AAWSAP DIRD, Invisibility Cloaking Theory and Experiments, March 2010

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dimensional plane, one can use the stereographic projection, the central ingredient of
the Mercator projection in cartography that is used to map the surface of a round
object, the Earth, onto a flat sheet of paper.
A line drawn from the North Pole of the sphere through a point on the surface intersects
the equatorial plane at one point. This point is the stereographic projection of the point
on the sphere. Figure 19 shows that the stereographic projection of a circle on the
sphere is a circle on the plane with a different radius. The stereographic projection
distorts the measure of space, but the distortion around any given point is the same in
all directions, because otherwise circles would be deformed. Therefore, an optical
material that implements the geometry of the sphere via the stereographic projection
must be isotropic. One can read off the required refractive index from the drawing as
follows: in virtual space, on the sphere, light propagates at the speed of light in vacuum
from a point to its infinitesimally close neighbor; in physical space the distance between
the two infinitesimal neighbors is modified-the speed of light is changed by the
refractive index that is given by the ratio between infinitesimal distances in virtual
space and the corresponding distances in physical space. For the stereographic
projection, the refractive index is smaller than 1 for points on the northern hemisphere
and larger than 1 on the southern hemisphere. This device is known as Maxwell's fish­
eye lens. 33 In this lens, light follows the great circles, light goes around in circles, and
light rays meet at antipodal points; the fish-eye makes a perfect lens (although a fairly
near-sighted one). It is possible to extend these ideas to three-dimensional curved
spaces. For example, the surface of the four-dimensional sphere is a three-dimensional
curved space, and the device implementing this hyperspace object is just a three­
dimensional fish-eye. Hyperspace is not out of this world; it can be built, and it turns
out to be practically useful for invisibility. 34
Figure 19. Stereographic Projection
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