AAWSAP DIRD, Invisibility Cloaking Theory and Experiments, March 2010

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Broadband Invisibility
To understand why non-Euclidean geometry comes to the rescue of invisibility, consider
the following two-dimensional example. 35 Imagine a virtual space made of a flat space,
a sheet of paper, and a curved space, the surface of a sphere. The two spaces touch at
one line. Consider the fate of light rays in this two-dimensional virtual world. Light rays
would either pass the sphere or enter, through the connecting line, the surface of the
sphere, whereupon, after one loop, they would continue in the same direction as they
entered, as if the tour on the sphere had never happened. The sphere is invisible; it
shows only as a time delay of the light ray. Although the sphere is invisible, it does not
make something else invisible yet. However, this is easily arranged. Imagine a mirror
around the equator of the sphere. The ray bounces off at the mirror, but, after another
bounce, is back on track. A mirror in this curved space reflects light back to itself! The
mirror creates the illusion that the light performs a full great circle, whereas in reality it
stays on one hemisphere. The other hemisphere is hidden. Alternatively, some lines on
the sphere are never crossed by light rays. Such lines can be opened like an eye; the
space they enclose is hidden from sight.
Figure 20. Non-Euclidean Cloaking Device in Two Dimensions. The device creates the illusion shown in A:
light propagates through a virtual space that consists of a plane and the surface of a sphere, a curved space, which
touch along a line. Some incident light rays venture from the plane to the sphere; they return after one loop and
continue in the same direction. Note that the rays never cross the red zigzag line on the sphere. Plane and sphere
carry a coordinate grid that is mapped onto physical space B. The magenta circle defines the boundary of the
device. Its interior has been expanded to make space for the grid of the sphere. In particular, the line where plane
and sphere touch has been opened like an eye (thick black lines) to include the sphere. This is not a cloaking
device yet, but one could place a mirror around the equator of the virtual sphere C, making the northern
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