  We introduce a class of conformal versions of the previously introduced
quasi-conformal carpet cloak, and show how to construct such conformal cloaks
for different cloak shapes. Our method provides exact refractive-index profiles
in closed mathematical form for the usual carpet cloak as well as for other
shapes. By analyzing their asymptotic behavior, we find that the performance of
finite-size cloaks becomes much better for metal shapes with zero average
value, e.g., for gratings.
