invisibility cloak may have intrigued many of J.K. Rowling’s fans, and now researchers have come a step closer to realising it, by explaining how mathematics can turn out to be a major tool in the developing field of cloaking.
Ranked the number five breakthrough of the year by Science magazine in 2006, cloaking involves making an object invisible or undetectable to electromagnetic waves.
And now scientists have presented a paper giving an overview of the theoretical developments in cloaking from a mathematical perspective.
One method involves light waves bending around a region or object and emerging on the other side, as if the waves had passed through empty space, creating an “invisible” region that is cloaked.
But for that, the object or region has to be concealed using a cloaking device, which must be undetectable to electromagnetic waves.
Manmade devices called metamaterials use structures having cellular architectures designed to create combinations of material parameters not available in nature.
Mathematics is essential in designing the parameters needed to create metamaterials, and to show that the material ensures invisibility.
The mathematics comes primarily from the field of partial differential equations, in particular from the study of equations for electromagnetic waves described by the Scottish mathematician and physicist James Maxwell in the 1860s.
One of the “wrinkles” in the mathematical model of cloaking is that the transformations that define the required material parameters have singularities, that is, points at which the transformations fail to exist or fail to have properties such as smoothness or boundness that are required to demonstrate cloaking.
But the singularities are removable, which means that the transformations can be redefined over the singularities to obtain the desired results. The authors of the paper describe this as “blowing up a point”.
The paper also suggests that if there are singularities along a line segment, it is possible to “blow up a line segment” to generate a “wormhole”.
The cloaking version of a wormhole allows for an invisible tunnel between two points in space through which electromagnetic waves can be transmitted.
Cloaking via electromagnetic wormholes has many possible applications, which include the creation of invisible fibre optic cables, for example for security devices, and scopes for MRI-assisted medical procedures for which metal tools would otherwise interfere with the magnetic resonance images.
The invisible optical fibres could even make three-dimensional television screens possible in the distant future.
The paper titled ‘Cloaking Devices, Electromagnetic Wormholes, and Transformation Optics’, has been published in SIAM Review.