I’m currently writing a textbook on Electromagnetic Waves for my graduate optics students. I was reading up on zero refractive index materials for a chapter section and thought it would be fun to write a popularized account of their fascinating and counterintuitive properties!

The past two decades have been a fascinating time to be an optics researcher. During that period, old rules about what light can and cannot do have been found in many cases to be more like guidelines, and ignoring those guidelines have led to some really astonishing new optical phenomena and devices.

One area where the rules have changed dramatically is in our understanding of the refractive index of light. The refractive index of a material, usually expressed in mathematical equations by the symbol n, represents the amount by which the speed of light is reduced in the medium from its vacuum value. If we label the speed of light in vacuum as c, then the speed of light in the medium is given by c/n. As an example, the refractive index of water in the visible light spectrum is roughly 1.33, which means that the speed of light in water is c/1.33, or 3/4ths the vacuum speed of light.

The most famous and most dramatic demonstration of seeming rule-breaking is the demonstration of materials with a negative index of refraction. When light travels from one medium to another, its direction changes according to the law of refraction, known as Snell’s law.

Mathematically, we write Snell’s law as

.

where “sin” represents the trigonometric sine function. This formula indicates that when light goes from a rare medium (low refractive index) to a dense medium (high refractive index), the light direction bends towards the perpendicular to the surface.

But what if the second medium has a negative index of refraction? Then Snell’s law would tell us that the light would bend on the opposite side of the perpendicular to the surface.

For centuries, this was assumed to be impossible, because among other things how could light have a negative speed? But in the 1960s, Russian physicist Victor Vesalago argued¹ that there is nothing in physics that prohibits a negative refractive index, and further argued that a negative index material could be used to make a flat lens, as illustrated below.

Veselago’s work went largely unnoticed until, in 2000, UK physicist John Pendry noted² that not only was Veselago’s lens possible, but it would in principle have perfect resolution, violating another long-held belief by optical scientists that imaging systems always have finite resolution.

Pendry’s result requires the fabrication of materials with optical properties that do not exist in nature, now called metamaterials. A metamaterial is a material that gets its optical properties from an artificial subwavelength-size structure. Many scientists initially scoffed at Pendry’s predictions, but materials with a negative refractive index³ were fabricated soon afterward, and rough experimental tests⁴ of the perfect lens prediction demonstrated that the principle is sound.

The introduction of negative refraction led physicists to ask: what other types of very unusual optical materials are possible, and what might they be used for? One obvious answer to the question was: we can make materials with a refractive index that is zero, or very close to zero! Such materials are known as “epsilon near zero” (ENZ) materials, and let’s take a look at what they can do.

Let’s get one of the biggest issues out of the way first: how is a zero or near-zero refractive index even possible, and doesn’t it violate Einstein’s relativity, which say that nothing travels faster than the vacuum speed of light? For if the speed of light in matter is , an index less than one indicates a velocity greater than c.

The answer is that Einstein’s theory has been clarified to say that nothing interesting or useful can travel faster than the speed of light. I’ve talked about this before in my series of posts on quantum entanglement, and gave a number of examples of things that appear to travel faster than c but that cannot be used to convey information faster than c. In the case of a zero index material, we note that the refractive index of a material also depends on the frequency of the light, and that a material can have a zero refractive index only for a small range of frequencies.

Why does this make a difference? If we have perfectly monochromatic light, i.e. light of a single frequency, its wave will wiggle up and down like a perfect sine wave, forever.

This wave is changing in time, but the form of the wave is the same that it was an infinitely long time ago, and will remain the same for an infinite amount of time. Even if the ripples of the wave are moving faster than the vacuum speed of light, there is no way to convey information on them, because they are always the same. If we tried to encode information on this wave, we would have to change it, for example by making one of the ripples bigger than then others. But then it would no longer be a monochromatic wave, and we would inevitably find that the speed of our data would move slower than the ripples of the perfect sine wave. The speed of the ripples is known as the phase velocity, and the speed of the data can be called the signal velocity. We are quite confident that the signal velocity of light in matter never exceeds the vacuum speed of light, though researchers continue to test experimentally whether there might be some sort of loopholes in Einstein’s theory.

This argument about phase and signal velocity can be extended over a small range of frequencies, so we can argue that it is possible to have a refractive index near zero in a material over such a small range. Since we are often performing experiments with lasers that have a very small frequency bandwidth anyway, we can say that it should be possible to make a material with a zero refractive index useful for practical applications.

In fact, ENZ materials were less controversial than negative index materials because we already knew of materials that exist that have a near-zero refractive index! In metals like gold or silver, there is a critical frequency of light, called the plasma frequency, above which the metal acts like an insulator and below which the metal acts like a conductor. At frequencies just above the plasma frequency, the refractive index is very close to zero. The plasma frequency for gold or silver in their natural forms, however, is in the high-frequency ultraviolet range. One of the earliest metamaterial designs was a collection of thin silver rods closely-spaced to each other, as illustrated below from a top-down view. Researchers showed⁵ that an appropriate spacing and thickness of these silver rods could shift the plasma frequency into the visible range, allowing optical experiments to be performed with zero index materials.

So what can we do with a near-zero index material? Let’s rewrite Snell’s law in a slightly different form:

.

Let us suppose, for a moment, that we have a light source inside a low-index material () that is radiating into a normal index material like air (). Snell’s law tells us that , which indicates , which means that all the light coming out of the low index material will be traveling in the same direction, perpendicular to the surface! We end up with a beam of light that is perfectly collimated, or directional, as illustrated in the figure below.

This result is quite striking because ordinary light sources, like light bulbs, do not produce collimated light without using some sort of optics. The most collimated light sources available are lasers, which use the quantum properties of light and matter to produce a collimated beam. Here we see that, in principle, a low-index material could convert an ordinary light source into one that is highly directional like a laser! This phenomenon was first proposed by Enoch et al. in 2002⁵.

However, things are not quite so easy, and now we can get at why such materials are called “epsilon near zero” materials. There are two quantities that characterize the optical response of a material, the permittivity and the permeability. The permittivity is written in equations with the Greek letter epsilon and describes how strongly the material responds to the electric part of a light wave. The permeability is written in equations with the Greek letter mu and describes how strongly the material responds to the magnetic part of a light wave.

Alternatively, we can describe a material by the effect it has on the behavior of a light wave. In this case, we can use the refractive index, which is given by the square root of the product of epsilon and mu, and the impedance, which is given by the square root of the ratio of mu over epsilon.

For visible light, almost all natural materials are effectively non-magnetic, i.e. they have the same permeability as empty space. In such a case, the refractive index is determined entirely by epsilon. Saying “refractive index near zero” is effectively the same as saying “epsilon near zero.” However, it can be shown that the amount of light reflected from a non-magnetic material also depends on the refractive indices of the two media, in what are known as the Fresnel equations. If one of the refractive indices is zero, the Fresnel equations indicate that all the light will be reflected at the boundary between the media!

In fact, creating a perfect mirror was the first proposed application of zero refractive index materials⁶. The Fresnel formulas also indicate that the amount of light reflected from an ordinary mirror will depend on the polarization of light as well as the direction of illumination. For a zero refractive index material, all light would be reflected.

This means that our light collimator discussed above would not transmit any light at all if epsilon was exactly zero! This is partly why we instead talk about “epsilon near zero” materials, where light is still transmitted through the surface but the refractive index is small enough to produce interesting effects.

There is a way around the low transmission efficiency of an epsilon near zero material. When a material has a magnetic response as well as an electric response, i.e. it has a non-trivial permeability as well as a permittivity, the Fresnel equations indicate that the amount of light transmitted depends on the impedance, rather than the permittivity alone. If two media have equal impedance values, then a significant amount of light will be transmitted through them. So if, for example, we want to make an ENZ material that is matched with air, we design the material to make epsilon and mu very close to zero, but in such a way that their ratio, the impedance, matches the impedance of air!

Such impedance-matched metamaterials have been shown to be possible, but nobody is quite sure how to make them yet. Ziolkowski⁷ was apparently the first researcher to theoretically propose zero refractive index, impedance matched materials, and show that they could be used to efficiently collimate a light source as we discussed earlier.

Since 2002, there have been many proposed applications of ENZ materials, but I would like to share one more fascinating possibility. The refractive index also dictates how much the wavelength of light changes when it enters a medium; the wavelength in the medium is (lambda)/(refractive index). If the refractive index is close to zero, the wavelength becomes extremely large, in principle larger than the size of the ENZ object itself. This means that the wave will stretch out to fill the entire interior of the object, making the phase of the wave constant throughout. If the output surface is curved, the wave coming out will be shaped exactly like the output surface. This is roughly illustrated below.

If the output surface is curved outward, for example, it will act like a diverging lens; if the output surface is curved inward, it will act like a converging lens. But unlike ordinary lenses, where the shape of the outgoing wavefront depends on the shape of the incoming wavefront, an ENZ lens would produce the same output wavefront regardless of what incoming wavefront is doing. It would provide a perfect control of the light incident upon it! This was proposed by Alù et al. in 2007⁸.

It is worth noting that these properties are not entirely speculative. In 2013, researchers demonstrated an ENZ material for visible light⁹. It remains to be seen if such materials will become widely used in optics applications, but it shows that there are still unexpected surprises to be found in the behavior of light.

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1. V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10 (4) (1968), 509–14.
2. J.B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85 (2000), 3966.
3. R. A. Shelby et al. “Experimental Verification of a Negative Index of Refraction,” Science 292 (2001),77-79.
4. Nicholas Fang et al. “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens.” Science 308 (2005), 534-537.
5. Enoch et al. “A metamaterial for emissive direction,” Phys. Rev. Lett. 89 (2002), 213902.
6. N. Garcia, E. V. Ponizovskaya, John Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett. 80 (2002), 1120–1122.
7. R. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E 70 (2004), 046608.
8. Andrea Alù, Mário G. Silveirinha, Alessandro Salandrino, and Nader Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern”, Phys. Rev. B 75 (2007), 155410.
9. Maas, R., Parsons, J., Engheta, N. et al. Experimental realization of an epsilon-near-zero metamaterial at visible wavelengths. Nature Photon 7 (2013), 907–912.