9 Aperiodic Order in Nanoplasmonics
333
diffraction diagram”, irrespective of spatial periodicity, thus shifting from direct to
the reciprocal Fourier space the defining aspect of a crystal structure [13]. According
to this new picture, the essential attribute of crystalline order, either periodic or quasiperiodic, is the presence of a discrete diffraction spectrum containing only isolated
Bragg peaks, namely a pure-point spectrum.
It has also been realized after the breakthrough discovery of quasi-crystals and
the fabrication of Fibonacci [22, 23] and Thue–Morse semiconductor heterostructures [30, 31], that physical systems can display singular-continuous energy spectra
featuring an infinite hierarchy of narrow gaps with vanishingly small widths (in the
limit of infinite-size systems), bridging a long standing gap between the theory of
spectral operators and condensed matter physics.
9.2.2 Aperiodic Order and Substitutions
We will now discuss how to manipulate aperiodic order beyond traditional quasicrystals. In optics and electronics, a very efficient algorithmic approach to generate
aperiodic structures with controlled Fourier spectral properties is provided by symbolic substitutions [7, 8, 13, 32]. Substitutions are an essential component of every
recursive symbolic dynamical system formally defined on a finite symbolic alphabet G = (a, b, c, . . .). In physical realizations, each letter in the alphabet can be
associated to a different type of building block (e.g., metal nanoparticle, dielectric
layer, etc). A specific substitution rule λ then replaces each letter in the alphabet
by a finite word, starting from a given letter called an axiom or initiator. An aperiodic sequence is then obtained by iterating the substitution rule λ multiple times
to any desired order, producing a symbolic string of arbitrary length. For instance,
one-dimensional Fibonacci quasicrystal structures can be simply generated by the
iteration of the rule λ F : a ≈ ab, b ≈ a with axiom a, as exemplified by the inflation
process: a ≈ ab ≈ aba ≈ abaab ≈ abaababa ≈ abaababaabaab ≈ . . .
A large number of substitution rules have been explored in the study of deterministic aperiodic optical systems [13, 16, 17], producing 1D structures with all three
primitive spectral measures, as shown in Fig. 9.1.
The diffraction spectrum of a Fibonacci quasicrystal is pure-point, featuring a
countable set of ε-like Bragg peaks at incommensurate intervals. More complex
structures displaying singular-continuous spectra 2 can also be conveniently generated by the symbolic substitution approach [13]. In such media, individual Bragg
peaks are no longer separated by well-defined gaps, but cluster to form “broad bands”
in reciprocal space.
The chief example of a deterministic sequence with a singular-continuous diffraction spectrum is the Thue–Morse sequence [8, 33], which is generated by the
substitution λ TM : a ≈ ab, b ≈ ba. This binary sequence was first studied by
2 Singular-continuous structures support Fourier spectra can be covered by an ensemble of open
intervals with arbitrarily small total length.
333
diffraction diagram”, irrespective of spatial periodicity, thus shifting from direct to
the reciprocal Fourier space the defining aspect of a crystal structure [13]. According
to this new picture, the essential attribute of crystalline order, either periodic or quasiperiodic, is the presence of a discrete diffraction spectrum containing only isolated
Bragg peaks, namely a pure-point spectrum.
It has also been realized after the breakthrough discovery of quasi-crystals and
the fabrication of Fibonacci [22, 23] and Thue–Morse semiconductor heterostructures [30, 31], that physical systems can display singular-continuous energy spectra
featuring an infinite hierarchy of narrow gaps with vanishingly small widths (in the
limit of infinite-size systems), bridging a long standing gap between the theory of
spectral operators and condensed matter physics.
9.2.2 Aperiodic Order and Substitutions
We will now discuss how to manipulate aperiodic order beyond traditional quasicrystals. In optics and electronics, a very efficient algorithmic approach to generate
aperiodic structures with controlled Fourier spectral properties is provided by symbolic substitutions [7, 8, 13, 32]. Substitutions are an essential component of every
recursive symbolic dynamical system formally defined on a finite symbolic alphabet G = (a, b, c, . . .). In physical realizations, each letter in the alphabet can be
associated to a different type of building block (e.g., metal nanoparticle, dielectric
layer, etc). A specific substitution rule λ then replaces each letter in the alphabet
by a finite word, starting from a given letter called an axiom or initiator. An aperiodic sequence is then obtained by iterating the substitution rule λ multiple times
to any desired order, producing a symbolic string of arbitrary length. For instance,
one-dimensional Fibonacci quasicrystal structures can be simply generated by the
iteration of the rule λ F : a ≈ ab, b ≈ a with axiom a, as exemplified by the inflation
process: a ≈ ab ≈ aba ≈ abaab ≈ abaababa ≈ abaababaabaab ≈ . . .
A large number of substitution rules have been explored in the study of deterministic aperiodic optical systems [13, 16, 17], producing 1D structures with all three
primitive spectral measures, as shown in Fig. 9.1.
The diffraction spectrum of a Fibonacci quasicrystal is pure-point, featuring a
countable set of ε-like Bragg peaks at incommensurate intervals. More complex
structures displaying singular-continuous spectra 2 can also be conveniently generated by the symbolic substitution approach [13]. In such media, individual Bragg
peaks are no longer separated by well-defined gaps, but cluster to form “broad bands”
in reciprocal space.
The chief example of a deterministic sequence with a singular-continuous diffraction spectrum is the Thue–Morse sequence [8, 33], which is generated by the
substitution λ TM : a ≈ ab, b ≈ ba. This binary sequence was first studied by
2 Singular-continuous structures support Fourier spectra can be covered by an ensemble of open
intervals with arbitrarily small total length.