Page 206 - Principles and Applications of NanoMEMS Physics

5. NANOMEMS APPLICATIONS: PHOTONICS 195

G
∂ 2 n δ + n ∇ ⋅ ∂ G ∂ 2 n δ + n ∇ § − e E · ¸ =
v =
¨ ⋅
t ∂ 2 0 t ∂ t ∂ 2 0 ¨ m ¸ ¹ 0 , (5)
©
and which, in turn, upon substituting (3) into (5) becomes,
∂ 2 n δ + 4π e 2 n 0 n δ =
t ∂ 2 m 0 . (6)

Eq. (6), being analogous to that of a harmonic oscillator, prescribes the
frequency of plasmon oscillation as,
4π n e 2
ω = 0 . (7)
p
m
Of particular interest in this chapter, is the concept of surface plasmons,
(SPs), Fig. 5-1, thoroughly reviewed by Raether [215]. These are elicited by
the interaction of external electromagnetic fields with surface electrons, and
are characterized by a dispersion relation, a spatial extension, and a
propagation length or lifetime.

5.2.1 Surface Plasmon Characteristics

The dispersion relation for SPs at the interface between a dielectric
characterized by ε , deposited on the plane surface of a semi-infinite metal
2
characterized by ε = ε ' + iε , is given by [215],
' '
1 1 1
Dielectric, ε
Dielectric, ε
Evanescent W ave
2 2 Evanescent W ave
x x

z z
M etal, ε
M etal, ε
1 1
Figure 5-1. Sketch of surface plasmon. The field accompanying a surface plasmon peaks at
the dielectric-metal interface and diminishes exponentially away from the interface.

§ ω · 2
k = ε ¸ − k 2 , i = 2 , 1 , (8)
zi i ¨ x
© c ¹