Read more: “Instant Expert: Metamaterials”
A mathematical tool tells us what kind of metamaterial will bend light along a desired curved path, making devices such as invisibility cloaks possible
We need a clever design tool to make the most of metamaterials’ potential. In conventional optics, light travels in a straight line until it hits the boundary between two transparent materials, at which point it abruptly changes direction. Metamaterials are much more sophisticated. They can force light to travel along a curved path, thereby opening up the possibility of devices such as invisibility cloaks.
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The mathematical tool that tells us what kind of metamaterial will bend light along the desired path is known as transformation optics.
A little help from Einstein
A remarkable experiment conducted during a solar eclipse in 1919 showed that the sun acts like a giant lens, bending starlight that passes close to the sun’s disc. It vindicated Albert Einstein’s prediction in his general theory of relativity that the sun’s gravitational field would distort space.
As far as light is concerned, the warped space near to the sun appears to have a large refractive index. Very helpfully for metamaterials, Einstein produced a formula relating the distortion of space to changes in effective refractive index. Furthermore, Einstein’s formula is an exact transformation of Maxwell’s equations, which govern all electromagnetic phenomena.
What transformation optics seeks to achieve is to take a ray of light and distort its trajectory in whatever way we please. We could do this by distorting space itself using extremely massive objects, but thanks to Einstein’s insight this is not necessary. Simply changing the refractive index will have the same effect as far as light is concerned.
Transformation optics can be used to calculate the required refractive indices. All that is then needed is to construct metamaterials that meet these specifications.
Negative space, perfect lens
General relativity predicts that extremely massive objects will cause severe distortions to the surrounding space. Perhaps the best known of these objects are black holes – singularities in space-time from which even light cannot escape. Some people have asked if metamaterials can be used to mimic black holes, but this is not possible as metamaterials lack the built-in energy source required to produce the “Hawking radiation” that real black holes emit.
But there is something metamaterials can do that is even more spectacular: they can create “negative optical space”. Think of space as a sheet of rubber that can be compressed. By pushing hard enough on the sheet, it should in principle be possible to fold space back on itself so that light moving in the folded space passes the same point three times. A fish swimming in this space would come into focus three times, the middle focus being inverted.
Transformation optics tells us that a portion of the folded space must have a negative refractive index. This negative optical space has a remarkable property. Light leaving an object can be thought of as defocusing as it travels away in all directions. Negative optical space cancels out this effect. As a result, a lens made of a metamaterial that creates negative optical space is optically perfect. It effectively eliminates the space separating the object and image, so that the object and image coincide.
Another limitation of ordinary lenses is that they can never resolve details smaller than the wavelength of the light being used. That’s because light diffracts or bends around objects of a similar size to its own wavelength, so it can never be focused to a sharp point. The lens is said to be “diffraction limited”.
When I calculated in 2000 that a negatively refracting lens breaks the diffraction limit, the result caused a furore, so entrenched was the notion that lenses cannot be used to see anything smaller than the wavelength of light. Several experiments have since shown my prediction to be correct. In particular, Nicholas Fang, working at the time with Xiang Zhang at the University of California, Berkeley, demonstrated a lens that could resolve details as small as one-sixth the wavelength of visible light. Subsequent work has improved resolution to 1/20th of the wavelength.
Such lenses are challenging to make, however, because they are extremely sensitive to imperfections.
A cloak of invisibility
To be invisible seems somehow magical. People have dreamed of the possibility for centuries, but only with the advent of transformation optics and metamaterials has the dream of an “invisibility cloak” approached reality.
Such a cloak must have two properties. First, it must reflect no light and ensure that no light is reflected from the object it is cloaking. This is relatively easy to achieve with a pot of black paint – or its equivalent for whatever wavelengths we want to be invisible in. In essence, this is how the stealth technology used by the military operates.
Second, the hidden object must cast no shadow. Removing a shadow is a much greater challenge, but one that transformation optics has successfully addressed.
Our brains assume that light travels in a straight line to reach our eyes, as if it travelled along a rigid rod. Now suppose that the rod were not rigid, but could be bent so that it curves around the object that we want to hide. Our brains would be none the wiser because the light rays would reach our eyes exactly as before with no hint of the curved path that they had in fact taken. Metamaterials can make light flow like water around the hidden object, smoothly closing in behind the object to leave no trace of its presence.
The eye is fooled in the same way by certain natural phenomena that cause light to travel in a curve. When the sun heats the desert sands, the air immediately above the sand is also heated. It becomes less dense and therefore less refracting. This effect tails off with height, creating a gradient in the air’s refractive index. As a result, light from the sky is bent so that it can appear to originate from the desert itself, as if part of the surface were covered with water. Hence the disappointment of a thirsty traveller who thinks he has seen an oasis. Making an invisibility cloak then becomes a matter of devising a suitable refractive index gradient to bend light in precisely the right way. Transformation optics enables us to work out the exact properties a metamaterial would need to achieve the required bending.
“An invisibility cloak prevents reflections and shadows from an object”
It is a huge challenge to actually build an invisibility cloak – particularly at visible wavelengths where interest is naturally strongest. But at radar frequencies, progress has been rapid. There have been more limited successes with visible light, with cloaks just a few wavelengths across.
Read more: “Instant Expert: Metamaterials”
