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L. D. Negro et al.
quasiperiodic crystals to more disordered “amorphous” materials with diffuse diffraction spectra. Moreover, structures featuring “mixed spectra”, containing both
discrete peaks and a diffuse background, also frequently occur in science and technology [13, 15]. Therefore, the rigid dichotomy between periodic and amorphous
structures is inadequate and needs to be surpassed. More rigorously, non-periodic
structures have been classified according to the nature of their Fourier and energy
spectra, which correspond to mathematical measures [8, 13]. In optics, these spectral measures are often identified with the characteristics of diffraction patterns and
optical mode spectra (i.e., Local Density of States).
According to the Lebesgue’s decomposition theorem [26], any measure can be
uniquely decomposed in terms of three primitive spectral components (or into a mixture of them), namely: pure-point (μ P ), singular continuous (μ SC ), and absolutely
continuous spectral components (μ AC ), with: μ = μ P ∪ μ SC ∪ μ AC . Based on this
result, Maciá Barber [13, 15] recently proposed to classify different types of structures according to a matrix with nine entries, corresponding to all the combinations
of the three fundamental types of spectral measures describing their spatial Fourier
and energy spectra.
Random media are characterized by large structural fluctuations modeled by
continuous (i.e., constant) spatial Fourier spectra. However, their energy spectra
are ideally discrete (i.e., pure-point), since disorder-induced localized states appear
at discrete resonant frequencies where the electronic/optical transport comes to a
halt. On the other hand, the diffraction patterns of periodic structures contain well
defined and sharp (i.e., ε-like) peaks in their spatial Fourier spectra corresponding
to the presence of periodic long-range order. Therefore, the reciprocal Fourier space
of periodic and multi-periodic lattices is discrete (i.e., pure-point), with peaks (i.e.,
Bragg peaks) positioned at rational multiples of primitive reciprocal lattice vectors.
However, their energy or optical transmission spectra consist of continuous functions
describing the different energy bands.
In between these two extremes lies the extremely rich spectral domain of
deterministic aperiodic systems. It was recently realized that the presence of sharp
peaks in the diffraction spectra of materials does not necessarily imply structural
periodicity. In 1984, Dan Shechtman et al. [27] when studying the electron diffraction spectra from certain metallic alloys (Al 6 Mn), discovered sharp diffraction peaks
arranged with icosahedral point group symmetry, which cannot be reconciled with
structural periodicity [6, 27]. However, the sharpness of the measured diffraction
peaks, which indicates the coherence of the spatial interference patterns, turned out
to be comparable with the one of ordinary periodic crystals. Stimulated by these
findings, Levine and Steinhardt promptly formulated the notion of aperiodic crystals
or quasicrystals in a seminal paper titled [28]: “Quasicrystals: a new class of ordered
structures”. The geometry of aperiodic crystals was already anticipated in the seminal work on aperiodic tilings by the mathematician Penrose, who discovered in 1974
the existence of two simple polygonal shapes (i.e., tilings) capable of exactly covering the infinite Euclidean plane without spatial periodicity [29]. In response to these
breakthrough discoveries, the International Union of Crystallography (IUCr) reformulated the concept of crystal structures as “any solid having an essentially discrete
L. D. Negro et al.
quasiperiodic crystals to more disordered “amorphous” materials with diffuse diffraction spectra. Moreover, structures featuring “mixed spectra”, containing both
discrete peaks and a diffuse background, also frequently occur in science and technology [13, 15]. Therefore, the rigid dichotomy between periodic and amorphous
structures is inadequate and needs to be surpassed. More rigorously, non-periodic
structures have been classified according to the nature of their Fourier and energy
spectra, which correspond to mathematical measures [8, 13]. In optics, these spectral measures are often identified with the characteristics of diffraction patterns and
optical mode spectra (i.e., Local Density of States).
According to the Lebesgue’s decomposition theorem [26], any measure can be
uniquely decomposed in terms of three primitive spectral components (or into a mixture of them), namely: pure-point (μ P ), singular continuous (μ SC ), and absolutely
continuous spectral components (μ AC ), with: μ = μ P ∪ μ SC ∪ μ AC . Based on this
result, Maciá Barber [13, 15] recently proposed to classify different types of structures according to a matrix with nine entries, corresponding to all the combinations
of the three fundamental types of spectral measures describing their spatial Fourier
and energy spectra.
Random media are characterized by large structural fluctuations modeled by
continuous (i.e., constant) spatial Fourier spectra. However, their energy spectra
are ideally discrete (i.e., pure-point), since disorder-induced localized states appear
at discrete resonant frequencies where the electronic/optical transport comes to a
halt. On the other hand, the diffraction patterns of periodic structures contain well
defined and sharp (i.e., ε-like) peaks in their spatial Fourier spectra corresponding
to the presence of periodic long-range order. Therefore, the reciprocal Fourier space
of periodic and multi-periodic lattices is discrete (i.e., pure-point), with peaks (i.e.,
Bragg peaks) positioned at rational multiples of primitive reciprocal lattice vectors.
However, their energy or optical transmission spectra consist of continuous functions
describing the different energy bands.
In between these two extremes lies the extremely rich spectral domain of
deterministic aperiodic systems. It was recently realized that the presence of sharp
peaks in the diffraction spectra of materials does not necessarily imply structural
periodicity. In 1984, Dan Shechtman et al. [27] when studying the electron diffraction spectra from certain metallic alloys (Al 6 Mn), discovered sharp diffraction peaks
arranged with icosahedral point group symmetry, which cannot be reconciled with
structural periodicity [6, 27]. However, the sharpness of the measured diffraction
peaks, which indicates the coherence of the spatial interference patterns, turned out
to be comparable with the one of ordinary periodic crystals. Stimulated by these
findings, Levine and Steinhardt promptly formulated the notion of aperiodic crystals
or quasicrystals in a seminal paper titled [28]: “Quasicrystals: a new class of ordered
structures”. The geometry of aperiodic crystals was already anticipated in the seminal work on aperiodic tilings by the mathematician Penrose, who discovered in 1974
the existence of two simple polygonal shapes (i.e., tilings) capable of exactly covering the infinite Euclidean plane without spatial periodicity [29]. In response to these
breakthrough discoveries, the International Union of Crystallography (IUCr) reformulated the concept of crystal structures as “any solid having an essentially discrete