\chapter{Choice of technique}\label{chapter_Introduction}

Two basic methods have been considered for producing photons of the
highest possible energy, flux and polarization from electrons of $12$~GeV.
The methods are bremsstrahlung and Compton scattering of light. Both
are well-established methods of producing photon beams, and both are
actually described by the same set of 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}\centering
\mbox{\epsfig{file=qedgraphs.eps,width=4in}}
\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 $80$\% $E_{0}$ with a significant degree of linear
polarization.  The energy resolution for individual photons in the beam
should be better than 0.5\%, ideally on the same order of magnitude
as the energy spread
of the electron beam itself.  It should be capable of producing intensities
up to $10^8$/s in the range 8-9~GeV, with the flux outside the desired
energy band 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.

\section{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 emerge
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~ps 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~m with a crossing angle of 1$^{\circ}$.
Both cavity and electron beam are focused to a tiny spot of 10~microns
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 achievable using high-reflectivity dielectric mirrors.
The wavelength of the light is 514~nm.  The rate spectrum of the 
back-scattered beam from this source is shown in Fig~\ref{comptonbs}a for 
a 1~$\mu$A electron beam at 12~GeV.

\begin{figure}\centering
\mbox{\epsfig{file=comptonbs.eps,width=5in}}
\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~GeV electron beam at $1$\miamps:  (a) cavity of gain
10000 driven by a 10~kW Argon-ion laser (514~nm) at 
100~MHz, (b) cavity of gain 250 driven by 3~kW frequency-doubler
(257~nm) pulsed at 100~MHz, and (c) cavity of gain 1 driven by a 
hypothetical FEL source operating at 20~eV with the same time 
structure as \cebaf{} beam, peak power 1~kW.}
\label{comptonbs}
\end{figure}

From the point of view of flux, this source is marginal.  With a few $\mu$A
of beam and mirror improvements, it might produce $10^{8}$ photons/s in the 
upper $\frac{1}{3}$ of its energy spectrum.  However, its maximum photon 
energy of 3.7~GeV is far below the 9~GeV needed for \GX{}.
To remedy this, a shorter wavelength light source is required.  This can 
be achieved by the use of a frequency-doubling crystal that absorbs 514~nm
light from a green laser and produces ultraviolet light at 257~nm.  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 $250$
imposed by the diminished reflectivity of mirrors in the UV.  Other loss
factors are the inefficiency of the doubling crystal, the factor two in rate 
from the doubling itself, and the decreasing Compton cross section with 
increasing energy.  The maximum photon energy is still under 6~GeV and the
flux is three orders of magnitude below the desired rate.

In order to reach photon energies of 9~GeV, a source of 20~eV light is
needed.  The brightest source of 20~eV photons would be synchrotron
radiation.  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~GeV beam 
before it enters the dump.  This light would have the same time structure
as the 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 of the cavity to 1, the wavelength to 62~nm, 
and assuming 1~kW peak (1~W average) of synchrotron light within the peak.
The back-scatter rate for this source is shown in Fig~\ref{comptonbs}c.
This plot shows that even if the full power of 1~$\mu$A in the 12~GeV 
electron beam were converted into 20~eV photons and back-scattered from
the incident beam, the rate of 9~GeV photons produced would still be 
less than 100~Hz, six orders of magnitude below the design intensity
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.

\section{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 transferred 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~$\mu$A 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 circularly polarized electron beam, but for the purposes of
\GX{}, linear polarization is essential.  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.  Photon tagging is helpful in
eliminating many of the false triggers in the detector that are produced
by background beam photons, but this technique is only effective during
offline analysis at rates above a few $10^7$ tagged photons/s.
For the typical large-acceptance experiment using tagged
bremsstrahlung, 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 \GX{} tagged photon beam
design pushes that limit up to $10^{8}$/s by taking advantage of
{\em coherent} bremsstrahlung with collimation.

\section{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}\centering
\mbox{\epsfig{file=bremsint.eps,width=5in}}
\caption
[Photon power spectrum from an oriented diamond radiator.]
{Photon power spectrum from an oriented diamond radiator.  The
$y$ axis is $dP/dE$ with power $P$ expressed in GeV/s and $E$ in GeV.  
The radiator thickness is 20 microns and the electron beam
current is 1~$\mu$A.  Shown is what emerges after the photon beam passes
through a collimator 3.4~mm in diameter located 75~m downstream from the
radiator.}
\label{bremsint}
\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 
energy 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.

