
\section{Introduction}
The search for the elusive top quark, the determination of its mass and the
study of its electroweak couplings constitutes an important part of the program
of presently operating accelerators and of those of the next generation. Top
quark production in hadronic collisions and in electron positron annihilation
(For a recent reviev see e.g. \cite{kz92}) has been discussed quite frequently
in the literature and will certainly be in the forefront of these studies. In
addition there is the option for high energy photon-photon collisions at a
future "Compton collider".
Such an experiment would make use of well focussed high energy photons produced
through backscattering of laser light from high energy electron beams.
Production of Higgs bosons \cite{richard} or of $W$ pairs
\cite{choi} in photon photon
fusion are reactions of great interest, $t$-$\bar{t}$--production is also
possible with large rates and could provide interesting information on the
top-photon coupling.

The production cross section of top quarks depends sensitively on the initial
photon polarization. In section 2 of this paper analytical formulae are derived
for the $\gamma$-$\gamma$--luminosity as a function of the
$\gamma$-$\gamma$--energy and of the initial electron and the laser
polarization 
and decomposed according to the energetic photon helicities. In section 3 the
Born cross section for $\gamma\gamma\rightarrow t\bar{t}$ will be folded with
these luminosities leading to a substantial effective cross section e.g.\  for
a $e^{+}e^{-}$ collider of 500 GeV. Strong dependence of the production rate on
laser and electron polarization is observed\footnote{These results in Born
approximation have been presented in a preliminary version in \cite{bern}}.

Only "direct" photon induced amplitudes will be considered. 
Contributions from the hadronic component of the photon are small. 
This is a consequence of the steeply falling quark and gluon 
distribution in the photon and the fact that $4m_t^2/s_{\gamma\gamma}$ 
is of order one.
QCD corrections to top quark production are evaluated in section 4 and found to
be large -- in particular in the threshold region. For the approximate choice
of 
$\alpha_{s}$ a smooth matching between continuum region and resonances is
observed, which allows to evaluate again an effective cross section by folding
the QCD corrected $\gamma\gamma\rightarrow t\bar{t}$ cross section with the
luminosity spectrum derived in section 2. Large corrections are observed also
after this convolution.



