562
V. N. Pustovit and T. V. Shahbazyan
G
0
μν (r, r
√
) =
δ μν −
1
k 2 ∞ μ ∞
√
ν
e ikr
r
1
4π
−
ikr √
3
m
Y 1m (ˆ r)Y
∗
1m (ˆ r
√
)
.
(15.93)
After differentiation, the far field asymptotics takes the form
G
0
μν (r, r
√
) =
e ikr
4πr
δ μν −
4π
3
m
ˆ
r μ Y 1m (ˆ r)χ
ν∗
1m (r
√
)
,
(15.94)
where we introduced χ
μ
lm (r) = ∞ μ [r l Y lm (ˆ r) and ψ
μ
lm (r) = ∞ μ [r −l−1 Y lm (ˆ r)].
Now turn to the scattered part of the Green dyadic derived from solution of Mie
problem for electromagnetic wave scattered on single sphere,
G
s
μν (r, r
√
, k) = ik
lm
a l N
μ
lm (r)N
ν
lm (r
√
)
+ b l M
μ
lm (r)M
ν
lm (r
√
)
,
(15.95)
where the first and second terms are electric and magnetic contributions and a l and
b l are the Mie coefficients. In the long wave approximation, k R ∝ 1, the magnetic
contribution in Eq.(15.95) can be neglected as b l ∝ 1 [24–27]. The Mie coefficient
a l has a form
a l =
ε 0 j l (ρ 0 )[ρ j l (ρ)] √ − εj l (ρ)[ρ 0 j l (ρ 0 )] √
ε 0 h l (ρ 0 )[ρ j l (ρ)] √ − εj l (ρ)[ρ 0 h l (ρ 0 )] √
(15.96)
where ρ i = k i R, k i =
ω
c
≥
ε i , and i = (ε, ε 0 ). For k R ∝ 1, it becomes
a l = −is l ˜
α l k
2l+1
, s l =
l + 1
l(2l + 1)[(2l − 1)!!] 2 ,
(15.97)
where
˜
α l =
α l
1 − is l k 2l+1 α l
,
(15.98)
is NP multipolar polarizability that accounts for plasmon radiative decay, and
α l = R
2l+1
l(ε − ε 0 )
lε + (l + 1)ε 0
,
(15.99)
is the standard NP polarizability. The function N lm (r) is given by
N lm (r) =
1
k
≥
l(l + 1)
∇ ×
h
(1)
l (kr)LY lm (ˆ r)
,
(15.100)
V. N. Pustovit and T. V. Shahbazyan
G
0
μν (r, r
√
) =
δ μν −
1
k 2 ∞ μ ∞
√
ν
e ikr
r
1
4π
−
ikr √
3
m
Y 1m (ˆ r)Y
∗
1m (ˆ r
√
)
.
(15.93)
After differentiation, the far field asymptotics takes the form
G
0
μν (r, r
√
) =
e ikr
4πr
δ μν −
4π
3
m
ˆ
r μ Y 1m (ˆ r)χ
ν∗
1m (r
√
)
,
(15.94)
where we introduced χ
μ
lm (r) = ∞ μ [r l Y lm (ˆ r) and ψ
μ
lm (r) = ∞ μ [r −l−1 Y lm (ˆ r)].
Now turn to the scattered part of the Green dyadic derived from solution of Mie
problem for electromagnetic wave scattered on single sphere,
G
s
μν (r, r
√
, k) = ik
lm
a l N
μ
lm (r)N
ν
lm (r
√
)
+ b l M
μ
lm (r)M
ν
lm (r
√
)
,
(15.95)
where the first and second terms are electric and magnetic contributions and a l and
b l are the Mie coefficients. In the long wave approximation, k R ∝ 1, the magnetic
contribution in Eq.(15.95) can be neglected as b l ∝ 1 [24–27]. The Mie coefficient
a l has a form
a l =
ε 0 j l (ρ 0 )[ρ j l (ρ)] √ − εj l (ρ)[ρ 0 j l (ρ 0 )] √
ε 0 h l (ρ 0 )[ρ j l (ρ)] √ − εj l (ρ)[ρ 0 h l (ρ 0 )] √
(15.96)
where ρ i = k i R, k i =
ω
c
≥
ε i , and i = (ε, ε 0 ). For k R ∝ 1, it becomes
a l = −is l ˜
α l k
2l+1
, s l =
l + 1
l(2l + 1)[(2l − 1)!!] 2 ,
(15.97)
where
˜
α l =
α l
1 − is l k 2l+1 α l
,
(15.98)
is NP multipolar polarizability that accounts for plasmon radiative decay, and
α l = R
2l+1
l(ε − ε 0 )
lε + (l + 1)ε 0
,
(15.99)
is the standard NP polarizability. The function N lm (r) is given by
N lm (r) =
1
k
≥
l(l + 1)
∇ ×
h
(1)
l (kr)LY lm (ˆ r)
,
(15.100)