Mathematical Details

The speed at which monochromatic light travels through a medium other than a vacuum will generally depend on the frequency of the light, ν. This variation can be characterised by describing the medium’s refractive index, n, as a function of ν; the speed of a pure sine wave is then equal to c, the speed of light in a vacuum, divided by n(ν). This speed is known as the phase velocity of the medium at the specified frequency. The electric field of a monochromatic wave in a medium with refractive index n(ν) takes the form:

E(x,t) = E0 sin(2πν (n(ν)x/ct)) (1)

At any given moment, two sine waves of slightly different frequency travelling together through the medium will be in phase with each other at a number of points. These points will move with time, and the speed at which they move can be found by equating the phases of the two waves. Assuming that the waves have frequencies of ν and ν+Δν:

ν (n(ν)x/ct) = (ν+Δν) (n(ν+Δν) x/ct)  
ν (n(ν)x/ct) = (ν+Δν) ((n(ν) + (dn(ν)/dν)Δν) x/ct)  
x = (c / (n(ν) + (ν+Δν) dn(ν)/dν)) t (2)

The coefficient of t in Equation (2), in the limit as Δν goes to zero, is known as the group velocity, vg.

vg = c / (n(ν) + ν dn(ν)/dν) (3)

The group velocity describes the speed of any feature of a wave that relies on different frequencies remaining in phase. For example, a pulse of finite width will contain a range of frequencies, and the centre of the pulse will occur where they are all in phase, so it will move with this velocity.

In most materials, the refractive index has a roughly saw-tooth shaped graph when plotted against frequency: long rises punctuated by sudden dips close to the resonant frequencies of the medium, which usually correspond to absorption bands. In more exotic materials, it’s possible for the refractive index to exhibit these dips at frequencies where the material is transparent. In either case, when the refractive index falls with increasing frequency — that is, whenever dn(ν)/dν is negative — the situation is known as anomalous dispersion.

If dn(ν)/dν is sufficiently negative, it can reduce the denominator in Equation (3) to less than one, yielding a group velocity greater than c. Why is this not a contradiction of special relativity? No energy or information needs to travel at the group velocity in order for the shape of the wave to exhibit features that move at that speed. If you tried to signal someone with a superluminal pulse by dropping a shutter in its path at the last moment, you’d find you were too late: the pulse would happily “pass through” the shutter, because every influence that was actually responsible for its appearance on the other side would have passed through already.

If dn(ν)/dν is even more negative, it can make the denominator in Equation (3) less than zero, yielding a negative group velocity. It’s apparent from Equation (1) that the wavelength, λ, of light of a given frequency, ν, is:

λ(ν) = c / (ν n(ν)) (4)

This means that the group velocity and the wavelength are related by the following equation:

1 / vg = d(1/λ(ν))/dν (5)

So, for the group velocity to be negative, it’s necessary for the reciprocal of the wavelength to fall as the frequency increases — and the wavelength itself must increase, the opposite of the usual situation.

If the refractive index takes the form:

n(ν) = a + b / ν (6)

for some constants a and b, it follows that:

dn(ν)/dν = b / ν2  
ν dn(ν)/dν = b / ν  
n(ν) + ν dn(ν)/dν = a + b / ν – b / ν  
  = a  
vg = c / a (7)

So the group velocity is independent of the frequency, and pulses can propagate without distortion. Equation (6) is unlikely to hold true over a broad range of frequencies in any real medium, but it can be approximately correct for a limited range. The applet assumes it is true for all of the frequencies used to construct the wave portrayed.

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Applets Gallery / Subluminal (Technical Notes) / created Wednesday, 16 August 2000
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