9 Aperiodic Order in Nanoplasmonics
339
the spectrum of this operator, which is determined by the aperiodic geometry of the
arrays for a given shape of the nanoparticles, explains the spectral positions of the
plasmonic pseudo-band gaps in terms of the singularities of the aperiodic Fourier
spectra. This analysis therefore establishes the validity of the gap-labeling theorem in the context of aperiodic nanoplasmonics. Therefore, the demonstration of
controllable plasmonic band-gaps and sub-wavelength localized plasmon fields in
finite-size chains of metal nanoparticles with aperiodic order suggests novel designs
for the implementation of plasmonic devices and functionalities.
9.2.4 Rotational Symmetry: Aperiodic Tilings and Vogel Spirals
One of the main features of aperiodic planar arrays is the ability to encode forbidden
rotational symmetries in their discrete or diffuse diffraction diagrams, displayed by
their Fourier spectra [6, 56]. It was recently discovered that aperiodic tilings can be
constructed with an arbitrary degree of rotational symmetry using a purely algebraic
approach [57]. In addition, deterministic tilings with full rotational symmetry up
to infinite order (i.e., circular symmetry) have been demonstrated [58] by a simple
procedure that iteratively decomposes a triangle into five congruent copies. The
resulting tiling, called Pinwheel tiling, has triangular elements (i.e., tiles) which
appear in infinitely many orientations and, in the limit of arrays with infinite-size,
the diffraction pattern displays continuous (“infinity-fold”) rotational symmetry.
Radin has shown that there are no discrete components in the Pinwheel diffraction
spectrum [58]. However, it is currently unknown if the spectrum is continuous or
singular continuous.
We recently engineered [59] Pinwheel arrays of resonant metallic nanoparticles
and reported on isotropic structural coloration of metal films using homogenized
Pinwheel patterns. In particular, following this approach we demonstrated bright
green coloration of Au films with greatly reduced angular sensitivity and enhanced
spatial uniformity of coloration compared to both periodic and random arrays [59].
In Fig. 9.4, we show three different types of deterministic aperiodic arrays of
particles with increasing degree of rotational symmetry in their diffraction spectra
(Fig. 9.4d–f). In particular, Fig. 9.4a shows a particle array with tenfold rotational
symmetry in the arrangement of its (interior) Bragg peaks. This array is obtained
by positioning particles at the vertices of a planar Penrose tiling. A Danzer particle
array [60] with sevenfold symmetry (Fig. 9.4b) and the Pinwheel array (Fig. 9.4c)
are also shown along with the corresponding diffraction spectra (Fig. 9.4d–f).
We can appreciate in Fig. 9.4 how by increasing the degrees of rotational symmetry the different spectra acquire a more diffuse spectral character. This behavior
is particularly evident in a broad class of finite-size deterministic aperiodic arrays,
known as Vogel’s spirals (Fig. 9.5), whose diffraction spectra do not posses any discrete component and display almost continuous circular symmetry.
Vogel’s structures have been investigated by mathematicians, botanists, and
theoretical biologists [61] in relation to the outstanding geometrical problems of
339
the spectrum of this operator, which is determined by the aperiodic geometry of the
arrays for a given shape of the nanoparticles, explains the spectral positions of the
plasmonic pseudo-band gaps in terms of the singularities of the aperiodic Fourier
spectra. This analysis therefore establishes the validity of the gap-labeling theorem in the context of aperiodic nanoplasmonics. Therefore, the demonstration of
controllable plasmonic band-gaps and sub-wavelength localized plasmon fields in
finite-size chains of metal nanoparticles with aperiodic order suggests novel designs
for the implementation of plasmonic devices and functionalities.
9.2.4 Rotational Symmetry: Aperiodic Tilings and Vogel Spirals
One of the main features of aperiodic planar arrays is the ability to encode forbidden
rotational symmetries in their discrete or diffuse diffraction diagrams, displayed by
their Fourier spectra [6, 56]. It was recently discovered that aperiodic tilings can be
constructed with an arbitrary degree of rotational symmetry using a purely algebraic
approach [57]. In addition, deterministic tilings with full rotational symmetry up
to infinite order (i.e., circular symmetry) have been demonstrated [58] by a simple
procedure that iteratively decomposes a triangle into five congruent copies. The
resulting tiling, called Pinwheel tiling, has triangular elements (i.e., tiles) which
appear in infinitely many orientations and, in the limit of arrays with infinite-size,
the diffraction pattern displays continuous (“infinity-fold”) rotational symmetry.
Radin has shown that there are no discrete components in the Pinwheel diffraction
spectrum [58]. However, it is currently unknown if the spectrum is continuous or
singular continuous.
We recently engineered [59] Pinwheel arrays of resonant metallic nanoparticles
and reported on isotropic structural coloration of metal films using homogenized
Pinwheel patterns. In particular, following this approach we demonstrated bright
green coloration of Au films with greatly reduced angular sensitivity and enhanced
spatial uniformity of coloration compared to both periodic and random arrays [59].
In Fig. 9.4, we show three different types of deterministic aperiodic arrays of
particles with increasing degree of rotational symmetry in their diffraction spectra
(Fig. 9.4d–f). In particular, Fig. 9.4a shows a particle array with tenfold rotational
symmetry in the arrangement of its (interior) Bragg peaks. This array is obtained
by positioning particles at the vertices of a planar Penrose tiling. A Danzer particle
array [60] with sevenfold symmetry (Fig. 9.4b) and the Pinwheel array (Fig. 9.4c)
are also shown along with the corresponding diffraction spectra (Fig. 9.4d–f).
We can appreciate in Fig. 9.4 how by increasing the degrees of rotational symmetry the different spectra acquire a more diffuse spectral character. This behavior
is particularly evident in a broad class of finite-size deterministic aperiodic arrays,
known as Vogel’s spirals (Fig. 9.5), whose diffraction spectra do not posses any discrete component and display almost continuous circular symmetry.
Vogel’s structures have been investigated by mathematicians, botanists, and
theoretical biologists [61] in relation to the outstanding geometrical problems of