  Cloaking devices are prescriptions of electrostatic, optical or
electromagnetic parameter fields (conductivity $\sigma(x)$, index of refraction
$n(x)$, or electric permittivity $\epsilon(x)$ and magnetic permeability
$\mu(x)$) which are piecewise smooth on $\mathbb R^3$ and singular on a
hypersurface $\Sigma$, and such that objects in the region enclosed by $\Sigma$
are not detectable to external observation by waves. Here, we give related
constructions of invisible tunnels, which allow electromagnetic waves to pass
between possibly distant points, but with only the ends of the tunnels visible
to electromagnetic imaging. Effectively, these change the topology of space
with respect to solutions of Maxwell's equations, corresponding to attaching a
handlebody to $\mathbb R^3$. The resulting devices thus function as
electromagnetic wormholes.
