\section{Basic studies}
\label{sec:basic}

In this study, photon-induced
EM showers are simulated using 
the GEANT4 program~\cite{ref:basic:geant4}.
The nominal ECAL geometry used has 30 layers
of 500\,$\mu$m thick silicon sensors, with 0.6 radiation
lengths ($X_0$) of tungsten in front of each of the first 
20 layers and $1.2X_0$ in front of each of the last 10 layers.
This geometry is typical of the designs being 
considered~\cite{ref:introduction:sid, ref:introduction:ild}.
To allow for the difference in tungsten thickness,
the information from
the last 10 layers is weighted by a factor of two compared to
the first 20 layers
for all energy and resolution estimates presented later,

Each layer is simulated to have a 1\,mm thick PCB between the
tungsten and the silicon sensors, assumed to be required
for the sensor readout. There is also an air gap of 1\,mm
between the sensors of one layer and the tungsten of the next.
While this represents a very simplified geometry compared
with a realistic ILC-type ECAL, the shower properties are
mainly dominated by the tungsten thickness and spacing and these
correspond to realistic values.

Shower simulation uses ILC physics list.

For the results shown in this section,
no experimental effects have been included in the
simulation. Specifically, the analogue energy deposits are assumed
to be measured perfectly, with no resolution 
effects, while
the number of charged particles is determined from the
perfect simulation information as the number of separate 
particles emerging from the bac surface of each silicon layer.
Also, there are no dead regions in the sensitive layers.

Typical shower event display.


The basic result is
illustrated in figure~\ref{fig:basic:mips}
which shows the distribution of the
energy deposited and the number
of charged particles for various incident photon
energies. It is clear than the spread in the deposited
energy is larger than than in the number of charged
particles, for all photon energies.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=3.5cm]{basic/Ebulk.png}
\includegraphics[width=3.5cm]{basic/Nepi.png}
\caption{\sl 
Comparison of the distributions of the quantity
measuring shower energy using (left) the energy deposited
in the sensitive layers and (right) the number of
charged particles crossing the sensitive layers.
Results for a range of incident photon energies
from 1 to 50\,GeV are shown.
}
\label{fig:basic:mips}
\end{center}
\end{figure}

This supports the basic contention that the number of
charged particles is an intrinsically better measure
of the shower energy than the deposited energy.
Figure~\ref{fig:basic:compaE} shows the average
value of each measure as a function of photon energy and
shows both estimates demonstrate good linearity.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=7cm]{basic/compaE.png}
\caption{\sl 
Comparison of the linearity as a function
of the incident photon energy, when
measuring shower energy using (left) the energy deposited
in the sensitive layers and (right) the number of
charged particles crossing the sensitive layers.
}
\label{fig:basic:compaE}
\end{center}
\end{figure}

Figure~\ref{fig:basic:ideal} shows the
root mean square (RMS) of each measure as an estimate
of the basic irreducible resolution in the two cases.
It is clear
the number of charged particles gives a very
significant improvement over the energy deposited as
an estimator of the incident photon energy. Fitting
these resolutions to a function of the form
\begin{displaymath}
\frac{\sigma_E}{E} = \frac{s}{\sqrt{E\ {\rm in\  GeV}}} \oplus c
\end{displaymath}
give values for the stochastic terms $s$ of 14.6\% for the
``analogue'' (energy deposited) case and 9.5\% for
the ``digital'' (number of charged particles) case,
and for the constant terms $c$ of 1.0\% and 0.8\%, respectively.
Hence, the analogue case is degraded by a factor of around 1.5
compared to the digital case. 

The above is for an ECAL without detector effects such as noise,
dead areas, etc. It is known that a realistic analogue
ECAL with low noise electronics can approach the intrinsic resolution
for the analogue case~\cite{ref:physics:calice}.
The purpose of the study reported in the rest of the paper
paper is to estimate the resolution of a realistic binary ECAL
and compare it with the intrinsic resolution shown above.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=8cm]{basic/resEsq.png}
\caption{\sl 
Comparison of the resolution dependence on 
the incident photon energy, when
measuring shower energy using (green) the energy deposited
in the sensitive layers and (red) the number of
charged particles crossing the sensitive layers.
The lines show the results of the fits to the
data points, as described in the text.
}
\label{fig:basic:ideal}
\end{center}
\end{figure}
