**Problem 1** - surface plasmons
If you went to the Dexter Lecture given by Prof. Capasso on March 3, you heard a lot about surface plasmons. In this problem you will derive the dispersion relation and other properties of the surface plasmon!

Consider a metal with a surface in the *xy* plane, i.e. *z* > 0 is metal and *z* < 0 is the vacuum. The surface plasmon is a solution to Maxwell's equations of the form,

*E*_{x} = *A*e^{iqx} e^{−Kz}e^{−iωt}, *E*_{y} =0, *E*_{z} = *B*e^{iqx}e^{−Kz}e^{−iωt} for *z* > 0

*E*_{x} = *C*e^{iqx} e^{K'z}e^{−iωt}, *E*_{y} =0, *E*_{z} = *D*e^{iqx}e^{K'z}e^{−iωt} for *z* < 0

with *q, K, K'* real and *K, K'* positive (so that the amplidue of the wave decays as *z* moves away from *z*=0).

Assume that there is no charge density induced in the bulk of the metal, i.e. **∇**⋅**E**=0, (but there may be induced surface charge).
By requiring the above expression to satisfy Maxwell's equation, and to satisfy the usual electromagnetic boundary conditions, tangential component of **E** continuous and normal component of ε**E** continuous, find equations relating the unknown amplitudes *A, B, C, D* to each other, and relating *q, K, K'* to each other and to the frequency ω.

Use for the metal the dielectric function ε(ω) = 1+4π*i*σ(ω)/ω, with σ(ω) the ac Drude conductivity. You may make the assumption that ωτ >> 1 so that ε(ω) ≈ 1 − (ω_{P}/ω)^{2}, where ω_{P}=4π*ne*^{2}/*m* is the plasma frequency.

Use your results above to solve for *q* as a function of ω. Plot *q*^{2}*c*^{2} vs ω^{2}, and indicate for what range of ω there is indeed a solution with the desired properties.

Compute the surface charge density induced on the surface of the metal? (Hint: remember your electrostatics!)

What is the dispersion relation ω(*q*) in the limit of small *q*? Show that as *q* gets very large (i.e. *qc* >> ω) there is a solution at frequency ω = ω_{P}/√2. What is the polarization of this wave when ω = ω_{P}/√2?