Page 207 - Principles and Applications of NanoMEMS Physics

196 Chapter 5

where the wave vector k , is given by,
x
ω ε ε
k = 1 2 . (9)
x
c ε + ε
1 2
Substituting the complex dielectric constant expression into (9), the wave
vector becomes k = k + ik , with components,
' '
'
x
x
x
ω ε ' ε
k = 1 2 , (10)
'
x '
c ε + ε
1 2
and
ω § ε ' ε · / 3 2 ε ' '
k = ¨ 1 2 ¸ 1 . (11)
' '
()
x ¨ ' ¸ 2
c ε 1 + ε 2 ¹ 2 ε 1 '
©
Since ω c < k , see (9), and ε ' < 0, (characteristic of the metal), both
x 1
k and k z 2 are imaginary. As a result, the SP field becomes evanescent.
1 z
The corresponding spatial decay of the field, away from the interface, is thus
proportional to exp ( k− ) z [215], and is characterized by the distance at
zi
which it has decreased into either medium by 1/e [215]. Thus is given by,
λ ε ' + ε
z = 1 2 , (12)
2π ε
2 2
2
into the medium with ε , and,
2
λ ε ' + ε
z = 1 2 , (13)
1 2
2π ' ε
1
into the medium with ε .
1
The propagation length L for SPs propagating along a smooth surface is
i
defined as the distance, away from the interface, at which their intensity,
which is proportional to ( 2exp − k ' ' ) x , has decreased by 1/e, namely,
x
L = 1 . (14)
i ' '
2 k
x