1 Nanoplasmonics: From Present into Future
11
γ
(r )
= 4ε
3/2
d
ω sp R
c
3 ∂Re ε m (ω sp )
∂ω sp
−1
.
(1.16)
Note that for losses not very large (which is the case in the entire plasmonic region
for noble metals), the Kramers-Kronig relations for ε m (ω) predict [30] that
∂Re ε m (ω sp )
∂ω sp
> 0,
(1.17)
which guarantees that γ (r ) > 0 in Eq. (1.16).
Comparing this expression to Eq. (1.3) [see also Eq. (1.49)], we immediately conclude that, in contrast to the internal (radiationless) loss rate γ, the radiative rate is
proportional to the volume of the system (i.e., the number of the conduction electrons
in it), which is understandable. Thus for systems small enough, the radiative rate can
be neglected. The quality factor of the SP resonance is actually defined by the total
decay rate γ (tot) [cf. Eq. (1.5)],
Q =
ω sp
2γ (tot) , γ
(tot)
= γ + γ
(r )
.
(1.18)
Therefore, Q is lower for larger nanoparticles, tending to a constant for small R. To
quantify it, we find a ratio
γ (tot)
γ
= 1 +
4
Im ε m (ω sp )
√
ε d ω sp R
c
3
.
(1.19)
We illustrate behavior of this rate ratio of the total to internal loss, γ (tot)
γ, in
Fig. 1.4. General conclusion is that the radiative loss for silver is not very important for
nanospheres in the true quasistatic regime, i.e., for R < l s ≈ 25 nm but is a dominant
mechanism of loss for R > 30 nm, especially in high-permittivity environments. In
contrast, for gold the radiative loss is not very important in the quasistatic regime
due to the much higher intrinsic losses, except for a case of a relatively high ambient
permittivity, ε d = 5.
Though it is outside of the scope of this chapter, we would like to point out that
there is a general approach to combat radiative losses in relatively large nanoparticles.
This is related to the well-known Fano resonances originally discovered by Ugo Fano
in atomic spectra [68]. These resonances can be described in the following way. In
certain cases of optical excitation, when two quantum paths lead to the same final
quantum state of the system, the resonance peaks have specific asymmetric line
shapes due to the interference of these quantum paths.
An analogous phenomenon is also known in nanoplasmonics and metamaterials
[69–77]. It can be explained in the following way [77]. Apart from bright plasmonic
resonances with high transitional dipole moment, there are also dark ones [78],
which by themselves are not very prominent in optical spectra. However, if a bright
11
γ
(r )
= 4ε
3/2
d
ω sp R
c
3 ∂Re ε m (ω sp )
∂ω sp
−1
.
(1.16)
Note that for losses not very large (which is the case in the entire plasmonic region
for noble metals), the Kramers-Kronig relations for ε m (ω) predict [30] that
∂Re ε m (ω sp )
∂ω sp
> 0,
(1.17)
which guarantees that γ (r ) > 0 in Eq. (1.16).
Comparing this expression to Eq. (1.3) [see also Eq. (1.49)], we immediately conclude that, in contrast to the internal (radiationless) loss rate γ, the radiative rate is
proportional to the volume of the system (i.e., the number of the conduction electrons
in it), which is understandable. Thus for systems small enough, the radiative rate can
be neglected. The quality factor of the SP resonance is actually defined by the total
decay rate γ (tot) [cf. Eq. (1.5)],
Q =
ω sp
2γ (tot) , γ
(tot)
= γ + γ
(r )
.
(1.18)
Therefore, Q is lower for larger nanoparticles, tending to a constant for small R. To
quantify it, we find a ratio
γ (tot)
γ
= 1 +
4
Im ε m (ω sp )
√
ε d ω sp R
c
3
.
(1.19)
We illustrate behavior of this rate ratio of the total to internal loss, γ (tot)
γ, in
Fig. 1.4. General conclusion is that the radiative loss for silver is not very important for
nanospheres in the true quasistatic regime, i.e., for R < l s ≈ 25 nm but is a dominant
mechanism of loss for R > 30 nm, especially in high-permittivity environments. In
contrast, for gold the radiative loss is not very important in the quasistatic regime
due to the much higher intrinsic losses, except for a case of a relatively high ambient
permittivity, ε d = 5.
Though it is outside of the scope of this chapter, we would like to point out that
there is a general approach to combat radiative losses in relatively large nanoparticles.
This is related to the well-known Fano resonances originally discovered by Ugo Fano
in atomic spectra [68]. These resonances can be described in the following way. In
certain cases of optical excitation, when two quantum paths lead to the same final
quantum state of the system, the resonance peaks have specific asymmetric line
shapes due to the interference of these quantum paths.
An analogous phenomenon is also known in nanoplasmonics and metamaterials
[69–77]. It can be explained in the following way [77]. Apart from bright plasmonic
resonances with high transitional dipole moment, there are also dark ones [78],
which by themselves are not very prominent in optical spectra. However, if a bright