Children should be heard, not seen. My daughter Maria's drawing of her invisible brother Max.
The bending of light in media  is the cause of many optical illusions. For example, in a mirage in the desert , light rays from the sky are bent above the hot sand where the air is thin and the refractive index is low. They are creating images of the sky that deceive the observer as illusions of water .
Why is light bent? Light rays obey Fermat's Principle of the least optical path . The refractive index n determines the optical length; distances in media with high n appear longer to light than distances in media with low n. When n varies, light rays try to stay as long as possible in regions of relatively low refractive index. In the mirage, light rays bend their paths to stay longer in the hot thin air above the desert where n is low.
Fermat's Principle also explains Snell's law of refraction (discovered by the Arabian scientist Ibn Sahl more than a millennium ago). Refraction is familiar to everyone who has tried to gauge directions underwater. Imagine an angler sees a fish. Is the fish where it appears to be? No, because the water surface refracts the light coming from the fish (the sunlight reflected off the fish). The image of the fish reaches the eye of the angler at a different direction than the "line of sight". The angler is misled, unless he has learned from experience. Light is refracted at the water surface, because the refractive index n of air is smaller than the refractive index of water and, according to Fermat's Principle, light tries to stay as long as possible in a region of low n. In media where n gradually varies, like in the desert air above the hot sand, light is not refracted, but bent.
Another example is the green flash. When the Sun is setting the Sun's rays are bent in the atmosphere, because the air is thinner at higher altitudes and so the refractive index is lower there. The light rays are bent in the opposite way to a mirage (but for the same reason). So, when the Sun appears to be setting, she has already set, because of the light bending in the atmosphere. The refractive index n of air is larger for higher frequencies of light; blue or green light is refracted stronger than orange or red light, so blue or green light is more strongly bent. The blue light, however, is scattered in the atmosphere, coloring the sky blue, due to Rayleigh scattering (conjectured by Alhazen of Basra during the early eleventh century and by Leonardo da Vinci ca. 1500). Therefore, the last light from the setting Sun is green and appears as a green flash when the atmospheric conditions are right.
Light propagation in an invisibility device [1,2].
Imagine a different situation, shown in the pictures above, where a medium guides light around a hole in it. Suppose that all parallel bundles of incident rays are bent around the hole and recombined in precisely the same direction as they entered the medium. An observer would not see the difference between light passing through the medium or propagating across empty space. Any object placed in the hole would be hidden from sight. The medium would create the ultimate optical illusion: invisibility .
Note that this idea of guiding light around an object differs from the principle behind H.G. Well's Invisible Man. The Invisible Man has made himself transparent by inventing a recipe to make his refractive index uniform, to remove the causes of scattering or absorption of light in his body. He has changed himself and pays a heavy price. In contrast, another fictitious character, the Invisible Woman creates a field that distorts the geometry of space around her, but not herself. Her field is cunningly designed to smoothly guide light around her. Whatever she does she stays hidden. My ideas [1-3] are related to recent independent work  on controlling electromagnetic fields with anisotropic metamaterials. Both concepts are based on geometry , because a medium changes the measure of space and time for light (or electromagnetic fields in general) and such a measure defines the geometry. (The word geometry comes from the Greek geometria; geo = earth, metria = measure.) The equivalence between geometries and media is also at the heart of my other research subject on artificial black holes.
Another invisibility idea is Optical Camouflage. Here the person to be camouflaged wears a retro-reflective cloak (retro-reflective like cat's eyes). The background scenery is filmed and projected onto the cloak. In this way one can hide the person, but probably not the equipment around him. My idea  also differs from plasmonic cover and anomalous localised resonance. These are methods [11-13] to make particles invisible that are smaller than the wavelength of light. Stealth technology is designed to make objects of military interest as black as possible to radar. Here the first line of defence is to reflect incoming radar waves off at odd angles. The waves that reach the object are then absorbed without reflection using impedance matching . They disappear without any echo detectable by radar.
A stealth plane is black (also black to radar waves), not invisible.
However, there is a problem in the shape of a formidable mathematical theorem: perfect invisibility devices with isotropic media are proven to be impossible due to the wave nature of light [15,16]. No illusion is perfect, or, expressed in mathematical terms, the inverse scattering problem for linear waves in isotropic media has unique solutions . But does it matter? The imperfections of invisibility might be invisibly small. Maybe the device would create a slight haze, instead of a perfect image, like the haze around Maria's brother Max in her cartoon. So one could imagine cloaking devices that are not quite perfect, but good enough in practice [1,2].
Another idea  for bending the rules of invisibility is to use anisotropic materials. In such materials the refraction of light depends on the polarization. For example, birefringent crystals are anisotropic. With anisotropic materials, one can create the optical illusion illustrated in the figures shown below:
Imagine the material appears to move the points of space to different locations: it performs a coordinate transformation from a virtual space (A) to physical space (B). It creates the illusion that light flows through the empty virtual space (A), whereas in reality it propagates in (B). In physical space, one point of virtual space has been expanded to finite size. As light would not notice a single point it flows around the expanded region without any distortion. So the interior of the expanded point is hidden and the act of hiding is concealed. In short, the material makes a cloaking device.
Once the mathematics is sorted out, how to build the invisibility device? The coordinate transformations  illustrated in the figure require media with unusually strong optical properties . Nevertheless, the very first prototype of a cloaking device  has been made by David Smith's group at Duke University.
Electromagnetic cloaking device.
How does it work? Let us take a step back. An ordinary medium such as glass consists of atoms. Each atom interacts with light like a mobile-phone antenna interacts with radio waves, receiving and sending waves. The strengths of the atomic antennas determine the optical properties of the medium and they are usually given by Nature. Recently, so-called metamaterials have been manufactured. These are materials with manmade structures that act like artificial atoms. The cloaking device consists of such a structure, a cleverly designed array of cells etched in quite ordinary circuit board. These structures are electromagnetic resonators, they are like musical instruments for electromagnetic waves, and, like musical instruments, they can be tuned. The structures are tuned such that the device performs the desired coordinate transformation of space. The electromagnetic cloaking device does not make things invisible from visible light - as you can clearly see the device, but from microwaves. Microwaves are electromagnetic waves like visible light, but they have much longer wavelengths, typically a few centimeters, whereas the wavelength of visible light lies around 500 nanometers. Microwaves are used in radar, mobile phones and wireless technology.
The cloaking device also works for only one frequency for a fundamental reason that is apparent in the coordinate transformation it performs. Have another look at the figure below:
The device changes the geometry of space. In particular, it creates the illusion shown in (A) that light propagates through space that is completely empty, apart from one point. In physical space (B) this point occupies an extended region. However, electromagnetic waves, light or microwaves, pass a point in infinitely short time. So the speed of light in the material of the device must approach infinity. As information cannot be transmitted faster than the speed of light in vacuum, electromagnetic waves get stuck in the cloaking device . Cloaking works only when no new information is transmitted, when nothing changes. So the waves have to be completely stationary - they must oscillate at only one frequency. This frequency is set by the cloaking device and can vary from device to device, but it always is a single frequency that must be known in advance. For visible light, the frequency appears as the colour of light. So for seeing things disappear, everyone has to wear tinted glasses and agree on their colour in advance, which obviously is not of much practical use.
The way to get around this problem  has been inspired by my paper Optical Conformal Mapping  that appeared at the same time as the blueprint  of the Duke cloaking device. Here I developed an alternative way of bending the rules of invisibility. The inspiration behind my idea  comes from the solution of another impossible task, a problem in cartography: Gerardus Mercator managed to map the Earth, a curved surface, onto a flat sheet of paper, while preserving the right angles between longitude and latitude.
One might object that the Mercator Projection is not really politically correct; areas in the polar regions appear to be much larger than the developing countries near the equator. This is the price to be paid for preserving the angles on the globe.
An optical medium changes the measure of distance perceived by light. Suppose that the medium preserves the right angles between light rays and wavefronts like the Mercator Projection preserves the angles between longitude and latitude. In this case, the medium could smoothly guide light without reflection. In mathematics, maps that preserve angles are known as conformal maps and their optical implementation with an appropriate refractive-index profile is an Optical Conformal Mapping . Most conformal maps require more then one sheet to represent them, but rather several or even infinitely many, sewn together at branch cuts between branch points. Some generate mind-boggling Riemann surfaces like the one that represents the tiling on the title page of the book on complex analysis shown below. The picture beside the book shows the tiling behind the scenes of the figure on light propagation . Riemann surfaces have inspired some of M.C. Escher's intriguing paintings. Their many branches can be used to hide regions of space, provided an optical medium facilitates the mapping [1,2]. If one has something to hide, one should hide it on Riemann sheets .
There is yet another obstacle: light that has passed the branch cut to another sheet of the optical Riemann surface will get stuck there and never return, as illustrated in example shown in the figures below . The left figure shows the light rays in real space, whereas the right figure shows how they would appear on the Riemann surface. In this imaginary map the rules of light propagation are much simpler than in the real medium that facilitates the optical conformal mapping. Light rays are straight here and their destiny is clear to see. Whenever light passes the branch cut to the lower sheet, the optical underworld that corresponds to the interior of the circle on the left picture, it approaches the singularity of the medium at the centre and gets stuck.
One should gently guide light back from the optical underworld. Here the inspiration comes from the tale  of the one-pound note that the Bank of England issued to commemorate the 300th anniversary of Sir Isaac Newton's Philosophiae Naturalis Principia Mathematica.
Besides Sir Isaac the note shows the Sun and the planets orbiting around her in ellipses. Imagine that the branch point of the invisibility device plays the role of the Sun with light circling around like the planets. A refractive index profile that mimics Newton's law of gravity on the Riemann sheet would do the task. Such devices, called Eaton lenses, have been conceived . Astronomy students might object that there is something distinctly wrong with the celestial ellipses on the one-pound note: the Sun is supposed to sit at the focal point, not the centre! Instead of Newton's ellipses, the Bank of England plotted the ellipses of Newton's rival, Robert Hooke. Yet a simple conformal transformation rectifies the blunder and transmutes Hooke's into Newton's force and vice versa [2,19]. Ironically, the laws of the two archrivals are essentially identical. Devices based on the analogue of Hooke's law are known in radar technology and are known as Luneburg lenses .
These ideas inspired Tomas Tyc and me to develop concepts for Broadband Invisibility by Non-Euclidean Cloaking . Let me try to explain. The potentials used for guiding light back from the underworld of the Riemann surface can be viewed as creating a Non-Euclidean geometry of curved space, another one of Riemann's visionary mathematical creations. A curved space for light is not as outlandish as it may sound, because ordinary materials create such optical spaces all the time. Take a simple lens. The lens focuses parallel light rays in a single point, the focal point. The lines the light rays draw are no longer parallel - the parallels meet - which is one possible feature of a Non-Euclidean geometry where Euclid's famous parallel postulate is no longer valid. Now, replace the previous Riemann surface by a combination of two spaces, a Euclidean space, the plane, and a Non-Euclidean space, the surface of a sphere. The Euclidean space is partly wrapped around the sphere like a sheet of paper, touching the sphere at one straight line segment, the branch cut.
Imagine both the sphere and the plane are covered by a joint coordinate grid, the thin black lines shown in the figure. We map the coordinate lines onto physical space that, in two-dimensions, is the plane shown below:
We simply flatten the sphere and stretch the entire geometric contraption as if it were made of rubber. The magenta circle defines the boundary of the device. Its interior has been expanded to make space for the grid of the sphere. In particular, the line where plane and sphere touch has been opened like an eye (thick black lines) to include the sphere. But the curvature of the sphere should be maintained as intrinsic curvature like in shape memory plastic. Note that it is possible, in principle, to create materials that implement such geometries for light . But how do they help in making a cloaking device? Everything in virtual space (A) has a faithful image in physical space (B). So we can discuss the propagation of light in virtual space (A) where we can use our intuition. What happens there? Most light rays pass the sphere, but some incident light rays venture from the plane to the sphere; they return after one loop and continue in the same direction. We see that all light rays leave in the same direction they came from - as if nothing has happened. The sphere is invisible. However, it does not make anything else invisible yet. But now imagine we place a mirror along the equator of the sphere:
Light rays are reflected at the mirror, creating the illusion that they propagate on the northern hemisphere, whereas they stay in southern regions. After another reflection they are back on track and perform the rest of their loops on the sphere as if the mirror were not there. Now one can hide things - behind the mirror! There is an alternative to doing it with mirrors: note that light rays on the sphere in (A) never cross the red zigzag line. One could expand the line that light never crosses to create a hidden space:
These geometrical ideas are useful for broadband invisibility, because their implementation does not demand extreme optical properties such as infinities or zeros of the speed of light, for the following reason. In virtual space, light propagates at the speed of light in vacuum. Physical space represents a deformed image of virtual space; the speed of light follows this deformation. Expressed in quantitative terms, if an infinitesimal line element in virtual space is n times longer than its image in physical space, then the refractive index in the corresponding direction in physical space is n. Our figures show that the ratio of the line elements is neither infinite nor zero. Even at a branch point the spatial deformation in any direction is finite, because here the coordinate grid is only compressed in angular direction by a finite factor, in contrast to optical conformal mapping .
So far, we discussed two-dimensional space, because here we can visualize a curved space, the surface of the sphere, embedded in three-dimensional space. In order to see a curved three-dimensional space we would need at least four dimensions, we would need to see it in hyperspace. Nevertheless, we can still use similar ideas as for our two-dimensional cloaking device, but we cannot visualize them as simply as in the two-dimensional case. We replace the three-dimensional sphere by a four-dimensional hypersphere and the branch cut by a suitable two-dimensional surface, the door to hyperspace. The surface of the hypersphere is a three-dimensional curved space. Such a space can be made in optics in a device known as Maxwell's Fish Eye . It all works out, and we get a cloaking device where light rays are bent like this:
With geometry, light and a wee bit of magic - aka technology - invisibility may become practical .