As I noted a couple of days ago, apparently there has been another significant experimental breakthrough in the development of dielectric cloaking devices. Researchers at UC Berkeley were responsible, though it is a little unclear what exactly the breakthrough is. The results will appear this week in Science and Nature. In the meantime, it seemed like a good time to review the two articles that started the whole cloaking craze.

As I’ve noted in a pair of previous posts (here and here), the search for objects which can be considered in some sense invisible goes back nearly a hundred years. For the most part, however, the idea that one could make a truly invisible object was considered impossible — and theory backed up that view.

This changed with the publication of two back-to-back theoretical papers in Science in 2006. The first, by U. Leonhardt, was titled “Optical conformal mapping,” and the second, by J.B. Pendry, D. Schurig and D.R. Smith, was titled “Controlling electromagnetic fields”. Both papers mapped out strategies — in a nearly literal sense — for creating what could be called a dielectric invisibility device. How would such a device work? Let’s recall a little basic optics that will help us understand the process…

When light travels in the absence of matter (vacuum), it travels at the speed of

m/s.

When it is traveling in a material medium, however, the light field interacts with the matter and (typically) travels at a slower speed. The slower speed is , where is called the index of refraction and depends on the specific material the light travels through, as well as the frequency of light. For visible light traveling in water, the refractive index is roughly 1.33, while diamond has an index of 2.41 (a list of typical indices can be found here).

What happens when light travels across a flat interface between media? The direction of the light is changed according to Snell’s law,

,

where and are the refractive indices of medium 1 and medium 2 and and are the angles which the light fields make with the normal to the surface. This is illustrated below.

If light travels in a gradient index medium, where the refractive index changes continuously as a function of position, the light ray will follow a curved path determined by the nature of the gradient; this is illustrated roughly below:

This is, in essence, the principle behind the dielectric invisibility devices. They are objects which possess a gradient refractive index chosen such that all light rays incident upon the object are guided completely around a central core and released from the other side moving along their original path. Below are illustrations of this effect as shown in each of the two papers:

These images show the effects of the invisibility devices on light rays passing through them. It is to be noted that the Leonhardt device is a two-dimensional device (a cylinder, being looked at from the top), while the Pendry et al. device is a three-dimensional sphere.

How does one determine the refractive index which will result in such a redirecting of light rays? Let us consider the following thought experiment. Suppose our light rays are stuck to a flat, elastic sheet with grid lines drawn on it, as illustrated in the picture on the left below (adapted from the Pendry et al. paper). We proceed to stretch the sheet outward in all directions (more accurately, we apply an ‘anti-pinch’ to the sheet, done with Adobe Photoshop). We have, in essence, ‘stretched space’, and the light rays have gone along for the ride! The rays now bend outward when they approach the central region, as illustrated in the picture on the right:

Mathematically, we have applied a coordinate transformation to space, stretching it out in all directions. In the Leonhardt paper, a special type of coordinate transformation known as a conformal mapping is applied, hence the title, “Optical conformal mapping.”

It turns out that, mathematically, a gradient refractive index has the same effect on light rays as a coordinate transformation of this sort. One can, with some effort, determine the relationship between the effective coordinate transformation and the refractive index that will create it. To theoretically design an optical cloak, then, one finds a coordinate transformation which makes all the light rays ‘avoid’ the central core of the cloaking device, and then works backward to find the refractive index that will create it. It is important to note, however, that space is not actually being distorted by the cloaking device. The refractive index has the same effect on the light field as such a distortion only in a mathematical sense.

This is a relatively simple idea, and it might seem somewhat surprising that nobody had thought of it before. There are two good reasons for this, one theoretical and one experimental.

Theoretically, it had been shown quite conclusively in the late 1980s and early 1990s that so-called ‘non-scattering scatterers’ (i.e. invisible objects) could not exist*. The theory was accepted by most researchers and therefore nobody bothered to look for such invisible objects.

The two cloaking papers have taken different approaches to addressing this concern. Dr. Leonhardt, aware of the earlier work on non-scattering scatterers, quite rightly pointed out that although the existing theory suggested that perfectly invisible objects could not exist, it said nothing about the possibility of objects that are effectively invisible, i.e. impossible to see with the relatively insensitive human eye. The group of Pendry et al., apparently unaware of the earlier literature, nevertheless found a different loophole in the earlier theory: their dielectric invisibility device is constructed out of optically inhomogeneous materials as well as magnetic materials.

In a previous optics basics post, I discussed the polarization of light fields. Light acts as a transverse wave, and any light beam has two possible polarization states, up/down oscillation or left/right oscillation. Most materials are optically homogeneous, which means that they are insensitive to the specific polarization state of the light field. Many crystals, however, are optically inhomogeneous: the crystal responds differently to light of different polarizations, resulting in light of different polarizations traveling at different speeds in the material. This is illustrated dramatically in a piece of calcite, shown below (from Wikipedia):

The two images one can see through the crystal are the result of two different light polarizations passing through the crystal.

Furthermore, at optical frequencies most materials are sensitive only to the electric field of the illuminating light, and the magnetic properties play no role. Materials which possess a significant response to the magnetic field of the light can have a very different response.

The theoretical work performed in the 80s/90s demonstrated that homogeneous materials could not make an invisible object, but left a big loophole: the possibility that inhomogeneous materials might work. As I have noted, the Pendry group seems to have been initially unaware of the earlier literature, and seems to have independently found this loophole. It is not clear that the Pendry group even thought that they could make a ‘perfect’ cloaking device. However, later theoretical work has demonstrated that their cloak in principle is perfectly invisible**.

Experimentally, the biggest obstacle to creating a cloaking device is the large range of refractive indices required. The response of both authors to this obstacle is to invoke the word metamaterials. Metamaterials are loosely defined as (to quote Wikipedia), “Macroscopic composites having a manmade, three-dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to specific excitation.” In essence, it has become possible in recent years for experimenters to fine-tune the structure of materials on the microscopic or nanoscopic level to make materials which are not seen naturally and possess both electric and magnetic responses to an illuminating light field. In principle, one could make materials with a refractive index of almost any value, even negative, though in practice it is still extremely difficult to make metamaterials which work for visible light.

That is the first of several major obstacles to a true invisibility device: to make a metamaterial for visible light (wavelength less than a micrometer), one must be able to control the structure of the material on the same scale. This is still a daunting experimental task. Furthermore, the invisibility cloaks developed so far, even theoretically, only work well for a small range of frequencies. In other words, the object may cloak green light but strongly scatter red and blue light! It is not clear that it is even possible to make a cloak which would work over the entire visible spectrum. Finally, making a cloak which is invisible for all directions of illumination, i.e. a fully three-dimensional cloak, is also a daunting experimental task.

Most of these concerns are illustrated by the first experimental test of an invisibility cloak, reported in late 2006***. The cloak was constructed to be invisible to microwaves operating at 8.5 GHz, which translates into a wavelength of about 3 cm. A two-dimensional cloak was constructed out of so-called split ring resonators which respond to both the electric and magnetic fields of an illuminating microwave. A picture of the device, which has a total diameter of about 12 cm, is shown below (from the paper):

The device, though far from perfect, was shown by experiment (as well as simulations) to provide an appreciable cloaking of the inner core.

So what comes next? The big obstacles seem to be: 1. Making a 3-D cloak (experimental), 2. Making a cloak which works for visible light (experimental), 3. Designing a cloak which works at multiple frequencies (theoretical). The news reports out of Berkeley suggest that the group there has managed to overcome obstacles 1 and 2. What exactly have they done? We’ll have to wait and see…

Finally, let me assuage the fears of those who might think that this technology will become some nefarious government spy tool. First, it is to be noted that the cloak in principle blocks all light from the interior region. In other words, someone can’t see the cloak wearer but the cloak wearer can’t see anything! In principle this could be overcome by the wearer using active sensing at a different frequency, but this different frequency would then also be detectable. Second, absorption of light limits the size of the cloak. All materials absorb light, and a large cloak would lose much of the light as it travels through it. In other words, the cloak would cast a shadow! It’s hard to imagine making a cloak much bigger than a desktop device, though this might be overcome by amplification of some sort. Third, all designs so far have been spherical. The math is complicated enough that it is not clear that a cloak could be easily made of any other shape, much less a bending, flexible shape like an actual cloak! Finally, it is to be noted that the frequency limitations of cloaking devices may prove an insurmountable barrier. Looking back at our ‘pinching’ picture above, the rays passing through the cloak have to travel a further distance than their counterparts outside. This implies that the speed of the rays inside the cloak must be greater than the speed outside! This is possible for a single frequency (some discussion about the speed of light is given here), but not generally possible for a range of frequencies.

In short, don’t worry too much yet about invisible government agents sneaking around in your bathroom. The technology has quite a few obstacles to overcome before it comes even close to being put in James Bond’s hands!

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* A.I. Nachman, “Reconstructions from boundary measurements,” Ann. Math. 128 (1988), 531-576, E. Wolf and T. Habashy, “Invisible bodies and uniqueness of the inverse scattering problem,” J. Mod. Opt. 40 (1993), 785-792.

** Hongsheng Chen, Bae-Ian Wu, Baile Zhang, and Jin Au Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99 (2007), 063903.

*** D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314 (2006), 977-980.
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Leonhardt, U. (2006). Optical Conformal Mapping. Science, 312(5781), 1777-1780. DOI: 10.1126/science.1126493

Pendry, J.B. (2006). Controlling Electromagnetic Fields. Science, 312(5781), 1780-1782. DOI: 10.1126/science.1125907 (The complete author list, truncated by ResearchBlogging software, is J.B. Pendry, D. Schurig and D.R. Smith.)