Metamaterials | Basic Homogenization

What is a Metamaterial?

This fundamental question has generated an enormous amount of discussion in the metamaterials community over the years and led to countless arguments and controversies! Initially, a metamaterial was defined simply as an artificial material, structured using arrays of macroscopic elements rather than the atoms and molecules of conventional materials. Of course, these macroscopic elements, or inclusions, are themselves composed of atoms and molecules, but by fashioning materials into larger constructs, geometry begins to play an important role in the properties of the composite; that is, the properties of a metamaterial are derived not only through the chemical bonds and composition of the constituent materials, but also by their shapes. We therefore can obtain new properties for a metamaterial by changing the shapes of its inclusions.

An illustration of the metamaterial concept is shown in the figure below, which shows a composite material formed out of some red dots. The dots might be little plastic spheres, for example, or maybe even molecules, but they are separated from each other by some amount of space. If we attempt to investigate this material with a wave (like light or electromagnetic waves), then the wavelength corresponding to the wave indicates how much of the detailed internal structure we can resolve. If the wavelength is on the order of or smaller than the inclusions, then we don’t really see a homogeneous material but rather a collection of objects. On the other hand, if the wavelength is much larger than the inclusions and their spacing, then the wave cannot resolve the internal structure of the material and effectively averages the properties. This is why x-rays can be used to study the internal structure of materials; x-ray wavelengths can be smaller than the distances between atoms in a solid and are thus able to ‘see’ the individual atoms. At the other end of the extreme, when the wavelength is much longer, it makes much better sense to pretend the collection of atoms or inclusions forms some continuous substance with averaged, macroscopic properties—at least in terms of understanding the effect that the material has on the wave. Clearly, the material properties of the inclusions play a role in determining the properties of the effective medium, but so do their shapes and arrangement.

In the illustration above, our atoms are represented by little red dots. For electromagnetic waves, the red color might indicate an index-of-refraction or could represent something a little more technical, like a polarizability. The polarizability of a material provides a measure of the degree to which a charge separation is induced across an object under the application of an electric field—more about this later. The white region in the figure represents the properties of space. To illustrate how the properties of the collection of inclusions averages, all we did was to blur the red dots, eventually mixing with the white region until we have a uniform light red color. So, we might say if the original red color represents some property of the inclusions, then the equivalent property of the effective medium for longer wavelengths lies somewhere between the white and the red colors, given by the light red result we see.

So, is it as simple as that? Can we just average the properties of molecules or other collections of objects, the way an artist might blend colors? Sometimes, in fact, it’s nearly that easy. But, in general, obtaining the homogenized properties from the microscopic properties can be tricky. In fact, many researchers have considered this problem, which is often called “effective medium theory,” or homogenization, in many different areas of physics. These theories often take the form of mathematical formulas that apply to fairly specific arrangements of objects—a cubic array of dielectric or metal spheres, for example.

To turn the metamaterial concept into more of an engineering tool, we need a technique that allows one to determine the homogenized properties of any collection of objects, regardless of their shape or material composition. Creating such a tool has been one of the goals of the metamaterials effort, and one of the particular interests of our group.

Metamaterial Homogenization: A Physical Approach

Materials have numerous macroscopic properties that are used to describe them. However, for electromagnetics and optics, there are really only two parameters of fundamental importance: The electric permittivity, ${\varepsilon}$, and the magnetic permeability, ${\mu}$, which are often referred to as the constitutive parameters. The constitutive parameters are important because they are the only material parameters that enter directly into Maxwell’s equations, which govern all of electromagnetic and optical wave propagation. So, the goal of homogenization in electromagnetics is to determine the parameters ${\varepsilon}$ and ${\mu}$ for any collection of atoms, molecules, or other macroscopic elements. For metamaterials, we want to find the effective ${\varepsilon}$ and ${\mu}$ for a collection of artificially structured elements—things like metal wires or loops, or any other set of objects that might provide interesting and useful properties. Once we have replaced the composite set of elements with an effective medium having values of ${\varepsilon}$ and ${\mu}$ obtained from a homogenization procedure, then we can easily predict how waves will interact with the composite by solving Maxwell's equations. Our tool allows us to avoid having to calculate the interaction of the wave with every single object in the composite.

To illustrate how a homogenization tool can be developed, consider a simple conceptual experiment where we seek to measure the capacitance of a parallel plate capacitor filled with some material. Capacitance is a measure of the ability of a set of conductors to store electric charge. So, if a voltage is applied across two conductors, equal and opposite charges accumulate on the conductors in proportion to the capacitance multiplied by the voltage difference, or

\[Q = CV\]

For a parallel plate capacitor, the capacitance $C$ has the simple form

\[C = {\varepsilon _0}\frac{A}{d}\]

where ${\varepsilon_0}$ is a constant known as the permittivity of free space. (Note that this expression is only accurate if the area is very large and the space between the plates very small. In an actual parallel plate capacitor, the fields fringe out the sides and the picture isn't quite so simple. However, for the purposes of this discussion, we pretend that the fields are uniform everywhere and ignore the edge effects.) If some uniform insulating material slab fills the volume within the plates, then the applied voltage causes the material to polarize, inducing equal and opposite charges to appear on either slide of the slab; the polarization charges must be balanced by even more charge accumulating on the conducting plates, and so the capacity of conductors to store charge increases when a dielectric material is present between the conductors. Suppose that the dielectric material has a dielectric constant of $K$; then, the capacitance increases proportionally:

\[C = {\varepsilon _0}K\frac{A}{d} = \varepsilon \frac{A}{d}\]

where $A$ is the area of the plates and $d$ is the distance between the plates. We can eliminate $C$ in the above equations to arrive at an expression for the effective permittivity of the material in between the plates:

\[\varepsilon = \frac{{Q/A}}{{V/d}}\]

This last equation is extremely useful: Since the applied voltage is constant regardless of the material in between the plates, the dielectric constant of the material slab can be determined by noting just how much charge accumulates on the plates when the slab is inserted.

Consider a first simple experiment. We take the dielectric slab in the picture above, and insert the slab in between the metal plates, applying a voltage from the voltage source as shown in the figure below. While we may not know the dielectric of the slab to start with, we know the voltage across the plates and we assume we can measure the charge that appears on the plates. Using the equation above, we have thus determined the dielectric constant of the material.

What would happen if we didn’t fill the two plates entirely with a slab of material? What if we only filled it partially, or put in various objects like the spheres illustrated above? Each of the elements would become polarized, and entire system would have an increased capacitance. Since we still have the same applied voltage, the total accumulated charge now is an indication of how much polarization charge is induced on the objects within the plates. The charge density is now non-uniform over the plates, but we assume that we can just somehow add it all up. The situation is illustrated in the figure below. On the left, the application of a voltage across two parallel conducting plates produces a uniform electric field (arrows), with a uniform charge density $+\sigma$ appearing on the upper plate, and an equal and opposite charge density $-\sigma$ appearing on the lower plate. If, instead, a sphere is placed inside the plates, then additional charge accumulates on the plates, $Q_{sphere}$, and the fields are disturbed.

One can now imagine the entire region in between the plates, dielectric sphere plus air, as some uniform material with an effective dielectric constant $\epsilon_{eff}$. What is that value? We can use the equation just found above:

\[\varepsilon_{eff} = \frac{{Q_{sphere}/A}}{{V/d}}\]

We can say we have homogenized the sphere and space into an effective material, like we showed conceptually in the very first figure above.

We can build on the idea of an effective medium by arranging the spheres shown in the first illustration above into a lattice between the plates. From what we've learned above, we know that the lattice of spheres can be thought of as a homogenized medium with an effective dielectric constant. Again, we can measure the entire charge that accumulates on the plates and use the same procedure as above to determine quantitatively the dielectirc constant. Although the distributions of the electric fields and the potential are non-uniform inside the plates, nevertheless the entire collection of spheres behaves like a homogeneous slab of material with dielectric constant $\varepsilon_{eff}$.

We can take one final step to complete our design tool. From the above discussion, we see that if we perform a measurement of an object inside a parallel plate capacitor, the capacitance of the system represents an averaging or homogenization of the structures between the plates. That is, we can obtain the same capacitance (and same total charge induced on the plates) from a uniform dielectric slab with an effective permittivity. But, as we mentioned earlier, we have a little problem in that the fields in a capacitor at the edges fringe outwards and the picture isn’t quite what we’d like. We can actually fix this in software with a simple trick. The top and bottom of the capacitor are conductors—specifically electric conductors. That means the tangential component of the electric field must be zero at that surface. If we now bound the other sides of the capacitor with magnetic conductors—for which the tangential component of the magnetic field goes to zero—then this arrangement is equivalent to simulating a periodic array of identical elements, as shown below. That now means we can perform a numerical simulation on just one element of our prospective metamaterial, and we end up with the effective properties of the composite. Note that there are a couple of assumptions included in this method: The electric field is assumed to be polarized perpendicular to the (electric) conductors, and the material inside the plates must be symmetry about the axis between the plates.

So far, we’ve talked about dielectric materials and creating materials with effective electric permittivity. What about magnetic materials? We can’t do the actual experiment as pictured above, because there is no such thing as a magnetic charge (like an electron). However, if we are interested in this approach as a computational design tool, then it is very easy to simulate a magnetic conductor and we can repeat the entire process above. Without going through all of the details, we can imagine two parallel plates formed of magnetic conductors with a bunch of magnetic objects in between that will become magnetized under the application of a magnetic field. Then, by measuring the amount of magnetic charge on the magnetic conductors, we can arrive at an effective magnetic permeability, or $\mu_{eff}$.

Finally, there is one more important point that needs to be mentioned regarding this method: The fields are static. That means we apply either an electrostatic field or a magnetostatic field, which is uniform everywhere and does not vary with time. We are interested, however, in electromagnetic waves that have fields that vary in both time and space. That variation adds some complexity to the technique described here, but much of the thought process above can still be used to gain intuition. Since metamaterials consist of elements that are much smaller than the wavelengths of operation, the fields actually don’t vary too much over one element and the homogenization approach provides a reasonable first approximation.

Metamaterial Homogenization: A Numerical Tool

In a numerical simulation, we can easily compute the charge density on a conducting plate and thus determine the total charge. However, there is an alternative way to get that information. It turns out that the charge density on a conductor is related to a field often referred to as the displacement field, or ${\bf{D}} \cdot \hat n = \rho $. This equation indicates that the displacement field measured at a point on the plate will correspond to the charge density; to get the total charge, we just add up (or integrate) the displacement field over the surface, or

\[Q = \int\limits_A {{\bf{D}} \cdot \hat n\,da} \]

Many computer programs will compute either the displacement field or the charge density directly, so that the total charge can be easily determined. The voltage between two oppositely charged conductors can also be related to a field—the electric field—according to

\[V = \int\limits_d {{\bf{E}} \cdot d{\bf{l}}}. \]

This is a line integral, so that it is the component of the electric field along the line connecting the plates that must be summed up. Since nearly all electromagnetic programs will find the $E$ and $D$ fields for arbitrary configurations of materials, we arrive at a very general form for the permittivity, or

\[\varepsilon = \frac{{\frac{1}{A}\int\limits_A {{\bf{D}} \cdot d{\bf{s}}} }}{{\frac{1}{d}\int\limits_d {{\bf{E}} \cdot d{\bf{l}}} }}\]

This final equation reveals a completely general approach to finding the dielectric constant of a material that only requires we integrate the displacement and electric fields over a surface and volume, respectively. In fact, if we allow a tiny gap to exist between the material and the metal plate, and use the relationship that ${\bf{D}} = {\varepsilon _0}{\bf{E}}$ , then we can perform a simulation just to find the electric field everywhere, and use

\[\varepsilon = {\varepsilon _0}\frac{{\frac{1}{A}\int\limits_A {{\bf{E}} \cdot d{\bf{s}}} }}{{\frac{1}{d}\int\limits_A {{\bf{E}} \cdot d{\bf{l}}} }}\]

Useful References

The dielectric constant of a composite material—a problem in classical physics
D. J. Bergman
Phys. Rep. 9, 377 (1978)

Direct calculation of permeability and permittivity for a left-handed metamaterial
D. R. Smith, D. C. Vier, N. Kroll, S. Schultz
Appl. Phys. Lett. 77, 2246 (2000)

Homogenization of metamaterials by field averaging
D. R. Smith, J. B. Pendry
J. Opt. Soc. Am. B 23, 321 (2006)