15 Cooperative Effects in Plasmonics
537
W ens =
kc
2
Q ens =
kc
2
3
μ=1
Γ r
Γ + 3Γ nr
0 /N λ μ
.
(15.42)
For high-yield (small Γ nr
0 ) emitters, we obtain Eq. (15.1) and hence W ens ∈ 3W . In
contrast, the radiated power,
P ens =
kc
2
3
μ=1
Q μ Γ
r
μ N
kc
2
QΓ
r
= N P,
(15.43)
scales with the ensemble size due to the shorter (by factor N /3) radiative lifetime of
superradiant states.
For low-yield emitters (large Γ nr
0 ), the relation Eq. (15.1) holds only approximately. However, it is evident from comparison of Eqs. (15.36) and (15.42) that here
the relative effect of internal relaxation is much weaker than for usual cooperative
emission. Numerical results for both high-yield and low-yield emitters are presented
below.
Let us now turn to the role of interactions between dipoles in the ensemble. Interactions play critical role in cooperative emission since they introduce a disorder into
system energy spectrum by causing random shifts of individual dipole frequencies
[7, 8, 34]. In the conventional cooperative emission, the main disorder effect is to
split the narrow subradiant peak in the ensemble emission spectra [8]. In the presence of metal nanostructure, radiation of subradiant states is expected to be quenched
by much faster nonradiative losses in the metal. The crucial question is, however,
whether interactions between closely spaced individual dipoles can significantly alter
the structure of collective eigenstates. Below we present the results of our numerical
simulations of cooperative emission fully incorporating both direct and plasmonmediated interactions.
We consider an ensemble of N molecular dyes attached to an Ag spherical particle
with radius R = 20 nm via molecular linkers with approximately same length. The
system is embedded in aqueous solution with dielectric constant ε 0 = 1.77, and
two types of dyes with quantum efficiencies q = 0.3 and q = 0.95 are used in
the calculations. A distinguishing feature of this system is a strong effect of interactions on its geometry [20]. The flexible linker molecules hold the attached dyes
with certain orientation of their dipole moments, so that repulsive inter-molecule
interactions compel the dyes to form a spatially ordered structure on spherical surface. In our simulations, the dyes with normal dipole orientations were located at
the sites of spherical lattice, specifically, fullerenes C20, C32, C60, and C80, and,
in some calculations, we included random deviations from the ideal lattice positions. The system eigenstates are found by numerical diagonalization of self-energy
matrix, Σ jk = Δ jk −
i
2 Γ jk , with its real and imaginary parts given by Eqs. (15.24)
and (15.23), respectively. Calculations were carried at the SP energy of 3.0 eV, the
size-dependent Landau damping was incorporated for all plasmon modes, and NP
537
W ens =
kc
2
Q ens =
kc
2
3
μ=1
Γ r
Γ + 3Γ nr
0 /N λ μ
.
(15.42)
For high-yield (small Γ nr
0 ) emitters, we obtain Eq. (15.1) and hence W ens ∈ 3W . In
contrast, the radiated power,
P ens =
kc
2
3
μ=1
Q μ Γ
r
μ N
kc
2
QΓ
r
= N P,
(15.43)
scales with the ensemble size due to the shorter (by factor N /3) radiative lifetime of
superradiant states.
For low-yield emitters (large Γ nr
0 ), the relation Eq. (15.1) holds only approximately. However, it is evident from comparison of Eqs. (15.36) and (15.42) that here
the relative effect of internal relaxation is much weaker than for usual cooperative
emission. Numerical results for both high-yield and low-yield emitters are presented
below.
Let us now turn to the role of interactions between dipoles in the ensemble. Interactions play critical role in cooperative emission since they introduce a disorder into
system energy spectrum by causing random shifts of individual dipole frequencies
[7, 8, 34]. In the conventional cooperative emission, the main disorder effect is to
split the narrow subradiant peak in the ensemble emission spectra [8]. In the presence of metal nanostructure, radiation of subradiant states is expected to be quenched
by much faster nonradiative losses in the metal. The crucial question is, however,
whether interactions between closely spaced individual dipoles can significantly alter
the structure of collective eigenstates. Below we present the results of our numerical
simulations of cooperative emission fully incorporating both direct and plasmonmediated interactions.
We consider an ensemble of N molecular dyes attached to an Ag spherical particle
with radius R = 20 nm via molecular linkers with approximately same length. The
system is embedded in aqueous solution with dielectric constant ε 0 = 1.77, and
two types of dyes with quantum efficiencies q = 0.3 and q = 0.95 are used in
the calculations. A distinguishing feature of this system is a strong effect of interactions on its geometry [20]. The flexible linker molecules hold the attached dyes
with certain orientation of their dipole moments, so that repulsive inter-molecule
interactions compel the dyes to form a spatially ordered structure on spherical surface. In our simulations, the dyes with normal dipole orientations were located at
the sites of spherical lattice, specifically, fullerenes C20, C32, C60, and C80, and,
in some calculations, we included random deviations from the ideal lattice positions. The system eigenstates are found by numerical diagonalization of self-energy
matrix, Σ jk = Δ jk −
i
2 Γ jk , with its real and imaginary parts given by Eqs. (15.24)
and (15.23), respectively. Calculations were carried at the SP energy of 3.0 eV, the
size-dependent Landau damping was incorporated for all plasmon modes, and NP