Mathematics of Invisibility Cloak Made Plain

December 5, 2008

In a theoretical study, researchers have laid out a plan for developing cloaking devices that could shield an object from detection by light in a variety of wavelengths. The key element is a mathematical way to describe such an invisibility cloak, developed by Ulf Leonhardt of St. Andrew’s University in Scotland and Tomás Tyc of Masaryk University in the Czech Republic.

From a mathematical viewpoint, an invisibility cloak is a transformation of flat space that makes light follow a curved path around an object. Instead of describing space in terms of perpendicular axes, Leonhardt and Tyc used non-Euclidean geometry based on the surface of a sphere, which they intersected with a Euclidean plane in an arrangement that resembles a globe partially wrapped by a piece of paper.

The plane represents the region away from the cloak containing a light source and an observer, while the spherical geometry contains the region to be cloaked. When the sphere is between the source and an observer, some light from the source will travel from the plane onto the sphere, where the light will follow a curved path. However, because of the way the plane intersects the sphere, the curved paths don’t cross in a small region on the sphere. According to the physicists, the secret is to use a coordinate transformation to expand this into a space that could enclose a cloaked object.

"Our method works for optical singularities which are the curse of physics, often seeming intractable, but we have found a way of transmuting optical singularities with just harmless crystal defects as a side-effect," Leonhardt and Tyc wrote.

The researchers reported their findings in the article "Broadband Invisibility by Non-Euclidean Cloaking," published online by Science on Nov. 20. Additional information on cloaking and transformation optics is available in a special issue (November) of the New Journal of Physics.

Source: EE Times, Nov. 26, 2008.