Two-dimensional Periodic Structures Research Articles

Study on In-Plane Band Gap Characteristics of a Circular Periodic Structure Using DQM

A new two-dimensional periodic structure is proposed. This structure consists of curved beams with different radii of curvature and lengths spirally connected to each other to form a circle. Geometrical parameter effects on the first three in-plane vibration band gaps of this structure are studied using the differential quadrature method. Results show that for each set of the radii of curvature, as the lengths of inner beam elements increase, all the bands become close to each other. These close band gaps can be considerably enlarged by increasing the difference between the radii of curvature. Having close and wide band gaps means that this structure absorbs in-plane vibrations over a very wide frequency range. The dimensions of this structure are much smaller than other periodic structures due to its unique shape. All the mentioned features make this periodic structure an efficient vibration absorber. Validation of the analytical results is provided through the forced vibration analysis via ANSYS.

Generating optical vortex beams by momentum-space polarization vortices centred at bound states in the continuum

Optical vortices, beams with spiral wavefronts and screw phase dislocations, have been attracting increasing interest in various fields. Here, we theoretically propose and experimentally realize an easy approach to generating optical vortices. We leverage the inherent momentum-space topological vortex-like response of polarization (strong polarization anisotropy) around bound states in the continuum of two-dimensional periodic structures, for example photonic crystal slabs, to induce Pancharatnam–Berry phases and spin–orbit interaction in the beams. This new class of optical vortex generators operates in momentum space, meaning that the structure is almost homogeneous without a real-space centre. In principle, any even-order optical vortex that is a diffraction-resistant high-order quasi-Bessel beam can be achieved at any desired working wavelength. The proposed approach expands the application of bound states in the continuum and topological photonics. Optical vortices can be generated by applying the winding behaviour of resonances in the momentum space of a photonic crystal slab, which naturally exists and is associated with bound states in the continuum, to modify the phase front of a beam.

High Refractive Index Electromagnetic Devices in Printed Technology Based on Glide-Symmetric Periodic Structures

We demonstrate the beneficial effects of introducing glide symmetry in a two-dimensional periodic structure. Specifically, we investigate dielectric parallel plate waveguides periodically loaded with Jerusalem cross slots in three configurations: conventional, mirror- and glide-symmetric. Out of these three configurations, it is demonstrated that the glide-symmetric structure is the least dispersive and has the most isotropic response. Furthermore, the glide-symmetric structure provides the highest effective refractive index, which enables the realization of a broader range of electromagnetic devices. To illustrate the potential of this glide-symmetric unit cell, a Maxwell fish-eye lens is designed to operate at 5 GHz. The lens is manufactured in printed circuit board technology. Simulations and measurements are in good agreement and a measured peak transmission coefficient of −0.5 dB is achieved.

Perturbation theories for symmetry-protected bound states in the continuum on two-dimensional periodic structures

On dielectric periodic structures with a reflection symmetry in a periodic direction, there can be antisymmetric standing waves (ASWs) that are symmetry-protected bound states in the continuum (BICs). The BICs have found many applications, mainly because they give rise to resonant modes of extremely large quality-factors ($Q$-factors). The ASWs are robust to symmetric perturbations of the structure, but they become resonant modes if the perturbation is non-symmetric. The $Q$-factor of a resonant mode on a perturbed structure is typically $O(1/\delta^2)$ where $\delta$ is the amplitude of the perturbation, but special perturbations can produce resonant modes with larger $Q$-factors. For two-dimensional (2D) periodic structures with a 1D periodicity, we derive conditions on the perturbation profile such that the $Q$-factors are $O(1/\delta^4)$ or $O(1/\delta^6)$. For the unperturbed structure, an ASW is surrounded by resonant modes with a nonzero Bloch wave vector. For 2D periodic structures, the $Q$-factors of nearby resonant modes are typically $O(1/\beta^2)$, where $\beta$ is the Bloch wavenumber. We show that the $Q$-factors can be $O(1/\beta^6)$ if the ASW satisfies a simple condition.

Perforation-rotation based approach for band gap creation and enlargement in low porosity architected materials

Artificially designed Perforated architected materials (PAMs) have attracted growing attention in recent years due to its relatively simple fabrication process and unusual physical properties for potential applications. Despite numerous works on configuration optimization, few studies have explored the role of perforation orientation on band gap structures of PAMs. In this study, based on adjusting the orientation of holes we propose a novel design approach for low porosity PAMs, which can effectively create and enlarge the band gaps. Both experiments and numerical analyses all demonstrate that the perforation rotation in a two-dimensional periodic structure of mutually orthogonal rectangular holes can be manipulated to form band gaps. Compared with the baseline configuration, a new wide band gap with the gap-to-midgap ratio up to 81% is achieved in the revolved configuration at a small pore volume fraction. This band gap can be created and enlarged by breaking of structural symmetry and the resulted destructive interferences modify the mode shapes and frequencies of the periodic architected materials. This approach may open promising avenues for designing metamaterials with broadband wave attenuation in low porosity.

A meshless collocation method for band structure simulation of nanoscale phononic crystals based on nonlocal elasticity theory

In this paper, the band structures of nanoscale phononic crystals based on the nonlocal elasticity theory are calculated by using a meshfree local radial basis function collocation method (LRBFCM). The direct method is applied to enhance the stability of the derivative calculations in the LRBFCM. A simple summation in the LRBFCM is proposed to deal with the integration related to the nonlocal stresses or tractions. The LRBFCM for the band structure calculations is validated by the results obtained with the first-principle and the transfer matrix (TM) method for one-dimensional (1D) phononic crystals, as well as the comparison of the frequency responses of the two-dimensional (2D) periodic structures.

Spectral variational multiscale model for transient dynamics of phononic crystals and acoustic metamaterials

In this manuscript, a spectral multiscale model is developed for transient elastic wave propagation in periodic composites. The basis of the multiscale approach is the variational multiscale enrichment method that permits analysis of wave propagation in the scale-inseparable condition, i.e., short wave propagation. A spectral coarse-scale representation is proposed to capture the salient transient wave phenomena, such as wave dispersion and band gaps that occur in the short wavelength regime. A material-phase-based model basis reduction strategy is devised at the fine scale to achieve computational efficiency. An important feature of the proposed method is that it does not rely on the classical assumption of separation of scales, which permits its application to a broad range of architectured composites including phononic crystals and acoustic metamaterials. Transient elastic wave propagation in two-dimensional periodic structures is investigated. The proposed multiscale approach is verified against direct numerical simulations. The accuracy and computational efficiency of the approach are demonstrated for both phononic crystals and acoustic metamaterials.

On-chip wavefront shaping with dielectric metasurface

Metasurfaces can be programmed for a spatial transformation of the wavefront, thus allowing parallel optical signal processing on-chip within an ultracompact dimension. On-chip metasurfaces have been implemented with two-dimensional periodic structures, however, their inherent scattering loss limits their large-scale implementation. The scattering can be minimized in single layer high-contrast transmitarray (HCTA) metasurface. Here we demonstrate a one-dimensional HCTA based lens defined on a standard silicon-on-insulator substrate, with its high transmission (<1 dB loss) maintained over a 200 nm bandwidth. Three layers of the HCTAs are cascaded for demonstrating meta-system functionalities of Fourier transformation and differentiation. The meta-system design holds potential for realizing on-chip transformation optics, mathematical operations and spectrometers, with applications in areas of imaging, sensing and quantum information processing.

Harnessing inclusions to tune post-buckling deformation and bandgaps of soft porous periodic structures

The emergence of phononic crystals paves a new way for manipulating elastic waves in structures for their particular bandgap properties. In this paper, the two-dimensional soft porous periodic structures that can be filled with hard inclusions are considered. Both numerical simulations and experiments are conducted to study the effects of inclusions on the buckling modes, post-buckling deformations, and band structures in soft porous periodic structures. It is found that either the number or the arrangement (i.e. filling pattern) of the inclusions has a great influence on the bandgap characteristics. Meanwhile, the material damping affects the wave propagation in soft phononic crystals significantly in the high frequency range. Compared with the unfilled soft porous structure, the sensitivity of the post-buckling deformation to the initial geometrical imperfections can be significantly reduced for the structure filled with inclusions. This means the post-buckling deformation could develop robustly. Further numerical study indicates that the bandgaps can be tuned in a versatile and reversible way when the structure undergoes a large deformation. A more fruitful manner to tune the bandgaps therefore can be achieved by changing the filling pattern of inclusions along with dramatically deforming the structure. The work provides a useful reference for the design of tunable phononic switches and acoustic filters.

The flow of a viscous fluid with a predetermined pressure gradient through periodic structures

In the Stokes approximation, the problem of viscous fluid flow through two-dimensional and three-dimensional periodic structures is solved. A system of thin plates of a finite width is considered as a two-dimensional structure, and a system of thin rods of finite length is considered as a three-dimensional structure. Plates and rods are periodically located in space with certain translation steps along mutually perpendicular axes. On the basis of the procedure developed earlier, the authors constructed an approximate solution of the equations for fluid flow with an arbitrary orientation of structures relative to a given vector of pressure gradient. The solution is sought in a finite region (cells) around inclusions in the class of piecewise smooth functions that are infinitely differentiable in the cell, and at the cell boundaries they satisfy the continuity conditions for velocity, normal and tangential stresses. Since the boundary value problem for the Laplace equation is solved, it is assumed that the solution found is unique. The type of functions allows us to separate the variables and to reduce the problem's solution to the solution of ordinary differential equations. It is found that the change in the flow rate of a fluid through a characteristic cross section is determined mainly by the geometric dimensions of the cells of the free liquid in such structures and is practically independent of the size of the plates or rods.

Band gap characteristics of flexural wave of two-dimensional periodic frame structure composed of locally resonant composite beam

Abstract In this paper, single material beams as the basic structural members in the traditional periodic frame structure were extended to the ones of multi-material composite beam with locally resonant (LR) unit. Based on the spectral element method and the topological characteristics of structure, an assembly method of the stiffness matrix for virtual full component model is presented. This method can be applied not only to the periodic structures, but also to the aperiodic structures, because the whole stiffness matrix of the structure is assembled by its topological information. The effects of the rest design parameters on the band gap behavior of two-dimension (2D) periodic frame structure were studied when the lattice constant was fixed. The results show that the Bragg band gaps can be adjusted by changing the lengths of structural members and the length ratio of the subunit A to the structural members. The results also show that the LR band gaps can be adjusted by adding LR units with different parameters into different members. The results in this work enrich and expand the modulatory pathway of the band gaps of periodic frame structure, and provide a modified approach to the design and correction of the passive vibration suppression system.

Tunable two-dimensional optical filter based on guided-mode resonance.

In this paper, a tunable, two-dimensional periodic structure based on guided-mode resonance (GMR) is proposed. This element can be employed as a tunable optical filter or a display pixel. Resonance tuning in this device is accomplished through perturbation in the refractive index profile and element's thickness, conceptually by a micro-electro-mechanical system mechanism. Simulation results show that resonance wavelength tunings of ∼48 nm (522-570nm) and ∼17 nm (526-543nm) are achieved by physical movement of 166nm horizontally and 300nm vertically, respectively. The proposed device can fairly maintain its tunability under ±5° deviations in the angle of incidence.

Observations of OH airglow from ground, aircraft, and satellite: investigation of wave-like structures before a minor stratospheric warming

Abstract. In January and February 2016, the OH airglow camera system FAIM (Fast Airglow Imager) measured during six flights on board the research aircraft FALCON in northern Scandinavia. Flight 1 (14 January 2016) covering the same ground track in several flight legs and flight 5 (28 January 2016) along the shoreline of Norway are discussed in detail in this study. The images of the OH airglow intensity are analysed with a two-dimensional FFT regarding horizontal periodic structures between 3 and 26 km horizontal wavelength and their direction of propagation. Two ground-based spectrometers (GRIPS, Ground-based Infrared P-branch Spectrometer) provided OH airglow temperatures. One was placed at ALOMAR, Northern Norway (Arctic Lidar Observatory for Middle Atmosphere Research; 69.28∘ N, 16.01∘ E) and the other one at Kiruna, northern Sweden (67.86∘ N, 20.24∘ E). Especially during the last third of January 2016, the weather conditions at Kiruna were good enough for the computation of nightly means of gravity wave potential energy density. Coincident TIMED-SABER (Thermosphere Ionosphere Mesosphere Energetics Dynamics–Sounding of the Atmosphere using Broadband Emission Radiometry) measurements complete the data set. They allow for the derivation of information about the Brunt–Väisälä frequency and about the height of the OH airglow layer as well as its thickness. The data are analysed with respect to the temporal and spatial evolution of mesopause gravity wave activity just before a minor stratospheric warming at the end of January 2016. Wave events with periods longer (shorter) than 60 min might mainly be generated in the troposphere (at or above the height of the stratospheric jet). Special emphasis is placed on small-scale signatures, i.e. on ripples, which may be signatures of local instability and which may be related to a step in a wave-breaking process. The most mountainous regions are characterized by the highest occurrence rate of wave-like structures in both flights.

Regulation of the sensitivity of measurements in holographic interferometry of dynamic periodic structures

This paper discusses the features of the recording of a hologram of a periodic structure that make it possible when implementing a method of either real-time or double-exposure holographic interferometry to adjust the measurement sensitivity when investigating the temporal behavior of the periodic structure of an object. Experimental test results of the real-time holographic interferometry method, devised while studying the dynamics of two-dimensional periodic structures in time, are presented.

Transmission properties of metal mesh filters at 90 GHz

Freestanding frequency-selective surfaces with cross-shaped aperture elements are investigated numerically and experimentally. Transmission characteristics of such two-dimensional periodic structures as functions of geometrical sizes are analyzed using the 3D finite element method. These structures find application in modern microwave and terahertz engineering as band-pass metal mesh filters. Utilizing the conjugate gradient method narrowband, 90 GHz filters have been developed and optimized numerically. The results of measurements carried out in the present study agree well with the obtained theoretical data. It is shown that some parameters of such metal mesh filters can be improved by the proper selection of structure periodicity. A single-layer filter with a fractional bandwidth of 5.3% and an insertion loss of − 0.5 dB at the central frequency is demonstrated.

Triangular nanoperforation and band engineering of InGaAs quantum wells: a lithographic route toward Dirac cones in III–V semiconductors

The design of two-dimensional periodic structures at the nanoscale has renewed attention for band structure engineering. Here, we investigate the nanoperforation of InGaAs quantum wells epitaxially grown on InP substrates using high-resolution e-beam lithography and highly plasma based dry etching. We report on the fabrication of a honeycomb structure with an effective lattice constant down to 23 nm by realising triangular antidot lattice with an ultimate periodicity of 40 nm in a 10 nm thick InGaAs quantum well on a p-type InP. The quality of the honeycomb structures is discussed in detail, and calculations show the possibility to measure Dirac physics in these type of samples. Based on the statistical analysis of the fluctuations in pore size and periodicity, calculations of the band structure are performed to assess the robustness of the Dirac cones with respect to distortions of the honeycomb lattice.

A regularized adaptive spatial resolution technique for fast and accurate analysis of metal–dielectric crossed gratings

The application of regularized jump points has been proposed [Khavasi and Mehrany in Opt Commun 284(13):3211–3215, 6] as an adaptive spatial resolution (ASR) technique to improve the convergence of the Fourier modal method (FMM) in the analysis of one-dimensional periodic structures. We extend this approach to obtain the optical response of metal–dielectric crossed gratings, two-dimensional periodic structures with a high-contrast permittivity distribution. A metallic grid and a slab of split-ring resonators are analyzed using the proposed ASR technique, and the results compared with those obtained using the conventional ASR technique and FMM. The results show that, by adequately choosing the discontinuity points of the dielectric function in the new coordinates, the convergence speed and accuracy of the proposed ASR technique are further increased compared with the conventional ASR technique.

Evaluating resonances in PCSEL structures based on modal indices

The frequently sought after combination of characteristics in semiconductor lasers of high power together with narrow beam divergence and monochromatic output is usually difficult to attain. The photonic crystal surface emitting laser (PCSEL) is one category of device, however, which tends to provide the above-mentioned desirable output features. The PCSEL uses a large area optically active surface but with a two-dimensional periodic structure that enables it to generate high power in a narrow vertically emitted beam yet maintaining single wavelength operation. A primary requirement to model PCSELs is to obtain the optical field resonances that identify the lasing mode. This study presents an alternative method for evaluating the resonances, based essentially on the transfer-matrix technique and wave propagation in multilayer medium, which is relatively easy to formulate, and has quite modest demands on computing requirements.

Diffraction from two-dimensional orthogonal nonseparable periodic structures: Talbot distance dependence on the number theoretic properties of the structures.

In this work, the diffraction-based discrimination of two-dimensional (2D) orthogonal separable and nonseparable periodic structures and prediction of the reduced Talbot distances for 2D orthogonal nonseparable periodic structures are presented. 2D orthogonal periodic structures are defined and classified into separable (multiplicative or additive) and nonseparable categories with the aid of a spatial spectrum lattice. For both the separable and nonseparable cases, the spatial spectra or far-field impulses are 2D orthogonal lattices. We prove that for a 2D orthogonal separable structure, in addition to the DC impulse, there are other impulses on the coordinate axes. As a result, if all the spectrum impulses of a structure on the coordinate axes, except for the DC impulse, vanish, we conclude that the structure is nonseparable. In the second part of this work, using a unified formulation, the near-field diffraction of the 2D orthogonal separable and nonseparable periodic structures is investigated. In general, the Talbot distance equals the least common multiple of the individual Talbot distances in the orthogonal directions, say, z t =z lcm . For the 2D orthogonal nonseparable periodic structures having Fourier coefficients only with odd indices, we have found surprising results. It is shown that for this kind of structure, the Talbot distance strongly depends on the number theoretic properties of the structure. Depending on the ratio of the structure's periods in the orthogonal directions, pxpy, the Talbot distance reduces to z lcm 2, z lcm 4, or z lcm 8. In addition, for the 2D orthogonal nonseparable sinusoidal grating, we show that, regardless of the value of pxpy, self-images are formed at distances smaller than the conventional Talbot distances attributed to px and py that we name the reduced Talbot (RT) distances. Halfway between two adjacent RT distances, the formation of negative self-images with a complementary amplitude of the self-images is predicted. Halfway between two adjacent self-image and negative-elf-image, subimages are formed. As another interesting result, we show that the intensity patterns of the subimages are 2D multiplicatively separable with halved periods in both directions. Finally, we show that 2D almost periodic structures with impulses on zone-plate-like concentric circles have self-images under plane wave illumination.

Sound transmission in two-dimensional periodic poroelastic structures

To obtain excellent sound reduction performance, in this paper we introduce a novel periodic poroelastic composite structure, which combines poroelastic material and periodic structure and aims at using the remarkable acoustic performance of these two. This periodic composite structure comprises three parts, i.e. the poroelastic domain, the elastic domain (thin plate), and the periodic resonators, which can be simple single-degree-of-freedom resonators (SRs) or composite two-degree-of-freedom resonators (CRs). A theoretical model is established by using Biot theory for the poroelastic domain, and by using the effective medium method for the resonator-plate coupling system, which is considered as an isotropic plate with an equivalent dynamic density. This method is validated with degenerated model in the literature; the results obtained by this method are in excellent consistence with the results in the literature. Parameter analyses are performed to test the influences of poroelastic addition and periodic resonator on the sound transmission loss (STL) of this periodic composite structure under two kinds of boundary conditions. The poroelastic addition is found to increase the STL while the influences of resonators are complicated. The STL increases notably in the frequency range bounded by the characteristic frequencies of these resonators, however, a decrease just follows when it exceeds these frequencies, which can be observed in both SR case and CR case under the two boundary conditions. In the meantime, when multiple SR is placed in a periodic lattice, it is found that different resonators with ascending mass and characteristic frequencies have superior STL to those with ascending characteristic frequencies but have equal mass. The case with CR, which is more complicated as expected, shows less STL decrease than the case with SR, but wider frequency range where the STL increases than a poroelastic composite structure without resonators. This results from the fact that the frequency band of vibration suppression in the CR case is wider than in the SR case. As a result, to achieve the desired STL performance in a frequency range, the proposed composite structure using SR with tuned characteristic frequencies is enough; however, if a wider frequency band is expected even if there is a slight STL tradeoff, the CR case is a better option. Though the method proposed is only valid in the low-to-medium frequency range, the results obtained can benefit theoretical development of low-to-medium sound modulation applications, they are also valuable and illuminating for investigating the broadband sound modulation.