  A precise and fast Fabry-Perot cavity polarimeter, installed in the HERA
tunnel in the summer of 2003, was used to measure the longitudinal polarisation
of the lepton beam. A complete theoretical model has been developed in order to
control at the per mill level the degree of circular polarisation of the laser
beam. The transport of this quantity within the whole optical setup has also
been performed and controlled at the same level of precision. This is the first
time that such a precision is achieved in the difficult, hostile and noisy
environment of a particle collider.
