Physical nature of critical wave functions in Fibonacci systems

Quasiperiodic metamaterials empowered non-metallic broadband optical absorbers

Realizing a polarization-insensitive broadband optical absorber plays a key role in the implementation of microstructure optoelectrical devices with on-demand functionalities. However, the challenge is that most of these devices involve the constituent metals, thus suffering from poor chemical and thermal stability and a complicated manufacturing process. In addition, the extreme contrast between the negative (metallic) and positive (dielectric) real parts of the constituent permittivities can cause additional problems in the design of structural devices. Based on these facts, this work proposes a design of planar broadband one-dimensional structure based on Fibonacci geometry. Experimental results show that the proposed planar structure exhibits high absorptivity behavior independent of polarization and angle in the wavelength range of 300-1000 nm. The absorptivity remains more than 80% when the incident angle is 60°. This proof-of-concept represents a new strategy for realizing non-metallic broadband optical absorbers with advantages of polarization-independence, low-cost, and wide-field-of-view and paves the way for light manipulation under harsh conditions.

Scaling Law, Confined and Surface Modes in Photonic Fibonacci Stub Structures: Theory and Experiment

We investigate both theoretically and experimentally the properties of electromagnetic waves propagation and localization in periodic and quasi-periodic stub structures of Fibonacci type. Each block constituting the Fibonacci sequence (FS) is composed of an horizontal segment and a vertical stub. The origin of the primary and secondary gaps shown in such systems is discussed. The behaviors and scattering properties of the electromagnetic modes are studied in two geometries, when the FS is inserted horizontally between two semi-infinite waveguides or grafted vertically along a guide. Typical properties of the Fibonacci systems such as the fragmentation of the frequency spectrum, the self-similarity following a scaling law are analyzed and discussed. It is found that certain modes inside these two geometries decrease according to a power law rather than an exponential law and the localization of these modes displays the property of self-similarity around the central gap frequency of the periodic structure where the quasi-periodicity is most effective. Also, the eigenmodes of the FS of different generation order are studied depending on the boundary conditions imposed on its extremities. It is shown that both geometries provide complementary information on the localization of the different modes inside the FS. In particular, in addition to bulk modes, some localized modes induced by both extremities of the system exhibit different behaviors depending on which surface they are localized. The theory is carried out using the Green’s function approach through an analysis of the dispersion relation, transmission coefficient and electric field distribution through such finite structures. The theoretical findings are in good agreement with the experimental results performed by measuring in the radio-frequency range the transmission along a waveguide in which the FS is inserted horizontally or grafted vertically.

Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization

Structural defects are inherent in solids at a finite temperature, because they diminish free energies by growing entropy. The arrangement of these defects may display long-range orders, as occurring in quasicrystals, whose hidden structural symmetry could greatly modify the transport of excitations. Moreover, the presence of such defects breaks the translational symmetry and collapses the reciprocal lattice, which has been a standard technique in solid-state physics. An alternative to address such a structural disorder is the real space theory. Nonetheless, solving 1023 coupled Schrödinger equations requires unavailable yottabytes (YB) of memory just for recording the atomic positions. In contrast, the real-space renormalization method (RSRM) uses an iterative procedure with a small number of effective sites in each step, and exponentially lessens the degrees of freedom, but keeps their participation in the final results. In this article, we review aperiodic atomic arrangements with hierarchical symmetry investigated by means of RSRM, as well as their consequences in measurable physical properties, such as electrical and thermal conductivities.

Connection of the nearest-neighbor spacing distribution and the local box-counting dimension for discrete sets

In a recent work [Phys. Rev. E 97, 030202(R) (2018)10.1103/PhysRevE.97.030202], Sakhr and Nieminen (SN) solved a hypothesis formulated two decades ago, according to which the local box-counting dimension D_{box}(r) of a given energy spectrum, or more generally of a discrete set, should exclusively depend on the nearest-neighbor spacing distribution P(s) of the spectrum (set). SN found analytically this dependence, which led them to obtain closed formulas for the local box-counting dimension of Poisson spectra and of spectra belonging to Gaussian orthogonal, unitary, and symplectic ensembles. Here, first, we present a different derivation of the equation establishing the connection of D_{box}(r) and P(s) using the concept of surrogate spectrum. Although our equation is formally different to the SN result, we prove that both are equivalent. Second, we apply our equation to solve the inverse problem of determining the functional form of P(s) for spectra with real fractal structure and constant box-counting dimension D_{box}, and we find that P(s) should behave as a power-law of the spacing, with an exponent given by -(1+D_{box}). Finally, we present four applications or consequences of this last result: First, we provide a simple algorithm able to generate random fractal spectra with prescribed and constant D_{box}. Second, we calculate D_{box} for the sets given by the zeros of fractional Brownian motions, whose P(s) is known to have a power-law tail. Third, we also study D_{box}(r) for the zeros of fractional Gaussian noises, whose P(s) in known to present fat (but not power-law) tails, and that could be misinterpreted as real fractals. And finally, we present the calculation of D_{box} for the spectra of Fibonacci Hamiltonians, known to have fractal properties, simply by fitting their corresponding P(s) to a power-law without the need of applying a box-counting algorithm.

Omnidirectional reflection from generalized Fibonacci quasicrystals

We determine the optimal thicknesses for which omnidirectional reflection from generalized Fibonacci quasicrystals occurs. By capitalizing on the idea of wavelength- and angle-averaged reflectance, we assess in a consistent way the performance of the different systems. Our results indicate that some of these aperiodic arrangements can largely over-perform the conventional photonic crystals as omnidirectional reflection is concerned.

Fractal spectrum of charge carriers in quasiperiodic graphene structures

In this work we investigate the interaction of charge carriers in graphene with a series of p-n-p junctions arranged according to a deterministic quasiperiodic substitutional Fibonacci sequence. The junctions create a potential landscape with quantum wells and barriers of different widths, allowing the existence of quasi-confined states. Spectra of quasi-confined states are calculated for several generations of the Fibonacci sequence as a function of the wavevector component parallel to the barrier interfaces. The results show that, as the Fibonacci generation is increased, the dispersion branches form energy bands distributed as a Cantor-like set. Besides, for a quasiperiodic set of potential barriers, we obtain the electronic tunneling probability as a function of energy, which shows a striking self-similar behavior for different generation numbers.

Electromagnetic wave propagation in quasi-periodic photonic circuits

We study theoretically and experimentally the properties of quasiperiodic one-dimensional serial loop structures made of segments and loops arranged according to a Fibonacci sequence (FS). Two systems are considered. (i) By inserting the FS horizontally between two waveguides, we give experimental evidence of the scaling behaviour of the amplitude and the phase of the transmission coefficient. (ii) By grafting the FS vertically along a guide, we obtain from the maxima of the transmission coefficient the eigenmodes of the finite structure (assuming the vanishing of the magnetic field at the boundaries of the FS). We show that these two systems (i) and (ii) exhibit the property of self-similarity of order three at certain frequencies where the quasiperiodicity is most effective. In addition, because of the different boundary conditions imposed on the ends of the FS, we show that horizontal and vertical structures give different information on the localization of the different modes inside the FS. Finally, we show that the eigenmodes of the finite FS coincide exactly with the surface modes of two semi-infinite superlattices obtained by the cleavage of an infinite superlattice formed by a periodic repetition of a given FS.

Light transport through the band-edge states of Fibonacci quasicrystals

The propagation of light in nonperiodic quasicrystals is studied by ultrashort pulse interferometry. Samples consist of multilayer dielectric structures of the Fibonacci type and are realized from porous silicon. We observe mode beating and strong pulse stretching in the light transport through these systems, and a strongly suppressed group velocity for frequencies close to a Fibonacci band gap. A theoretical description based on transfer matrix theory allows us to interpret the results in terms of Fibonacci band-edge resonances.