My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
To look for solutions propagating in a direction parallel to the slab, we set
E(r, t) = E(z)e i(ωt−kx) , (8.13)
which, on substitution into equation (8.12) leads to the set of equations
−ik dE z
dz − d2 E x
dz 2
k 2 E y − d2 E y
dz 2
= ω2 − ω 2 p
c 2 E x (8.14a)
= ω2 − ω 2 p
c 2 E y (8.14b)
−ik dE x
dz + k2 E z = ω2 − ω 2 p
c 2 E z (8.14c)
The y equation is decoupled from the others, and has solutions
where
E y = E 0 e ±Kyz (8.15)
k 2 − K 2 y = ω2 − ω 2 p
c 2 . (8.16)
If k = 0, the solution is symmetrical in x and y; equation (8.14a) corresponds
to equation (8.14b), with E x replacing E y . Equation (8.14c) shows that there is no
field in the z-direction in this case, unless ω = ω p .
When k ≠ 0 and ω ≠ ω p , equation (8.14c) may be rearranged to give
E z =
−ikc 2
ω 2 − ω 2 p − k 2 c 2 dE x
dz , (8.17)
which, on substitution into equation (8.14a), leads to
)
d 2 E x
=
(k 2 − ω2
dz 2 c + ω2 p
E 2 c 2 x . (8.18)
The fields may then be calculated:
E x = E 0 e ±Kz (8.19)
E z = ±E 0
ik
K e±Kz (8.20)
where
K =
√
k 2 − ω2
c 2 + ω2 p
c 2 (with ω p = 0 in vacuum). (8.21)
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