Phase velocity and group velocity

[nuttyquark]

I am kind of stuck in a question relating to phase velocity and group velocity.

I have been given that the index of refraction of a media is inversely propotional to the vacuum wavelength. And we are supposed to show the group velocity is half the phase velocity.

Now, the work I have done thus far is to have a relation between phase velocity and group velocity consisting of (dn/d{lambda}) where n is index of refraction. I don't know how to proceed from there..

Please guide me in the right direction.

 

[DivGradCurl]

I'm not sure I have the right idea in mind, but here you go...

Group Velocity:

$$v_{\mbox{group}} = \frac{d\omega}{dk}$$

Phase Velocity:

$$v_{\mbox{phase}} = \frac{\omega}{k}$$

So, according to what you said: "we are supposed to show the group velocity is half the phase velocity", we have:

$$\frac{d\omega}{dk} = \frac{1}{2} \left( \frac{\omega}{k} \right)$$

Consider the following:

$$\omega = 2\pi f = 2\pi \left( \frac{c}{\lambda} \right) = 2\pi \left[ \frac{c}{\left(\ \frac{2\pi}{k} \right)} \right] = ck$$

Then:

$$\frac{d\omega}{dk} = c = n \left( \frac{\omega}{k} \right)$$

We obtain

$$n \left( \frac{\omega}{k} \right) = \frac{1}{2} \left( \frac{\omega}{k} \right)$$

and so

$$n = \frac{1}{2}$$

Again, this is just a shot in the dark...

 

[nuttyquark]

You might have misunderstood the question thiago..

we are not given that group velocity is half phase velocity..i need to prove that equality using the fact the n=A/L0 where LO is vacuum wavelength..