Fibonacci Arrays Research Articles

Palindromes in involutive Fibonacci arrays

Palindromes in involutive Fibonacci arrays

Comparative study of a perovskite solar cell using different aperiodic arrangements of plasmonic nanoparticles

In this research work, the optimized Fibonacci, Thue-Mors, and Gaussian-based aperiodic arrays of plasmonic nanoparticles were studied to improve the performance of thin perovskite solar cells. Here, the effect of different types of deterministic aperiodic plasmonic arrays on the optical properties of perovskite absorbers is analyzed. At first, different polytypes of perovskite material consistent (C6H5C2H4NH3)2PbI4, CH3NH3FeI2Br, CH3NH3FeI3, CH3NH3PbBr3, CH3NH3PbCl3, and CH3NH3PbI3 as the active layer of a solar cell were analyzed. The results show that CH3NH3PbI3 has higher absorption spectra than other cases. Then, different types of metals (Ag, Au, Cu, Al) as the form of periodic arrays are simulated and Ag was selected to form our arrays according to the Fibonacci, Thue-Mors, Gaussian, and periodic array arrangement. To form such arrays, 36 nanoparticles per cell were used and the effect of network constant (closest distance between the nanoparticles) was evaluated. Our results confirm that all cases of the Fibonacci, Thue-Mors, and Gaussian arrangement improve the photocurrent, while the photocurrent of the Thue-Mors arrangement is higher than in other cases. These findings reveal that it is possible to improve the performance of perovskite solar cells using a realistic plasmonic array.

Fibonacci and Lucas Riordan arrays and construction of pseudo-involutions

Riordan arrays, denoted by pairs of generating functions , are infinite lower-triangular matrices that are used as combinatorial tools. In this paper, we present Riordan and stochastic Riordan arrays that have connections to the Fibonacci and modified Lucas numbers. Then, we present some pseudo-involutions in the Riordan group that are based on constructions starting with a certain generating function . We also present a theorem that shows how to construct pseudo-involutions in the Riordan group starting with a certain generating function whose additive inverse has compositional order 2. The theorem is then used to construct more pseudo-involutions in the Riordan group where some arrays have connections to the Fibonacci and modified Lucas numbers. A MATLAB algorithm for constructing the pseudo-involutions is also given.

Combinatorial properties of Fibonacci partial words and arrays

Combinatorial properties of Fibonacci words and Fibonacci arrays play a vital role in formal languages. The origination of partial words came through the behaviour of DNA strands. In this paper, we introduce Fibonacci partial words and Fibonacci partial arrays. We have also established their combinatorial properties such as palindromes and primitivity. Furthermore, we prove that certain properties that are satisfied by Fibonacci words and Fibonacci arrays are not satisfied by Fibonacci partial words and Fibonacci partial arrays if they are not palindromic.

Localized quantum walks in quasi-periodic Fibonacci arrays of waveguides.

New quasi-periodic arrays of waveguides (AWs) constructed with Fibonacci sequences are proposed to realize localized quantum walks (LQWs). The proposed Fibonacci arrays of waveguides (FAWs) are simple and straightforward to make, but have a rich set of properties that are of potential use for applications in quantum communication. Our simulations show that, in contrast with randomly disordered AWs, LQWs in FAWs are highly controllable due to the deterministic disorder nature of quasi-periodic systems. Furthermore, unique LQWs with symmetrical probability distribution can be conveniently realized in the FAWs.

Fibonacci and golden ratio in interdisciplinary teaching

Golden ratio and Fibonacci numbers appear oft en in nature. It is known in Mathematics that any potential of the golden number fi (φ) we can get by addition of the two previous potentials. In biology, for example, it is seen in organism growth, as well as adding a number to of units into the existing population. Sequences of numbers known as Fibonacci's serial can be recognized in many creations in nature, and very oft en among living organism and natural occurrences, there are patterns of three, five or even eight golden spirals. The topic of golden ratio is very appropriate as the model for achieving interdisciplinary segments in teaching. Interdisciplinary teaching opens possibilities towards widening teachers' competencies and contributes to exchanging ideas and experience. Nevertheless, in recent time, it has been insisted on this way of work at schools, teachers are reluctant to realize this kind of taching in practice, because of organisation and the number of classes. Taking into account problems, which teachers face, thematic interdisciplinary seminar Spirals and rabbits has been organised at the Faculty of Biology, aimed primarily at primary school teachers. According to attitudes of the participants, application of Fibonacci's arrays, by their simplicity and being so obvious, can contribute to raising students' interest for school curriculum, and it is a good way of connecting knowledge from different subject and popularization of science in general.

Enhancement of the 1.54 μm Er3+ emission from quasiperiodic plasmonic arrays

Periodic and Fibonacci Au nanoparticle arrays of varying interparticle separations were fabricated on light emitting Er:SiNx films using electron beam lithography. A 3.6 times enhancement of the photoluminescence (PL) intensity accompanied by a reduction in the Er3+ emission lifetime at 1.54 μm has been observed in Fibonacci quasiperiodic arrays and explained with radiating plasmon theory. Our results are further supported by transmission measurements through the Fibonacci and periodic nanoparticle arrays with interparticle separation in the 25–500 nm range. This work demonstrates the potential of quasiperiodic nanoparticle arrays for the engineering of light emitting devices based on the silicon technology.

Dipolar mode localization and spectral gaps in quasi-periodic arrays of ferromagnetic nanoparticles

In this paper we study the spectral, localization, and dispersion properties of the ferromagnetic dipolar modes around a stable, saturated, and spatially uniform equilibrium in quasi-periodically modulated arrays of ferromagnetic nanoparticles based on the Fibonacci sequence. The Fibonacci sequence is the chief example of deterministic quasi-periodic order. The problem is reduced to the study of a linear-generalized eigenvalue equation for a suitable Hermitian operator connected to the micromagnetic effective field, which accounts for the magnetostatic, anisotropy, and Zeeman interactions. The coupling with a weak applied magnetic field, varying sinusoidally in time, is dealt with and the role of the losses is highlighted. By calculating the resonance frequencies and eigenmodes of the Fibonacci arrays we demonstrate the presence of large spectral gaps and strongly localized modes and we evaluate the pseudodispersion diagrams. The magnetization oscillation modes in quasi-periodic arrays of magnetic nanoparticles show, at microwave frequencies, behaviors that are very similar to those shown, at optical frequencies, by plasmon modes in quasi-periodic arrays of metal nanoparticles. The presence of band gaps and strongly localized states in magnetic nanoparticle arrays based on quasi-periodic order may have an impact in the design and fabrication of new microwave nanodevices and magnetic nanosensors.

Light transmission through Fibonacci and periodic sub-wavelength slit arrays

In this paper, the light transmission spectra of quasi-periodic and periodic sub-wavelength slit arrays are studied. The results show that the spectral shape and transmission efficiency depend on the reciprocal vectors as well as the Fourier coefficients. The analysis suggests that the transmission minimum can be attributed to the surface plasmon polaritons (SPP) whereas the transmission maximum is associated with the Fabry–Perot-like resonance.

Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles

In this paper we investigate for the first time the near-field optical behavior of two-dimensional Fibonacci plasmonic lattices fabricated by electron-beam lithography on transparent quartz substrates. In particular, by performing near-field optical microscopy measurements and three dimensional Finite Difference Time Domain simulations we demonstrate that near-field coupling of nanoparticle dimers in Fibonacci arrays results in a quasi-periodic lattice of localized nanoparticle plasmons. The possibility to accurately predict the spatial distribution of enhanced localized plasmon modes in quasi-periodic Fibonacci arrays can have a significant impact for the design and fabrication of novel nano-plasmonics devices.

Fractal feature of localized electronic states in Fibonacci arrays of Aharonov-Bohm rings

The property of electronic transport in the Fibonacci array of ideal one-dimensional Aharonov-Bohm rings is studied utilizing the Landauer formalism and by analyzing the quantity called ``the Fibonacci invariant,'' which is derived from renormalization-group ideas. In contrast to previous studies, our invariant is not independent of the Fibonacci generation number $j$ in a limited sense despite its expression having the same form as the previous ones. Even so, this ``$I$-function,'' which is a $j$-dependent invariant, is shown to preserve its importance in the study of the transport properties of a quasiperiodic system. The line shape of the $I$-function at $j\ensuremath{\ge}15$ exhibits a fractal-like behavior within the transmission rift (i.e., fine transmission gap). This fractal-like behavior of the line shape of the $I$-function is characterized by a scaling law. Self-similarity appears in the trace of the transmission probability, when the scaling index is in good agreement with the scaling index at another $j$.

Magnetotransport in periodic and quasiperiodic arrays of mesoscopic rings

We study theoretically the transmission properties of serially connected mesoscopic rings threaded by a magnetic flux. Within a tight-binding formalism we derive exact analytical results for the transmission through periodic and quasiperiodic Fibonacci arrays of rings of two different sizes. The role played by the number of scatterers in each arm of the ring is analyzed in some detail. The behavior of the transmission coefficient at a particular value of the energy of the incident electron is studied as a function of the magnetic flux (and vice versa) for both the periodic and quasiperiodic arrays of rings having different number of atoms in the arms. We find interesting resonance properties at specific values of the flux, as well as a power-law decay in the transmission coefficient as the number of rings increases, when the magnetic field is switched off. For the quasiperiodic Fibonacci sequence we discuss various features of the transmission characteristics as functions of energy and flux, including one special case where, at a special value of the energy and in the absence of any magnetic field, the transmittivity changes periodically as a function of the system size.

Fibonacci arrays and their two-dimensional repetitions

Notions related to repetitive substructures in two-dimensional arrays are introduced and studied in an attempt to parallel some of the analogous developments already known for strings. In particular, sequences of “Fibonacci arrays” are defined, capable of exhibiting extremal properties in terms of certain repetitive subpatterns called “tandems”. Two types of tandems are considered. For one type, it is shown that the number of occurrences in an m×n Fibonacci array attains the general upper bound of O (m 2n logn) .