  Is it possible to actually make Harry's invisibility cloaks? The most
promising approach for realizing such magical cloaking in our real world would
be to use transformation optics, where an empty space with a distorted geometry
is imitated with a non-distorted space but filled with transformation medium
having appropriate permittivity and permeability. An important requirement for
true invisibility cloaks is nonreciprocity; that is, a person in the cloak
should not be seen from the outside but should be able to see the outside. This
invisibility cloak, or a nonreciprocal shield, cannot be created as far as we
stay in conventional transformation optics. Conventional transformation optics
is based on Riemann geometry with a metric tensor independent of direction, and
therefore cannot be used to design the nonreciprocal shield. To overcome this
problem, we propose an improved theory of transformation optics that is based
on Finsler geometry, an extended version of Riemann geometry. Our theory shows
that nonreciprocal shielding can be realized by covering cloaking space with
transformation medium having anisotropic, nonreciprocal permittivity and
permeability. This theory includes conventional transformation optics as
special cases. We show the method for designing the spatial distribution of the
permittivity and permeability required to make the nonreciprocal shield.
