Transformation Optics | Making Things Invisible

Cloaking with Transformation Optics

There are many possible routes to making objects seem invisible--after all, magicians have been doing this for centuries! The magician can make an object invisible to an audience by cleverly positioning mirrors around the object to deflect the light away from the object. The mirrors trick our eyes, making us believe that light from somewhere in the distance passed through empty space. The trick works because we intuitively understand that light travels in a straight line; when something causes light to bend or be rerouted, we can get confused and fooled by the light that finally reaches our eyes.

The diagram below shows a simple example of a light source whose beam is rerouted around an object--in this case a rabbit, seeming as if it passed through empty space. While not the grand stage illusion a magician might concoct, this example shows how light can be managed to make an object seem as if it isn’t there.

Cloaking with Transformation Optics - While the use of mirrors provides one scheme for achieving invisibility, there are a number of problems with this approach that can be easily pointed out. Most notably, the invisibility effect, as pictured, works with the observer situated at only one position. If the observer moves around, he or she will quickly discover they've been tricked, since they'll see the mirrors and maybe other gizmos where only the rabbit should be. Magicians go to great lengths to ensure the audience has a limited perspective of the illusion, keeping their apparatus out of view and hidden in the shadows.

It is a very challenging task to truly make an object appear invisible, regardless of where the observer and the light source are located. Assuming that we can't change the optical properties of the object we are trying to conceal, we must instead try to imaging wrapping the object with some sort of cloak that will render both the hidden object and itself invisible, appearing as if light had passed through empty space. It's what the magician tried to do, but could only achieve with a bunch of heavy mirrors, and only if the observer were fixed in one location. It's very hard to imagine what sort of collection of mirrors or materials could allow light to detour around the concealed object, looking as though it had passed through space, no matter what direction it came from.

Fortunately, it turns out we don't have to sort out all of those details to figure out how to make an invisibility cloak. At least conceptually, the design process turns out to be remarkably simple, and starts just with a little bit of imagination.

First, we need a way to visualize empty space. We can do this by imagining a set of points fixed in space, which we connect by lines to form a grid. Space, of course, is continuous, so that there are an infinite number of points that can be specified in Cartesian coordinates by $(x, y, z)$; but we visualize space by plotting lines of $x$ along intervals of $y$, and lines of $y$ along intervals of $x$. Since light travels in a straight line we can easily depict the trajectory of a ray of light in space by just plotting a line. We've indicated a ray by the blue line on the grid, shown below, which represents the path that a ray of light will take in space.

We often think of rays when we think of light, but in reality light is an electromagnetic wave, which varies throughout space. We can depict a wave by indicating its oscillations throughout space, as shown on the plot below (right). Here, the wave is a simple sinusoidal variation, with the wavelength being twice the distance between the white lines. This sort of depiction of a wave is more common at lower frequencies, where the earlier experiments first took place.

Although it makes great sense to visualize empty space with a simple grid as above, we don't have to do it that way. We can create a coordinate transformation of any sort, and make the same sort of grid using the new transformed variables. Why would we want to do that? Let's not answer that right away, but just show the result first.

We initially create a function that transforms us from our usual coordinates $(x, y)$ to a new set of coordinates $(x', y')$. We can write down any transformation we like that allows us to compute $x'(x, y)$ and $y'(x, y)$. If we want to create a cloak, it's important that the transformed coordinates become identical to the original coordinates at some point, otherwise we have to transform all of space. A common type of transformation is a radial transformation, that pushes all of the space within some circular region into a shell. The transformation can be visualized by plotting the same lines of constant $x$ or constant $y$, but in the new frame $(x', y')$. It's sometimes useful to write the transformation as a matrix equation relating the new coordinates to the old, like ${\bf{x'}} = {\bf{\Lambda x}}$.

The result of this little mathematical exercise is the plot shown below. We haven't done anything real, we just are now looking at space in a different but very interesting way. First off, there are no grid lines in the core of the space. That means if the grid lines were to represent paths that light can take, no light can seemingly reach a portion of space. It's as if we created a hole in space. But, again, this is just an illusion of plotting space in a different set of coordinates!

Now, if we actually could warp space physically, then this grid shows what would happen. Our original ray of light, shown by the straight blue line above, would take a curved trajectory in the transformed space. In fact, all rays of light entering the transformed region would be pushed away from the center. Anything placed within the center region would effectively be invisible, since no light could ever come into contact with it.

The same sort of cloaking picture is true for a wave. If we plot the sinusoidal wave in the new coordinates, then we get the sort of picture shown above and to the right. Here, the flat wavefronts that enter the transform region are bent away from the core, and are then restored as they exit the other side.

Comparing these visualizations of warped space to the mirror analogy at the top of the page, we can see that we have achieved invisibility, at least in principle. Rays of light are redirected around the object or region to be concealed, in this case by opening up a hole in space and pushing all the space from a circular region into a shell. Outside the shell, space is not affected. In addition, because we are just transforming space, we have created a shell that does not scatter light or waves—nothing reflects from the shell, and light that passes through returns to its trajectory as if it had passed through empty space. Unlike the magicians approach with mirrors, our potential invisibility cloak works no matter where the observer is and no matter where the light comes from.

Transformation Media - Our proposed invisibility cloak is an intriguing device, but suffers from a major implementation problem: We cannot warp space. One might be inclined to dismiss the entire concept as unreasonable, except that there is an interesting way to exploit the transformations and the very elegant means of managing light that is implied.

Maxwell's equations in media, the equations that govern light and other electromagnetic waves, can be written to have the same form in any set of coordinates. The trick is that the presence of a medium enters Maxwell's equations as a couple of parameters that indicate the electric and magnetic response of a medium. When we transform our coordinate system, we can apply the transformation to these parameters such that the rest of the equations look identical as they did before the transformation.

So, what changed?

When we apply the transformations to the material parameters, we end up creating the specification for a new material. That material accomplishes exactly our goal, causing light to behave as if we had actually warped space! It's actually a stunning result, and results in a remarkably simple recipe. One doesn't have to understand the details and all the symbols to appreciate just how simple the recipe actually is. If our coordinate transformation is expressed as ${\bf{x'}} = {\bf{\Lambda x}}$, then the material parameters required to actually implement the transformation are specified by

$${\bf{\varepsilon '}} = \frac{{\bf{\Lambda}}{\bf{\varepsilon}}{\bf{\Lambda}}^T}{\left| {\bf{\Lambda }} \right|}     {\bf{\mu '}} = \frac{{\bf{\Lambda}}{\bf{\mu}}{\bf{\Lambda}}^T}{\left| {\bf{\Lambda }} \right|}$$

That's all there is to it. Just from those relatively simple equations, we can obtain the design of a material that will cause light to propagate as if we had warped space. Now it's fair to ask, what's the catch? The catch is that the required material generally must be anisotropic (having different properties along different directions), and must have both electric and magnetic response. Moreover, those properties must vary throughout space--or at least over the transformed region. So a transformation optics solution requires a very complicated medium, or what we might call a transformation medium, to achieve the desired management of light. Trying to achieve a general transformation optical medium with conventional materials would be a difficult, if not an impossible task.

However, we are not bound to using conventional materials when we create a transformation medium. Modern artificial materials, now often called metamaterials, have provided us with enough new capabilities that we can actually often achieve the incredibly demanding transformation optical properties. Metamaterials, combined with the transformation optical design approach, have now provided us with a new route to invisibility!