  We consider the geodesic flow defined by periodic Eaton lens patterns in the
plane and discover ergodic ones among those. The ergodicity result on Eaton
lenses is derived from a result for quadratic differentials on the plane that
are pull backs of quadratic differentials on tori. Ergodicity itself is
concluded for $\mathbb{Z}^d$-covers of quadratic differentials on compact
surfaces with vanishing Lyapunov exponents.
