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Phase velocity and group velocity

  1. Nov 21, 2004 #1
    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.
  2. jcsd
  3. Nov 21, 2004 #2
    I'm not sure I have the right idea in mind, but here you go....

    Group Velocity:

    [tex] v_{\mbox{group}} = \frac{d\omega}{dk} [/tex]

    Phase Velocity:

    [tex] v_{\mbox{phase}} = \frac{\omega}{k} [/tex]

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

    [tex] \frac{d\omega}{dk} = \frac{1}{2} \left( \frac{\omega}{k} \right) [/tex]

    Consider the following:

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


    [tex] \frac{d\omega}{dk} = c = n \left( \frac{\omega}{k} \right) [/tex]

    We obtain

    [tex] n \left( \frac{\omega}{k} \right) = \frac{1}{2} \left( \frac{\omega}{k} \right) [/tex]

    and so

    [tex] n = \frac{1}{2} [/tex]

    Again, this is just a shot in the dark...
  4. Nov 21, 2004 #3
    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..
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