## “Faster than light”? A Brief Introduction to Lower-than-unity Refractive Index — Lin Sen

Many of us would be familiar with the definition of refractive index:

This primarily treats n as a ratio of the speed of light in vacuum to the “apparent” speed of light in some material. It is understood that the speed of light is still c even when light is passing through these materials, and that the decrement to a lower “apparent” speed only results from the continual absorption and re-emission of light by charges in the material.

Now, the most commonly taught interpretation is that the lag results from time lag in each absorption-emission cycle. However, this understanding is intuitively limited by the fact that the range of n is restricted to ≥ 1. Contrary to common belief, this is not true. The value of n can indeed be lower than unity!

The more complete understanding stems from viewing v term as an expression of the phase velocity of the EM wave. We will embark on an elementary discussion on this phenomenon, not because I think you are not pro enough to grasp the mathematics, but because I think I am not.

## What actually happens when light enters a non-vacuum medium?

If anyone ever told you that light entering glass is like a man running into the sea, he might be (Kewei or Mr Wee) trying to illustrate Fermat’s Principle of Least Time. I think it is wise to restrict the analogy to that matter.

In this discussion, we see the material as planar layers of charges which can oscillate when subjected to the oscillating E field in the EM wave. The equation for a sheet of oscillating charges has been kindly provided by Feynman Almighty:

When light enters a boundary from vacuum, it induces oscillation in the first layer of charges it meets, which subsequently causes minute oscillations of the E field following the equation above. This oscillation then superposes with the part of the original EM wave which was transmitted without interference with the layer of charges.

According to Mathematics[1], the additional oscillations induced (Let’s just call it Ea)  lags the original wave (Es) by 90o[2]. Hence the vector addition of these 2 phasors would cause a phase change shown in the diagram below.

Consequently, we see that the resultant wave, Eafter plate, now lags the original wave, which is to be expected.

The diagram expresses the phase change in terms of the thickness of the layer of charge (∆z) and the refractive index (n). We can also express this phase change in terms of Ea, which we can find, according to Mathematics, by:

1.       Modeling the layer of charge as undergoing forced oscillations due to the incoming electromagnetic wave, hence finding a relevant equation of the oscillations of charges within the layer.

2.       From this oscillation, we find Ea with the equation above.

Then, through identity of the 2 forms, we can find the equation expressing n in terms of the parameters of the layer of charge, where N is the number of charges per unit volume:

Now, before we get too carried away with the mathematics, this is Quanta, not MathSoc, and I dare not impeach the territory of King Kang. We must appreciate the significance of this equation through the term  — (Natural frequency of the layer of charge)2 – (Frequency of incoming light)2.

## Non-conventional values of n

We have the following 2 cases for discussion:

1.

2.       just a bit smaller than

For the first case, we can see that indeed n < 1 as  becomes insignificant. This is what we would observe if we shine x-ray into graphite. The problem arises with the conventional understanding when v seemingly exceeds c, which Einstein has identified and JiaHuang has categorically prohibited.

In actuality, the lower-than-unity n simply means that the phase of the final wave now leads the original wave. However, the speed of the signal carried by the EM wave still travels with a speed less than c, as illustrated below. Therefore, the phase velocity did exceed c, but the actual velocity of the travelling wave did not. So you see, I scammed you in my title to get your attention. Muhahaha!

The second case is quite tricky, because that would open up the possibility of a negative refractive index. In most practical cases, we must consider the damping factor in the original solution to oscillations in the layer of charges, which would correspond to absorption of the EM wave being dominant when  . However, there is no intrinsic law prohibiting the existence of n < 0 values as the damping factor varies with each material. It is in the current frontier of material sciences that researchers are trying to create usable samples of these metamaterials.

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Just to spark your interest, metamaterials (n < 0) may potentially create Halo like Invisibility Camouflage suits: http://en.wikipedia.org/wiki/Metamaterial_cloaking

For further references: Feynman Lectures Volume 1 Chapter 30-31

[1] This means there is potential for substantial mental pain.

[2] Only in case of 1st degree approximation and with no damping factor.