Trying to derive a group velocity equation?

In summary, the conversation involves trying to derive the group velocity equation, which is the inverse of a given equation involving refractive index and light wavelength. The speaker has been using chain and product rules to manipulate derivatives and has come up with several equations, but is struggling to find a simple solution. Another person suggests double checking how derivatives are being taken and points out a discrepancy in the speaker's equations. Finally, a suggestion is made to calculate a specific term in order to complete the derivation.
  • #1
jeebs
325
4
hi,
I am trying to derive the group velocity equation (actually its inverse):

[tex] \frac{1}{v_g} = \frac{1}{c}(n(\lambda) - \lambda\frac{dn}{d\lambda}) [/tex]
where n is the refractive index and depends on the light's wavelength.

I have started by saying that if [tex]v_g = \frac{\partial\omega}{\partial k} [/tex] then I can do [tex] \frac{dk}{d\omega} = \frac{dk}{d\lambda}\frac{d\lambda}{d\omega} [/tex].

I then say that [tex] k=\frac{2\pi}{\lambda} [/tex] and [tex] \omega=\frac{2\pi c}{n\lambda} [/tex], also that [tex] k=\frac{n\omega}{c} [/tex] .

When I've done the differentiation I have been using chain rules and product rules.

After messing around with this stuff all bloody weekend and calculating every possible derivative a dozen times I am certain that (among other things):

[tex] \frac{d\lambda}{d\omega} = 2\pi c(\frac{-1}{n^2\omega^2} - \frac{1}{n^2\omega}\frac{dn}{d\omega}) [/tex]

[tex] \frac{dk}{d\omega} = \frac{4\pi^2c}{\lambda^2n^2\omega^2}(n + \omega\frac{dn}{d\omega}) [/tex]

[tex] \frac{dn}{d\omega} = c(\frac{1}{\omega}\frac{dk}{d\omega} - \frac{k}{\omega^2}) [/tex]

[tex] \frac{dk}{d\lambda} = \frac{-2\pi}{k^2}[/tex]

there are more things I have calculated but I always seem to be going in circles, working out things that end up being in terms of just another "dx/dy".

after going through about 40 trees'worth of paper the closest I have got with this is to find that [tex] \frac{1}{v_g} = \frac{1}{c}(n - (\frac{4\pi^2 c}{n^2\omega^2\lambda}\frac{d\omega}{d\lambda} + \frac{4\pi^2 c}{n^3\omega\lambda}\frac{dn}{d\lambda})\frac{dn}{d\lambda})[/tex]

and to finish I'd need to show that the bit in the brackets is equal to [tex]\lambda[/tex] but I just can't. the fact that this is such a simple looking equation and that I have managed to spend over 12 hours in total tearing my hair out over this tells me I am probably overcomplicating things. Can anyone save my sanity here? I know this will be some simple trick that I'll feel like an idiot for not seeing.

thanks.
 
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  • #2
Double check how you are taking those derivatives. For instance,

[tex] \frac{\partial}{\partial x} \left[ \alpha f(x) \beta g(x) \right] = \alpha \beta \left ( \frac{\partial f}{\partial x}g(x)+\frac{\partial g}{\partial x}f(x) \right ) [/tex]

where alpha and beta are constants
 
  • #3
i'm pretty sure that's the way I was doing them.
 
  • #4
You say that

[tex] k=\frac{n\omega}{c} \implies \frac{dk}{d\omega}=\frac{1}{c} \left ( n + \omega \frac{dn}{d\omega} \right) [/tex]

yet also that

[tex]
\frac{dk}{d\omega} = \frac{4\pi^2c}{\lambda^2n^2\omega^2}(n + \omega\frac{dn}{d\omega})
[/tex]
 
  • #5
Notice that

[tex]\frac{1}{v_g} = \frac{dk}{d\omega} = \frac{d}{d\omega} \frac{n\omega}{c} = \frac{1}{c} \left ( n + \omega \frac{dn}{d\omega} \right) [/tex]

[tex]\omega \frac{dn}{d\omega} = \omega \frac{dn}{d\lambda} \frac{d \lambda}{d\omega}[/tex]

now calculate this last term and you should be done.
 

1. What is group velocity?

Group velocity is a measure of how fast a wave packet, or a group of waves, propagates through a medium. It takes into account both the individual wave velocities and the frequency of the waves in the group.

2. Why is it important to derive a group velocity equation?

Deriving a group velocity equation allows us to understand how waves behave and interact in different mediums. It also helps us predict and control the propagation of waves, which has practical applications in fields like optics, acoustics, and signal processing.

3. What factors affect the group velocity of a wave?

The group velocity of a wave is affected by the dispersion of the medium, which is the dependence of the wave velocity on its frequency. It is also influenced by the shape and characteristics of the wave packet, as well as the properties of the medium, such as its refractive index or density.

4. How is the group velocity equation derived?

The group velocity equation can be derived using mathematical techniques such as the Fourier transform and the Euler-Lagrange equation. It involves finding the first derivative of the wave's phase with respect to its wave number, or the rate of change of the wave's frequency with respect to its wavenumber.

5. Can the group velocity be greater than the speed of light?

No, the group velocity cannot be greater than the speed of light in a vacuum. This is because the group velocity is a derived quantity that takes into account the velocity of individual waves, which cannot exceed the speed of light. However, in certain materials with high refractive indices, the group velocity can be slower than the speed of light.

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