554
V. N. Pustovit and T. V. Shahbazyan
turn, the solution of Maxwell’s equation with dipole sources [33]
E(r, ω) =
4πω 2
c 2
β
G(r, r β ; ω) · p β (ω),
(15.68)
where index β = (a, j) runs over all dipoles positions, G(r, r √ ; ω) is Maxwell’s
equation Green’s dyadic, satisfying ∇ × ∇ × ˆ
G − ε(r, ω)(ω/c) 2 ˆ
G = ˆ
I, and ε(r, ω)
equals metal permittivity, ε(ω), inside the metal region and that of outside medium,
ε 0 , otherwise. The quantity of interest is energy absorbed by the acceptor in unit
frequency interval,
dW ad
dω
= −
ω
π
Im
p
∗
a (ω) · E(r a , ω)
=
ωα √√
a
π
p a
α a
2
,
(15.69)
where we used E(r a , ω) = p a (ω)/α a (ω) from Eq. (15.67). A closed system for
p j (ω) is obtained by using Eq. (15.68) to eliminate the electric field from Eq. (15.67),
p a + α a D aa p a + α d
j
D aj p j = 0,
(15.70)
p j + α d D ja p a + α d
k
D jk p k = p
0
j ,
where we introduced frequency-dependent matrix
D ββ √ (ω) = −
4πω 2
c 2 e β · G(r β , r β √ ; ω) · e β √ .
(15.71)
Expressing p a from Eq. (15.70),
p a = −
α a
1 + α a ˜
D aa
jk
D aj S jk p
0
k ,
(15.72)
where
S
−1
jk (ω) = δ jk + α d D jk (ω)
(15.73)
and
˜
D aa = D aa − α d
jk
D aj S jk D ka .
(15.74)
Averaging over random phases ϕ j , we obtain
dW
dω
=
ωE 2
0
π
|α d | 2 α √√
a
1 + α a ˜
D aa
2
jkl
D aj S jk S
†
kl D
†
la .
(15.75)