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Negative Index of Refraction
Mary (Betsey) Mathew
La Rosa, Winter 2006
ABSTRACT
In recent years, the advances of John Pendry have allowed physicists to examine the characteristics of negative index of refraction materials. Negative index of refraction was an idea first theorized by Victor Veselago in 1967. This report investigates Pendry’s discovery and describes the experiments that were carried out as a result of Pendry’s finding.
INTRODUCTION
I decided to research negative index of refraction because I was amazed that such an apparently simple discovery could have been made only so lately. Many of the optical concepts that govern negative index of refraction, such as Snell’s law and the Poynting vector, are those that are taught in a general physics course. I was curious to discover what had stopped this concept from being fully explored decades or even centuries ago.
CONTENT
Much of the study of optics is how light moves when it transitions between one material and another. For instance, it is this change of movement that allows us to construct lenses. In order to describe how materials affect the light that passes through them, physicists came up with the idea of assigning materials a value that would describe how the speed of light was changed in a material relative to the speed of light in a vacuum. This value is known as the index of refraction (n) and is described by the following equations:
Equation 1
Equation 2
Equation 3
The first equation explicitly links the index of refraction to the permeability (μ) and permittivity (ε) of the material. The variables ε0 and μ0 are the permittivity of free space and permeability of free space. When the permittivity and permeability are divided by ε0 and μ0, respectively, the results are the relative permittivity (εr) and the relative permeability (μr). εwill always have the same sign as εr, and if the magnitude of one ε is larger than another, the magnitude of the associated εr will be larger as well. For these reasons, we can treat ε and εr very similarly. I may choose to make a claim about one variable for simplicity’s sake, but most often, the claim will be true of the other variable as well. Now let us examine the permittivity and permeability. The permittivity of a material can be described by the equation:
Equation 4
Permittivity describes how the E field affects a material electrically. The D field in the equation is known as electric displacement. It describes a secondary electric field (sometimes called the auxiliary electric field) that is set up in the vicinity of the material. Because of this, ε can be seen to govern how the E field moves through a material. If ε is small, then a low strength D field will be established. If ε is negative, then the D field will oppose the direction of the E field. This negative ε case is found in metals and the positive case is found in dielectrics.
The permeability of a material is similar to the permittivity, except it describes the magnetic aspects of a material. Permeability can be described by the equation:
Equation 5
As one can see from the previous equation, permeability determines how the magnetic field B will be manifested in the auxiliary magnetic field H. If µ of one material is larger than that of another, then H will be smaller in the first material than in the second. Likewise if µ of one material is smaller than that of another, then H will be larger in the first material than in the second. Perhaps most important to this paper is one other fact: if µ is negative then H will be antiparallel to B. This means that the auxiliary magnetic field will point opposite to the magnetic field that created it. Until 1999 however, negative µ had never been observed in any material. This changed when J. B. Pendry et al.[i]
In this experiment, Pendry’s team described the method for constructing a material with a negative effective permeability. Pendry defines an effective permeability based on the following equation:
Equation 6
For ease of understanding I will explain that by comparing the previous equation to eqn. 5 we can see that Pendry’s µeff is basically relative permeability when the B and H fields are averaged over the unit cell. This strongly implies that if a material exhibits a negative µeff, then, when viewed on a scale that does not enter the unit cell, µr and therefore µ will both be negative as well. To restate this, if Pendry can show that his material exhibits a negative µeff then as long as we are treating each cell of the material as a unit, and we do not take measurements or make conclusions on any activity inside the cell then we can treat the material as though it has a negative µ. So, on to Pendry’s proposed –µeff material.
Pendry’s initial material consists of two concentric cylinders, each with a slit down its side, arranged so that one slit is 180° from the other (Fig. 1). When a magnetic field is initiated parallel to the shared axis of the cylinders, Faraday’s law tells us that a current will be induced in each cylinder. The split in the rings prevents the circuit from being completed for either of the rings, but the orientation of the rings along with the fact that the current flows clockwise in each ring, creates a large capacitance between the cylinders, which allows the current to continue to flow (see Fig. 1). Based on Pendry’s definition of µeff, in this situation we can graph the effective permeability as shown in Fig. 2
Looking at this graph we can see that µeff is negative and large at frequency values just above the resonant frequency ω0. Pendry goes on to show that for a split ring structure like those on each side of the cube in Fig. 3, the ability to create a negative µeff at higher frequencies persists. These structures are known as split-ring resonators (SRRs). For an array such as that in Fig 3, the resonant frequency is around 13.5 GHz, so µeff will remain negative in the range of 13.5-14.2 GHz; this is in the microwave range of the EM spectrum.
Pendry and his team were not specifically interested in creating a negative n material. Others have since combined Pendry’s advances with the ideas of Russian physicist Veselago. Veselago proposed that if a material existed that had both a negative ε and a negative μ, it would also have a negative index of refraction. This is based on the fact that, as seen in eqn. 2, n combines εr and μr to form one value. The index of refraction is the square root of εr×μr. Square roots are conventionally positive, but if one rewrites eqn. 2 as eqn.3 then it becomes obvious that if n had a negative value, it would still produce the same result as it would had n been positive but had the same numerical value. Based on this Veselago theorized that if both ε and μ were negative, then the negative root should be taken to determine n.
Prior to looking at the experiments into negative index of refraction materials (NIMs), a basis in theory and terminology should be established. A negative index of refraction material is just that: a material with a negative n. According to Veselago, this material would have a negative ε and a negative μ, but at this point in the paper Veselago’s claim has not been proven. Another term that is often mentioned is left-handed material (LHM). This term can be explained more fully through the aid of the Figure 4. The image Figure 4.a depicts a normal right-handed material. The vectors shown are E, H, k (the propagation vector), and P (the Poynting vector). The Poynting
vector is determined by E × H, and the propagation vector is E × B. B is not shown in this diagram, but from eqn. 5 we know that if μ is positive B will point the same direction as Hand if μ is negative B will point the opposite direction as H. We can therefore infer that since k and P are pointing opposite directions in figure4.b, a left-handed material has a negative μ. This is indeed the distinction between left and right-handed materials. Incidentally, ε must also be negative in a left-handed material because otherwise, n would be imaginary and all the light would be absorbed, not refracted.
One of the first groups to examine -µeff materials as they relate to –n materials was lead by D. R. Smith[ii]. Smith et al. first performed numerical simulations using the EM solver MAFIA. They performed four simulations, two with only SRRs each in a different orientation, and the other two examining these same orientations but with the addition of copper wires (see the inserts of figures 5a-d)). The the setup depicted in figure 5a produced a phase advance bandgap similar to the one setup of Fig 5.b. Both discontinuities signify that either the µeff is negative and εeff is resonating or εeff is negative and µeff is resonating. Unfortunately, because of the symmetry of the
equation, we cannot be sure which of the above cases is true based simply on the curves that have already been created. To get more information, Smith performed the second part of the simulation and experiment, which involved placing copper wires as shown in the inserts of Figs. 5c and d. The purpose of these wires is to add a material with a well understood ε and µ characteristics. As mentioned earlier, metals have –ε, and by placing a wire with its axis parallel to the E field, Smith assured that as far as his propagation direction is concerned, ε has been defined. The next step is to see how the phase advance curve changes when microwaves are sent through this new composite structure. The results with the wires are shown as dashed lines in the curves of Figs. 5c and d. If we look at the blow-up of the graph we can see that there is actually a forbidden region in the graph on the left. This means that for certain frequencies, there was no phase advance; the light was absorbed. When the wire is added, it connects the discontinuity in the earlier curve to fill in a phase advance for those frequencies. In the graph on the right, there is simply a sudden jump in the phase advance in the bold line curve. The dashed curve simply signifies that the wire, with its negative permittivity, was less than the negative permittivity of the SRR for these frequencies.
This experiment was followed the next year by nearly the same team looking at a two-dimensional LHM[iii]. The specific equipment used by this team was designed to detect X-band microwave frequencies. Because of this, the team had to redesign the SRRs to have a specific resonance frequency. This is somewhat of a side issue, but Shelby et al. have shown by redesigning the SRR the validity of Pendry’s equations. This experiment was very similar to the previous one, so I will not go into too much detail. This experiment is two-dimensional so the unit cell looks different (Fig 6). Using similar numerical simulations and experiments Shelby’s team produced the curves in Fig 7. The agreement between these curves indicates that we can continue to treat LHMs as homogeneous material and that the numerical electromagnetic-mode solver GDFIDL is appropriate to model LHMs.
These first experiments have not shown much about the actual properties of LHMs. They have mainly served to establish that Pendry’s -µeff material can be combined with –ε wires to create a LHM, and to create equations (based on Maxwell’s equations) that effectively model these materials. The following experiments actually begin looking into some of the claims made by Veselago about the nature and properties of negative index of refraction materials.
One of the predictions made by Veselago was that Snell’s law
Equation 7
would hold for NIMs. To test this prediction, Shelby, Smith et al.[iv] used their LHM material from the previous experiment (fig. 6). They constructed a wedge of this material (see fig. 8), and one of Teflon that was the same shape and size. The n of Teflon is known to be around 1.35-1.4, so Teflon was being used a control sample in this experiment. In the actual experiment, a microwave beam was shone on one side of the wedge, and a microwave detector senses the wave after it has been deflected by the other side of the wedge (see fig. 8). The detector moves in 1.5° steps around the sample and collects data at each point. The data is graphed in fig. 9. As we can see from the graph, the NIM sample did indeed have a negative angle of refraction, as predicted by Snell’s law. We can examine these angles using Snell’s law to determine the experimental n of both materials. The Teflon has an experimental n of 1.4, and the NIM has an experimental n of -2.7. The Teflon experimental n matches nicely the accepted n value for Teflon of 1.35-1.4.
The next experiment is simply a follow up to the previous one. C. G. Parazzoli[v] and his team designed and executed this experiment. It was set up slightly differently from that of Shelby and Smith’s and so yielded new results. First, the microwaves are focused on the near face of the sample wedge using a lens (fig. 10). Second, the detector took two sets of measurements. The first set was at 33 cm from the sample and the second set was at a distance of 66 cm from the sample. Finally, this team also performed numerical simulations using MAFIA. These simulations beautifully match the experimental data of both the 33 cm and 66 cm sets (fig. 11).
CONCLUSION
My initial interest in the topic of NIMs was to determine why such an apparently simple concept was not made until the 21st Century. As far as I can tell, the stumbling block was a simple lack of manufacturing techniques and powerful numerical simulators. One of the reasons that Pendry made his discovery was the fact that he knew that others could actually produce the SRR structure that he had envisioned.
Negative index of refraction has many probable applications, although few of them have been explored as of now. Pendry has proposed that a perfect lens could be created using NIMs. This lens would not be bound by current diffraction limits and could therefore get a clearer view of the specimen. There are also proposals to use NIMs in the creation antennas, microwave devices, dispersion-compensating interconnects, and radar devices. Unfortunately, there are still some obstacles that must be overcome. For instance, current NIMs only show left-handed behavior when the EM frequencies used are in the microwave range. Also, there are currently huge power losses that as of yet have not been overcome. Figure 12 depicts these losses. All in all though, the future appears bright for this new advance in optical technology.
REFERENCES
[i] Pendry, J. B., A. J. Holden, D. J. Robbins and W. J. Stewart. “Magnetism from conductors and enhanced nonlinear phenomena” IEEE Transactions on Microwave Theory and Techniques, Vol. 47, No. 11. November 1999
[ii]Smith, D. R., Willie J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz. “Composite Medium with Simultaneously Negative Permeability and Permittivity” Physical Review Letters Vol. 84, No. 18. May 2000
[iii]Shelby, R. A., D. R. Smith, S. C. Nemat-Nasser, S. Schultz. “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial” Applied Physics Letters Vol. 78, No. 4 January 2001
[iv] Shelby, R. A., D. R. Smith, S. Schultz. “Experimental Verification of a Negative Index of Refraction” Science Vol. 292 April 2001
[v] Parazzoli, C. G., R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian. “Experimental Verification and Simulation of Negative Index of Refraction using Snell’s Law” Physical Review Letters Vol. 90, No. 10. March 2003