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What’s a zero refractive index materials?

What’s a zero refractive index materials?

2023-12-10 02:28:35

I’m presently writing a textbook on Electromagnetic Waves for my graduate optics college students. I used to be studying up on zero refractive index supplies for a chapter part and thought it might be enjoyable to write down a popularized account of their fascinating and counterintuitive properties!

The previous twenty years have been an enchanting time to be an optics researcher. Throughout that interval, outdated guidelines about what gentle can and can’t do have been discovered in lots of circumstances to be extra like pointers, and ignoring these pointers have led to some actually astonishing new optical phenomena and units.

One space the place the foundations have modified dramatically is in our understanding of the refractive index of sunshine. The refractive index of a fabric, often expressed in mathematical equations by the image n, represents the quantity by which the pace of sunshine is decreased within the medium from its vacuum worth. If we label the pace of sunshine in vacuum as c, then the pace of sunshine within the medium is given by c/n. For example, the refractive index of water within the seen gentle spectrum is roughly 1.33, which signifies that the pace of sunshine in water is c/1.33, or 3/4ths the vacuum pace of sunshine.

Essentially the most well-known and most dramatic demonstration of seeming rule-breaking is the demonstration of supplies with a unfavorable index of refraction. When gentle travels from one medium to a different, its path modifications in accordance with the legislation of refraction, referred to as Snell’s legislation.

Mathematically, we write Snell’s legislation as

n_1sintheta_1 = n_2sintheta_2.

the place “sin” represents the trigonometric sine perform. This components signifies that when gentle goes from a uncommon medium (low refractive index) to a dense medium (excessive refractive index), the sunshine path bends in direction of the perpendicular to the floor.

However what if the second medium has a unfavorable index of refraction? Then Snell’s legislation would inform us that the sunshine would bend on the alternative aspect of the perpendicular to the floor.

For hundreds of years, this was assumed to be unimaginable, as a result of amongst different issues how might gentle have a unfavorable pace? However within the Nineteen Sixties, Russian physicist Victor Vesalago argued1 that there’s nothing in physics that prohibits a unfavorable refractive index, and additional argued {that a} unfavorable index materials may very well be used to make a flat lens, as illustrated beneath.

Veselago’s work went largely unnoticed till, in 2000, UK physicist John Pendry famous2 that not solely was Veselago’s lens doable, however it might in precept have excellent decision, violating one other long-held perception by optical scientists that imaging methods all the time have finite decision.

Pendry’s consequence requires the fabrication of supplies with optical properties that don’t exist in nature, now referred to as metamaterials. A metamaterial is a fabric that will get its optical properties from a synthetic subwavelength-size construction. Many scientists initially scoffed at Pendry’s predictions, however supplies with a unfavorable refractive index3 had been fabricated quickly afterward, and tough experimental checks4 of the right lens prediction demonstrated that the precept is sound.

The introduction of unfavorable refraction led physicists to ask: what different sorts of very uncommon optical supplies are doable, and what would possibly they be used for? One apparent reply to the query was: we are able to make supplies with a refractive index that’s zero, or very near zero! Such supplies are referred to as “epsilon close to zero” (ENZ) supplies, and let’s check out what they’ll do.

Let’s get one of many largest points out of the way in which first: how is a zero or near-zero refractive index even doable, and doesn’t it violate Einstein’s relativity, which say that nothing travels quicker than the vacuum pace of sunshine? For if the pace of sunshine in matter is v=c/n, an index lower than one signifies a velocity better than c.

The reply is that Einstein’s principle has been clarified to say that nothing attention-grabbing or helpful can journey quicker than the pace of sunshine. I’ve talked about this earlier than in my collection of posts on quantum entanglement, and gave a lot of examples of issues that seem to journey quicker than c however that can not be used to convey info quicker than c. Within the case of a zero index materials, we word that the refractive index of a fabric additionally is dependent upon the frequency of the sunshine, and {that a} materials can have a zero refractive index just for a small vary of frequencies.

Why does this make a distinction? If we have now completely monochromatic gentle, i.e. gentle of a single frequency, its wave will wiggle up and down like an ideal sine wave, eternally.

This wave is altering in time, however the type of the wave is identical that it was an infinitely very long time in the past, and can stay the identical for an infinite period of time. Even when the ripples of the wave are shifting quicker than the vacuum pace of sunshine, there is no such thing as a method to convey info on them, as a result of they’re all the time the identical. If we tried to encode info on this wave, we must change it, for instance by making one of many ripples larger than then others. However then it might now not be a monochromatic wave, and we might inevitably discover that the pace of our knowledge would transfer slower than the ripples of the right sine wave. The pace of the ripples is called the part velocity, and the pace of the information could be referred to as the sign velocity. We’re fairly assured that the sign velocity of sunshine in matter by no means exceeds the vacuum pace of sunshine, although researchers continue to test experimentally whether or not there may be some form of loopholes in Einstein’s principle.

This argument about part and sign velocity could be prolonged over a small vary of frequencies, so we are able to argue that it’s doable to have a refractive index close to zero in a fabric over such a small vary. Since we are sometimes performing experiments with lasers which have a really small frequency bandwidth anyway, we are able to say that it must be doable to make a fabric with a zero refractive index helpful for sensible purposes.

In actual fact, ENZ supplies had been much less controversial than unfavorable index supplies as a result of we already knew of supplies that exist which have a near-zero refractive index! In metals like gold or silver, there’s a important frequency of sunshine, referred to as the plasma frequency, above which the metallic acts like an insulator and beneath which the metallic acts like a conductor. At frequencies simply above the plasma frequency, the refractive index may be very near zero. The plasma frequency for gold or silver of their pure types, nonetheless, is within the high-frequency ultraviolet vary. One of many earliest metamaterial designs was a group of skinny silver rods closely-spaced to one another, as illustrated beneath from a top-down view. Researchers confirmed5 that an applicable spacing and thickness of those silver rods might shift the plasma frequency into the seen vary, permitting optical experiments to be carried out with zero index supplies.

Prime-down illustration of a metamaterial consisting of a bunch of silver rods spaced shut collectively. One can set the plasma frequency to lie at a frequency within the optical regime by an applicable selection of the thickness and spacing of the rods.

So what can we do with a near-zero index materials? Let’s rewrite Snell’s legislation in a barely totally different type:

displaystyle sin(theta_2) = frac{n_1}{n_2}sin(theta_1).

Allow us to suppose, for a second, that we have now a light-weight supply inside a low-index materials (n_1=0) that’s radiating into a traditional index materials like air (n_2=1). Snell’s legislation tells us that sin(theta_2)=0, which signifies theta_2=0, which signifies that all the sunshine popping out of the low index materials shall be touring in the identical path, perpendicular to the floor! We find yourself with a beam of sunshine that’s completely collimated, or directional, as illustrated within the determine beneath.

This result’s fairly hanging as a result of abnormal gentle sources, like gentle bulbs, don’t produce collimated gentle with out utilizing some form of optics. Essentially the most collimated gentle sources accessible are lasers, which use the quantum properties of sunshine and matter to supply a collimated beam. Right here we see that, in precept, a low-index materials might convert an abnormal gentle supply into one that’s extremely directional like a laser! This phenomenon was first proposed by Enoch et al. in 20025.

Nevertheless, issues are usually not fairly really easy, and now we are able to get at why such supplies are referred to as “epsilon close to zero” supplies. There are two portions that characterize the optical response of a fabric, the permittivity and the permeability. The permittivity is written in equations with the Greek letter epsilon and describes how strongly the fabric responds to the electrical a part of a light-weight wave. The permeability is written in equations with the Greek letter mu and describes how strongly the fabric responds to the magnetic a part of a light-weight wave.

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Alternatively, we are able to describe a fabric by the impact it has on the conduct of a light-weight wave. On this case, we are able to use the refractive index, which is given by the sq. root of the product of epsilon and mu, and the impedance, which is given by the sq. root of the ratio of mu over epsilon.

For seen gentle, virtually all pure supplies are successfully non-magnetic, i.e. they’ve the identical permeability as empty area. In such a case, the refractive index is decided fully by epsilon. Saying “refractive index close to zero” is successfully the identical as saying “epsilon close to zero.” Nevertheless, it may be proven that the quantity of sunshine mirrored from a non-magnetic materials additionally is dependent upon the refractive indices of the 2 media, in what are referred to as the Fresnel equations. If one of many refractive indices is zero, the Fresnel equations point out that each one the sunshine shall be mirrored on the boundary between the media!

In actual fact, creating an ideal mirror was the primary proposed utility of zero refractive index supplies6. The Fresnel formulation additionally point out that the quantity of sunshine mirrored from an abnormal mirror will rely on the polarization of sunshine in addition to the path of illumination. For a zero refractive index materials, all gentle can be mirrored.

Which means our gentle collimator mentioned above wouldn’t transmit any gentle in any respect if epsilon was precisely zero! That is partly why we as an alternative speak about “epsilon close to zero” supplies, the place gentle continues to be transmitted via the floor however the refractive index is sufficiently small to supply attention-grabbing results.

There’s a means across the low transmission effectivity of an epsilon close to zero materials. When a fabric has a magnetic response in addition to an electrical response, i.e. it has a non-trivial permeability in addition to a permittivity, the Fresnel equations point out that the quantity of sunshine transmitted is dependent upon the impedance, reasonably than the permittivity alone. If two media have equal impedance values, then a big quantity of sunshine shall be transmitted via them. So if, for instance, we wish to make an ENZ materials that’s matched with air, we design the fabric to make epsilon and mu very near zero, however in such a means that their ratio, the impedance, matches the impedance of air!

Such impedance-matched metamaterials have been proven to be doable, however no person is kind of certain the best way to make them but. Ziolkowski7 was apparently the primary researcher to theoretically suggest zero refractive index, impedance matched supplies, and present that they may very well be used to effectively collimate a light-weight supply as we mentioned earlier.

Since 2002, there have been many proposed purposes of ENZ supplies, however I wish to share yet one more fascinating chance. The refractive index additionally dictates how a lot the wavelength of sunshine modifications when it enters a medium; the wavelength within the medium is (lambda)/(refractive index). If the refractive index is near zero, the wavelength turns into extraordinarily massive, in precept bigger than the dimensions of the ENZ object itself. Which means the wave will stretch out to fill all the inside of the article, making the part of the wave fixed all through. If the output floor is curved, the wave popping out shall be formed precisely just like the output floor. That is roughly illustrated beneath.

Illustration of wavefront shaping with an ENZ materials, roughly exhibiting an incident wave being transformed right into a diverging wave via an ENZ materials the place the part is fixed.

If the output floor is curved outward, for instance, it’s going to act like a diverging lens; if the output floor is curved inward, it’s going to act like a converging lens. However not like abnormal lenses, the place the form of the outgoing wavefront is dependent upon the form of the incoming wavefront, an ENZ lens would produce the identical output wavefront no matter what incoming wavefront is doing. It could present an ideal management of the sunshine incident upon it! This was proposed by Alù et al. in 20078.

It’s price noting that these properties are usually not fully speculative. In 2013, researchers demonstrated an ENZ materials for seen gentle9. It stays to be seen if such supplies will grow to be extensively utilized in optics purposes, nevertheless it exhibits that there are nonetheless sudden surprises to be discovered within the conduct of sunshine.


  1. V.G. Veselago, “The electrodynamics of gear with concurrently unfavorable values of ε and μ,” Sov. Phys. Usp. 10 (4) (1968), 509–14.
  2. J.B. Pendry, “Unfavorable refraction makes an ideal lens,” Phys. Rev. Lett. 85 (2000), 3966.
  3. R. A. Shelby et al. “Experimental Verification of a Unfavorable Index of Refraction,” Science 292 (2001),77-79.
  4. Nicholas Fang et al. “Sub-Diffraction-Restricted Optical Imaging with a Silver Superlens.” Science 308 (2005), 534-537.
  5. Enoch et al. “A metamaterial for emissive path,” Phys. Rev. Lett. 89 (2002), 213902.
  6. N. Garcia, E. V. Ponizovskaya, John Q. Xiao, “Zero permittivity supplies: Band gaps on the seen,” 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 part sample”, Phys. Rev. B 75 (2007), 155410.
  9. Maas, R., Parsons, J., Engheta, N. et al. Experimental realization of an epsilon-near-zero metamaterial at seen wavelengths. Nature Photon 7 (2013), 907–912.

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