18
M. I. Stockman
(ψ 2 | ψ 1 ) =
V
∂
∂r
ψ
∗
2 (r)
∂
∂r
ψ 1 (r)
d
3 r,
(1.29)
This construct possesses all the necessary and sufficient properties of a scalar product:
it is a binary, Hermitian self-adjoined, and positive-defined operation. It is easy
to show that the eigenfunctions of Eqs. (1.25)–(1.26) are orthogonal. They can be
normalized as
(ϕ n | ϕ m ) = δ nm ,
(1.30)
1.3.2 Inhomogeneous Localization of SPs and Hot Spots of Local
Fields
One of the most fundamental properties of eigenmodes is their localization. By
nature, the SP eigenmodes of small nanoplasmonic systems are localized and nonpropagating. This generally follows from the fact that the eigenproblem (1.25) is real
and has real eigenvalues, implying time-reversal invariance and, consequently, zero
current carried by any eigenmode.
From the early days of nanoplasmonics, there has been keen attention paid to
the localization of SP eigenmodes, because it was immediately clear that absence
of any characteristic wavelength of the localized SPs leads to the possibility of
their concentration in nanoscopic volumes of the space [117, 120, 149]. Many early
publications claimed that the SPs in disordered nanoplasmonics systems, e.g., fractal
clusters, experience Anderson localization [150–156].
However, a different picture of the SP localization, named inhomogeneous localization, has been introduced [78, 157–160]. In this picture of inhomogeneous localization, eigenmodes of very close frequencies with varying degree of localization,
from strongly localized at the minimum scale of the system to delocalized over the
entire nanosystem coexist. This phenomenon of inhomogeneous localization has
been experimentally confirmed recently [161]. The eigenmodes experiencing the
Anderson localization are dark, corresponding to dipole-forbidden transitions, and
thus can only be excited from the near field [78].
A related phenomenon is the formation of hot spots in local fields of nanoplasmonic system that we introduced in Refs. [157, 158, 162, 163]. As characteristic of
the inhomogeneous localization, the energy is localized by different SP eigenmodes
at vastly different scales. However, it is the localization at the minimum scale that
gives the highest local fields and energy density; these tightly-localized modes are
the most conspicuous in the near-field intensity distributions as the hot spots. The
hot spots exist in all kind of nanoplasmonic system but they are especially strongly
pronounced in disordered and aperiodic systems [164].
We will illustrate the hot spots and the inhomogeneous localization of the SP
eigenmodes using the results of the original works that established the phenomena
[157, 158] using plasmonic-metal fractal clusters as objects. The model of these
fractals were the so-called cluster-cluster aggregates (CCA) [165, 166]. In Fig. 1.5,
M. I. Stockman
(ψ 2 | ψ 1 ) =
V
∂
∂r
ψ
∗
2 (r)
∂
∂r
ψ 1 (r)
d
3 r,
(1.29)
This construct possesses all the necessary and sufficient properties of a scalar product:
it is a binary, Hermitian self-adjoined, and positive-defined operation. It is easy
to show that the eigenfunctions of Eqs. (1.25)–(1.26) are orthogonal. They can be
normalized as
(ϕ n | ϕ m ) = δ nm ,
(1.30)
1.3.2 Inhomogeneous Localization of SPs and Hot Spots of Local
Fields
One of the most fundamental properties of eigenmodes is their localization. By
nature, the SP eigenmodes of small nanoplasmonic systems are localized and nonpropagating. This generally follows from the fact that the eigenproblem (1.25) is real
and has real eigenvalues, implying time-reversal invariance and, consequently, zero
current carried by any eigenmode.
From the early days of nanoplasmonics, there has been keen attention paid to
the localization of SP eigenmodes, because it was immediately clear that absence
of any characteristic wavelength of the localized SPs leads to the possibility of
their concentration in nanoscopic volumes of the space [117, 120, 149]. Many early
publications claimed that the SPs in disordered nanoplasmonics systems, e.g., fractal
clusters, experience Anderson localization [150–156].
However, a different picture of the SP localization, named inhomogeneous localization, has been introduced [78, 157–160]. In this picture of inhomogeneous localization, eigenmodes of very close frequencies with varying degree of localization,
from strongly localized at the minimum scale of the system to delocalized over the
entire nanosystem coexist. This phenomenon of inhomogeneous localization has
been experimentally confirmed recently [161]. The eigenmodes experiencing the
Anderson localization are dark, corresponding to dipole-forbidden transitions, and
thus can only be excited from the near field [78].
A related phenomenon is the formation of hot spots in local fields of nanoplasmonic system that we introduced in Refs. [157, 158, 162, 163]. As characteristic of
the inhomogeneous localization, the energy is localized by different SP eigenmodes
at vastly different scales. However, it is the localization at the minimum scale that
gives the highest local fields and energy density; these tightly-localized modes are
the most conspicuous in the near-field intensity distributions as the hot spots. The
hot spots exist in all kind of nanoplasmonic system but they are especially strongly
pronounced in disordered and aperiodic systems [164].
We will illustrate the hot spots and the inhomogeneous localization of the SP
eigenmodes using the results of the original works that established the phenomena
[157, 158] using plasmonic-metal fractal clusters as objects. The model of these
fractals were the so-called cluster-cluster aggregates (CCA) [165, 166]. In Fig. 1.5,