Two-dimensional Periodic Structures Research Articles

Two-dimensional periodic structures modeling based on second strain gradient elasticity for a beam grid

Higher order gradient elasticity theories are widely applied to determine the wave propagation characteristics of micro-sized structures. The novelty of this paper, firstly, is using the Second Strain Gradient (SSG) theory to explore the mechanism of a micro-sized 2D beam grid. The strong formulas of continuum model including governing equations and boundary conditions are derived by using the Hamilton principle. Then, a valuable long-range Lattice Spring Model (LSM) is elaborated, providing a reasonable explanation for the model based on SSG theory. The dynamic continuum equations from LSM are calculated through the Fourier series transform approach. Finally, the dynamic properties of 2D beam grid are analyzed within the Wave Finite Element Method (WFEM) framework. The band structure and slowness surfaces, confined to the irreducible first Brillouin zone, are studied in frequency spectrum. The energy flow vector fields and wave beaming effects are discussed through SSG theory and Classical Theory (CT) of elasticity. The results show that the proposed approach is of significant potential for investigating the 2D wave propagation characteristics of complex micro-sized periodic structures.

Transformation from nano-ripples to nano-triangle arrays and their orientation control on titanium surfaces by using orthogonally polarized femtosecond laser double-pulse sequences

One-dimensional and two-dimensional laser-induced periodic structures (nano-ripples and nano-triangle arrays) are fabricated on titanium surfaces by direct scanning with orthogonally polarized and equal-energy femtosecond laser double-pulse sequences (OP pulses). Clear and regular nano-ripples and nano-triangle arrays are formed when the laser fluence is slightly higher than the ablation threshold of titanium and the time delay is approximately 2 ps or shorter. By changing only the scanning speed, we achieve the transformation from nano-ripples to nano-triangle arrays. The orientation of nano-ripples and that of one of the three sides of nano-triangle arrays are always perpendicular to the scanning direction rather than the laser polarization. The mechanism underlying this phenomenon is as follows. First, the nano-ripples are formed due to periodic energy deposition along the direction of surface waves (surface plasmon polaritons; SPPs), which is along the scanning direction during the scan process. Subsequently, the preformed nano-ripples irradiated with subsequent laser pulses undergo hexagonal convection flow and are transformed into nano-triangle arrays. The preformed nano-ripples perpendicular to the scanning direction have a positioning effect on the formation of nano-triangle arrays, as verified in two-step fast scan experiments. Large-area patterning of nano-triangle arrays producing three-directional structural color is demonstrated in this paper.

Two-dimensional quasi periodic structures for large-scale light out-coupling with amplitude, phase and polarization control.

Chip-scale light-atom interactions are vital for the miniaturization of atomic sensing systems, including clocks, magnetometers, gyroscopes and more. Combining as many photonic elements as possible onto a photonic chip greatly reduces size and power consumption, where the critical elements are those interfacing between the 2D circuit and the 3D vapor cell. We introduce a new design method for large scale two-dimensional converter structures, enabling out-coupling of radiation from the photonic chip into the atomic medium. These structures allow light intensity and phase spatial distribution and polarization control, without external light-manipulating elements. Large, 100 × 100 µm2 structures were designed generating low divergence optical beams with high degree of circular polarization. Simulations obtain mean circular polarization contrast of better than 30 dB.

A virtual microscope for simulation of Nanostructures

Light-matter interplay is widely used for analyzing the topology of surfaces on small scales for use in areas such as nanotechnology, nanoelectronics, photonics, and advanced materials. Conventional optical microscope imaging methods are limited in resolution to a value comparable to the wavelength, the so-called Abbe limit, and cannot be used to measure nano-sized structures. Scatterometry is an optical method that can measure structures smaller than the wavelength. However, the relative uncertainties of the structure dimensions measured with scatterometry increase with decreasing structure size, and the industry is therefore looking for replacing simple intensity based scatterometry with a phase-sensitive measurement method such as coherent Mueller ellipsometry. In this work, we present a virtual microscope capable of simulating the coherent Mueller ellipsometry and scatterometry response from one-dimensional and two-dimensional periodic structures. Furthermore, we argue that coherent nonnormalized Mueller ellipsometry gives results with less uncertainties than standard normalized Mueller ellipsometry.

High-Q refractive index sensors based on all-dielectric metasurfaces.

Possessing fantastic abilities to freely manipulate electromagnetic waves on an ultrathin platform, metasurfaces have aroused intense interest in the academic circle. In this work, we present a high-sensitivity refractive index sensor excited by the guided mode of a two-dimensional periodic TiO2 dielectric grating structure. Numerical simulation results show that the optimized nanosensor can excite guided-mode resonance with an ultra-narrow linewidth of 0.19 nm. When the thickness of the biological layer is 20 nm, the sensitivity, Q factor, and FOM values of the nanosensor can reach 82.29 nm RIU−1, 3207.9, and 433.1, respectively. In addition, the device shows insensitivity to polarization and good tolerance to the angle of incident light. This demonstrates that the utilization of low-loss all-dielectric metasurfaces is an effective way to achieve ultra-sensitive biosensor detection.

Double negative media based on antiferromagnetic metamaterials

The paper presents the theoretical study results of the dispersion characteristics of electromagnetic waves existing in antiferromagnetic (AFM) metamaterial. The AFM metamaterial consists of a transversely magnetized antiferromagnet, inside of which a two-dimensional periodic structure of thin conducting wires surrounded by insulators is placed. It has been established that the AFM metamaterial has two frequency ranges, in which there are backward waves, and the material pa-rameters of the medium are twice negative. The indicated areas are located in the terahertz range. Keywords: metamaterials, left-handed medium, antiferromagnetics, spin waves.

Solvability of Singular Equations of Peridynamics on Two-Dimensional Periodic Structures

Solvability of Singular Equations of Peridynamics on Two-Dimensional Periodic Structures

Complete vibration band gap characteristics of two-dimensional periodic grid structures

In this article, the spectral element method combined with Bloch theorem is applied to calculate complete vibration band gap characteristics of two-dimensional periodic grid structures, which is verified by structural vibration transmission results based on spectral element method and finite element method. According to the differential equation of motion, complete spectral stiffness matrix of beam element is established at first, then stiffness matrix of grid structure can be obtained by coordinate transformation and assembly method, so as to form governing equation of motion based on Bloch boundary conditions. The band gap characteristics of periodic structures can be calculated by solving motion equation. Through band gap distribution analysis of two-dimensional periodic grid structures, the dispersion relations of different unit arrangements and effects of material and structural parameters on band gap characteristics are obtained which can be applied to acoustic design or vibration and noise reduction areas.

Direct Calculation of the Group Velocity for Two-Dimensional Complex, Composite and Periodic Structures Using a Wave and Finite Element Scheme

Guided waves have immense potential for structural health monitoring applications in numerous industries including aerospace. It is necessary to evaluate guided wave dispersion characteristics, i.e., group velocity and phase velocity profiles, for using them effectively. For complex structures, the profiles can have highly irregular shapes. In this work, a direct method for calculating the group velocity profiles for complex, composite, and periodic structures using a wave and finite element scheme is presented. The group velocity calculation technique is easy to implement, highly computationally efficient, and works with the standard finite element formulation. The major contribution is summarised in the form of a comprehensive algorithm for calculating the group velocity profiles. The method is compared with the existing analytical and numerical methods for calculation of dispersion curves. Finally, an experimental study in a multilayered composite plate is conducted and the results are found to be in good agreement. The technique is suitable to be used in all guided wave application areas such as material characterisation, non-destructive testing, and structural health monitoring.

Deep-Subwavelength 2D Periodic Surface Nanostructures on Diamond by Double-Pulse Femtosecond Laser Irradiation.

Two-dimensional laser-induced periodic surface structures with a deep-subwavelength periodicity (80 nm ≈ λ/10) are obtained for the first time on diamond surfaces. The distinctive surface nanotexturing is achieved by employing a single step technique that relies on irradiation with two temporally delayed and cross-polarized femtosecond-laser pulses (100 fs duration, 800 nm wavelength, 1 kHz repetition rate) generated with a Michelson-like interferometer configuration, followed by chemical etching of surface debris. In this Letter, we demonstrate that, if the delay between two consecutive pulses is ≤2 ps, the 2D periodicity of nanostructures can be tuned by controlling the number of pulses irradiating the surface. Under scanning mode, the method is effective in treating uniformly large areas of diamond, so to induce remarkable antireflection properties able to enhance the absorptance in the visible up to 50 times and to pave the route toward the creation of metasurfaces for future diamond-based optoelectronic devices.

Shape transformers for phononic band gaps tuning in two-dimensional Bloch-periodic lattice structures

In this paper, shape transformers, a set of non-dimensional geometrical parameters that define beam cross-section efficiencies based on their size and shape, are employed in the wave propagation analysis of two-dimensional periodic lattice structures. Here, shape transformers of lattice unit-cell beams are used as design variables to tailor the frequency band gaps of the lattice structure, in the form of their mid frequencies and aspect ratios, by means of numerical Bloch-periodic simulation. Case studies are presented for lattices with square and hexagon Bravais symmetries where a total of six different unit-cell topologies are studied, considering 12 classes of different cross-section shapes. It is observed that a significant shift of the waveband branches towards lower frequencies are obtained with the variation of shape transformers from a void to a filled envelope configuration. Moreover, while changing the cross-section geometric variables, it is possible to determine associated values of maximum aspect ratios of band gap frequencies, thus identifying cross-section shapes that have better performance with respect to lattice dynamic characteristic. It is also found that the “H” and “I” cross-section shapes, in comparison to other shape families presented, resulted in the maximization of the band gap aspect ratio while maintaining a low frequency range, hence being ideal candidates for designing lattice metamaterials with lower frequency ranges of elastic wave attenuation. Furthermore, among all lattice topologies investigated, it is noted that the employment of diamond cross-section and its derivatives resulted in the greatest sensitivity of the mid-frequency shift of the band gap with respect to variations in cross-section geometric efficiency. It is also observed, as a general behavior for all unit-cells studied, that when scaling up the in-plane height of the cross-section, the aspect ratio of the band gap was maximized. Finally, using the Modal Assurance Criterion, we demonstrated that the variation in the shape transformers conserves the consistency between the eigenwaves of unit-cells at different characteristic points of the Irreducible Brillouin Zone. The presented study paves the way for the development and the systematic implementation of a novel wave attenuation tuning technique.

Shock wave filtering of two-dimensional CFRP X-lattice structures: A numerical investigation

A composite X-lattice structure, comprising corrugated carbon-fiber-reinforced plastic ribs in a periodic pattern, is considered a superior light-weight aerospace structure. It is expected to exhibit good wave attenuation properties owing to the frequency and spatial filtering characteristics of a two-dimensional periodic structure. Wave propagation analysis using the theory of a two-dimensional periodic structure and finite element method (FEM) was introduced to numerically evaluate the wave filtering properties with fewer computational resources. The results obtained using the periodicity were validated by comparing them with the frequency responses of the full-scale FEM simulation. The range of the wave filtering properties was presented by changing the geometrical properties of the lattice structure. An approach to design a composite lattice structure with broader and desirable wave filtering properties was proposed by combining several types of lattice structures with different wave filtering properties.

Raman Spectroscopy of Twisted Bilayer Graphene

When two periodic two-dimensional structures are superposed, any mismatch rotation angle between the layers generates a Moiré pattern superlattice, whose size depends on the twisting angle θ. If the layers are composed by different materials, this effect is also dependent on the lattice parameters of each layer. Moiré superlattices are commonly observed in bilayer graphene, where the mismatch angle between layers can be produced by growing twisted bilayer graphene (TBG) samples by CVD or folding the monolayer back upon itself. In TBG, it was shown that the coupling between the Dirac cones of the two layers gives rise to van Hove singularities (vHs) in the density of electronic states, whose energies vary with θ. The understanding of the behavior of electrons and their interactions with phonons in atomically thin heterostructures is crucial for the engineering of novel 2D devices. Raman spectroscopy has been often used to characterize twisted bilayer graphene and graphene heterostructures. Here, we review the main important effects in the Raman spectra of TBG discussing firstly the appearance of new peaks in the spectra associated with phonons with wavevectors within the interior of the Brillouin zone of graphene corresponding to the reciprocal unit vectors of the Moiré superlattice, and that are folded to the center of the reduced Brillouin Zone (BZ) becoming Raman active. Another important effect is the giant enhancement of G band intensity of TBG that occurs only in a narrow range of laser excitation energies and for a given twisting angle. Results show that the vHs in the density of states is not only related to the folding of the commensurate BZ, but mainly associated with the Moiré pattern that does not necessarily have a translational symmetry. Finally, we show that there are two different resonance mechanisms that activate the appearance of the extra peaks: the intralayer and interlayer electron–phonon processes, involving electrons of the same layer or from different layers, respectively. Both effects are observed for twisted bilayer graphene, but Raman spectroscopy can also be used to probe the intralayer process in any kind of graphene-based heterostructure, like in the graphene/h-BN junctions.

Reactive-ion-etched glass surface with 2D periodic surface texture for application in solar cells

Thin-film silicon solar cells are promising candidates for the low-cost production of electricity from sunlight. However, their efficiency remains relatively low, which is one of the major issues for their commercialisation. In the thin-film silicon technology, light trapping is an important design consideration for achieving a higher device efficiency. An advanced light-trapping technology is expected to boost the efficiency of solar cells further. Therefore, we investigated the development of an advanced light-trapping scheme in a two-dimensional periodic surface structure produced via inductively coupled plasma reactive-ion etching. The texture of the glass surface was varied under various plasma conditions. For the various shapes and sizes of the obtained surface texture, the increased height and base angle of the individual texture showed an overall increase in the diffused transmission of light. In some cases, the diffused transmission was as high as 89 % of the total light transmitted through the glass. The effect of the textured glass on a solar cell was investigated through simulation. Results show that the efficiency of a single-junction solar cell can be enhanced significantly when such a textured glass is used instead of a non-textured glass.

Nonlinear Talbot self-healing in periodically poled LiNbO3 crystal [Invited

The nonlinear Talbot effect is a near-field nonlinear diffraction phenomenon in which the self-imaging of periodic objects is formed by the second harmonics of the incident laser beam. We demonstrate the first, to the best of our knowledge, example of nonlinear Talbot self-healing, i.e., the capability of creating defect-free images from faulty nonlinear optical structures. In particular, we employ the tightly focused femtosecond infrared optical pulses to fabricate LiNbO3 nonlinear photonic crystals and show that the defects in the form of the missing points of two-dimensional square and hexagonal periodic structures are restored in the second harmonic images at the first nonlinear Talbot plane. The observed nonlinear Talbot self-healing opens up new possibilities for defect-tolerant optical lithography and printing.

Дважды отрицательные среды на основе антиферромагнитных метаматериалов для терагерцевого диапазона частот

The paper presents the theoretical study results of the dispersion characteristics of electromagnet-ic waves existing in antiferromagnetic (AFM) metamaterial. The AFM metamaterial consists of a transversely magnetized antiferromagnet, inside of which a two-dimensional periodic structure of thin conducting wires surrounded by insulators is placed. It has been established that the AFM metamaterial has two frequency ranges, in which there are backward waves, and the material pa-rameters of the medium are twice negative. The indicated areas are located in the terahertz range.

A hybrid symplectic and high-frequency homogenization analysis for the dispersion property of periodic micro-structured thin plate structures

A hybrid homogenization analysis approach is developed for the dispersion of one- and two-dimensional periodic micro-structured thin plate structures by combining the high-frequency homogenization method and the symplectic method. In the present method, the displacement field of the unit cell which is needed in the high frequency homogenization method is expressed in terms of wave components which are obtained by solving an eigenvalue problem which is first proposed. The eigenproblem developed based on the symplectic method allows the displacement function of the unit cell with arbitrary simple boundary conditions to be obtained rationally, so that the high-frequency homogenization analysis can be more conveniently applied to the dispersion analysis of the periodic micro-structured plate structures. Numerical examples of one-dimensional and two-dimensional periodic structures with various boundary conditions for the unit cell are provided to confirm the utility of the proposed method.

Effective Conductivity of Densely Packed Disks and Energy of Graphs

The theory of structural approximations is extended to two-dimensional double periodic structures and applied to determination of the effective conductivity of densely packed disks. Statistical simulations of non-overlapping disks with the different degrees of clusterization are considered. The obtained results shows that the distribution of inclusions in a composite, as an amount of geometrical information, remains in the discrete corresponding Voronoi tessellation, hence, precisely determines the effective conductivity for random composites.

How surface stress transforms surface profiles and adhesion of rough elastic bodies.

The surface of soft solids carries a surface stress that tends to flatten surface profiles. For example, surface features on a soft solid, fabricated by moulding against a stiff-patterned substrate, tend to flatten upon removal from the mould. In this work, we derive a transfer function in an explicit form that, given any initial surface profile, shows how to compute the shape of the corresponding flattened profile. We provide analytical results for several applications including flattening of one-dimensional and two-dimensional periodic structures, qualitative changes to the surface roughness spectrum, and how that strongly influences adhesion.

Realization of an ultrawide stop band in a 2-D elastic metamaterial with topologically optimized inertial amplification mechanisms

The aim of this study is to design a two-dimensional solid structure with embedded inertial amplification mechanisms that shows an ultrawide stop band (band gap) at low frequencies. First of all, a unique compliant inertial amplification mechanism is suggested. The compliant (flexure) hinge connections are designed and the topology of the beams in the unit cell mechanism are optimized to achieve the maximum possible stop band width. Then, a two-dimensional periodic structure is formed by using this topologically optimized inertial amplification mechanism. Thereafter, the formed periodic structure is manufactured. Experimental and finite element analyses show that an ultrawide stop band between 29 Hz and 590 Hz is obtained for both longitudinal and transverse excitations. This outcome reveals a phononic gap whose upper and lower limits have a ratio that exceeds 20 (i.e., arithmetic mean normalized bandwidth of 181% or geometric mean normalized bandwidth of 429%). This much bandwidth has not been achieved in the literature for two dimensions, so far.