One-dimensional Periodic Structure Research Articles

Photonic Bandgap Properties of One-Dimensional Bilayer Periodic Structure Composed of Silicon Dioxide and Gallium Phosphide Material Layers

Photonic Bandgap Properties of One-Dimensional Bilayer Periodic Structure Composed of Silicon Dioxide and Gallium Phosphide Material Layers

Non-reciprocity and selective wave amplification in nonlinear inertia-induced autoparametric periodic structures

Inspired by the spatiotemporally modulated metamaterials, we propose a one-dimensional autoparametric periodic structure induced by nonlinear inertia that is realized simply by constructing an array of pendula coupled to local oscillators. The mechanism of autoparametric modulation is revealed using a unit cell. The evolution equations for a superlattice describing the global dynamics of the wave-vibration coupling are obtained analytically by combining the Floquet-Bloch theorem and the multiple-time scales perturbation technique. The band structure exists corresponding to the fixed point in the phase space, wherein large amplitude local vibrations with distributed traveling phases modulate the inertial properties of the host pendula propagating with a small amplitude signal wave by playing the role of a pump wave. Non-reciprocal dispersion relations are demonstrated both by numerical results and perturbation results with two-order accuracy. Selective wave amplification with two-mode directional emission is observed in a two-port system which is composed of a sandwich composite medium. The present research based on the proposed autoparametric structure paves a way towards building tunable wave manipulators, vibration energy migrators and mechanical amplifiers.

Computation of the dynamic scalar response of large two-dimensional periodic and symmetric structures by the wave finite element method

In the past, the study of periodic media mainly focused on one-dimensional periodic structures (meaning periodic along one direction), on the one hand to determine the dispersion curves linking the frequencies to the wavenumbers and on the other hand to obtain the response of a structure to an external excitation, both for bounded or unbounded structures. In the latter case, effective approaches have been obtained, based on methods such as the Wave Finite Element (WFE). Two-dimensional periodic media are more complex to analyse but dispersion curves can be obtained rather easily as in the one-dimensional case. Obtaining the steady state response of two-dimensional periodic structures to time-harmonic excitations is much more difficult than for one-dimensional media and the results mainly concern infinite media. This work is about this last case of the steady state response of finite two-dimensional periodic structures to time-harmonic excitations by limiting oneself to structures described by a scalar variable (acoustic, thermal, membrane behaviour) and having symmetries compared to two orthogonal planes parallel to the edges of a substructure. Using the WFE for a rectangular substructure and imposing the wavenumber in one direction, we can numerically calculate the wavenumbers and mode shapes associated with propagation in the perpendicular direction. By building solutions with null forces on parallel boundaries, we can decouple the waves in the two directions parallel to the sides of the rectangle. The solution of each of these two problems is obtained by a fast Fourier transformation giving the amplitudes associated with the waves. By summing the contributions of all these waves we obtain the global solution for a two-dimensional periodic medium with a large number of substructures and a low computing time. Examples are given for the case of a two-dimensional membrane with many substructures and different types of heterogeneities.

An array of QDs on a one-dimensional periodic structure of graphene nanoribbons as a nanoscale plasmonic grating

An array of QDs on a one-dimensional periodic structure of graphene nanoribbons as a nanoscale plasmonic grating

A novel efficient method for transient heat analysis of cylindrical periodic structure based on the physical features and group theory

In this paper, a novel efficient and accurate numerical method is developed for analyzing the transient temperature responses of cylindrical periodic structures. By exploiting the structure’s circumferential cyclic periodic property and leveraging group theory, a circumferential decomposition strategy is presented to transform the temperature response analysis of the cylindrical periodic structure into the analyses of a series of one-dimensional periodic structures. Then, based on the physical nature of the transient heat conduction, an axial decomposition strategy is developed to convert the computations of the temperatures of these one-dimensional periodic structures into the calculations of the temperatures of a series of small-scale structures. The computational cost of these small-scale structures is further reduced by using the group theory. Several numerical examples demonstrate that the proposed method has higher accuracy and computational efficiency in comparison with the Crank-Nicolson method. When the Crank-Nicolson method attains the acceptable results, the proposed method is about 10 and 20 times faster than the Crank-Nicolson method with direct and preconditioning conjugate gradient solvers, respectively.

Metamaterials of Auxetic Geometry for Seismic Energy Absorption.

The propagation of earthquake energy occurs primarily through elastic waves. If the seismic force input to a structure can be directly reduced from the source, then the structure can be protected from seismic wave energy. Seismic metamaterials, regarded as periodic structures with properties different from conventional materials, use wave propagation characteristics and bandgaps to dissipate seismic wave energy. When the seismic wave is located in the bandgap, the transmission of seismic wave energy is effectively reduced, which protects the structure from the damage caused by seismic disturbance. In practical application, locating seismic frequencies below ten Hz is a challenge for seismic metamaterials. In the commonly used method, high-mass materials are employed to induce the effect of local resonance, which is not economically feasible. In this study, a lightweight design using auxetic geometry is proposed to facilitate the practical feasibility of seismic metamaterials. The benefits of this design are proven by comparing conventional seismic metamaterials with metamaterials of auxetic geometry. Different geometric parameters are defined using auxetic geometry to determine the structure with the best bandgap performance. Finite element simulations are conducted to evaluate the vibration reduction benefits of auxetic seismic metamaterials in time and frequency domains. Additionally, the relationship between the mass and stiffness of the unit structure is derived from the analytical solution of one-dimensional periodic structures, and modal analysis results of auxetic metamaterials are verified. This study provides seismic metamaterials that are lightweight, small in volume, and possess low-frequency bandgaps for practical applications.

Peculiarities of the Resonant Response of a Subwavelength Double Grating with Optical PT-Symmetry

In this paper, we study the features of the resonant response of a system consisting of two subwavelength one-dimensional periodic structures, considering the dispersion of the refractive index in the presence of optical PT-symmetry for TM polarization. For the considered structure in the green wavelength range, two possible resonance lines were identified at 514.86 nm and 518.5 nm. Ultra-narrow resonances (FWHM of 0.00015 nm) have been obtained for transmitted and reflected waves, and a significant enhancement of the resonant response has been achieved (up to 105 times). The dependence of the system’s optical response on the relative position of its two sub-wavelength gratings and the magnitude of the amplification coefficient of the active part was investigated. This can be used to tune the spectral characteristics of filters, modulate the optical radiation, and create optomechanical sensors such as strain gauges.

Bandgaps of microwave photonic crystals: Study of quasi-periodic metamaterial multilayers

A lossy double-negative index and air are used to investigate one-dimensional periodic and quasi-periodic multilayered structure using the transfer-matrix method. The transmittance of the lossy double-negative index and air quasi-system has been shown using the Fibonacci, Thue–Morse, double-period, Octonacci and Dedocanacci quasi-sequence. For the periodic structure, which is used here as the refereed structure for the comparison, a bandgaps called zero-n¯ appears near the frequency at which the permittivity and permeability of double-negative material are negative. Apart from this, zero-ϵ and zero-μ bandgaps have been found except for normal incidence along with the Bragg bandgaps.

Morphology dependent excitation of hybrid Tamm-bandedge state in metal coated opal three-dimensional photonic crystal

The optical Tamm state is a type of electromagnetic mode that exists at the interface between a periodic dielectric structure and a metal layer. While optical Tamm states have recently gained attention as a potential replacement for surface plasmon polaritons in various applications, most of these demonstrations have been limited to one-dimensional periodic dielectric structures, such as Bragg mirrors. In this article, we present a detailed study on the existence of optical Tamm states at the interface between a metal layer and opal-like three-dimensional photonic crystal (3D-PhC) structures, with a focus on two types of devices with different cross-sectional morphologies. One device consisted of a corrugated silver layer on self-assembled PS opals, while the second one was realized by self-assembling PS opals on a thin flat gold film. Our experimental and numerical observations revealed a dip in the reflection spectra for both configurations, but a hybrid Tamm-PhC bandedge state was found to exist only between the corrugated metal layer and 3D-PhC. Importantly, the nature of the hybrid Tamm-PhC bandedge states was found to strongly depend on the thickness of the metal layer. In contrast, no mode hybridization was observed at the interface between the flat metal and 3D-PhC, and the reflection dip in this case corresponded to a PhC edge mode. Our demonstration highlights the potential for realizing cost-effective and integrated hybrid plasmonic-photonic crystal structures for sensitive nanophotonic sensors.

Wave propagation analysis in laminated composite periodic frame structures using spectral finite element method

In this article, wave propagation in a one-dimensional (1-D) composite periodic frame structure is analyzed using spectral finite elements (SFE) method together with Bloch’s theorem. Each element in the periodic frame structure is modeled as a first-order shear deformation theory (FSDT) beam element and wave equations for the beam elements are derived using FSDT considering axial, flexural, and shear deformations. Next, these equations are solved using the frequency domain SFE method to obtain an elemental dynamic stiffness matrix which on assembly gives the dynamic stiffness matrix for the unit cell of the periodic structure. A polynomial eigenvalue problem is formulated thereafter using Bloch’s theorem to evaluate the dispersion coefficients and wave amplitude ratios. Finally, the entire periodic frame is modeled as homogenized two-noded elements capturing the wave response of the PS with a drastic reduction in computational cost. The results obtained using the present method are validated with finite element simulation using ANSYS mechanical Ansys Parametric Design Language (APDL). The frequency band gaps observed in the frequency domain responses are investigated and corroborated with those predicted by the dispersion relations. This is followed by a detailed parametric study on the influence of different parameters on the band gap characteristics. The present method provides a scheme of wave propagation analysis of composite periodic frames with substantial computational efficiency.

Numerical and experimental analysis of the stiffness and band-gap properties of shell structures with periodically variable cross sections

This paper describes one-dimensional periodic shell structures that have variable cross sections, a new type of periodic shell structures made from photopolymer. This paper will discuss the stiffness of periodic sub-cells that have variable cross sections and the band gaps of Bragg scattering shell structures based on numerical analysis and a series of experiments. This paper uses the Bloch theorem and lumped-mass method to create a band gap model for periodic shell structures. In this paper, an equivalent stiffness model for sub-cells is also created based on the principle of superposition and validated by experiments. Numerical studies and experiments are conducted to examine the effects of geometrical parameters, number of sub-cells, and stiffness of sub-cells on band gaps of one-dimensional periodic shell structures and to test the effectiveness of the models. The findings in this paper prove that by varying the stiffness of sub-cells under a fixed lattice constant, band gaps of one-dimensional periodic shell structures can be decreased. The findings also confirmed that the initial band gap of one-dimensional periodic shell structures can be lowered by increasing the number of sub-cells in a period. Unlike other types of Bragg scattering periodic structures, one-dimensional periodic shell structures allow their longitudinal band gaps to be adjusted under a fixed lattice constant. Those findings serve as a theoretical foundation for the application of Bragg scattering periodic shell structures in low-frequency vibration.

Free Vibration Characteristics of CFRP Laminate with One-Dimensional Periodic Structures.

This paper proposes an approach of stacking prepreg periodically for carbon fiber-reinforced polymer composites (CFRP) laminate. This paper will discuss the natural frequency, modal damping, and vibration characteristics of CFRP laminate with one-dimensional periodic structures. The damping ratio of CFRP laminate is calculated using the semi-analytical method which combines modal strain energy with the finite element method. The finite element method is used to calculate the natural frequency and bending stiffness which are verified with experiments. The numerical results of the damping ratio, natural frequency, and bending stiffness are in good agreement with the experiment results. Finally, the bending vibration characteristics of CFRP laminate with one-dimensional periodic structures and traditional CFRP laminate are investigated with experiments. The finding confirmed that the CFRP laminate with one-dimensional periodic structures exists band gaps. This study provides theoretical support for the promotion and application of CFRP laminate in the field of vibration and noise.

Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends

Intermode interactions in one-dimensional nonlinear periodic structures have been studied by many authors, starting with the classical work by Fermi, Pasta, Ulam, and Tsingou (FPUT) in the middle of the last century. However, symmetry selection rules for the energy transfer between nonlinear vibrational modes of different symmetry, which lead to the possibility of excitation of some bushes of such modes, were not revealed. Each bush determines an exact solution of nonlinear dynamical equations of the considered system. The collection of modes of a given bush does not change in time, while there is a continuous energy exchange between these modes. Bushes of nonlinear normal modes (NNMs) are constructed with the aid of group-theoretical methods and therefore they can exist for the case of large amplitude atomic vibrations and for any type of interatomic interactions. In most publications, bushes of NNMs or similar dynamical objects in one-dimensional systems are investigated under periodic boundary conditions. In this paper, we present a detailed study of the bushes of NNMs in monoatomic chains for the case of fixed boundary conditions, which sheds light on a series of new properties of the intermode interactions in such systems. We prove some theorems that justify a method for constructing bushes of NNMs by continuation of conventional normal modes to the case of large atomic oscillations. Our study was carried out for FPUT chains, for the chains with the Lennard-Jones interatomic potential, as well as for the carbon chains (carbynes) in the framework of the density functional theory. For one-dimensional bushes (Rosenberg nonlinear normal modes), the amplitude–frequency diagrams are presented and the possibility of their modulational instability is briefly discussed. We also argue in favor of the fact that our methods and main results are valid for any monoatomic chain.

A diffraction problem for the biharmonic wave equation in one-dimensional periodic structures

This work concerns the propagation of flexural waves through one-dimensional periodic structures embedded in thin elastic plates. We show that the out-of-plane displacement of the plate only contains the Helmholtz wave component and the modified Helmholtz wave component is not supported when the Navier boundary condition is imposed. An adaptive finite element method with transparent boundary condition is developed for solving the associated boundary value problem. Numerical results show that the method is effective to solve the diffraction grating problem of the biharmonic wave equation.

Size effect on light propagation modulation near band edges in one-dimensional periodic structures

Periodic photonic structures can provide rich modulation in propagation of light due to well-defined band structures. Especially near band edges, light localization and the effect of near-zero refractive index have attracted wide attention. However, the practically fabricated structures can only have finite size, i.e., limited numbers of periods, leading to changes of the light propagation modulation compared with infinite structures. Here, we study the size effect on light localization and near-zero refractive-index propagation near band edges in one-dimensional periodic structures. Near edges of the band gap, as the structureʼs size shrinks, the broadening of the band gap and the weakening of the light localization are discovered. When the size is small, an added layer on the surface will perform large modulation in the group velocity. Near the degenerate point with Dirac-like dispersion, the zero-refractive-index effects like the zero-phase difference and near-unity transmittance retain as the size changes, while absolute group velocity fluctuates when the size shrinks.

Transmittance properties of one-dimensional metamaterial nanocomposite photonic crystal in GHz range

We have theoretically demonstrated and explored the transmittance characteristics of a one-dimensional binary photonic crystal composed of metamaterial (MM) and nanocomposite (NC) layers. The NC layer was designed from silver nanoparticles (Ag-NPs) in a host material as Yttrium oxide (Y2O3). Using the transfer matrix approach (TMM), the optical properties of a one-dimensional binary periodic structure having MM and NC layers in the Giga Hertz (GHz) range were examined. The filling fractions of nanoparticles have been explored to see their effect on the effective permittivity of NC materials. Furthermore, the transmittance properties of the suggested structure were investigated at various incident angles for the transverse electric (TE) polarization. In addition to that, different parameters, such as the thickness of the MM layer, the permittivity of the host dielectric material, the filling fraction, and the thickness of the NC layer are also taken into account. We also discussed the effect of these parameters on the width of the photonic bandgap (PBG). With the optimum values of the optical parameters of NC layer, this research could open the way for better photonic crystal circuits, splitters, switches and others.

Femtosecond Laser Fabrication of Anisotropic Structures in Phosphorus- and Boron-Doped Amorphous Silicon Films.

Femtosecond laser-modified amorphous silicon (a-Si) films with optical and electrical anisotropy have perspective polarization-sensitive applications in optics, photovoltaics, and sensors. We demonstrate the formation of one-dimensional femtosecond laser-induced periodic surface structures (LIPSS) on the surface of phosphorus- (n-a-Si) and boron-doped (p-a-Si) amorphous silicon films. The LIPSS are orthogonal to the laser polarization, and their period decreases from 1.1 ± 0.1 µm to 0.84 ± 0.07 µm for p-a-Si and from 1.06 ± 0.03 to 0.98 ± 0.01 for n-a-Si when the number of laser pulses per unit area increases from 30 to 120. Raman spectra analysis indicates nonuniform nanocrystallization of the irradiated films, with the nanocrystalline Si phase volume fraction decreasing with depth from ~80 to ~40% for p-a-Si and from ~20 to ~10% for n-a-Si. LIPSS' depolarizing effect, excessive ablation of the film between LIPSS ridges, as well as anisotropic crystalline phase distribution within the film lead to the emergence of conductivity anisotropy of up to 1 order for irradiated films. Current-voltage characteristic nonlinearity observed for modified p-a-Si samples may be associated with the presence of both the crystalline and amorphous phases, resulting in the formation of potential barriers for the in-plane carrier transport and Schottky barriers at the electric contacts.

Transmission properties of one-dimensional symmetric ternary structures of metamaterials and dielectric materials with two different configurations

In this work, we have investigated the tunable transmission properties of a one-dimensional ternary periodic structure designed with meta- and dielectric-materials in two different configurations. To the study the optical properties of considered periodic structure, we have used the transfer matrix method. The two configurations of considered periodic structure show tunable transmission properties with change in the type of configuration, number of layers, and incident angle of the electromagnetic wave. It is noted that the broadband width and the number of peaks and transmittance increase with an increase in the incident angle, where the peaks becomes more confined in a narrow region. An increase in number of unit cells reduces the transmission in the peaks. Comparing the order of the symmetry, we find that the transmissions in the peaks are more in the second sequence as compared to the first one, and hence the second one is more suitable for multichannel filters. On the basis of the results, the proposed structures can be used in the large variety of opto-electronic and photonic applications, including multichannel filter, broadband reflectors, and switches.

Synergistic interference effect of femtosecond laser direct writing

With the increasing functional applications of periodic micro/nanostructures, it is necessary to provide simple and fast micromachining methods. It is very common to fabricate one-dimensional periodic structures based on one-step machining. However, it is difficult to rapidly process two- dimensional (2D) structures with multiple periods in one step. In this letter, we proposed a processing method based on laser-induced backward transfer (LIBT) technique, in which the sample is placed in a bound state, resulting in equal-thickness interference. Therefore, laser direct writing (LDW) ablation and interference effect are combined to induce 2D biperiodic microstructures. The effects of laser power and interference pattern on the 2D microstructures are studied. This is a promising method for fabricating 2D multiperiodic micro/nanostructures.

Anisotropic Behaviour and Elastic Moduli of Average Isotropic and Anisotropic Elastic Media

Using matricant method elastic moduli of occasionally heterogeneous isotropic and anisotropic elastic media were received. Anisotropic behaviour and conditions for change in anisotropy of media associated with averaging of one-dimensional periodic structures was determined.