\section{Choice of technique}

Two basic methods have been considered for producing photons of the
highest possible energy, flux and polarization from electrons of 
$E_{0}=12$ \eunit.
The methods are bremsstrahlung and Compton scattering of light. Both
are well-established methods of producing photon beams.  Both techniques
are actually described by the same Feynman diagrams, shown in
Fig.~\ref{qedgraphs}.  In the case of Compton scattering the incoming
photon is real, whereas it is virtual for the case of bremsstrahlung.

\begin{figure}[h!]\centering
\includegraphics[width=0.75\textwidth]{ch4_qedgraphs.eps}
\caption[Generic diagrams for hard photon production.]{\label{qedgraphs}
Generic diagrams for hard photon production from a high energy
electron beam. The symbol $ \times $ represents either a static charge
distribution, in the case of virtual photons in the initial state
(i.e. bremsstrahlung), or an optical cavity, in the case of real photons
in the initial state (i.e. Compton scattering).}
\end{figure}

Each of these techniques has its own limitations and advantages. In order
to be suitable for \GX, the photon source must be capable of producing
photons of energy at least $80$\% $E_{0}$, ($9\, GeV$).  
The photon beam should have linear
polarization.  The energy resolution for individual photons in the beam
should be as high as possible, {\em i.e.} on the order of the energy spread
of the electron beam itself.  It should be capable of producing intensities
up to $10^8$/s.  The contamination of the beam with photons outside the
desired energy band should be as low as possible.  It is also important
that the source be reliable and require a minimum of down-time for
maintenance.  The suitability of each approach is discussed below in the
light of these criteria.

\subsection{Compton back-scatter}

A Compton source begins with a beam of visible or ultraviolet light,
typically from a laser that is aligned to intersect the incident electron
beam at close to $180^{\circ}$.  Some of the photons undergo Compton scattering
with the beam electrons.  In the lab frame, the scattered photons come out
in a narrow cone about the incident electron direction and carry a
significant fraction of the electron energy.

The basic design of the Compton back-scatter source for this study
was put forward by  C. Keppel and R. Ent \cite{Keppel}.
The design entails the use of a four-mirror high-gain cavity pumped by
a 10~kW argon-ion laser putting out $2$~\pstunit\ pulses at a frequency
of $100$~\mhz. The pulses in the cavity are synchronized so that the light
pulses intercept an electron bucket each time they pass through the beam.
The total length of the cavity is $2$~\mlunit\ 
with a crossing angle of $ 1^{\circ}$.
Both cavity and electron beam are focused to a tiny spot of $10$~\micron{} 
r.m.s.\ radius at the crossing point.  A small spot size is necessary in order
to get as high a scattering rate as possible.  The gain of the cavity is
$10^4$, which is conservative in view of recent advances in mirror technology.
The wavelength of the light is $514$~\nmlunit.  The rate spectrum of the 
back-scattered beam from this source is shown in Fig~\ref{comptonbs}a for 
a $1$~\miamps{} electron beam at $12$~\eunit.

\begin{figure}[h!]\centering
\includegraphics[width=0.75\textwidth]{ch4_comptonbs.eps}
\caption[Photon energy spectrum from the Compton back-scatter source]
{Photon energy spectrum from the Compton back-scatter source described
in the text and a $12$~\eunit\ electron beam at $1$~\miamps.  (a) cavity of gain
$10000$ driven by a $10$~\kpwunit{} Argon-ion laser ($514$~\nmlunit) at 
$100$~\mhz. (b) cavity of gain $250$ driven by $3$~\kpwunit{} frequency-doubler
($257$~\nmlunit) pulsed at $100$~\mhz. (c) cavity of gain $1$ driven by a 
hypothetical FEL source operating at $20$~\eeunit\ with the same time 
structure as \cebaf{} beam, peak power $1$~\kpwunit.}
\label{comptonbs}
\end{figure}

From the point of view of flux, this source is marginal.  With a few \miamps{}
of beam and mirror improvements, it might produce $10^{8}$ photons/s in the 
upper third of its energy spectrum.  However, its maximum photon 
energy of $3.7$~\eunit\ is far short of the $80$\% $E_{0}$ needed for \GX.
To remedy this one must decrease the wavelength of the laser beam.  This can 
be done by the use of a frequency-doubling crystal that absorbs the green 
light from the laser and produces ultraviolet light at $257$~\nmlunit.  Storing
this light in a cavity of similar design to that described above yields the 
back-scatter rate spectrum shown in Fig.~\ref{comptonbs}b.  The major reason 
for the drop in rate is the decrease in the cavity gain from $10000$ to $250$.
This is imposed by the diminished reflectivities of mirrors in the UV.  Other 
factors are the inefficiency of the doubling crystal, a factor of two in rate 
from the doubling itself, and the decreasing Compton cross section with 
energy.  The maximum photon energy is still under $50$\% $E_{0}$ and the flux 
is three orders of magnitude below the desired rate.

In order to reach photon energies of $80$\% $E_{0}$, initial photons of 
$20$~\eeunit\ are
needed.  The brightest source of these would be a synchrotron light source 
or a free electron laser (FEL).  Mirrors that operate at these wavelengths
typically have reflectivities around $70$\%.  With these one could conceive of
a scheme that uses a wiggler to extract energy from the $12$~\eunit\ beam 
before it
enters the dump.  This light would have the same time structure as incident
beam, and so it could be reflected back and made to cross the incident beam
at a small angle for a Compton back-scatter source. An indication of the
level of flux that could be achieved with such a source can be obtained by
using the laser cavity model described above, setting the gain to $1$, the
wavelength to $62$~\nmlunit, 
and assuming $1$~\kpwunit{} peak ($1$~\pwunit{} average) of synchrotron light.
The back-scatter rate is shown in Fig~\ref{comptonbs}c.  This plot shows that
even if the full power of a $1$~\miamps{} on a $12$~\eunit{} beam were converted
into $20$~\eeunit{} photons and back-scattered from the incoming beam, the 
rate would still fall far short of the requirements for \GX.

From the point of view of polarization, the Compton back-scatter source would
be ideal.  The polarization of the back-scattered beam is controlled by that
of the laser, and can be essentially $100$\%.  This source is also virtually
background-free because the spectrum below any desired cutoff can be 
eliminated by collimation.  The energy of the remaining beam can be measured
to within the resolution of the electron beam by tagging.  However the
the combination of sufficient energy and sufficient flux for the purposes
of the \GX{} experiment in \hd{} cannot be achieved using this source.

\subsection{Tagged bremsstrahlung}

A bremsstrahlung source consists of a thin piece of material (the radiator)
that is placed in the electron beam and converts part of the energy of the
beam into bremsstrahlung radiation.  Bremsstrahlung offers the only practical
way, starting with an electron beam at \cebaf{} energies, to produce a photon
beam with a significant flux in the vicinity of the end point.  It produces
a naturally collimated photon beam with a characteristic angular spread of
$m/E_{0}$.  This allows the low emittance of the \cebaf{} beam to be 
effectively transfered into the secondary photon beam.

Bremsstrahlung does not suffer from the kind of flux limitations that were
encountered in the examination of Compton back-scatter sources.  The radiator
thickness must be kept below $1$\% of a radiation length in order to maintain
good energy resolution in the tagger.  Keeping the thickness below $10^{-3}$ 
radiation lengths ensures that multiple scattering in the radiator does not
significantly broaden the divergence angle of the photon beam.  A $10^{-3}$
radiator and $1$~\miamps{} of electrons would produce much more than sufficient
flux for \GX.

A bremsstrahlung source is, however, deficient in some other respects.
Averaged over the bremsstrahlung cone, the photon beam has zero 
linear
polarization.  Circular polarization can be achieved by polarization
transfer from a polarized electron beam, but for the purposes of
\GX{} it is linear
polarization that is desired.  A bremsstrahlung source
also suffers from a large low-energy flux in the beam.  The power spectrum
of a bremsstrahlung beam is approximately uniform from zero up to the energy
of the incident electrons.  This means that an experiment that uses the 
high-energy part of the beam must operate in a background of low-energy
photons that are many times more frequent.  The tagger is helpful in
eliminating many of the false starts in the detector that arise from the
background, but this technique becomes ineffective at rates above a few
$10^7$ tagged photons/s.  For the typical experiment using tagged
bremsstrahlung and open detector geometry, background from low-energy
beam particles limits the rate at which the experiment can run to less
than $5\cdot 10^{7}$ tagged photons/s.  The goal for \GX{} pushes that limit
to $10^{8}$/s by employing tagged {\em coherent} bremsstrahlung.

\subsection{Coherent bremsstrahlung}

The source described in the previous section meets most of the requirements
for \GX, but is deficient in the areas of polarization and backgrounds.
Both of these deficiencies can be remedied by replacing the conventional
amorphous or polycrystalline radiator with a thin mono-crystalline wafer.
At special settings for the orientation of the crystal, the atoms in the
radiator can be made to recoil together from the radiating electron. When
they do this they produce a coherent enhancement at particular energies
in the radiation spectrum, which correspond to the reciprocal lattice
vectors of the crystal.  The kinematics are such that a randomly oriented
lattice vector would make a tiny peak located up at the end point of the
energy spectrum, where the coherent gain factor is negligible.  By careful
orientation of the crystal, however, one of the lattice vectors can be
aligned with the favored kinematics for bremsstrahlung, at which point its
coherent peak appears well below the end point, and its coherent gain can
be large enough that it contributes a large fraction of the total radiated
power.

\begin{figure}[h!]\centering
\includegraphics[width=0.65\textwidth]{ch4_bremsint.eps}
\caption[Photon power spectrum from an oriented diamond radiator.]
{\label{bremsint}
Photon power spectrum from an oriented diamond radiator.  The
$y$ axis is $dP/dE$ with power $P$ expressed in \eunit/s and $E$ in \eunit.  
The
radiator thickness is $10^{-4}$ radiation lengths and the electron beam
current is $1$~\miamps.  Shown is what emerges after the photon beam passes
through a collimator $3.4$~\mmlunit\ in diameter located $80$~\mlunit\
downstream from the
radiator.}
\end{figure}

This is illustrated in Fig.~\ref{bremsint}.  This plot shows the intensity
($dP/dE$) or power spectrum of the coherent bremsstrahlung beam after
collimation.  The sequence of secondary peaks above the primary correspond
to integral multiples of the fundamental reciprocal lattice vector and so they
are always present.  By careful choice of orientation angles it is possible
to suppress all other vectors and isolate just one primary peak in the 
enegy band of interest, as shown in the figure.  By a small rotation of
the crystal, the position of the peak can be moved from one end of the
spectrum to the other.  Note that the coherent peaks appear as
enhancements on top of the incoherent bremsstrahlung continuum.

Unlike those from the incoherent process, coherent bremsstrahlung photons have
significant net linear polarization in the plane given by the beam direction
and the crystal lattice vector.  This polarization is enhanced by
collimating the photon beam below its intrinsic angular spread, as
discussed in the next section.  The loss in flux from collimation
can be recovered by increasing  the electron
beam current.  As will be shown in the following section, even in the case
of very thin crystals and severe collimation, quite modest electron beam
currents are needed to produce the required photon flux.

The use of coherent bremsstrahlung improves the background conditions of
the beam by enhancing the spectral intensity in the desired energy band 
relative to the incoherent continuum.  For measurements
that do not require polarization, a crystal radiator can be used without
collimation to reduce the low-energy beam background for a given rate
of tagged photons.  Where polarization is required, coherent bremsstrahlung
is indispensable.

