Transformation Optics | The First Cloak

The First Metamaterial Cloak: 2006

In 2006, the concept of controlling electromagnetic fields using transformation optics became a reality. It was startlingly easy to write down the recipe for unprecedented structures that would manage light in unusual and exotic ways. Just a few years before, the material properties required for these structures might have been considered far too difficult to ever achieve in practice; but by 2006, advances in metamaterials research suggested that transformation optical devices could be a reality. Of course, the most immediately appealing transformation optics device to make was the 'invisibility cloak'. Could we actually make a metamaterial cloak designed by transformation optics?

The Cloak Experiment Concept - One of the interesting signatures of the cloak is the way that it bends waves as they enter then exit the device. When a wave enters, its phase front is distorted, looking as though the wave is avoiding the central cloaking region, then recombining into a planar front on the other side. The wave fronts actually contain the geometry of the transformation used to design the cloak, as can be seen by comparing the transformation grid with the field pattern in the plots below.

Although the purpose of an invisibility cloak is to make things invisible, one might anticipate that the very first cloak might not work perfectly. The conceptual picture above suggests, however, that if we could take a look at the fields propagating inside the cloak-if we could actually see how the wave fronts were behaving-we could at least verify the underlying cloaking mechanism. That is, we could confirm that a transformation optics design could be achieved, even if only approximately.

So, if we were to make a transformation optical cloak using metamaterials, how could we 'see' inside of it?

It turns out that, at microwave frequencies, there is an easy way to 'look inside' a material. You can sandwich the material in between two large metal plates. Microwaves that are injected into the region between the plates are guided, propagating in the two dimensional plane. By poking a tiny hole in one of the plates and inserting a probe, it's possible to measure the field at that point without disturbing the field pattern. Scanning the probe over an area allows a picture to be created of the wave interacting with the material. This technique had been successfully used in our lab for years to measure other interesting samples, and seemed like a great way to see if a metamaterial cloak would show the predicted field pattern.

A couple of illustrations of the field mapping chamber are shown below, along with a photograph of the actual chamber. The speckled dark gray material forming a toothed circle around the edges is a special material that absorbs microwaves; the microwave absorber is used to create a channel to input a beam into the chamber, and also to reduce reflections from microwaves that are scattered from the object under test.

So, the sample we decided on needed to work at microwave frequencies--where it would be easy to fabricate and work with our existing equipment--and would need to be shaped something like a flat pancake with a hole in the center where an object might be hidden. These constraints allowed us to take the next step of designing the material.

The Cloak Design - Because of the boundary conditions imposed by the parallel plate waveguide geometry, the flat metamaterial disk approximates an infinite cylinder. That means that we should make use of a two dimensional transformation. For simplicity, we decided to use a symmetric, radial transformation that could easily be written down in the cylindrical coordinate system as follows:

\[\rho ' = \left\{ {\begin{array}{*{20}{c}} {\rho \frac{{{R_2} - {R_1}}}{{{R_2}}} + {R_1}}&{\rho < {R_2}}\\ \rho &{\rho \ge {R_2}} \end{array}} \right.\]

In cylindrical coordinates, the transformation is easy to picture. ${R_1}$ is the inner radius of the cloak, while ${R_2}$ is the outer radius. The transformed angle remains the same in both coordinate systems. To actually make use of this transformation from one set of cylindrical coordinates to the other, we must first transform from and then back to Cartesian coordinates. We won't show the details here, but the result is that the constitutive parameters for the cloak can be written as

\[{\varepsilon _\rho } = {\mu _\rho } = \frac{{\partial \rho '}}{{\partial \rho }}\frac{\rho }{{\rho '}},\,\,\,\,\,\,{\varepsilon _\varphi } = {\mu _\varphi } = \frac{{\partial \rho }}{{\partial \rho '}}\frac{{\rho '}}{\rho },\,\,\,\,\,{\varepsilon _z} = {\mu _z} = \frac{{\partial \rho }}{{\partial \rho '}}\frac{\rho }{{\rho '}}\]

While we might have used any radial transform to achieve the cloak parameters, it was the simple transformation above that was applied. Using that transformation leads to the following constitutive parameters for the cloak:

\[{\varepsilon _{\rho '}} = {\mu _{\rho '}} = \frac{{\rho ' - {R_1}}}{{\rho '}}\]

\[{\varepsilon _{\varphi '}} = {\mu _{\varphi '}} = \frac{{\rho '}}{{\rho ' - {R_1}}}\]

\[{\varepsilon _z '} = {\mu _z '} = {\left( {\frac{{{R_2}}}{{{R_2} - {R_1}}}} \right)^2}\frac{{\rho ' - {R_1}}}{{\rho '}}\]

While it is a relatively straightforward mathematical exercise to arrive at these material specifications, actually making the material is a far more difficult proposition. The parallel plate waveguide forces the electric field of the propagating waves to be perpendicular to the plates, or along the z-direction, with the magnetic field lying in the plane. Since we have only three field components to worry about, we can reduce the set of six constitutive parameters down to three: ${\bf{\varepsilon _z}}$, ${\bf{\mu _{\rho}}}$, and ${\bf{\mu _{\varphi}}}$. Still, even controlling those three represents a challenge, since they all vary as a function of the cloak radius, and take extreme values (from zero to infinity).

A plot of the required transformation optical parameters reveals just how demanding a material condition the cloak requires. As shown in the figure below, two of the components, ${\bf{\varepsilon _z}}$ and ${\bf{\mu _{\rho}}}$ approach zero at the inner radius of the cloak, while ${\bf{\mu _{\varphi}}}$ tends towards infinity. All three parameters vary widely throughout the cloak, and presumably must be controlled to a fairly tight tolerance.

Before we might rush off to try and build this cloak, we might first ask if, assuming everything were to go perfectly and we achieved the right material parameters everywhere, would it actually work? One way to answer this question is to directly simulate the structure. That is, take the recipe for the material found above, and put it into a computer program that solves for electromagnetic fields in the presence of materials. At this point, we don't need to think about transformations or any deeper physics associated with the recipe; we just take the formulas for the material parameters, program them into a simulator and see what the wave does.

An example simulation of the ideal cloak is shown below. This particular simulation was performed using COMSOL Multiphysics by Steve Cummer at Duke, in 2006. As predicted by the transformation optical approach, the full electromagnetic simulation reveals the expected cloaking behavior. Even though we might call this the 'ideal' cloak, there are some compromises necessary to actually perform the simulation. First, we cannot put in infinite values for any of the material parameters. We have to cap the value of ${\bf{\mu _{\varphi}}}$ at something large (but finite). Second, COMSOL is a finite element program, meaning that the cloak is broken up into finite sized cells, so that the material parameters don't vary smoothly over the cloak region. Yet, both of these imperfections are hardly noticeable in the final simulation, which shows that there is very minimal reflection, and almost no shadow on the opposite side from where the wave is incident. (The wave is incident from the left in the figure below.)

The Reduced Parameter Cloak - The simulation shown above was very exciting to see, since it confirmed the predictions of transformation optics. But, it wasn't an experiment; the material still had to be created or fabricated somehow. And, still, the variation of three components of the constitutive parameters would have to be controlled to--at the time--unknown precision. It seemed like a daunting task.

Supposing that we wished to pursue a metamaterial version of the ideal cloak, what would that look like? Well, we know that split rings can provide a magnetic response, and other structures like short wires can provide an electric response. And we know that the constitutive parameters only vary as a function of radius, not as a function of angle. So, the structure that comes to mind consists of concentric layers of cells, each cell containing metamaterial elements that are used to control ${\bf{\varepsilon _z}}$, ${\bf{\mu _{\rho}}}$ and ${\bf{\mu _{\varphi}}}$. All of the elements in the cells at a given radius are identical, just rotated; and all of the elements will be slightly different as a function of the radius to achieve the variations in the effective medium parameters described above.

The metamaterial layout that comes to mind, after considering the constraints imposed, is shown schematically in the figure below. The cylindrical cloak consists of a set of concentric layers broken into identical (but rotated) cells. Inside each cell might be a set of components, like the split ring resonators and electric resonators shown in the figure. To achieve a magnetic response along two axes, as required by the transformation optics specification, two split ring resonators are needed in different (orthogonal) directions. We might also need some elements to provide electric response that contributes to the permittivity component. As the schematic diagram shows, the cell quickly fills up to satisfy the full cloaking requirements!

While the transformation optical specification is easy to write down mathematically, the figure above indicates the difficulty in creating a metamaterial version. Each of the elements within the cell must be precisely designed, and the geometry will change in every layer to achieve the variation in the effective constitutive parameters. Complicating the picture is that the elements will also interact with each other strongly, so that the properties of the composite structure are not predicted by the properties of each element computed separately.

Faced with this dilemma in 2006, we knew the full cloak design would take a considerable effort and would not be quick. Was there a simpler route to demonstrating the cloaking mechanism and the transformation optics approach, even without implementing the full design?

The full parameter cloak is an exact solution for invisibility, at least at one frequency, eliminating both reflection as well as any shadow. What if we were to back away from the ideal parameters, giving up some cloaking performance to arrive at a simpler structure? One idea that came to mind was to consider a cloak that would work for ray optics. In optics, magnetic response is often so small it is neglected, and the material characterized only by its index-of-refraction. The refractive index can be used to predict the propagation of rays through a medium. By focusing only on the refractive index, we might find more flexibility in the cloaking parameters.

For the orientation of the fields within the parallel plate waveguide, the refractive index is anisotropic with two components:

\[n_\rho ^2\left( \rho \right) = {\varepsilon _z}\left( \rho \right){\mu _\rho }\left( \rho \right) = {\left( {\frac{{\rho - {R_1}}}{\rho }} \right)^2}{\left( {\frac{{{R_2}}}{{{R_2} - {R_1}}}} \right)^2}\]

\[n_\varphi ^2\left( \rho \right) = {\varepsilon _z}\left( \rho \right){\mu _\varphi }\left( \rho \right) = {\left( {\frac{{{R_2}}}{{{R_2} - {R_1}}}} \right)^2}\]

If we only care about the components of the index, then we only care about the products of the permeability components and the z-component of the permittivity. We are thus free to choose the parameters however we want, as long as the index components are correct. Consider the following:

\[{\mu _\rho }\left( \rho \right) = {\left( {\frac{{\rho - {R_1}}}{\rho }} \right)^2}\]

\[{\mu _\varphi } = 1\]

\[{\varepsilon _z} = {\left( {\frac{{{R_2}}}{{{R_2} - {R_1}}}} \right)^2}\]

These parameters provide the same values for the index components, but two of the parameters are just constants! In fact, ${\bf{\mu _{\varphi}}}$ is just equal to one, so we can forget about it entirely. Achieving a set of parameters like these is a much easier task, requiring possibly the design of a single element. While the reduced parameter cloak might not function perfectly, at least it could possibly show the transformation optical mechanism, with waves being directed around a central cloaked region. It was the reduced parameter cloak that we decided to build in 2006.

A plot of the reduced parameters is shown below. Compared with the full parameter cloak, the reduced parameter cloak is a far more achievable structure, requiring only one of the material parameters to possess a gradient in the radial direction. The permittivity is held at a constant value of 4--a value similar to many insulating materials. All of the parameters are bounded; that is, none of the parameters tends to infinity as in the ideal case above.

How well should a reduced parameter cloak perform? As we did for the ideal cloak, we can put the material parameters into a computer program and see what happens when a wave interacts with the structure. The result is shown in the figure below. As expected, there is a considerable distortion in the incident wave, with both reflections observed back toward the source of the wave, and a shadow observed on the opposite side. But, also as expected, the waves are curved around the cloak in a way very similar to what would happen in a full parameter cloak; this occurs because the wave trajectories are controlled predominantly by the anisotropic refractive index. The cloak even appears to restore the wave fronts after they have passed through the structure.

So, the reduced parameter cloak vastly simplifies the potential metamaterial design, yet retains the basic expected behavior of the transformation optical design. But what we've seen thus far is a simulation, and have not yet actually fabricated any structure. If we were to map the fields of an actual metamaterial cloak, would it look anything like the simulated field plot?