  Let $\mathcal{L}$ be a measured geodesic lamination on a complete hyperbolic
surface of finite area. Assuming $\mathcal{L}$ is not a multicurve, our main
result establishes the existence of a geodesic ray which has finite
intersection number with $\mathcal{L}$ but is not asymptotic to any leaf of
$\mathcal{L}$ nor eventually disjoint from $\mathcal{L}$. In fact, we show that
the endpoints of such rays, when lifted to the universal cover $\mathbb{H}^2$
of $X$, give an uncountable set $h\tilde{\mathcal{L}}\subset S^1$ (called the
halo of $\tilde{\mathcal{L}}$), which is disjoint from the endpoints of leaves
of the lifted lamination $\tilde{\mathcal{L}}$.
