\section{Differential $\gamma\gamma$-luminosities}
There are copious ways that can lead to photon-photon collisions at an
$\eplus\eminus$ collider. Apart from collisions of slightly virtual
photons, beamstrahlung and Compton scattering are the most promising
sources \cite{telnov:2}.
Backscattering a laser beam off energetic electrons offers an efficient
mechanism to convert electrons into photons and reach thus the highest
possible photon-photon luminosity and energy
\cite{akerlof,ginzburg:1,telnov:1,richard}.
In the following we shall concentrate on photons from this source.
Reactions induced by beamstrahlung will typically arise from lower
energy photons and hence can easily be distinguished from $t\bar{t}$
production with its large $s_{\gamma\gamma}$ required to cross the
top quark threshold.
The compton cross section ${\sigma}_c$ depends on the initial and final
state polarization \cite{ginzburg:2}. For collinear electron and laser beams
the differential cross section is given by
\input{compton}
\noindent
with E, ${\omega}_0$ and ${\omega}$ as the energies of the incoming
electron
and photon and of the final state photon in the laboratory frame respectively.
$\pee,\pge$ and $\pgse$ denote the helicities of
the initial electron,
laser and scattered photon with $-1\leq\pee,\pge,\pgse\leq+1$.
For backscattering $y$ denotes the fraction of electron energy transfered to the
photon.
In the case of opposite helicities of electron and laser
the spectrum $\frac{dN}{dy}=\frac{1}{{\sigma}_c}\frac{d{\sigma}_c}{dy}$
is peaked at high energies
and depleted in the medium energy range (Fig. \ref{compton_1}).
Full (or vanishing) longitudinal polarization both for electrons and
incoming photons is assumed and, if not stated otherwise, an electron
energy E = 250 GeV
and a laser energy $\omega_0$ = 1.26 eV.
These values imply $x\approx 4.83$ close below
the threshold for the undesirable production of an
${\eplus\eminus}$ pair in the collision of a high energy and a laser
photon \cite{telnov:1,borden}.
Choosing equal helicities of incoming electrons and photons
enhances the spectrum for intermediate energies with a corresponding
depletion towards the high end.
\begin{figure}
\begin{center}
\mbox{\epsfig{file=bild1.eps,width=12cm}}
\end{center}
\caption{Energy spectrum of the backscattered photons for unpolarized beams,
equal and opposite initial state helicities $({\omega}_0=1.26\,\mbox{eV}, E=250\, \mbox{GeV})$.}
\label{compton_1}
\end{figure}
The spectrum is decomposed according to the helicities of the backscattered
photon in Fig. \ref{compton_2}. For high energies the helicity flip
contribution dominates, below $y\approx 0.55$ the helicity conserving part
is more important.
\begin{figure}
\begin{center}
\mbox{\epsfig{file=bild2.eps,width=12cm}}
\end{center}
\caption{Energy spectrum of the backscatetred photons for opposite initial
state helicities and helicity flipped (solid) respectively helicity
conserved (dashed) final state $({\omega}_0=1.26\, \mbox{eV}, E=250\, \mbox{GeV})$.}
\label{compton_2}
\end{figure}
These photon energy distributions allow to calculate normalized spectral $\gamma\gamma$-luminosities
$\lgg$ through\footnote{The tilde denotes quantities
related to the second beam} \cite{ginzburg:2}
\input{lumifml}
The luminosity depends on the helicities of incoming as well as outgoing
particles. Since all $\gamma\gamma$ reactions of interest are
dependent on the helicities of the photons it is mandatory to
decompose $dL/dz$ according to helicity components.
The fraction of $\eplus\eminus$ cms energy transformed to
$\gamma\gamma$ cms energy is denoted by $z$. The differential luminosities
can be calculated in closed analytical form:
\input{l}
%%%%%\input{la}
%%%%%\input{lb}
%%%%%\input{lc}
The cross sections in this formula are normalized relative to
${2\pi\alpha^2}/{xm_e^2}$. The functions $\Phi_i$ are listed in the appendix.
From the requirements of a luminosity peaked toward high energies it is
ideal to choose for the incoming electrons and photons $\pee\pge=\pez\pgz=-1$.
The difference between unpolarized and fully polarized initial states is
shown in Fig. \ref{lumibild_1} after summation over final state helicities.
\begin{figure}
\begin{center}
\mbox{\epsfig{file=bild3.eps,width=12cm}}
\end{center}
\caption{Spectral luminosity for $\pee=\pge=\pez=\pgz=0$ (dashed) and
\mbox{$\pee\pge=\pez\pgz=-1$ (solid), (${\omega}_0$ = 1.26 eV, E = 250 GeV).}}
\label{lumibild_1}
\end{figure}
For the evaluation of the top quark pair production rate (like for any other reaction) it is interesting to
distinguish between
the four distributions corresponding different polarisations
of the scattered photons. This decomposition is shown for ideal polarization
in Fig. \ref{lumibild_2}, where $h_{\gamma\gamma}
=\pgse-\pgsz$ denotes the total helicity of the photon-photon system.
The case $h_{\gamma\gamma}=0$ is appropriate for the production of $\esn$
toponium resonances so that $\pee=\pez$ and $\pge=\pgz$ is a suitable choice
for the incoming helicities.
The contributions from $\pgse=-1,\pgsz=1$ and $\pgse=1,\pgsz=-1$ are identical
and therefore added in one curve.
\begin{figure}
\begin{center}
\mbox{\epsfig{file=bild4.eps,width=12cm}}
\end{center}
\caption{Spectral luminosity dependence on final state photon polarisation,
\mbox{(${\omega}_0$ = 1.26 eV, E = 250 GeV).}}
\label{lumibild_2}
\end{figure}