  This paper discusses Floquet engineering with arbitrary polarization in
$\alpha$-RuCl$_3$. We describe the influence of arbitrary polarization and the
limiting cases of linear and circular polarization. The corresponding model is
derived via perturbation theory up to fourth-order. Starting from linear and
circular polarization we bridge the gap between those two limiting cases. We
then we study more complex Lissajous figures and general trends arising for
them.
