My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
The left-hand side is the modified Helmholtz equation in one dimension, which has
the Green’s function
Adapting this to equation (8.49) gives the full solution
The electric field is then easy to calculate:
where
G(z, z ′ ) = 1
2k e−k|z−z′| . (8.50)
φ z (z) = − ρ 0
2kɛ 0
(
e −k|z| ± e −k|z−s|) . (8.51)
E = −∇φ (8.52)
= − ρ 0
2ɛ 0
e i(ωt−kx) [f x (z)ˆx + f z (z)ẑ] (8.53)
f x (z) = i ( e −k|z| ± e −k|z−s|) (8.54)
f z (z) = sgn(z) e −k|z| ± sgn(z − s) e −k|z−s| (8.55)
The components of the field are plotted in figure 8.4.
For all surface plasmons, ω < ω p .
8.1.5 Bulk plasmons
Solutions also exist for the bulk plasmons. When ω = ω p , the boundary condition
on E z (equation (8.28c)) implies that at the interface E (v)
z
the field in the vacuum is
E (v)
z =
E (v)
x =
⎧
⎪⎨ E 0 e −Kvz
with K v given by equation (8.32). If E (v)
z
when z > s
⎪⎩ E 1 e Kvz when z < 0
⎧
⎪⎨ ik
K v
E 0 e −Kvz when z > s
⎪⎩ − ik
K v
E 1 e Kvz when z < 0
= 0. The general form of
(8.56)
(8.57)
= 0 at z = s and z = 0 then both E 0 and
E 1 are also zero, and consequently there is no field in the vacuum. Consideration of
equation (8.23) shows that this is also true in the special case k = k p = ω p /c.
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