26
M. I. Stockman
where γ n is given above by Eqs. (1.48) or (1.49). This expression constitutes what
is called the singular approximation or pole approximation of the Green’s function.
When an excitation frequency is in resonance with an SP frequency, i.e., ω = ω n ,
the Green’s function (1.52) increases in magnitude by ∼ω n /γ n ∼ Q times, where
the quality factor Q is given by Eq. (1.5).
Below, for the sake of reference, we give a modal expansion for the polarizability
α of a nanoplasmonic system as a tensor,
α αβ = −
ε d
4π
n
1
s n (s − s n )
M nα M
∗
nβ ,
(1.53)
where the indexes α, β denote Cartesian components, and M n is a coupling vector
defined as
M n = −
V
Θ(r)
∂ϕ n (r)
∂r
d
3 r.
(1.54)
Near a SP frequency, ω ≈ ω n , a singular part of the polarizability (1.53) acquires
a form
α αβ = −
ε d
4π s
n s n
M nα M ∗
nβ
ω − ω n + iγ n
.
(1.55)
Also, for the reference sake, we give a general expression for the SP radiative decay
rate, γ
(r )
n . This can be obtained from Eq. (1.55) taking into account Eqs. (1.10) and
(1.15) as
γ
(r )
n =
ε
3/2
d ω 3 |M n | 2
9π c 3 s
n s n
.
(1.56)
Note that |M n | 2 ∼ V n , where V n is the modal volume of the n-th eigenmode. Thus
Eq. (1.56) is consistent with Eq. (1.16) obtained earlier in this chapter.
1.3.5 Examples of Local Fields and Their Hot Spots
Let us give an example of local fields computed using Eq. (1.39). We start with
the results of the original publications Ref. [157, 158] where the hot spots of the
plasmonic local fields have been predicted. This prediction was made for fractal
clusters because the fractals were expected to possess highly inhomogeneous and
fluctuating local optical fields as was shown in pioneering papers in a subfield of
physical optics that today is called nanoplasmonics [117, 149, 177].
In Fig. 1.8 adapted from Ref. [157], we illustrate the local-field hot spots for a
silver CCA cluster of N = 1500 identical nanospheres embedded in water. We show
local field intensity I = |E(r, ω)|
2 relative to the excitation field intensity I 0 at
M. I. Stockman
where γ n is given above by Eqs. (1.48) or (1.49). This expression constitutes what
is called the singular approximation or pole approximation of the Green’s function.
When an excitation frequency is in resonance with an SP frequency, i.e., ω = ω n ,
the Green’s function (1.52) increases in magnitude by ∼ω n /γ n ∼ Q times, where
the quality factor Q is given by Eq. (1.5).
Below, for the sake of reference, we give a modal expansion for the polarizability
α of a nanoplasmonic system as a tensor,
α αβ = −
ε d
4π
n
1
s n (s − s n )
M nα M
∗
nβ ,
(1.53)
where the indexes α, β denote Cartesian components, and M n is a coupling vector
defined as
M n = −
V
Θ(r)
∂ϕ n (r)
∂r
d
3 r.
(1.54)
Near a SP frequency, ω ≈ ω n , a singular part of the polarizability (1.53) acquires
a form
α αβ = −
ε d
4π s
n s n
M nα M ∗
nβ
ω − ω n + iγ n
.
(1.55)
Also, for the reference sake, we give a general expression for the SP radiative decay
rate, γ
(r )
n . This can be obtained from Eq. (1.55) taking into account Eqs. (1.10) and
(1.15) as
γ
(r )
n =
ε
3/2
d ω 3 |M n | 2
9π c 3 s
n s n
.
(1.56)
Note that |M n | 2 ∼ V n , where V n is the modal volume of the n-th eigenmode. Thus
Eq. (1.56) is consistent with Eq. (1.16) obtained earlier in this chapter.
1.3.5 Examples of Local Fields and Their Hot Spots
Let us give an example of local fields computed using Eq. (1.39). We start with
the results of the original publications Ref. [157, 158] where the hot spots of the
plasmonic local fields have been predicted. This prediction was made for fractal
clusters because the fractals were expected to possess highly inhomogeneous and
fluctuating local optical fields as was shown in pioneering papers in a subfield of
physical optics that today is called nanoplasmonics [117, 149, 177].
In Fig. 1.8 adapted from Ref. [157], we illustrate the local-field hot spots for a
silver CCA cluster of N = 1500 identical nanospheres embedded in water. We show
local field intensity I = |E(r, ω)|
2 relative to the excitation field intensity I 0 at