simply sets
the scale for the response $y$ and has nothing to do with calorimeter resolution.
\begin{equation}
g \equiv \frac{1}{E}\sum_{i=1}^N g_i \left< E_i\right>
\label{eq:gdef}
\end{equation}
By simply rescaling $y$ one may arbitrarily set $g=1$.  Now it is convenient
to express the $g_i$ in terms of gain mismatch variables $\delta_i$.
\begin{equation}
g_i = 1+\delta_i
\label{eq:deldef}
\end{equation}
which obey the relation
\begin{equation}
\sum_{i=1}^N \delta_i \left< E_i\right> = 0
\label{eq:delmean}
\end{equation}
as follows from Eq.~\ref{eq:gdef}.

These results may be collected to calculate
the mean and variance of the $y$ energy measurement.
\begin{eqnarray}
\left< y\right> &=& E
\label{eq:ymean}
\\
V(y) &=& \sum_{i=1}^{N}\left(\left< y_i^2\right> - \left< y_i\right>^2\right) +
         \sum_{i\ne j}^{N}\left(\left< y_i y_j\right> -
                                \left< y_i\right>\left< y_j\right>\right)
\nonumber\\
     &=& \sum_{i=1}^{N}(1+\delta_i)^2\left(\left< s_i^2\right>-
                                           \left< s_i\right>^2\right) +
         \sum_{i\ne j}^{N}(1+\delta_i)(1+\delta_j)\left(\left< s_i s_j\right> -
                                \left< s_i\right>\left< s_j\right>\right)
\nonumber\\
     &=& \sum_{i=1}^{N}(1+\delta_i)^2\left(\alpha^2\left< E_i\right> + 
                            \left< E_i^2\right> - \left< E_i\right>^2\right)
\nonumber\\
     & & + \sum_{i\ne j}^{N}(1+\delta_i)(1+\delta_j)\left(\left< E_i E_j\right> -
                                \left< E_i\right>\left< E_j\right>\right)
\nonumber\\
     &=& \alpha^2 E + 2\alpha^2\sum_{i=1}^N\delta_i\left< E_i\right> + 
                  \alpha^2\sum_{i=1}^N\delta_i^2\left< E_i\right>
\nonumber\\
     & & + 2\sum_{i=1}^N\delta_i\left(\left< E_i^2\right> -
                                    \left< E_i\right>^2\right) +
         2\sum_{i\ne j}^N\delta_i\left(\left< E_i E_j\right> -
                                \left< E_i\right>\left< E_j\right>\right)
\nonumber\\
     & & + \sum_{i=1}^N\delta_i^2\left(\left< E_i^2\right> -
                                     \left< E_i\right>^2\right) +
         \sum_{i\ne j}^N\delta_i\delta_j\left(\left< E_i E_j\right> -
                                \left< E_i\right>\left< E_j\right>\right)
\label{eq:vary_ij}
\end{eqnarray}
The assumption that there are a large number of cells and that the gain
mismatch parameters average to zero allows us to neglect terms that contain
a single power of the gain mismatch parameters $\delta_i$ because they cancel
in the sum over many cells.  This leaves only the intrinsic terms and terms
containing two $\delta_i$ factors.
\begin{eqnarray}
V(y) &=& \alpha^2 E + \alpha^2\sum_{i=1}^N\delta_i^2\left< E_i\right> 
      +  \sum_{i=1}^N\delta_i^2\left(\left< E_i^2\right> -
                                     \left< E_i\right>^2\right)
\nonumber\\
     & & + \sum_{i\ne j}^N\delta_i\delta_j\left(\left< E_i E_j\right> -
                                \left< E_i\right>\left< E_j\right>\right)
\label{eq:vary}
\end{eqnarray}
The final term proportional to $\delta_i\delta_j$ only sums to zero if the
gain mismatches between different cells are really uncorrelated between the
cells.  This gives a physical significance to the choice of the size of the
unit cells in the
calorimeter that are used for computing results in the following sections.
Choosing a calorimeter cell size that matches the scale of local variations in
the gain mismatch allows one to consider the gains to be uniform within a cell,
but uncorrelated between different cells.  If the cell size is chosen smaller
than this, then the last term in Eq.~\ref{eq:vary} will need to be included
in computing $V(y)$.  If the cell size is chosen larger than this, then the
assumption that the gain is uniform within each cell is violated.  Assuming
that an appropriate choice for the cell size is made so that the $g_i$ are
uncorrelated\footnote{Actually when offline gain corrections are applied at
the level of readout channels, this introduces anti-correlations between the
$\delta_i$ within a single readout channel, so that the term containing
the product $\delta_i\delta_j$ in Eq.~\ref{eq:vary} has a negative sign
that cancels some of the excess error from the last term in
Eq.~\ref{eq:vary_trunc}.  This fact is ignored in Eq.~\ref{eq:vary_trunc},
but is re-introduced later in the section entitled {\em Offline Calibration}},
\begin{eqnarray}
V(y) &=& \alpha^2 E + \alpha^2\sum_{i=1}^N\delta_i^2\left< E_i\right>
      + \sum_{i=1}^N\delta_i^2\left(\left< E_i^2\right> -
                                     \left< E_i\right>^2\right)
\label{eq:vary_trunc}
\end{eqnarray}
The physical interpretation of the terms in Eq.~\ref{eq:vary_trunc} is as
follows.  The first term proportional to $\alpha^2$ is the intrinsic calorimeter
energy resolution that includes both sampling fluctuations and photon detection
statistics effects.  The second term proportional to $\alpha^2$ accounts for how
the intrinsic uncertainties are amplified by gain non-uniformities\footnote{
As an extreme example, imagine that the gains are zero
past a certain depth in the calorimeter.  This can be compensated
by increasing the global gain factor, but the sampling fluctuations
in the parts that do produce a response are amplified by the
larger gain factor.}.
The last term is typically more significant than the second.  It reflects the
fact that no two showers deposit energy the same way in the calorimeter, so
each one is sampled by cells with different gains.  Note that this
term is independent of the intrinsic calorimeter resolution parameter $\alpha$,
and depends only on the segmentation of the calorimeter.  Also note that it
scales with $E^2$, so that it introduces a so-called {\em floor
term} in the energy resolution formula
\begin{equation}
\frac{\sigma_y}{\left< y\right>} = \frac{a}{\sqrt{E}}\oplus b
\label{eq:sigmay}
\end{equation}
where $a$ is the intrinsic resolution parameter $\alpha$ amplified by gain
mismatch effects, and $b$ is the floor term.  The following sections are
concerned with estimating the values of the parameters $a$ and $b$.

\section{A Concrete Case}
To make the model more concrete, an ansatz is needed for the distribution of
the gain mismatch parameters $\delta_i$.  Suppose they are distributed according
to a quasi-normal distribution\footnote{The normal distribution will not serve
the purpose because gains must be positive definite,
whereas normal random variables cover the entire real axis.
A chi-square random variable with a suitably chosen
number of degrees of freedom $D$ has the advantage of being
positive definite and asymptotically normal for large $D$.  For this example,
$\delta_i = (\hat{x_i}/D) -1$ where
$\hat{x_i} \sim \chi^2(D)$ so that the $\delta_i$
have zero mean and standard deviation $\sqrt{2/D}$.  $D$ was chosen to give
the desired standard deviation $\sigma_{\delta}$ in the values for $\delta_i$.}
with mean zero and standard deviation $\sigma_{\delta}$, and that they are
distributed among the cells in a way that is independent of the way that energy
is typically shared within a shower.  Within this statistical model for the
gain mismatch distribution, Eq.~\ref{eq:vary_trunc} can be evaluated.
\begin{equation}
V(y) = \alpha^2(1+\sigma_{\delta}^2) E
      + \sigma_{\delta}^2 \sum_{i=1}^N\left(\left< E_i^2\right> -
                                     \left< E_i\right>^2\right)
\label{eq:vary_conc1}
\end{equation}
The magnitude of the last term in Eq.~\ref{eq:vary_conc1} can be estimated by
considering that variations in the depositions $E_i$ come about principally
because every shower hits the calorimeter in a different place. 

Consider some two-dimensional coordinate $\vec{x}$
that measures the impact point of each gamma ray of
energy $E$ in the sample, as it enters the calorimeter.  The energy
$E_i$ deposited in cell $i$ by any given shower depends principally on
the energy and position of the shower, and where the cell is located in the
calorimeter (eg.\ front, middle, back).  Let the coordinate $\vec{x_i}$
represent the value of shower coordinate $\vec{x}$ which maximizes the
average value of $E_i$ in cell $i$, and call that maximum value $\epsilon_i$.
All other showers with different impact coordinates $\vec{x}$ produce
reduced $E_i$ values.  Let the systematic variation of $E_i$ with $\vec{x}$
be presented by the shower shape function $f(\vec{x})$ which is defined to
have a maximum of 1 at $\vec{x}=0$ and go to zero as
$|\vec{x}|\rightarrow\infty$.  Let ${\cal A}$ represent the finite area
in $\vec{x}$ that describes the calorimeter acceptance.  Furthermore assume
that the function $f$ falls off sufficiently fast as
$|\vec{x}|\rightarrow\infty$ so that the integrals
\begin{eqnarray}
f_0(i) &=& \frac{1}{\cal A}\int d^2x \;{\cal I}(\vec{x})
\nonumber\\
f_1(i) &=& \frac{1}{\cal A}\int d^2x f(\vec{x}-\vec{x}_i)\;{\cal I}(\vec{x})
\nonumber\\
f_2(i) &=& \frac{1}{\cal A}\int d^2x f^2(\vec{x}-\vec{x}_i)\;{\cal I}(\vec{x})
\end{eqnarray}
are well-defined. Here ${\cal I}(\vec{x})$ represents the distribution of
the shower sample across the acceptance of the calorimeter.  The constants
$f_n(i)$ depend on the shower spatial distribution but not on the shower
energy $E$.\footnote{There are logarithmic corrections to this statement,
which must be included if one wants a consistent description over many
orders of magnitude.  These introduce logarithmic corrections to the
standard statistical + floor term energy resolution formula, and are
beyond the scope of this study.}
For a large sample, it follows that
\begin{eqnarray}
\left< E_i\right> &=& \frac{\epsilon_i}{f_0(i)}f_1(i)
\nonumber\\
\left< E_i^2\right> &=& \frac{\epsilon^2_i}{f_0(i)}f_2(i)
\nonumber
\end{eqnarray}
so that
\begin{equation}
\left< E_i^2\right> -\left< E_i\right>^2 = \epsilon^2_i
\left(\frac{f_2(i)}{f_0(i)} - \frac{f^2_1(i)}{f^2_0(i)}\right)\;.
\label{eq:varEi_as_E2}
\end{equation}
Note that the variance of the $E_i$ is proportional to $\epsilon^2_i$
independent of what is taken for the shower shape model $f(\vec{x})$.
Since $\epsilon_i$ is proportional to the total shower energy
$E$,\footnote{\em ditto}
it follows that {\em Var}$(E_i)$ is proportional to $E^2$.

As a concrete example, consider the GlueX barrel calorimeter and a sample
with fixed polar angle, so that $x\equiv \phi$ varies only in one dimension.
Taking a smooth Gaussian model for the transverse profile of a shower,
and assuming a uniform azimuthal shower distribution leads
to the following expression for the deposition in cell $i$ as a function of
shower centroid position $x$.
\begin{equation}
E_i = \epsilon_i e^{-\frac{(x-x_i)^2}{2r_E^2}}
\end{equation}
where $\epsilon_i$ is the maximum energy deposited in cell $i$,
which occurs when the shower axis passes through the cell,
and $r_E$ is the transverse rms radius of a shower of energy $E$. 
Such a simple one-dimensional model might apply to showers distributed
uniformly in azimuthal angle $\phi$ in the GlueX barrel calorimeter.
Averaging over the ensemble implies an average over $x$.
\begin{eqnarray}
\left< E_i\right> &=& 
\frac{\epsilon_i}{L}\int_0^L e^{-\frac{(x-x_i)^2}{2r_E^2}} dx
\nonumber\\
&=& \sqrt{2\pi}\,\frac{r_{_E}}{L}\,\epsilon_i
\end{eqnarray}
where $L$ represents the extent of generalized coordinate $x$ that lies within
the calorimeter acceptance.
\begin{eqnarray}
\left< E_i^2\right> &=& 
\frac{\epsilon_i^2}{L}\int_0^L e^{-\frac{(x-x_i)^2}{r_E^2}} dx
\nonumber\\
&=& \sqrt{\pi}\,\frac{r_{_E}}{L}\,\epsilon_i^2
\nonumber\\
&=& \frac{1}{\sqrt{2}}\left< E_i\right>\epsilon_i
\end{eqnarray}
\begin{eqnarray}
\left< E_i^2\right> - \left< E_i\right>^2 &=& 
\frac{1}{\sqrt{2}}\left< E_i\right>\epsilon_i\left(1-\sqrt{\pi}\frac{2r_{_E}}{L}\right)
\end{eqnarray}
Summing this expression over all cells leads to the result
\begin{equation}
V(y) = \alpha^2(1+\sigma_{\delta}^2) E
      + \sigma_{\delta}^2 E \frac{\overline{\epsilon}}{\sqrt{2}}
        \left(1-\sqrt{\pi}\frac{2r_{_E}}{L}\right)
\label{eq:vary_conc2}
\end{equation}
An estimate for the last factor can be obtained in the case of the
GlueX barrel calorimeter by estimating the transverse shower size to be one
third of the width of one module, so that $2r_E/L = 1/144$.
In a course-grained test of the formalism, one may divide the module into
$p$ cell layers in depth with layer thicknesses chosen to equalize the
value of $\epsilon_i$ in each layer.  In that case, one finds 
\begin{equation}
V(y) = \alpha^2(1+\sigma_{\delta}^2) E + \frac{\sigma_{\delta}^2}{\sqrt{2}p} E^2
\end{equation}
for the GlueX barrel calorimeter.  To test this, a Monte Carlo simulation of
the GlueX barrel calorimeter was carried out consisting of 1,000,000 events
with single showers of 1 GeV at polar angle 60$^{\circ}$ uniformly distributed
in azimuth.  The readout scheme used $p=9$ and $\sigma_{\delta}=0.15$, and
$\alpha=0$ in order to isolate the term of interest.  The Monte Carlo result for
$V(y)$ was $1.67\times 10^{-3}$, as compared with $1.75\times 10^{-3}$ computed
using the above model.  Translated into calorimeter energy resolution, this
corresponds to a floor term $b=4.2\%$.  This is a very large effect for such a
modest rms gain mismatch of 15\%, which demands the more refined treatment
presented in the next section.

\section{Offline Calibration}
The simple uncorrelated model of random gain mismatches distributed to
every cell in the calorimeter independently ignores the fact that these
gain shifts are partially suppressed by gain corrections applied to each
digitization channel in the offline data analysis.  Gain constants are
extracted from the data that allow the average gain mismatch in each
readout channel to be reduced to near zero.  Suppose that the cells are
grouped into readout channels such that the set $Q_r$ of cell indices
identifies the cells that belong to readout channel $r$.  Then
\begin{equation}
\sum_{i\,\in Q_r} \delta_i = 0
\label{eq:gmatch}
\end{equation}
for all readout channels $r$.  In a similar manner to Eq.~\ref{eq:varE},
this implies that
\begin{equation}
\sum_{i\,\in Q_r} \delta_i^2 +
\sum_{i,j\,\in Q_r}\!\!\!'\, \delta_i\delta_j
= 0
\label{eq:covar}
\end{equation}
where the prime on the second sum indicates that it excludes terms with
$i=j$.  In the limit of large $N$, the first sum in Eq.~\ref{eq:covar}
can be replaced with $n_r\sigma_{\delta}^2$.
Suppose that the $\delta_i$ have not been arranged in any special
order, so that there are no additional correlations between them
other than what is required by the constraint Eq.~\ref{eq:gmatch}.
In that case, Eq.~\ref{eq:covar} provides an expression for the
mean value of $\delta_i\delta_j$ for pairs $i\ne j$, after the gains have
been calibrated.  This allows Eq.~\ref{eq:vary_trunc} to be corrected
for the effects of offline gain calibration.
\begin{eqnarray}
V(y) &=& \alpha^2 E + \alpha^2\sum_{i=1}^N\delta_i^2\left< E_i\right> 
      +  \sum_{i=1}^N \delta_i^2\left(\left< E_i^2\right> -
                                     \left< E_i\right>^2\right)
\nonumber\\
     & & + \sum_r\sum_{i,j\,\in Q_r}\!\!\!'\,
            \delta_i\delta_j\left(\left< E_i E_j\right> -
                            \left< E_i\right>\left< E_j\right>\right)
\label{eq:vary_gmatch}
\end{eqnarray}
Within a single readout channel, the different $E_i$ are highly
correlated, whereas the $\delta_i$ are anti-correlated as required by 
Eq.~\ref{eq:covar}.  Thus the last sum in Eq.~\ref{eq:vary_gmatch}
has a negative sign and partially cancels the sum that precedes
it.  If the readout cells are smaller than the size of a typical
shower then one may approximate the correlation in the $E_i$ between
different cells within a single readout channel as nearly perfect,
in which case $\left< E_i E_j\right> - \left< E_i\right>
\left< E_j\right>\rightarrow \left< E_i^2\right> - \left< E_i\right>^2$
and the last two terms in Eq.~\ref{eq:vary_gmatch} completely cancel.
However, in practice shower energy deposition is grainy and the correlations
are never perfect.  Examination of the Monte Carlo results reported in
the previous section suggest that intra-channel correlations of the
$E_i$ are on the order of 80\%-85\% for cell sizes that correspond
to the GlueX barrel calorimeter readout segmentation. Thus
\begin{eqnarray}
V(y) &=& \alpha^2 E + \alpha^2\sum_{i=1}^N\delta_i^2\left< E_i\right> 
      +  (1-\rho)\sum_{i=1}^N \delta_i^2\left(\left< E_i^2\right> -
                                     \left< E_i\right>^2\right)
\nonumber\\
    &=& \alpha^2(1+\sigma_{\delta}^2) E
      + (1-\rho) \sigma_{\delta}^2 E^2 \frac{1}{\sqrt{2}p}
\label{eq:vary_corr}
\end{eqnarray}
where $\rho$ is the correlation coefficient between $E_i$ and $E_j$
\begin{equation}
\rho_{ij} = \frac{\left< E_iE_j\right>-\left< E_i\right>\left< E_j\right>}
{\sqrt{\left[\left< E_i^2\right>-\left< E_i\right>^2\right]
       \left[\left< E_j^2\right>-\left< E_j\right>^2\right]}}
\end{equation}
for cells $i\ne j$ within the same readout channel.  In writing
Eq.~\ref{eq:vary_corr} it was assumed that {\nobreak
$(\left< E_i^2\right>-\left< E_i\right>^2) =
(\left< E_j^2\right>-\left< E_j\right>^2)$} for $i,j$ in the
same readout channel, and that $\rho_{ij}$ is approximately the same for
all $i,j$ and all channels $r$, so that the subscripts can be dropped.
The factor $(1-2\sqrt{\pi}\rho/L)$ in Eq.~\ref{eq:vary_conc2} has been
dropped in Eq.~\ref{eq:vary_corr} because it is within a few percent
of unity for large calorimeters like those in GlueX, and the uncertainty
in the factor $(1-\rho)$ makes it irrelevant.
The value $\rho=0.80\pm 0.05$ is a good estimate for the GlueX barrel
calorimeter.  The offline gain calibration reduces the size of the floor
term of the resolution formula by a factor $\sqrt{1-\rho} \approx 0.4$ 
relative to the case where no channel-by-channel gain match is done.
For the case examined in the previous section, the model predicts
$b=1.8\%$ instead of $b=4.1\%$ once the gain calibration is taken
into account.

Of course, one might like to tolerate intra-channel gain mismatch factors
significantly larger than $\sigma_{\delta} = 15$\%.  Eq.~\ref{eq:vary_corr}
shows that the floor term is proportional to $\sigma_{\delta}$.  This means
that relaxing the uniformity specification on $\sigma_{\delta}$ from 15\% to
33\% would bring the floor term back up the to level of the original
estimate $b = 0.041$.  A floor term of 4\% is higher than is usually
considered tolerable for electromagnetic calorimeters.   An acceptable
design goal for the GlueX barrel calorimeter might be $b\le 0.02$, which
corresponds to $\sigma_{\delta}\le 0.17$.  How this upper bound translates
into specifications for the calorimeter readout components is the subject
of the next section.

\section{Light Guide Mixing}
Naively it may appear that $\sigma_{\delta}$ refers to the standard deviation
in the gain of the photon detector that is observed by scanning a point
light source across its surface.  This would be the case if the light
collection system entailed a one-to-one map between cells in the
calorimeter and segments of the photodetector surface, eg.\ if the
fibers were directly coupled to the detectors.  However in the GlueX
barrel calorimeter there is a light guide between them, as is the case
in most electromagnetic calorimeter designs.  The primary purpose for
the light guide is to reduce the required photodetector area, but it
also has the important side effect of reducing the impact of local
variations in the photodetector gain across its surface on the
calorimeter energy resolution.  The effects of light guide mixing
on the energy resolution are taken into account as follows.

Consider that both the
calorimeter and the photon detector each have $n_r$ independent cells
within readout channel $r$, and that the coupling between them
allows any calorimeter cell to illuminate any detector tile.  Let the
coupling between the calorimeter cells and the photodetector tiles be
described by a matrix $C_{ij}$, such that
\begin{equation}
y_i = \sum_{j=1}^N \gamma_j\, C_{ij}\, s_i
\label{eq:yfromC}
\end{equation}
where $\gamma_j$ is the exact analog of $g_i$ in Eq.~\ref{eq:g_i}, except
that it refers to the readout gain for light collected by tile $j$ on the
photodetector instead of for light produced in cell $i$.
The coupling matrix $C_{ij}$ is a $N$ x $N$ matrix in block-diagonal form,
with positive definite elements located in $n_r$ x $n_r$ blocks along the
diagonal.  It follows from comparing Eq.~\ref{eq:g_i} with
Eq.~\ref{eq:yfromC} that
\begin{equation}
g_i = \sum_{j=1}^N C_{ij}\gamma_j
\label{eq:gfromC}
\end{equation}
which shows that the effect of matrix $C$ being different from the unit
matrix is to make the $\{g_i\}$ more uniform than the $\{\gamma_j\}$.
Suppose that the $\gamma_j$ can be described by a quasi-normal distribution
centered on 1 with some standard deviation $\sigma_d$, and that
the gains $\gamma_j$ fluctuate independently from tile to tile across
the photodetector surface.  Let
\begin{equation}
\gamma_j = 1+d_j
\label{eq:ddef}
\end{equation}
for the $\{\gamma_j\}$ in analogy with Eq.~\ref{eq:deldef} for the $\{g_i\}$.
It is convenient to normalize the coupling matrix $C_{ij}$ by the relations
\begin{equation}
\sum_{i,j\in Q_r} C_{ij} = n_r
\label{eq:Cnorm}
\end{equation}
for all readout channels $r$,
consistent with the limiting case of the unit matrix $C_{ij} = \delta_{ij}$.
The offline calibration guarantees that the net response in each readout
channel $r$ is correct, that is
\begin{equation}
\left< y_r\right> = \sum_{i\in Q_r} \left< E_i\right> \equiv \left< E_r\right>
\label{eq:E_r}
\end{equation}
where $E_r$ is the total energy deposition within the cells subtended by
readout channel $r$.
Eq.~\ref{eq:E_r} can be rewritten in terms of the matrix $C$ by assuming
that the group of cells subtended by readout cell $r$ is small enough that
the $\left< E_i\right>$ are approximately equal for all $i\in Q_r$.
\begin{eqnarray}
\sum_{i,j\in Q_r} C_{ij} d_j &=& 0
\label{eq:dmean}
\end{eqnarray}

It is natural to suppose that the light-guide couplings of all of the
readout channels in the calorimeter have the same physical structure.  Then
all of the $n_r$ x $n_r$ sub-matrices along the diagonal in $C_{ij}$ are
identical, and can be denoted by the lower-case matrix $c_{ij}$.  This
assumption is not essential to the following treatment, but it is adopted
because it allows the some of the formulas to be written in a simplified way.
None of the final results depend on all of the cells actually having an
identical light-guide coupling scheme.

Eq.~\ref{eq:E_r} implies that the total measured energy has the 
correct mean $\left< y\right> = E$.
The remaining task is to compute its variance.  Beginning from
Eq.~\ref{eq:yfromC}, it may be written as
\begin{equation}
V(y) = \sum_{i,i'}^N\sum_{j,j'}^N C_{ij}C_{i'j'}\gamma_j\gamma_{j'}
\left(\left<s_i s_{i'}\right> - \left< s_i\right>\left< s_{i'}\right>\right)
\end{equation}
where all four indices $i,i',j,j'$ sum from 1 to N, but the factors
$C_{ij}$ and $C_{i'j'}$ guarantee that only pairs $i,j$ and $i',j'$
within the same readout channel actually contribute to the sum.
Eqs.~\ref{eq:vars}-\ref{eq:covars} allows this to be rewritten as
\begin{equation}
V(y) = \sum_{i,i'}^N\sum_{j,j'}^N C_{ij}C_{i'j'}\gamma_j\gamma_{j'}
\left(\alpha^2\left< E_i\right>\delta_{ii'} +
\left< E_i E_{i'}\right> - \left< E_i\right>\left< E_{i'}\right>\right)
\end{equation}
Replacing $\left< E_i\right>$ with $\left< E_r\right>/n_r$ leads to
\begin{eqnarray}
V(y) &=& \alpha^2\frac{1}{n_r}\sum_r \left< E_r\right> \sum_{i\in Q_r}
\sum_{j,j'} C_{ij}C_{ij'}(1+d_j)(1+d_{j'})
\nonumber\\
& & + \sum_{i,i'}^N\sum_{j,j'}^N C_{ij}C_{i'j'}\gamma_j\gamma_{j'}
\left( \left< E_i E_{i'}\right> - \left< E_i\right>\left< E_{i'}\right>\right)
\nonumber\\
&=& \alpha^2 E\, M_1
 + \alpha^2\frac{1}{n_r}\sum_r \left< E_r\right> \sum_{i\in Q_r}
\sum_{j,j'} C_{ij}C_{ij'}d_jd_{j'}
\nonumber\\
& & + \sum_{i,i'}^N\sum_{j,j'}^N C_{ij}C_{i'j'}(1+d_j)(1+d_{j'})
\left( \left< E_i E_{i'}\right> - \left< E_i\right>\left< E_{i'}\right>\right)
\label{eq:vary_red1}
\end{eqnarray}
where in the last step terms have been dropped that involve a single
power of the factors $d_j$ because they average to zero.  The factor $M_1$
introduced in Eq.~\ref{eq:vary_red1} is one of several bilinear sums
of the $c_{ij}$ that are introduced to abbreviate the formulas.
\begin{eqnarray}
M_0 &=& \frac{1}{n_r}\sum_{i=1}^{n_r}\sum_{j=1}^{n_r} c_{ij}^2
\label{eq:M0}
\\
M_1 &=& \frac{1}{n_r}\sum_{i=1}^{n_r}\sum_{j=1}^{n_r}\sum_{j'=1}^{n_r} c_{ij}c_{ij'}
\label{eq:M1}
\\
M_2 &=& \frac{1}{n_r}\sum_{i=1}^{n_r}\sum_{i'=1}^{n_r}\sum_{j=1}^{n_r} c_{ij}c_{i'j}
\label{eq:M2}
\end{eqnarray}
All three parameters $M_0$, $M_1$, and $M_2$ are normalized to equal
1 for the special case $c_{ij} = \delta_{ij}$.

The statistical model for the gain mismatch parameters $d_j$ allows
the remaining sums in Eq.~\ref{eq:vary_red1} to be evaluated in the
large-$N$ limit.  Each of them contain products 
of the form $d_jd_{j'}$.  The $d_j$ are assigned randomly to the
individual tiles of the readout, without regard to what the $C_{ij}$
are for that tile or what the $\left< E_i\right>$ values are for the
cells that couple to it.  Because of that, their values can be
replaced with their averages in the large-$N$ limit.
The only restriction that ties them together for different
$j$ is Eq.~\ref{eq:dmean} that expresses the overall channel
gain match constraint of the offline gain calibration.
If $j$ and $j'$ belong to different readout
channels then their $d_j$ and $d_{j'}$ are completely uncorrelated, in
which case they average to zero just like the factors with a solitary
$d_j$ do.  However terms with $j=j'$ do not average to zero because
it is given that $\sum_{j=1}^N d_j^2 \rightarrow N\sigma_d^2$
in the large $N$ limit.
\begin{equation}
\overline{d^2} = \sigma_d^2
\label{eq:d2mean}
\end{equation}
Squaring Eq.~\ref{eq:dmean} can be used to obtain an average value
$\overline{dd'}$ for $d_jd_{j'}$ with $j\ne j'$ within the same readout cell.
\begin{eqnarray}
0 &=& \sum_{i\in Q_r}\sum_{i'\in Q_r}\sum_{j,j'} C_{ij} C_{i'j'} d_j d_{j'}
\nonumber\\
&=& \sum_{i\in Q_r}\sum_{i'\in Q_r}\sum_{j} C_{ij}C_{i'j} d_j^2
+ \sum_{i\in Q_r}\sum_{i'\in Q_r}\sum_{j\ne j'} C_{ij}C_{i'j'} d_jd_{j'}
\nonumber\\
&=& n_r\,\sigma_d^2\, M_2 + \overline{dd'}
\left[\left(\sum_{i,j\in Q_r} C_{ij}\right)
\left(\sum_{i',j'\in Q_r} C_{i'j'}\right)-n_r\,M_2\right]
\nonumber\\
\overline{dd'} &=& -\;\frac{\sigma_d^2\,M_2}{n_r-M_2}
\label{eq:ddmean}
\end{eqnarray}
Substitution of Eqs.~\ref{eq:d2mean}-\ref{eq:ddmean} into
Eq.~\ref{eq:vary_red1} leads to
\begin{eqnarray}
V(y) &=& \alpha^2 E\, M_1
 + \alpha^2\,E \left[\sigma_d^2\, M_0 + \overline{dd'}(M_1-M_0)\right]
\nonumber\\
& & + \sum_{i,i'}^N\sum_{j,j'}^N C_{ij}C_{i'j'}(1+d_j)(1+d_{j'})
\left( \left< E_iE_{i'}\right> - \left< E_i\right>\left< E_{i'}\right>\right)
\nonumber\\
&=& \alpha^2\,E\, M_1
 + \alpha^2\,E\,\sigma_d^2 \left[M_0 -
\frac{M_2(M_1-M_0)}{n_r-M_2}\right]
\nonumber\\
& & + \sum_{i,i'}^N\sum_{j,j'}^N C_{ij}C_{i'j'}(1+d_j)(1+d_{j'})
\left( \left< E_iE_{i'}\right> - \left< E_i\right>\left< E_{i'}\right>\right)
\nonumber\\
&=& \alpha^2\,E\, \left[ M_1 + \sigma_d^2\;
\left( \frac{n_r M_0 - M_1 M_2}{n_r - M_2}\right)\right]
\nonumber\\
& & + (1-\rho)\sum_{i=1}^N\sum_{jj'}^N C_{ij} C_{ij'} d_j d_{j'}
\left( \left< E_i^2\right> - \left< E_i\right>^2\right)
\nonumber\\
& & + \sum_{i,i'}^N\sum_{j,j'}^N C_{ij} C_{i'j'}
\left( \left< E_iE_{i'}\right> - \left< E_i\right>\left< E_{i'}\right>\right)
\label{eq:vary_red2}
\end{eqnarray}
Further work is required to evaluate the two remaining sums in 
Eq.~\ref{eq:vary_red2}.  The first can be simplified by substitution
of the average values for the products $d_j^2$ and $d_jd_{j'}$.
\begin{eqnarray}
& & \sum_{i=1}^N\sum_{jj'}^N C_{ij} C_{ij'} d_j d_{j'}
\left( \left< E_i^2\right> - \left< E_i\right>^2\right)
\nonumber\\
&=& \sum_{i=1}^N\sum_{jj'}^N C_{ij} C_{ij'} \left[\sigma_d^2\delta_{jj'}
- \left(\frac{\sigma_d^2\, M_2}{n_r-M_2}\right)(1-\delta_{jj'})\right]
\left( \left< E_i^2\right> - \left< E_i\right>^2\right)
\nonumber\\
&=& \sigma_d^2\left(M_0 - \left(\frac{M_2}{n_r-M_2}\right)(M_1-M_0)\right)
\sum_{i=1}^N\left( \left< E_i^2\right> - \left< E_i\right>^2\right)
\nonumber\\
&=& \sigma_d^2\left(\frac{n_r\,M_0 - M_1 M_2}{n_r-M_2}\right)
\sum_{i=1}^N\left( \left< E_i^2\right> - \left< E_i\right>^2\right)
\end{eqnarray}
To aid in the evaluation of the second sum in Eq.~\ref{eq:vary_red2},
it is helpful to introduce the new parameters $e_i$ defined as
\begin{equation}
\sum_{j=1}^N C_{ij} = 1+e_i
\end{equation}
which measure the deviation from uniform net coupling between cell $i$ 
and the calorimeter readout.  Eq.~\ref{eq:Cnorm} requires that the $e_i$
average to zero across a readout channel.
\begin{eqnarray}
&& \sum_{i,i'}^N\sum_{j,j'}^N C_{ij} C_{i'j'}
\left( \left< E_iE_{i'}\right> - \left< E_i\right>\left< E_{i'}\right>\right)
\nonumber\\
&=& \left< \left[ \sum_{i=1}^N (1+e_i) \left( E_i - \left< E_i\right>\right)
\right]^2 \right>
\nonumber\\
&=& \sum_{i,i'}^N e_i\,e_{i'}\,
\left( \left< E_iE_{i'}\right> - \left< E_i\right>\left< E_{i'}\right>\right)
\nonumber
\end{eqnarray}
The last step above follows from Eq.~\ref{eq:varE}.  The terms $i\ne i'$
in this sum contain factors $e_ie_{i'}$ which have both signs and average
to zero for large $N$, leaving only the terms with $i=i'$, so that
\begin{eqnarray}
&& \sum_{i,i'}^N\sum_{j,j'}^N C_{ij} C_{i'j'}
\left( \left< E_iE_{i'}\right> - \left< E_i\right>\left< E_{i'}\right>\right)
\nonumber\\
&=& \sum_{i}^N e_i^2
\left( \left< E_i^2\right> - \left< E_i\right>^2\right)
\nonumber\\
&=& \left(\frac{1}{N}\sum_{i}^N e_i^2\right)
\sum_{i}^N \left( \left< E_i^2\right> - \left< E_i\right>^2\right)
\nonumber\\
&=& (M_1-1)
\sum_{i}^N \left( \left< E_i^2\right> - \left< E_i\right>^2\right)
\end{eqnarray}
Substitution of these results back into Eq.~\ref{eq:vary_red2} leads to
a final result for the variance of the measured energy for a sample of
showers with fixed contained energy $E$.
\begin{eqnarray}
V(y) &=& \alpha^2\,E\, \left[ M_1 + \sigma_d^2\;
\left( \frac{n_r M_0 - M_1 M_2}{n_r - M_2}\right)\right]
\nonumber\\
& & + (1-\rho)\left[M_1 - 1 + \frac{n_r M_0 - M_1 M_2}{n_r - M_2}\right]
\sum_{i=1}^N\left( \left< E_i^2\right> - \left< E_i\right>^2\right)
\label{eq:vary_red3}
\end{eqnarray}
Comparison with Eq.~\ref{eq:vary_corr} allows shows that this result
agrees with the basic model, with the identification
\begin{equation}
\sigma_{\delta}^2 = M_1 - 1 + \sigma_d^2
\left(\frac{n_r M_0 - M_1 M_2}{n_r - M_2}\right)
\label{eq:vardel-vard}
\end{equation}
Eq.~\ref{eq:vardel-vard} shows how specific design choices for the
light-guides and photodetector uniformity translate into a floor term
in the energy resolution formula.

\section{Examples}
Some examples of how to use the results derived in this report are
useful to illuminate their meaning.  In all of the following examples,
fixed values are taken for the intrinsic calorimeter resolution parameter
$a=0.050$ and shower auto-correlation parameter $\rho=0.080$.

\subsection{Direct coupling}
In the direct coupling example $C_{ij}=\delta_{ij}$, corresponding to
photodetectors that are glued directly to the ends of the fibers, with
100\% coverage.  In this case, Eqs.~\ref{eq:M0}-\ref{eq:M2} give
$M_0=M_1=M_2=1$ and $\sigma_{\delta}=\sigma_d$.  This is a worst-case
scenario in the sense that variability in the local photodetector
sensitivity translates directly into variability in the calorimeter
sensitivity to local shower energy deposition fluctuations.

\subsection{Perfect light guides}
At the other extreme from direct coupling is the limit where the light
guides couple all fibers in a readout channel to the photodetector
surface with equal efficiency, but there is no correlation between
which fiber produced the photon and which photodetector tile collected it.
In this case all of the rows of the matrix $c_{ij}$ are identical,
ie.\ $c_{ij}$ does not depend upon $i$, leading to $M_1=1$ and $M_0=M_2$,
which gives $\sigma_{\delta}=0$.  This means that there is no floor term at
all for perfect light guides within this model, no matter how non-uniform
the photodetectors turn out to be.  Of course, this assumes that the
offline gain calibration can be done perfectly, which is an idealization.

Note that in order to be {\em perfect} in this model, the
light guides do not need to illuminate the photodetector uniformly --
$c_{ij}$ can still depend on photodetector tile index $j$ -- but they
do need to collect light with uniform efficiency at the collection end
and there can be no gaps between the light guides.
Eqs.~\ref{eq:vary_red3}-\ref{eq:vardel-vard} provide a
quantitative means to evaluate a concrete design to check how close
it comes to the ideal coupling limit, in terms of the final energy
resolution.

\section{Summary}
The final energy resolution of a calorimeter with a segmented
photodetector readout is given by
\begin{equation}
V(y)= \alpha^2(1+\sigma_{\delta}^2) E
      + (1-\rho) \sigma_{\delta}^2 E^2 \frac{1}{\sqrt{2}p}
\end{equation}
where $y$ is the total measured energy after calibrated gain corrections
have been applied, $\alpha$ is the intrinsic calorimeter resolution
for 1~GeV showers, $\rho$ is the shower deposition auto-correlation 
between independent sampling cells within one readout channel, and
$p$ is the number of independent sampling cells that lie within the
core of a shower using energy-weighted counting with the maximum cell
counting as 1.

The size of an independent sampling cell is set by the self-correlation
of the readout efficiency described by the light coupling matrix $C$.  For
the GlueX calorimeter readout a conservative scale is obtained by dividing
each readout light guide into 4 zones of roughly equal area (corners,
edges, center, and mid-zone) which characterize the coupling across the
entrance face of the light guide.  Taking 4 independent cells per light
guide and approximately 9 light guides for the energy-weighted channel
count for a 1~GeV shower\footnote{The value of $p$ depends on shower energy,
but only weakly like $log(E/E_C)$, leading to logarithmic corrections to
the standard energy resolution formula Eq.~\ref{eq:sigmay}.  These are
not considered here.} gives $p=36$.  Taking the value $\rho=0.80$ leads to 
\begin{equation}
V(y)= \alpha^2(1+\sigma_{\delta}^2) E
      + (0.063\; \sigma_{\delta})^2\; E^2
\end{equation}
for this readout geometry.  The value of $\sigma_{\delta}$ is given
by Eq.~\ref{eq:vardel-vard}, with parameters $M_0$, $M_1$, and $M_2$
defined in Eqs.~\ref{eq:M0}-\ref{eq:M2}.  How large a tolerance $\sigma_d$
can be allowed for intra-channel gain variations within a photodetector
module depends on the uniformity of light guide coupling described by
the matrix $C$ and encapsulated in the sums $M_0$, $M_1$, and $M_2$.

In the worst-case scenario where the photodetectors are glued directly
onto the fibers, one obtains $\sigma_{\delta}=\sigma_d$.  In this case,
a specification $\sigma_d=0.25$ would lead to a floor term in the
energy resolution of $b=1.6\%$, which might be considered acceptable.
A more realistic light-guide scenario would lead to a significantly
less stringent specification for the photodetector gain variability,
for a given floor term. 

\section{Recommendations}

Based on the results of the above study, it is recommended that a
ray-tracing Monte Carlo study be made of the light-coupling efficiency
function of the light guides to be used with the GlueX barrel 
calorimeter.  A 4 x 4 grid on the input and on the output faces might
be used, leading to a 16 x 16 matrix $C$.  The parameters $M_0$, $M_1$,
and $M_2$ defined in Eqs.~\ref{eq:M0}-\ref{eq:M2} can then be used
to evaluate the rms spread in local readout sensitivity 
$\sigma_{\delta}$ for a given rms spread in local photodetector gain
$\sigma_d$.  These results can then be used to derive a specification
for $\sigma_d$ that can be given to the photodetector manufacturer.
If this specification results in significant extra production costs,
the design of the light guide might be refined to further reduce
$\sigma_{\delta}$ for fixed $\sigma_d$, so that the specification
on $\sigma_d$ can be further relaxed.

\begin{thebibliography}{10}
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi

\bibitem{bootstrap-04}
R.T.~Jones et.al., ``A Bootstrap Method for Gain Calibration and Resolution
Determination of a Lead-Glass Calorimeter'', Nucl. Instr. Meth. {\bf A 566}
(2006) 366.

\end{thebibliography}
\end{document}
