20
M. I. Stockman
One of the plasmonic system models studied in significant detail is a random planar
composite (RPC) also called a semi-continuous metal film [78, 128, 148, 155, 161,
167–170]. This is a planar system where metal occupies a given fill fraction f of the
system’s volume. At a low f, the RPC is a system of remote randomly positioned
metal particles. For high values of f, it is an almost continuous film with rare holes in
it. For f ≈ 0.5, there are percolation phenomena: there is a large connected random
cluster of the metal extending between the boundaries of the system [171]. This
connected percolation cluster is known to possess a fractal geometry.
To consider statistical measures of the SP localization, we introduce the localization radius L n of an eigenmode, which is defined as the gyration radius of its electric
field intensity |E n (r)| 2 , where
E n (r) = −
∂
∂r
ϕ n (r)
(1.32)
is the eigenmode electric field, as
L
2
n =
V
r
2
|E n (r)|
2 d
3 r −
V
r|E n (r)|
2 d
3 r
2
.
(1.33)
We remind that due to Eq. (1.30), the eigenmode fields are normalized
V
|E n (r)|
2 d
3 r = 1,
(1.34)
so Eq. (1.33) is a standard definition of the gyration radius.
In Fig. 1.6a, we show the smoothed, discretized nanostructure of one particular
sample of a RPC. This system is generated in the following way. We consider a
volume of size, in our case, 32 × 32 × 32 grid steps. In the central xz plane of this
cube we randomly fill a cell of size 2 × 2 grid steps with metal with some probability
0
10
20
30
x
0
10
20
30
z
0
5 10 15 20
L n
10
14
10
11
10
8
10
5
10
2
F n
(a
(
)
b )
Fig. 1.6 For a planar random composite (in the xz-plane), the density of the metal component
(panel a) and all eigenmodes plotted in the coordinates of oscillator strength F n versus localization
radius L n (panel b)
M. I. Stockman
One of the plasmonic system models studied in significant detail is a random planar
composite (RPC) also called a semi-continuous metal film [78, 128, 148, 155, 161,
167–170]. This is a planar system where metal occupies a given fill fraction f of the
system’s volume. At a low f, the RPC is a system of remote randomly positioned
metal particles. For high values of f, it is an almost continuous film with rare holes in
it. For f ≈ 0.5, there are percolation phenomena: there is a large connected random
cluster of the metal extending between the boundaries of the system [171]. This
connected percolation cluster is known to possess a fractal geometry.
To consider statistical measures of the SP localization, we introduce the localization radius L n of an eigenmode, which is defined as the gyration radius of its electric
field intensity |E n (r)| 2 , where
E n (r) = −
∂
∂r
ϕ n (r)
(1.32)
is the eigenmode electric field, as
L
2
n =
V
r
2
|E n (r)|
2 d
3 r −
V
r|E n (r)|
2 d
3 r
2
.
(1.33)
We remind that due to Eq. (1.30), the eigenmode fields are normalized
V
|E n (r)|
2 d
3 r = 1,
(1.34)
so Eq. (1.33) is a standard definition of the gyration radius.
In Fig. 1.6a, we show the smoothed, discretized nanostructure of one particular
sample of a RPC. This system is generated in the following way. We consider a
volume of size, in our case, 32 × 32 × 32 grid steps. In the central xz plane of this
cube we randomly fill a cell of size 2 × 2 grid steps with metal with some probability
0
10
20
30
x
0
10
20
30
z
0
5 10 15 20
L n
10
14
10
11
10
8
10
5
10
2
F n
(a
(
)
b )
Fig. 1.6 For a planar random composite (in the xz-plane), the density of the metal component
(panel a) and all eigenmodes plotted in the coordinates of oscillator strength F n versus localization
radius L n (panel b)