Though the introduction of optical invisibility cloaks in 2006 caused a huge sensation around the world in both the media and the general public, arguably even more significant to the optical science community is the technique used to design cloaks.
The original cloaking papers took advantage of the observation that a physical warping of space can be simulated by an appropriately-designed material medium. To design a cloak, we first figure out how we want to bend space around a cloaked region, and then it is relatively straightforward to figure out the medium that simulates that bent space.
How does this work? Let us imagine that we have an ordinary, unbent, region of space. To design an invisibility cloak, we imagine poking a pointlike hole in that space and stretching it to make a void. Physically, that void is completely inaccessible — it lies outside of ordinary space. This is illustrated below. Ordinary space is on the left, with a ray of light and the point that gets stretched into a void on the right. The ray gets bent away from the void thanks to the distortion of space.
Once a (hypothetical) warping of space is designed to achieve the desired effect, in this case cloaking, there is a systematic process to determine what sort of material will provide the same effect in our ordinary (unwarped) space.
This strategy of designing optical devices by a virtual warping of space is known as transformation optics, and it has been implemented for a number of crazy applications, from optical illusions to optical black holes. It can also be used to make more mundane but highly important devices, such as 90-degree bends in fiber optic cables that produce no loss. An optical fiber “traps” light inside of it by total internal reflection (see here for explanation), which works very well except at sharp bends or “kinks” in the fiber, where light can leak out. If we create a 90-degree bend by warping space, however, as illustrated below, the light will in principle be redirected without loss!
The tools of transformation optics were readily taken from those of Einstein’s theory of general relativity, where gravitational fields actually induce a warping and distortion of space and time. With this connection, it was quite natural for researchers to ask whether various types of astrophysical phenomena, both real and hypothetical, could be simulated for light using exotic optical materials.
In 2007, the most spectacular of these possibilities was proposed* by a group of researchers from the U.S., the U.K. and Finland. They suggested that it is possible to use transformation optics to design an optical wormhole — a tunnel for light between distant points in space! A longtime staple of science fiction stories, such wormholes (also known as Einstein-Rosen bridges) would provide a hidden tunnel for light that allows it to travel from one region to another. At first glance, as we will see, this would seem impossible, as a wormhole is an extra-dimensional region of ordinary space, and we can’t add extra dimensions to our three-dimensional space just by the use of weird materials. Or can we? It turns out that it is not only possible, but that the construction is far simpler than you might imagine.
But what is a wormhole? The name comes from the actual holes that worms burrow through apples, as might appear below.
If you were an insect that lived entirely on the surface of the apple, a wormhole presents a definite advantage for travel: the path through the wormhole from A to B is much shorter than the path along the outside of the apple. Similarly, if wormholes actually exist in our universe, they could present a shortcut between distant points in space, which is why they are especially attractive for use in science fiction.
Travel through a wormhole would likely be an exceedingly strange experience, however. Let us restrict ourselves to a world that is wholly two-dimensional at first, such as presented in the classic 1884 book Flatland, and take a closer look at such two-dimensional wormholes.
An illustration of one is shown below. To emphasize the two-dimensional nature of the world, I chose Pac-Man as a resident.
First of all, it can be seen that we can enter the wormhole from any direction in two-dimensional space. That is, Pac-Man can approach it from North, South, East or West, and enter it. The view ahead of him as he approaches the wormhole will be strange indeed; if he approaches the entrance from the direction between the two, he will actually see his own rear-end! The picture below shows how a ray of light emanating from his rear could be seen at his front.
The situation is even more potentially bizarre, as is best illustrated within the hole itself. When Pac-Man is inside, he lies on the surface of a long cylinder. Because light will travel a full circle around the circumference of the cylinder, he will actually see an infinite number of exact duplicates of himself, doing exactly the same thing that he is doing, on either side! Even more surprising, he can play catch with himself, throwing a ball to one of these images, and catching it from the other side.
The easiest way to imagine constructing a wormhole for Pac-Man is to use two flat sheets of paper. Take one sheet and cut two circular holes out of it; take the other sheet and tape two edges together to form a tube. Now tape the two edges of the two to the two circular holes, leaving no gap, and you will crudely get Pac-Man’s wormhole illustrated above.
A similar construction can be done for a wormhole in three-dimensional space by analogy, although it is much more difficult to visualize the result. For the 2-D case, we start with a plane and cut out two circles from it; for the 3-D case, we start with a volume and cut out two spheres from it. For the 2-D case, we connect the two holes with a three-dimensional cylinder formed from a surface; for the 3-D case, we connect the two spherical voids with a four-dimensional cylinder formed from a volume!
If this is a hard concept to grasp, don’t worry too much about it right now! More important for our discussion is the question of how to create a virtual 3-D wormhole for light. Here we run into what appears to be an insurmountable problem, as can be seen by considering the 2-D wormhole model constructed above. Using the construction above, we see that we have to create the model in three-dimensions; that is, a 2-D wormhole seems to require 3-D space to make it. By analogy, it would seem that the creation of a 3-D wormhole requires working in 4-D space! Since we live in a world of only three dimensions, it would seem at first that it is impossible to make a wormhole using transformation optics.
But let’s consider the 2-D case again, and look at a single sheet of paper. Now we take the following steps (and I use a real sheet of paper because illustrations become really tricky to make).
First, draw two finite parallel lines on the interior of your sheet of paper, and use scissors to cut only along those lines, leaving two slits in the sheet.
Now comes the tricky part: take the inner two edges of the cuts, between the two slits, and pinch them underneath and tape together. Then take the outer two edges of the cuts and tape them together.
If you’ve done it right, you end up with something that looks as follows:
The red and green lines highlight that one can travel along the tube that has been created (red), or one can pass under the tube (green). With a single sheet of paper, we have created our 2-D wormhole! Here’s the view from the other side:Remarkably, with just two cuts and some tape, we can take a single sheet of paper and turn it into a 2-D wormhole! To put it another way, a two-dimensional space with two parallel “cuts” in it is almost exactly the same thing as a wormhole.
Notice I said almost exactly the same thing. If we wanted to create a fake wormhole in 2-D space, we could easily do it by having two parallel impenetrable barriers that would act as the “cuts” in our sheet of paper. However, there’s no way in 2-D space to actually tape those cuts together, and so our sheet of paper with two barriers is topologically equivalent to an actual wormhole with two impenetrable barriers connecting the two entrances. The red and purple lines below represent the cuts, or barriers, that cannot be crossed.
I used the phrase “topologically equivalent” above, and it is worth a brief explanation. The mathematical discipline of topology is concerned with the equivalence of shapes; two objects are said to be topologically equivalent if one can be made into the shape of the other without creating holes, removing holes, or otherwise tearing the material. Therefore, topologically, a solid sphere is equivalent to a solid cube. A coffee mug is equivalent to a donut, because each has only a single hole:
So we can make an “almost wormhole” in 2-D with a single sheet of paper; it stands to reason that we should be able to make an “almost wormhole” in 3-D space as well. The topology of this wormhole can be correctly guessed by making the right observations about the 2-D case. In 2-D, the “almost wormhole” involves making a tunnel that connects two parts of the paper together. In three dimensions, a similar tunnel is simply… a cylinder!
Imagine that this cylinder is completely impenetrable by light. Then the inside surface of the cylinder is equivalent to the impenetrable barrier of the 2-D “almost wormhole” in purple, and the outside surface of the cylinder is equivalent to the impenetrable barrier in red.
So an “almost wormhole” in three dimensional space is simply… a cylinder? You will be forgiven if you feel a little cheated at this point. ”Surely,” you say, “There must be more to a wormhole than simply buying a piece of pipe at the local hardware store?”
Well, yes! A wormhole is a passage between two separate locations in space, and one should be able to pass between the two openings without bumping into a solid cylinder along the way. This is where transformation optics comes into play: we can use essentially the same trick used to design a cloak to design a wormhole.
In designing a cloak, we pushed space away from a point to make a spherical void; in designing a wormhole, we can roughly imagine pushing space away from a line in order to make a cylindrical void. Rays approaching the cylinder from the side will be bent around it and carry on as if they had run into nothing at all. Additionally, however, we recall that a genuine wormhole must accept rays approaching it from all directions; the transformation must also, therefore, take rays approaching the opening and bend them to pass through. A variety of possibilities are summarized in the picture below.
So creating an optical wormhole does not just involve buying a cylinder, but creating a cylindrically-shaped material that acts both as a cloak along the tube and an attractor of light, of sorts, at the ends. Also, some optically opaque material would need to be placed on the inside of the cylinder to simulate the “impenetrable barrier.”
We have noted how things could potentially look really weird through a 2-D wormhole, and the same is true for a 3-D version. The authors of the 2007 wormhole paper generated a simulation of what an infinite checkerboard plane would look like through a virtual wormhole oriented parallel to the plane.
For a short wormhole, the image is not terribly distorted, but for a long wormhole, multiple reflections of light off of the inner surface result in multiple images of the checkerboard beyond.
It is important to note that there are some significant limitations to the design of “optical wormholes”, albeit the same ones that plague most optical cloak designs. From a practical point of view, the materials required to design such a wormhole — materials not found in nature known as metamaterials — are extremely complex and cannot be made at optical frequencies. Even if the materials could be made in principle, the wormhole would only function effectively over a very narrow range of frequencies — in essence, for a single color!
Nevertheless, such a virtual wormhole could have a number of potential applications. The most interesting of these involves another exotic, theoretical and unobserved physical phenomenon: a wormhole could be used to make a magnetic monopole!
All known magnets, such as an ordinary bar magnet, have two sources, or poles, of magnetic fields; these magnets are therefore known as “dipoles.” The fields of a bar magnet are illustrated below, and it can be seen that the fields emanate from a north and south pole.
You might think that we could get individual north and south poles if we break the magnet in half, but in fact we’ll simply get two dipole magnets, each with its own north and south.
Though single poles of magnets have never been seen, there are good theoretical reasons why at least one magnetic monopole should exist in the universe. Being able to create a “fake” monopole would be an excellent opportunity to understand how to search for the real thing better.
Now suppose we stick one end of a bar magnet into the end of a really long wormhole. The fields will emerge from the other end all in the same direction, making that end a virtual magnetic monopole!
Other, more practical, possibilities exist for electromagnetic wormholes, including serving as shielding for fiber optic or electrical cables. Surprisingly, though, there has been very little follow-up on the original paper, save for a couple of other papers by the original researchers.**
Nevertheless, virtual wormholes show just how crazy the possibilities are when one mixes optics with the space-bending (and mind-bending) ideas of general relativity!
* A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett. 99 (2007), 183901.
** A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, “Electromagnetic wormholes via handlebody constructions,” Commun. Math. Phys. 281 (2008), 369-385.
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, “Cloaking Devices, electromagnetic wormholes, and transformation optics,” SIAM Review 51 (2009), 3-33.
Postscript: My choice of Pac-Man to represent an inhabitant of a wormhole-ridden space was not random! In fact, Pac-Man’s maze has its very own wormhole in it — I drew it explicitly on the classic game screen below.