This is a repost of an article I wrote at my old blog. It’s still fairly relevant, and metamaterials is a topic I hope to spend more time discussing.
The most intuitive place to start discussing metamaterials is generally index of refraction and Snell’s Law. Most people are familiar with the refraction of light by lenses…or at least I certainly hope so! If not, you may want to check out this tutorial on lenses and come back when you’re done.
One of the formulas the defines the refraction of light at an interface (for linear optics anyway) is Snell’s Law. Mathematically, this is written as:
What this means is that there is a relationship between the angle of the light leaving one material and the angle at which is enters the second material. When the light is in the first material, we call it in the incident beam and give subscripts of i to those values. The light leaving is called the transmitted beam, with subscripts of t. The angle is given by θ. And then there’s that n in there. We have to look closer at that because n is a really cool value in metamaterials.
n is called the index of refraction for a material. In vacuum, n is generally one. In glass, it’s around 1.5. In diamond, it’s 2.4. The index of refraction indicates how much light will bend as it goes from one substance to another.
If you can imagine a block of glass sitting in air, the plane where the air ends and the glass begins is the interface. Now imagine balancing a candle on top of the block of glass. The direction the candle points (hopefully straight up) is called orthogonal to the plane. Now imagine a second candle (candle #2) that points into the glass in exactly the opposite direction from the first candle.
We shine a light from the air toward the block of glass. It hits near the base of candle #1 but is at an angle to the candle. The light that shines inside the block of glass is bent also, but because glass has a higher index of refraction than air, it will be bent toward the candle. In other words, the angle between the light in the glass and candle #2 will be smaller than the angle formed between the light in air and candle #1. However, the light in glass will be on the opposite side of the candle from the light in air.
Metamaterials have a little twist to this scenario. If we replace our glass block with a block of metamaterials, the light would still be bent closer (assuming the magnitude of the index of refraction is higher than that of air), but it would be on the same side of the candle as the beam coming from the light.
Figure 1: Light shown into glass (left) and metamaterials (right). The red beam is the incident light, the blue beam is the reflected light, and the cyan beam is the refracted light.
This is because the index of refraction for metamaterials is negative. The index of refraction is actually determined by two other values. The first is the permittivity of the material, ε, which, in a nutshell, describes how an electric field behaves in the material. The second is the permeability, μ, which describes the behavior of the magnetic field. To get the index of refraction, we take the square root of the product of μ and ε. Mathematically, this is just
For most ordinary materials, you take the positive root. In some cases, when either μ or ε are negative, you have an imaginary value. For metamaterials, we take the negative root.
Even though the term metamaterials is often used to describe these materials in general, metamaterials is actually anything that is a material that is constructed and cannot be found in nature. For this reason, it is more precise to call them double negative metamaterials (because of the fact that μ and ε are negative)…or any of the other various names they have been given (left-handed metamaterials, backward wave media, etc.).
I won’t go into the details, but if you’ve stuck with it this far, we now get to the fun stuff. (I could probably say more, and undoubtedly will at some point, but I want you to stick with me.) One of the consequences of this behavior is that you can create a flat lens rather than your typical optic lens which requires curvature to bend light.
The Veselago lens is a flat lens which can focus electromagnetic energy from a point source to a point inside the lens. It’s named after the person who did the pioneering work on how left-handed metamaterials behave.
The image below is a simulation of a Veselago lens created in Ansoft HFSS. The source is at the bottom, and you can see a point inside the lens where the wave is focusing. If you look very carefully, you’ll notice that while waves radiate outward from the source, they seem to radiate toward the focusing point in the lens (the lens being the square grid). I’ll explain that behavior another time. But for now, you can look at pretty pictures. (Warning…it may take a bit to load because it’s large file.)