1 Nanoplasmonics: From Present into Future
21
f (fill factor or filling factor). Then we repeat this procedure with other 2 × 2 cells
in that central xz plane. As a result, we arrive at a thin planar layer of thickness 2
grid steps in the y direction and fill factor of f in the central xz plane.
In Fig. 1.6b, we display all of the eigenmodes (SPs) of the above-described RPC
in a plot of oscillator strength F n versus localization length L n . These eigenmodes
are strikingly unusual.
First, there is a large number of eigenmodes with negligible oscillator strengths
F n 10 −5 . Note that the rounding-up relative error in our computations is ∼10 −6 ,
so these eigenmodes’ oscillator strengths do not significantly differ from zero. Such
eigenmodes do not couple to the far-field electromagnetic waves, and they can be
neither observed nor excited from the far-field (wave) zone. We call them dark modes.
They can, however, be excited and observed by NSOM (near-field scanning optical
microscope) type probes in the near-field region. Such eigenmodes are also important
from the computational-mathematical point of view because they are necessary for
the completeness of the eigenmode set.
Second, in Fig. 1.6b, there also are many eigenmodes with relatively large oscillator strengths, F n 10 −4 , which we call luminous or bright modes. These do couple
efficiently to the far-zone fields.
Third, both the luminous and the dark modes have localization radii L n with all
possible values, from zero to one half of the diagonal system size, and with very
little correlation between F n and L n , except for the superlocalized (zero-size) eigenmodes that are all dark. This wide range of L n shows that the Anderson localization
does not occur for most of the modes, including all the luminous modes. Similar to
these findings in certain respects, deviations from the simple Anderson localization
have been seen in some studies of the spatial structure of vibrational modes [172,
173], dephasing rates [174] in disordered solids induced by long-range (dipole- type)
interactions. A direct confirmation of this picture of the inhomogeneous localization
has been obtained in experiments studying fluctuations of the local density of states
of localized SPs on disordered metal films [161].
To gain more insight, we show in Fig. 1.7 the local electric field intensities |E n (r)| 2
for particular eigenmodes of four extreme types, all with eigenvalues very close to
s n = 0.2. As a measure of the eigenmode oscillator strength, we show a normalized
oscillator strength F n . The data of Fig. 1.7 confirm the above-discussed absence of
correlation between the localization length and oscillator strength, and also show
that there is no correlation between the topology of the local field intensity and
the oscillator strength—compare the pairs of eigenmodes: s n = 0.1996 with s n =
0.2015, and s n = 0.2 with s n = 0.2011. Note that the large and random changes
of the intensities between the close eigenmodes evident in Fig. 1.7 is an underlying
cause of the giant fluctuations [175] and chaos [157–159] of local fields.
A fundamental property of the SP eigenmodes, whether localized or delocalized,
is that they may be thought of as consisting of hot spots. While the localized eigenmodes possess a single tight hot spot, the delocalized ones consist of several or many
host spots. Note that the fields in the hot spots constituting a single eigenmode are
coherent. In a sense, the hot spots are somewhat analogous to speckles produced by
laser light scattered from a random system. However, such speckles are limited by
21
f (fill factor or filling factor). Then we repeat this procedure with other 2 × 2 cells
in that central xz plane. As a result, we arrive at a thin planar layer of thickness 2
grid steps in the y direction and fill factor of f in the central xz plane.
In Fig. 1.6b, we display all of the eigenmodes (SPs) of the above-described RPC
in a plot of oscillator strength F n versus localization length L n . These eigenmodes
are strikingly unusual.
First, there is a large number of eigenmodes with negligible oscillator strengths
F n 10 −5 . Note that the rounding-up relative error in our computations is ∼10 −6 ,
so these eigenmodes’ oscillator strengths do not significantly differ from zero. Such
eigenmodes do not couple to the far-field electromagnetic waves, and they can be
neither observed nor excited from the far-field (wave) zone. We call them dark modes.
They can, however, be excited and observed by NSOM (near-field scanning optical
microscope) type probes in the near-field region. Such eigenmodes are also important
from the computational-mathematical point of view because they are necessary for
the completeness of the eigenmode set.
Second, in Fig. 1.6b, there also are many eigenmodes with relatively large oscillator strengths, F n 10 −4 , which we call luminous or bright modes. These do couple
efficiently to the far-zone fields.
Third, both the luminous and the dark modes have localization radii L n with all
possible values, from zero to one half of the diagonal system size, and with very
little correlation between F n and L n , except for the superlocalized (zero-size) eigenmodes that are all dark. This wide range of L n shows that the Anderson localization
does not occur for most of the modes, including all the luminous modes. Similar to
these findings in certain respects, deviations from the simple Anderson localization
have been seen in some studies of the spatial structure of vibrational modes [172,
173], dephasing rates [174] in disordered solids induced by long-range (dipole- type)
interactions. A direct confirmation of this picture of the inhomogeneous localization
has been obtained in experiments studying fluctuations of the local density of states
of localized SPs on disordered metal films [161].
To gain more insight, we show in Fig. 1.7 the local electric field intensities |E n (r)| 2
for particular eigenmodes of four extreme types, all with eigenvalues very close to
s n = 0.2. As a measure of the eigenmode oscillator strength, we show a normalized
oscillator strength F n . The data of Fig. 1.7 confirm the above-discussed absence of
correlation between the localization length and oscillator strength, and also show
that there is no correlation between the topology of the local field intensity and
the oscillator strength—compare the pairs of eigenmodes: s n = 0.1996 with s n =
0.2015, and s n = 0.2 with s n = 0.2011. Note that the large and random changes
of the intensities between the close eigenmodes evident in Fig. 1.7 is an underlying
cause of the giant fluctuations [175] and chaos [157–159] of local fields.
A fundamental property of the SP eigenmodes, whether localized or delocalized,
is that they may be thought of as consisting of hot spots. While the localized eigenmodes possess a single tight hot spot, the delocalized ones consist of several or many
host spots. Note that the fields in the hot spots constituting a single eigenmode are
coherent. In a sense, the hot spots are somewhat analogous to speckles produced by
laser light scattered from a random system. However, such speckles are limited by