How Does the Refractive Index Change with Frequency in Plasma Mode?

[v_pino]

Homework Statement

Attached as pdf.

Homework Equations

Attached as pdf.

The Attempt at a Solution

I know that refractive index is given by $$n=\sqrt{\varepsilon}$$ normally. But is it still the case when asked for $$n( \omega)$$?

If so, I've tried rearranging equation 3 for $$\varepsilon$$. Which gives $$\varepsilon = -k_m \varepsilon_0 / k_v$$, where the subscript v and m denote metal and vacuum. How does this help in finding $$n (\omega) = \sqrt{ \frac{\varepsilon( \omega)}{\varepsilon ( \omega) + \varepsilon_0}}$$?

 

[v_pino]

Please do suggest reading materials on this topic as I don't think I fully understand it from my lectures. Thank you.

 

[v_pino]

I went through the algebra and got this equation:

$$\frac{c^2}{\omega^2}k_x^2=\frac{(1-\varepsilon_0^3/\varepsilon(\omega))}{(1-\varepsilon_0^4/\varepsilon(\omega)^2)}$$

And I know that:

$$n(\omega)=\frac{c}{v_x}=\frac{ck_x}{\omega}$$

Is there a way in which I can arrange equation 1 into:

$$\frac{\varepsilon(\omega)}{\varepsilon(\omega)+ \varepsilon_0}$$

?