15 Cooperative Effects in Plasmonics
563
where L = −i(r × ∇) is angular momentum operator. Using the following identity,
∇ ×
h
(1)
l (kr)LY lm (ˆ r)
= irk
2 h
(1)
l (kr)Y lm (ˆ r)
+ i∇
[kr h
(1)√
l (kr) + h
(1)
l (kr)]Y lm (ˆ r)
,
(15.101)
prime standing for derivative, and expanding h
(1)
l (kr) = j l (kr) + in l (kr) in kr as
h
(1)
l (kr) =
(kr) l
(2l + 1)!!
− i
(2l − 1)!!
(kr) l+1 ,
(15.102)
we obtain
N lm (kr) = −
1
k
≥
s l (2l + 1)
∇[ϕ l (kr)Y lm (ˆ r)],
(15.103)
where
ϕ l (kr) =
1
(kr) l+1 − is l (kr)
l
.
(15.104)
Thus, for kr ∝ 1 and kr √ ∝ 1, the scattered part of the Green dyadic has the form
G
s
μν (r, r
√
, k) ∈ ik
lm
a l N
μ
lm (kr)N
ν
lm (kr
√
)
(15.105)
∈
lm
k 2l ˜
α l
2l + 1
∞ μ
ϕ l (kr)Y lm (ˆ r)
∞
√
ν
ϕ l (kr
√
)Y
∗
lm (ˆ r
√
)
.
This expression can be further simplified by substituting ˜
α l = ¯
α l + is l k 2l+1 | ˜
α l | 2 ,
where ¯
α l = α l |1 − is l k 2l+1 α l | −2 , and keeping the first two powers of k
G
s
μν (r, r
√
) =
1
k 2
lm
¯
α l
(2l + 1)
ψ
μ
lm (r)ψ
ν∗
lm (r
√
)
− i
ks 1
3
1
m=−1
˜
α 1
ψ
μ
1m (r)χ
ν∗
1m (r
√
) + χ
μ
1m (r)ψ
ν∗
1m (r
√
)
− | ˜
α 1 |
2
ψ
μ
1m (r)ψ
ν∗
1m (r
√
)
,
(15.106)
which, after adding the free-space part of the Green dyadic and neglecting plasmon
radiative decay, leads to Eq. (15.12).
For kr ≈ 1, with help of Eqs. (15.92), (15.100), and (15.101), we easily obtain
N lm (r) = −
(−i) l+1 e ikr
k
≥
l(l + 1)
∇Y lm (ˆ r),
(15.107)