Mathematics - Story Archives: Invisible Objects

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The Mathematics of Full-Wave Invisibility of Active Devices
    at All Frequencies:
    Cloaking Cell Phones and Computers

    (Professor Allan Greenleaf)

     by Lois H. Gresh

November 30, 2006: The invisibility cloak recently built by scientists at Duke University and Imperial College in London has received enormous attention all over the world. This breakthrough cloaking device makes a copper disk invisible at specific microwave frequencies. As it turns out, Mathematics Professor Allan Greenleaf of the University of Rochester along with Matti Lassas, who is now at the Helsinki University of Technology, and Gunther Uhlmann of the University of Washington, started the mathematics behind invisibility several years ago in the context of medical imaging and quantum mechanics. Joined by fourth member Yaroslav Kurylev of Loughborough University, the team has announced the mathematics of full-wave invisibility at all frequencies (http://www.arxiv.org/abs/math.AP/0611185). In their latest work, they examine the problem of cloaking not just passive objects, but also active devices that are emitting electromagnetic waves, and show that this requires modifying the original constructions.

Professor Allan Greenleaf first started working on the mathematics of invisibility in 2001. In 2003, with Professors Lassas and Uhlmann, he published two papers about the mathematics that could make objects with special properties invisible ("On Nonuniqueness for Calderon's Inverse Problem," Mathematical Research Letters 10, 685-693, 2003, and "Anisotropic Conductivities that cannot be Detected by EIT," Physiological Measurement 24, 413-419, 2003).

These papers were concerned with the mathematics of electrical impedance tomography (EIT), a medical imaging technique that uses measurements of electric voltages and currents to obtain images of the interior of the body. It was previously known that, under certain assumptions, the measurements uniquely determine the electric conductivity function, which varies from point to point and whose behavior might indicate the presence of tumors. In the two papers of 2003, Greenleaf, Lassas, and Uhlmann showed that, if the conductivity function is allowed to be a tensor, which means that it can depend on direction as well as location, then there are recipes for mathematically constructing different conductivities that are indistinguishable from each other by EIT measurements. At the time, this seemed like something of a curiosity, since it was highly unlikely that the exact mathematical structures needed to produce this "invisibility" would occur in nature, for example, on the exterior of a tumor.

This same mathematics surfaced again in 2006 when Sir John Pendry of Imperial College, Dr. David Schurig and Professor David Smith of Duke University, and Professor Ulf Leonhardt of St. Andrew's University, wrote papers theoretically proposing the construction of cloaking devices, rendering the objects inside them invisible from observation using electromagnetic (EM) waves. In October 2006, physicists and engineers led by the Duke University team built a device from man-made metamaterials that essentially cloaks a passive copper disk or rod. Based on a mathematical calculation very similar to one proposed in 2003, the metamaterials have special properties in the way that microwaves propagate through, or in this case, around them. The cloaking device bends microwaves around an object and does not cast a shadow. The copper disk or rod inside the cloaking device is invisible. It is as if absolutely nothing is there.

In ordinary light, the copper disk or rod, and the cloaking device, are visible. It is only at specific microwave frequencies that the copper disk or rod is invisible. (The EIT work of 2003 can be interpreted as addressing this question at frequency zero.)

The current work of Greenleaf, Kurylev, Lassas, and Uhlmann mathematically proves full-wave invisibility of active devices at all frequencies -- if suitable modifications are made to the original constructions.

Rather than cloaking an inactive object, which does not generate EM waves, the new mathematics allows for cloaking active devices, such as cell phones, computers, and anything else that might generate signals. In addition, the mathematics works (theoretically) at all wavelengths.

One way to modify the original constructions and obtain invisibility at all wavelengths is to insert a lining material inside the cloaking device. Or, rather than use a single coat of metamaterials, Greenleaf describes another method: "My group suggests that there's another construction of the cloaking device that we call the double coat. In this case, we basically put the metamaterials on the outside of the device, and we make a similar 'matched' construction inside the device and point it inwards."

Using Maxwell's Equations, which describe electric and magnetic fields and how they interact with matter, the mathematics, with either the lined single coating or the double coating, provides strong confirmation of cloaking.  It proves full invisibility with sources (an example is a cell phone transmitting radiation) and sinks (an example is a cell phone receiving and absorbing radiation) at every frequency.

In the future, if physicists and engineers build lined single-coated or double-coated cloaking devices, then regardless of frequency, the devices will cloak any object, even those that propagate signals.

For more information, please contact:
   Professor Allan Greenleaf
   Email: allan@math.rochester.edu
   Faculty Webpage: http://www.math.rochester.edu/people/faculty/allan/

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