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L. D. Negro et al.
individual transfer matrices:
u N +1
u N
= T N T N −1 T N −2 . . . T 2 T 1
u 1
u 0
= Q
u 1
u 0
(9.3)
Imposing the fixed boundary condition u 0 = u N+1 = 0, which describes an Nparticle chain, we finally obtain that the vibration frequencies of the plasmonic chain
satisfy the following eigenvalue equation:
Q 11 (λ) = 0
( 9 . 4 )
which can readily be solved numerically for an arbitrary aperiodic geometry to yield
the oscillation amplitudes of the dipole chain (i.e., the eigenvectors) and the oscillation frequencies (i.e., the eigenvalues) [32, 53].
In addition, Dal Negro et al. [32, 53] have shown, based on the eigenmode statistics, that it is possible to calculate the Integrated Density of States (IDS) and the
participation ratio of each dipolar eigenstate, which allows to quantify the degree of
spatial localization of the plasmonic modes. Based on this approach, we have computed the pseudo-dispersion λ − k diagram of periodic and Fibonacci-modulated
plasmonic chains of finite size (i.e., 144 nanoparticles) corresponding to longitudinal and transverse oscillation modes, and demonstrated the presence of large spectral
gaps in one-dimensional Fibonacci structures, as summarized in Fig. 9.3. We subsequently extended this transfer matrix approach to the analysis of the spectral and
localization properties of dipolar modes in nanoparticle chains based on Thue–Morse
and Rudin–Shapiro aperiodic sequences [32]. We have shown that the aperiodic modulation of the particle positions leads to the formation of sub-wavelength plasmon
band-gaps in Fibonacci and Thue–Morse structures, while Rudin–Shapiro structures
are characterized by vanishingly small gaps and a singular density of states, akin
to random systems. In addition, we have demonstrated a characteristic power-law
scaling in the localization degree of the eigenstates of all the investigated deterministic aperiodic structures, which is a manifestation of the multifractal nature of their
density of states spectra.
A more general approach for the calculation of dipolar modes and energy spectra of
aperiodic arrays of metal nanoparticles with ellipsoidal shapes, which also includes
their electromagnetic coupling with external fields, was developed by Forestiere
et al. [54]. The equations governing the plasmon oscillations were formulated in
such a way as to highlight the role of the geometrical arrangement of the particles on
one side, the particles shape, the dielectric response, and the incoming field polarization on the other side, enabling the accurate design of aperiodic plasmonic devices
with controlled sub-wavelength gaps and spectral positions of localized states. The
work in Ref. [54] rigorously demonstrates that the spectral and localization properties of dipolar modes in aperiodic chains of resonant nanoparticles are determined
by the mathematical spectral properties of a symmetric, positive-definite operator
relating the electric field along the chain to the electric dipole moments, within the
electric quasi-static approximation. In addition, Forestiere et al. [54] showed that
L. D. Negro et al.
individual transfer matrices:
u N +1
u N
= T N T N −1 T N −2 . . . T 2 T 1
u 1
u 0
= Q
u 1
u 0
(9.3)
Imposing the fixed boundary condition u 0 = u N+1 = 0, which describes an Nparticle chain, we finally obtain that the vibration frequencies of the plasmonic chain
satisfy the following eigenvalue equation:
Q 11 (λ) = 0
( 9 . 4 )
which can readily be solved numerically for an arbitrary aperiodic geometry to yield
the oscillation amplitudes of the dipole chain (i.e., the eigenvectors) and the oscillation frequencies (i.e., the eigenvalues) [32, 53].
In addition, Dal Negro et al. [32, 53] have shown, based on the eigenmode statistics, that it is possible to calculate the Integrated Density of States (IDS) and the
participation ratio of each dipolar eigenstate, which allows to quantify the degree of
spatial localization of the plasmonic modes. Based on this approach, we have computed the pseudo-dispersion λ − k diagram of periodic and Fibonacci-modulated
plasmonic chains of finite size (i.e., 144 nanoparticles) corresponding to longitudinal and transverse oscillation modes, and demonstrated the presence of large spectral
gaps in one-dimensional Fibonacci structures, as summarized in Fig. 9.3. We subsequently extended this transfer matrix approach to the analysis of the spectral and
localization properties of dipolar modes in nanoparticle chains based on Thue–Morse
and Rudin–Shapiro aperiodic sequences [32]. We have shown that the aperiodic modulation of the particle positions leads to the formation of sub-wavelength plasmon
band-gaps in Fibonacci and Thue–Morse structures, while Rudin–Shapiro structures
are characterized by vanishingly small gaps and a singular density of states, akin
to random systems. In addition, we have demonstrated a characteristic power-law
scaling in the localization degree of the eigenstates of all the investigated deterministic aperiodic structures, which is a manifestation of the multifractal nature of their
density of states spectra.
A more general approach for the calculation of dipolar modes and energy spectra of
aperiodic arrays of metal nanoparticles with ellipsoidal shapes, which also includes
their electromagnetic coupling with external fields, was developed by Forestiere
et al. [54]. The equations governing the plasmon oscillations were formulated in
such a way as to highlight the role of the geometrical arrangement of the particles on
one side, the particles shape, the dielectric response, and the incoming field polarization on the other side, enabling the accurate design of aperiodic plasmonic devices
with controlled sub-wavelength gaps and spectral positions of localized states. The
work in Ref. [54] rigorously demonstrates that the spectral and localization properties of dipolar modes in aperiodic chains of resonant nanoparticles are determined
by the mathematical spectral properties of a symmetric, positive-definite operator
relating the electric field along the chain to the electric dipole moments, within the
electric quasi-static approximation. In addition, Forestiere et al. [54] showed that