15 Cooperative Effects in Plasmonics
551
radiative FRET theory [38, 61] to plasmonic systems. In fact, the PERT mechanism
can dominate FRET even in the near field, as our numerical calculations below
demonstrate.
15.2.3 Numerical Results for Plasmon-Assisted Single-Donor
Energy Transfer Near Metal Nanoparticle
As an example, consider a donor and acceptor near spherical Ag NP in water with normal dipole orientations (see Fig. 15.1). The near field matrix D jk is readily obtained
from the Mie’s theory Green’s dyadic [25] as D jk = D 0
jk + D r
jk + D nr
jk , where [13]
D
r
jk = − i
2
3
k
3
1 + 2α 1
1
r 3
j
+
1
r 3
k
+
4|α 1 | 2
r 3
j r 3
k
(ˆ r j · ˆ
r k ),
D
nr
jk = −
l
α l (l + 1) 2
r
l+2
j r
l+2
k
P l (ˆ r j · ˆ
r k )
(15.66)
are NP-induced radiative and nonradiative terms, α l = R 2l+1 l(ε−ε 0 )
lε+(l+1)ε 0
is NP polarizability, P l (x) is a Legendre polynomial, D 0
ad = (1 + sin 2 θ/2)/r 3
ad , ˆ
r a · ˆ
r d = cos θ ,
and angular momenta up to l = 50 are included. Full decay rates are Γ j =
−
3/2k 3 γ r
j D √√
j j . We consider, for simplicity, a high-yield donor with a broad emission band due to the vibrational modes. Molecules optical bands are Lorentzians
of width 0.05 eV centered at 2.95 eV and 3.2 eV with maximal overlap at about
SP energy of 3.08 eV (see inset in Fig. 15.12a), σ a (ω) was normalized to its total
dωσ a (ω), and modified ¯
σ a , ˜
f d , and ˜
D ad were found using Eq. (15.66).
In Fig. 15.11, we plot W ad versus the molecule’s distance d from the R = 30
nm NP surface at θ = π/3 with equal d a = d d = d and with changing d d = d
at fixed d a . Three models—the full Eq. (15.61), its nonradiative part only, and the
Gersten–Nitzan model [62, 63]—are compared to Förster’s transfer Eq. (15.45).
For d d = d a , W ad is about three times larger than W F
ad and rapidly decays with
d, while for d/R ∝ 1 it is quenched by metal. There is no enhancement if only
the nonradiative channel is included in Eq. (15.61). In contrast, the Gersten-Nitzan
model yields much greater enhancement (up to 10 5 ) for d/R ∝ 1 since it includes
no quenching effects. However, at fixed d a and d d /R 1, the full W ad is the largest
one (see Fig. 15.11b) due to the dominant role of the PERT mechanism, as discussed
above.
The interplay of different FRET contributions is shown in Fig. 15.12 featuring
spectral density Eq. (15.59) together with quenching factor γ r
d /Γ d and coupling
|D ad | 2 at fixed d. dW ad /dω has a sharp plasmon peak which disappears if only the
nonradiative channel is included (see Fig. 15.12a). PERT channel reduces γ r
d /Γ d
due to SP-enhanced radiation but it strongly enhances D ad (see Fig. 15.12b), the
net result being FRET increase, while in the nonradiative channel the enhancement
551
radiative FRET theory [38, 61] to plasmonic systems. In fact, the PERT mechanism
can dominate FRET even in the near field, as our numerical calculations below
demonstrate.
15.2.3 Numerical Results for Plasmon-Assisted Single-Donor
Energy Transfer Near Metal Nanoparticle
As an example, consider a donor and acceptor near spherical Ag NP in water with normal dipole orientations (see Fig. 15.1). The near field matrix D jk is readily obtained
from the Mie’s theory Green’s dyadic [25] as D jk = D 0
jk + D r
jk + D nr
jk , where [13]
D
r
jk = − i
2
3
k
3
1 + 2α 1
1
r 3
j
+
1
r 3
k
+
4|α 1 | 2
r 3
j r 3
k
(ˆ r j · ˆ
r k ),
D
nr
jk = −
l
α l (l + 1) 2
r
l+2
j r
l+2
k
P l (ˆ r j · ˆ
r k )
(15.66)
are NP-induced radiative and nonradiative terms, α l = R 2l+1 l(ε−ε 0 )
lε+(l+1)ε 0
is NP polarizability, P l (x) is a Legendre polynomial, D 0
ad = (1 + sin 2 θ/2)/r 3
ad , ˆ
r a · ˆ
r d = cos θ ,
and angular momenta up to l = 50 are included. Full decay rates are Γ j =
−
3/2k 3 γ r
j D √√
j j . We consider, for simplicity, a high-yield donor with a broad emission band due to the vibrational modes. Molecules optical bands are Lorentzians
of width 0.05 eV centered at 2.95 eV and 3.2 eV with maximal overlap at about
SP energy of 3.08 eV (see inset in Fig. 15.12a), σ a (ω) was normalized to its total
dωσ a (ω), and modified ¯
σ a , ˜
f d , and ˜
D ad were found using Eq. (15.66).
In Fig. 15.11, we plot W ad versus the molecule’s distance d from the R = 30
nm NP surface at θ = π/3 with equal d a = d d = d and with changing d d = d
at fixed d a . Three models—the full Eq. (15.61), its nonradiative part only, and the
Gersten–Nitzan model [62, 63]—are compared to Förster’s transfer Eq. (15.45).
For d d = d a , W ad is about three times larger than W F
ad and rapidly decays with
d, while for d/R ∝ 1 it is quenched by metal. There is no enhancement if only
the nonradiative channel is included in Eq. (15.61). In contrast, the Gersten-Nitzan
model yields much greater enhancement (up to 10 5 ) for d/R ∝ 1 since it includes
no quenching effects. However, at fixed d a and d d /R 1, the full W ad is the largest
one (see Fig. 15.11b) due to the dominant role of the PERT mechanism, as discussed
above.
The interplay of different FRET contributions is shown in Fig. 15.12 featuring
spectral density Eq. (15.59) together with quenching factor γ r
d /Γ d and coupling
|D ad | 2 at fixed d. dW ad /dω has a sharp plasmon peak which disappears if only the
nonradiative channel is included (see Fig. 15.12a). PERT channel reduces γ r
d /Γ d
due to SP-enhanced radiation but it strongly enhances D ad (see Fig. 15.12b), the
net result being FRET increase, while in the nonradiative channel the enhancement