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Rochester Researchers Develop Mathematical Theory For Cloaking

The Oct. 26, 2006 RF Report described a demonstration of cloaking--making an object invisible--conducted at Duke University and Imperial College in London.

Allan Greenleaf, professor of mathematics at the University of Rochester noticed that the Duke University and Imperial College researchers used equations that were nearly identical to equations Greenleaf and his colleagues developed for finding hidden tumors.

Greenleaf said, "We were working on improving the mathematics behind tumor detection. In the final section to one paper, we spelled out a worst-case scenario where a tumor could be undetectable. We then wrote a couple of additional articles describing when this could happen. At the time, we didn't think further about it because it seemed extremely unlikely that any tumor would be covered with the necessary material to be hidden that way."

The University of Rochester researchers realized that their results could also be used to show how to "hide" an object and decided to analyze and improve the proposed cloaking device. Greenleaf and his colleagues found that everything seemed fine when they applied the Helmholtz equation, which is widely used to solve problems involving the propagation of light, but didn't work as well as Maxwell's equations, which deal with electromagnetic waves. Anything that emitted electromagnetic waves, even something as simple as an electric flashlight, caused the cloaking device to "go seriously awry," as the mathematics predicted the size of the electromagnetic fields would go to infinity at the surface of the cloaked region and possibly impair invisibility.

The equations also revealed an interesting side effect to the cloaking. Assuming the cloak worked over the full spectrum of light, a person trying to look out of the cloak would be faced with a mirror in every direction. According to the University of Rochester, "Greenleaf's team determined that a more complicated phenomenon arises when using Maxwell's equations, leading to a 'blow up' (an unexpected infinite behavior) of the electromagnetic fields. They determined that by inserting conductive linings, whose properties depend on the specific geometry of the cloak, this problem can be resolved. Alternatively, covering both the inside and outside surfaces of the cloaked region with carefully matched materials can also be used to bypass this problem."

Greenleaf said, "We should also keep in mind that, given the current technology, when we talk about invisibility, we're talking only about being invisible at just a narrow range of wavelengths. For example, an object could be rendered invisible at just a specific wavelength of red; it would be visible in nearly every other color."

In early December, the leader of the team that demonstrated cloaking, David R. Smith, associate professor of electrical and computer engineering at Duke University, met with Greenleaf. Smith said, "Allan has been looking at the problem much more generally, and deriving conditions for when true invisibility is or is not possible. We are very interested in what he and his colleagues come up with."

See The Mathematics of Full-Wave Invisibility of Active Devices at All Frequencies: Cloaking Cell Phones and Computers and the PDF of the paper (complete with formulas) for additional information.