Maxwell's Equations - Wavelength Dependance derivation from Group Velocity

[leoflindall]

Homework Statement

For a particular Dieletric it is observed that over a range of frequencies, the group velocity varies exponetinally with wave number:

v$$_{g}$$=ae$$^{bk}$$ , where a,b are constants. *(PLease not that the superscript g on v on the LHS side is meant to be subscript, however it won't let me change it...)*

Derive the wavelength dependence of the refractive index for this material.

Homework Equations

Maxwell's equations in dieletric media...?

The Attempt at a Solution

This is a revision question for an upcoming exam. I don't really have any idea at how to approach this question. The only thing that comes to mind is that the refractive index is the ratio of the velocity in two media. I think this is quite an easy question but don't know how to do it! Any help or guidance would be greatly appreciated.

Thanks, Leo

 

[Maxim Zh]

The definition of refractive index is

$$n = \frac{c}{v_{phase}}.$$ ----- (1)

The definitions of phase and group velocities are

$$v_{phase} = \frac{\omega}{k};$$ ----- (2)

$$v_{group} = \frac{\partial\omega}{\partial k}.$$ ----- (3)

Since the $$v_{group}(k)$$ function is given we can derive the $$\omega(k)$$ function, using (3):

$$\omega(k) = \frac{a}{b}e^{bk} + d,$$ ----- (4)

where $$d$$ is a constant.
(2) and (4) will give us:

$$v_{phase} = \frac{a}{bk}e^{bk} + \frac{d}{k},$$

or

$$n(k) = \frac{kc}{(a/b)\exp(bk) + d}.$$

It's known that

$$k = \frac{2\pi}{\lambda}$$

so the wavelength dependence will be

$$n(\lambda) = \frac{c}{\lambda[(a/b)\exp(b/\lambda) + d]}.$$

 

[leoflindall]

Cheers buddy that makes sense, quitre simple really just didn't think about w(k). Thanks for your help!