# Help understanding the speed of light in media with refractive index < 1

I have read the standard explanations on this, but I still have trouble with convincing myself that information is not propagating at faster than the speed c in media of refractive index less then unity (where c is the speed of light in the vacuum). I run into problems when I try to think about it mathematically. Here my query relates to phase velocity. I understand the argument that a given wave speed corresponds to monochromatic light and that this implies an infinitely long sinusoidal wave and thus no information is transmitted. But I have had a thought experiment that naively seems to conflict with that argument and I need help understanding where I am going wrong. Consider a wave pulse of the form:

$$f(x,t)=\int_{-\infty}^{\infty}g\left (k \right )e^{i\left ( kx-\omega(k) t \right )}dk$$

(equation 1)

And let us say at time t=0, we were to play God and somehow we created a light pulse exactly one full period of a sinusoidal wave:

$$f(x,0)=1+cos(kx)$$

between $$-\frac{\pi}{k} \leq x \leq \frac{\pi}{k}$$

and

$$f(x,0)=0$$

for $$x <-\frac{\pi}{k}$$ and $$\frac{\pi}{k} < x$$

Mathematically we should be able to do this as we have:

$$f(x,0)=\int_{-\infty}^{\infty}g\left (k \right )e^{i\left ( kx \right )}dk$$

and we should be able to find $$g(k)$$ by taking the Fourier transform of $$f(x,0)$$

The point is that this is a localised pulse. And next we then let this pulse propagate in time t. According to equation 1 we effectively have the sum of infinitely wide pure monochromatic waves, where in each case , their speed of propagation is determined by:

$$v_{p}=\omega(k)/k$$

where k is the wave number and $$\omega$$ is the angular frequency which depends on k. And for at least some range of k we have $$v_{p}\geq c$$

My problem is this:

We can state that each wave is monochromatic and infinitely wide and thus carries no information. But we started with localised pulse, and this pulse will begin moving in the +x direction (albeit dispersed). Then at some point $$X > \frac{\pi}{k}$$, and a certain time T we will find that

$$f(X,T) \neq 0$$

in a time $$T < \frac{X-\frac{\pi}{k}}{c}$$

So isn't information actually being propagated? And isn't some of it propagating at speeds > c ?

Where am I getting it wrong? Maybe the answer could be that it is physically nonsense to construct a wave in this way. Also I realize that in reality there are two field $$B(x,t)$$ and $$E(x,t)$$, and not a scalar function $$f(x,t)$$

A localized pulse does not propagate at the phase velocity, $$v_p=\omega/k$$, but at the group velocity, $$v_g=\frac{d\omega}{dk}$$.
• Hi. Thanks for that. I was aware of the $v_{g}$, but I never thought of it for localised pulses. But it makes sense. Can you answer a related question for me? Is it the case still that actual information can only propagate at speed c through the medium? In the wave analysis I only see two wave speeds, $v_{p}$ and $v_{g}$ as you have just described. I cant see a wave speed = c anywhere. Is it the case that for refractive index < 1 materials that $v_{g} \leq c$ Jun 16 '19 at 10:47
• @RoryCornish, For example, in communications systems information is transmitted using pulses, and these propagate at the group velocity, $v_g \le c$. Also, in dispersive media, pulses or wave packets are distorted, and attenuated if the medium introduces loss. Details depend on the specific dispersion relation $\omega(k)$. Jun 16 '19 at 14:20