# Surface plasmon polaritons at metal / insulator interfaces

1. Oct 21, 2016

### Monster1771

1. The problem statement, all variables and given/known data
Consider the metal-vacuum interface located at z = 0,the metal filling the entire half-space z ≥ 0, vacuum filling (!?) the half-space z < 0. The dielectric function in the metal in the long-wavelength limit is given by:

where ε0 is the vacuum permittivity. In the metal a solution of Laplace’s equation ∇2φ = 0 is

2. Relevant equations

3. The attempt at a solution
Tried to solve this problem for 8 hours, still no result. Maybe some of you might help?

2. Oct 21, 2016

### Twigg

a) How are E and the scalar potential related? Note: the tangential derivative in this case is simply $\frac{\partial}{\partial x}$, and the normal derivative is $\frac{\partial}{\partial z}$. Check to see if the solutions provided satisfy the boundary condition that the tangential component of the electric field is continuous at the interface (z=0).

b) Similar to (a), but now you check the normal direction and use the macroscopic formalism (D as opposed to E). The problem tells you that the normal component of D will be continuous (means 0 free charge at the interface). Use the given formula for the dielectric function of the metal and solve for $\omega$. What can you conclude about the optically-active oscillations at the interface?