Cantor Sequence Research Articles

Gamma radiation detector using Cantor quasi-periodic photonic crystal based on porous silicon doped with polymer

The measurement of patients’ dosages of radiation caused by medical diagnostics continues to be challenging. A Cantor sequence photonic crystal structure using porous silicon doped with a polymer of polyvinyl alcohol, carbol fuchsin and crystal violet (DPV) is proposed. The influence rules of geometrical and optical parameters such as the radiation doses, number of periods, porosity of porous layers, incident angle and thickness of layers are investigated using MATLAB based on the transfer matrix method. The transmittance of the Cantor sequence of a defective photonic crystal sensor under different conditions is investigated to select the optimum conditions. The proposed system recorded the accepted sensitivity of 0.265[Formula: see text]nm/Gy, FoM of 6.5[Formula: see text]Gy[Formula: see text], Q of 12,701, RS of [Formula: see text] and LoD of [Formula: see text] for gamma radiation. The suggested detector has simple design, accurate monitoring efficiency and immense potential for gamma radiation sensing.

Tunable and giant spatial Goos–Hänchen shifts in a parity-time symmetric Cantor photonic crystals incorporated with a centered graphene layer

In this study, we present the spectral features of a one-dimensional parity-time symmetric layered structure was composed of two quasi-photonic crystals which submit to the Cantor sequence and a graphene layer is embedded in the center of the quasi-crystals. Exceptional points, reflection and transmission spectra and the spatial Goos-Hänchen (GH) shifts are investigated at two distinct terahertz regions in the presence and absence of the graphene layer and compared them. The effect of the modification of imaginary part of refractive index of constituting gain and loss media are also examined. Our results show that, the proposed structure display giant enhanced GS shifts which are tunable with the chemical potential of embedded graphene layer, while GH shifts are weak in the absence of graphene layer. Results display different value and sign of GH shifts for the zero and nonzero chemical potentials. Very extreme GH shifts are obtained by judicious choice of chemical potential and imaginary value of the refractive index of constituting materials. Our results display that not only the photonic bandgap edge modes, but also bandgap modes can support giant GH shifts at Terahertz frequencies. Functionally, these types of structures are very desirable for designing optoelectronic devices that can be adjusted by the amount of chemical potential.

Comparative Study of Numerical Methods for Solving the Fresnel Integral in Aperiodic Diffractive Lenses

In this work, we present a comparative analysis of different numerical methods to obtain the focusing properties of the zone plates based on Fibonacci and Cantor sequences. The Fresnel approximation was solved numerically in order to obtain the axial irradiance provided by these diffractive lenses. Two different methods were applied. The first one is based on numerical integration, specifically the Simpson integration method and the two-dimensional Gaussian quadrature. The second consisted in the implementation of the Fast Fourier Transform in both one and two dimensions. The axial irradiance of the lenses, the relative error with respect to the analytical solution, and the calculation time required by each method are analyzed and compared. From this analysis it was concluded that the Gauss method presents the best balance between accuracy and computation time. This analysis could be useful to decide the most convenient numerical method to be used for the study of more complex diffractive structures.

Tunable Dual- and Multi-Channel Filter Based on Cantor Photonic Crystals Embedded With Graphene

We theoretically investigate the tunable dual- and multi-channel filter in one-dimensional aperiodic photonic crystals (PCs) embedded with graphene. The dielectrics slabs arrayed alternately submit to the Cantor sequence rule and graphene sheets are local at the interfaces of dielectrics layers. Cantor PCs are fractal structures and support a series of discrete resonant channels in transmission spectra, viz. fractal states. Consequently, dual- and multi-filtering channels could be achieved in the compound systems. The central wavelength of each channel and its value of transmission could be modulating by the chemical potential of grapheme and the incident angle as well. It shows that the dual-channel amplitude of transmission can be varied by over 100% and the multiple channels could be switched on and off by the chemical potential of graphene. This study will be very helpful in designing tunable optical filters.

On the 2-abelian complexity of generalized Cantor sequences

In this paper, we study the generalized Cantor sequence c, which is an ℓ-automatic sequence. We prove that the abelian complexity of the 2-block sequence of c is ℓ-regular if the factor set of the sequence c is mirror invariant. As a consequence, we show that the 2-abelian complexity of a generalized Cantor sequence satisfying certain conditions is ℓ-regular.

Effect of geometrical and physical properties of cantor structure for gas sensing applications

In this article, the effects of geometrical and Physical properties of gyroidal graphene and porous silicon of Cantor sequence terminated by gyroidal layer are studied for Gas sensing applications. The study is investigated in THz frequencies with transverse electric polarized light. Gyroidal graphene layers have good tunable optical properties and can excite Tamm resonance due to the metallic property of graphene in the THz range. The recorded spectral sensitivity of the proposed detector is 2431 THz/RIU when the analyte index of refraction is in the range from 1.000 to 1.002. Our research will be highly useful in the development of a high-sensitivity, multi-parameter Tamm resonance sensor.

Photonic passbands induced by optical fractal effect in Cantor dielectric multilayers.

We investigate the splitting and incorporation of optical fractal states in one-dimensional photonic quasi-crystals. The aperiodic crystals which are composed of two different dielectrics submit to Cantor sequence. Defects in Cantor crystals can greatly enhance the localization of electric field, which induces the optical fractal effect. The number of optical fractal states increases exponentially with the generation number of Cantor sequence. Moreover, the optical fractal characteristics depend on the incident angle of light, of which the fractal states may split/incorporate by modulating the value of incident angle. This study could be utilized for band-pass filters and reflectors.

Investigation of optical density in photonic quasicrystals for filtering applications: Effect of composite layers

The optical density in one-dimensional (1D) photonic quasicrystals, such as the symmetric alternating sequence, Thue–Morse sequence, Fibonacci sequence, double-periodic sequence, and Cantor sequence, is investigated in the framework of the transfer matrix method. In order to describe and measure the transmission of light through a photonic quasicrystal, we will study the optical density, which is an important physical property for designing optical filters and can be used to tune the operational wavelength of optical filters and sensors. The effect of composite layers consisting of small inclusion particles (metal fillers) embedded in a host medium on the optical density is shown and discussed by using the Maxwell Garnett effective medium approximation. The influence of the volume filling fraction of metal nanoparticles on the optical density spectrum of photonic quasicrystals and their spectral characteristics is demonstrated. In other words, we discuss the impact of individual components and their fractions in the composite on the optical density and compare the optical density spectra for different 1D photonic quasicrystals.

Unidirectional Invisibility in PT-Symmetric Cantor Photonic Crystals

In this paper, we investigate the nonreciprocity of reflection in parity-time−symmetric (PT-symmetric) Cantor photonic crystals (PCs). Two one-dimensional PCs abiding by the Cantor sequence are PT-symmetric about the center. The PT symmetry and defect cavities in Cantor PCs can induce optical fractal states which are transmission modes. Subsequently, the left and right reflectionless states are located on both sides of a transmission peak. The invisible effect depends on the incident direction and the invisible wavelength can be modulated by the gain–loss factor. This study has potential applications in tunable optical reflectors and invisible cloaks.

On the boundary sequence of an automatic sequence

In this paper, we prove a conjecture of Chen and Wen that the boundary sequence of an automatic sequence is also automatic. In particular, we study the boundary sequences of the generalized Cantor sequences, and give a complete characterization of the periodic boundary sequences. As an application, for a class of automatic sequences, we prove that their abelian complexities are also automatic.

Fractal spiral zone plate with high-order harmonics suppression.

We extend the concept of fractal spiral zone plates and define a new family of Cantor sequence spiral zone plates (CSSZPs) by removing the interference of high-order harmonics. In this typical design, apart from combining the spiral zone plates and Cantor fractal structure together, the desired physical properties have been realized by using a two-parameter modified sinusoidal apodization window along the azimuthal direction to eliminate the high-order harmonics. Numerical simulation reveals that the intensity of high diffraction orders of the CSSZPs can be effectively suppressed by at least 3 orders of magnitude, while the shapes of the sequences of focused optical vortices surrounding the first primary focal length are maintained, similar to those of the fractal spiral zone plates. The demonstration experiment, based on a spatial light modulator, has been also carried out to confirm the desired characteristics. This new kind of diffractive elements may offer potential alternatives for 3D optical tweezers, optical imaging, and lithography.

The Highest and Lowest Photonic Bandwidths, Absorption Coefficients and Field Localizations among Common 1D Quasiperiodic Structures Containing Graphene and Silicon Dioxide

By employing the transfer matrix method, we report the terahertz spectral properties of graphene induced photonic band gap in four types of common one-dimensional quasiperiodic structures based on double-period, Fibonacci, Thue-Morse and Cantor sequences with approximately the same number of layers. In this paper, a graphene monolayer is sandwiched between two adjacent dielectric slabs. It is found that due to the presence of graphene in each structure, an additional gap (GPBG) is emerged. The upper band edge of GPBG moves to the higher frequencies and gets enlarged, as the chemical potential and incident angle increase. The biggest and the smallest bandwidths belong to the Cantor and double-period sequences. Also, the height of the absorption peak at the GPBG edge goes up as the temperature, graphene thickness and damping factor increase. Moreover, the highest and lowest field localizations exist in the Cantor and Fibonacci structures, respectively. Some other features are presented in the results section.

On the acoustic invisibility

Over the past decade a great attention has been paid to the modeling and simulation of making objects invisible or deaf to noise. The recent results have enlightened the mode of control and handling the invisibility and cloaking to noise using the metamaterials engineered at the subwavelength scale in order to interact with acoustic field in a way that traditional materials do not. The acoustic invisibility is based on the property of acoustic equations to be invariant under a spatial compression that means a manipulation of the material parameters. In this paper, the sound invisibility performance is discussed for spherical cloaks. The original domain consists of an alternation of layers made from piezoelectric ceramics and epoxy resin, following a triadic Cantor sequence. The spatial compression, obtained by applying the concave-down transformation, leads to a metamaterial layer with an inhomogeneous and anisotropic distribution of the parameters.

On the k-abelian complexity of the Cantor sequence

In this paper, we prove that for every integer k≥1, the k-abelian complexity function of the Cantor sequence c=101000101⋯ is a 3-regular sequence.

Optical Fabry–Perot filters using hybrid Periodic, Fibonacci and Cantor photonic structures

In this numerical investigation, the design of new optical Fabry–Perot filters according to hybrid Periodic/Non-Periodic, one-dimensional photonic structures are proposed. The purpose of the paper is to design and optimize new Fabry–Perot filters to be used in optical systems and instruments. The hybrid structures studied are Periodic/[Fibonacci]P/Periodic and Periodic/[Cantor]P/Periodic, where P is the repetitive number of Fibonacci or Cantor sequence. The materials constituting the layers of the hybrid structure are the SiO2 and the TiO2. The transmission spectra are determined using the Transfer Matrix Method (TMM). We show that the number, the position of the transmission peaks, the finesse coefficient (F) and the quality factor (Q) depend on the repetitive number (P), the light-incidence angle (θ) and the Polarization modes (TE and TM). All of these parameters are optimized to obtain a Fabry–Perot filters with high finesse coefficient (F), high quality factor (Q), minimum number of layers and minimum thickness. By comparing with other research works, the hybrid Periodic/Cantor structure is the best one which permits to obtain a Fabry–Perot filter with high Finesse coefficient F=17300. On the other hand, the hybrid Periodic/Fibonacci structure makes it possible to achieve the best high quality factorQ=311600.

Algebraic independence of certain Mahler numbers

In this note, we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue–Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain the algebraic independence of the values of these functions at every non-zero algebraic point in the open unit disk. The proof uses results on Mahler's method.

Enhancement of Light Localization in Hybrid Thue-Morse/Periodic Photonic Crystals

The electric field intensity in one-dimensional (1D) quasiperiodic and hybrid photonics band-gap structures is studied in the present paper. The photonic structures are ordered according to Fibonacci, Thue-Morse, Cantor, Rudin-Shapiro, Period-Doubling, Paper-Folding, and Baum-Sweet sequences. The study shows that the electric field intensity is higher for the Thue-Morse multilayer systems. After that the Thue-Morse structure will be combined with a periodic structure to form a hybrid photonic structure. It is shown that this hybrid system is the best for a strong localization of light. The proposed structures have been modeled using the Transfer Matrix Method.

Self-similar transmission properties of aperiodic Cantor potentials in gapped graphene

We investigate the transmission properties of quasiperiodic or aperiodic structures based on graphene arranged according to the Cantor sequence. In particular, we have found self-similar behaviour in the transmission spectra, and most importantly, we have calculated the scalability of the spectra. To do this, we implement and propose scaling rules for each one of the fundamental parameters: generation number, height of the barriers and length of the system. With this in mind we have been able to reproduce the reference transmission spectrum, applying the appropriate scaling rule, by means of the scaled transmission spectrum. These scaling rules are valid for both normal and oblique incidence, and as far as we can see the basic ingredients to obtain self-similar characteristics are: relativistic Dirac electrons, a self-similar structure and the non-conservation of the pseudo-spin. This constitutes a reduction of the number of conditions needed to observe self-similarity in graphene-based structures, see D\'iaz-Guerrero et al. [D. S. D\'iaz-Guerrero, L. M. Gaggero-Sager, I. Rodr\'iguez-Vargas, and G. G. Naumis, arXiv:1503.03412v1, 2015].

A multilayer sonic film

A non-periodic multilayer film was analyzed to show that, despite its non-periodicity, the film exhibits full band-gaps and localized modes at its interfaces, as well as in the sonic composites. The film consists of alternating layers of two different materials that follow a triadic Cantor sequence. The Cantor structure shows extremely low thresholds for subharmonic generation of ultrasonic waves, compared with homogeneous and periodic structures. The coupling between the extended-mode (phonon) and the localized-mode (fracton) vibration regimes explains the generation of full band-gaps, for which there are no propagating Lamb waves. The large enhancement of the nonlinear interaction results from a more favorable frequency and spatial matching of coupled modes. A full band-gap that excludes Love waves is also analyzed.

The role of fractal aperiodic order in the transmittance, conductance and electronic structure of graphene-based systems

In this work we investigate the peculiar tunneling characteristics of Dirac electrons through quasi-regular or aperiodic single-layer graphene structures. In particular, we have implemented the aperiodic order to the barriers and potential wells arranged according to the Cantor sequence. We deal with two types of quasi-regular systems: the first one Electrostatic Cantor Graphene Structures (ECGSs), structures formed with electrostatic potentials, and the second one Substrate Cantor Graphene Structures (SCGSs), obtained by alternating substrates that can open and non-open, such as SiC and SiO2, an energy bandgap on graphene. We have used the transfer matrix method to compute the transmission probability, linear-regime conductance and spectrum of bound states of both systems, which are calculated for different combinations of the fundamental parameters: starting width, sequence generation number, electron incident energy and electron incident angle. The transmission spectra of ECGSs and SCGSs show self-similar patterns, which are more remarkable when the generation number increases. The conductance of ECGSs becomes smoother as the generation increases. Furthermore, in the case of SCGSs the conductance exhibits self-similar features as a function of the generation number. In general, the conductance of both systems presents an oscillatory behaviour that can be described directly by means of the bound states.