How Do Surface Plasmons Behave at a Metal Interface?

[Elfrae]

Homework Statement

Consider the plane interface z=0 between metal 1 at z>0 and metal 2 at z<0. Metal 1 has bulk plasmon frequency [tex]\omega_{p1}[/tex]; metal 2 has [tex]\omega_{p2}[/tex]. The dielectric constants in both metals are those of electron gases. Show that surface plasmons associated with the interface have the frequency [tex]\omega[/tex] = ([tex]\frac{1}{2}[/tex]([tex]\omega_{p1}^{2}[/tex] + [tex]\omega_{p2}^{2}[/tex]))[tex]^{\frac{1}{2}}[/tex]

Homework Equations

I have these, but I'm not sure whether they're relevant:

[tex]\epsilon[/tex]([tex]\omega[/tex]) = 1 - [tex]\frac{\omega_{p}^{2}}{\omega^{2}}[/tex]

[tex]\omega_{s}^{2}[/tex] = [tex]\frac{1}{2}[/tex][tex]\omega_{p}^{2}[/tex]

The Attempt at a Solution

I don't know how to begin this question. A pointer on what I need to think about would be appreciated. Thanks.

Edit: I've tried equating the surface plasmon frequency for each metal, but I'm not getting anywhere. I'm not sure if I'm going in the right direction with that.
Some of my previous questions were to do with components of the electric field being continuous at the boundary. Is that likely to be what I need to use?

 

[Elfrae]

Ok, I think I've solved this now. I'm just not sure about one of the steps I used to get to the answer.

I assumed that the normal component of the electric displacement D is continuous at the boundary, and hence

[tex]\epsilon_{1}[/tex] = - [tex]\epsilon_{2}[/tex]

Bearing in mind that the boundary is between two metals, is this correct?