Two-dimensional Periodic Structures Research Articles

Talbot-effect-based multiplication of Laguerre–Gaussian beams with non-zero radial indices: From theory to experimental realization

We present a Talbot self-imaging inspired method for multiplying Laguerre–Gaussian (LG) beams of non-zero radial indices. We explore theoretically and experimentally the LG beam diffraction from two-dimensional (2D) periodic structures. In particular, we determine near-field LG beam diffraction patterns from a 2D binary grating of a small opening ratio (OR). We demonstrate that incident LG beam intensity patterns are reproduced as images of individual square apertures of the grating in certain Talbot planes. We also investigate the impact of the number of fractional-Talbot and Talbot planes on the quality of the generated 2D array of multiplexed LG modes. Our theoretical predictions are in good agreement with our experiments. Our method can find applications in optical communication, optical tweezing, multi-particle trapping, screening, and micro-manipulation.

Nanoripples evolution on tungsten surface induced by two-pulse configuration

Tuneable shapes and uniformity of the laser-induced periodic surface structures (LIPSS) attract interest because of their diverse applications in both scientific research and technological advancements. In this work, we investigate the progression of regular one-dimensional (1D) LIPSS on a tungsten surface, examining its evolution based on the time delay between two laser pulses that initiate the formation of nanoripples. 1D-LIPSS were formed in the case of single-beam laser ablation with approximately 84 laser pulses. Two-dimensional (2D) LIPSS, including triangle, hexagonal, and square shapes, were generated by employing two cross-polarized laser beams (with wavelengths of 1030 nm and pulse durations of 40 fs) with no delay between the pulses. However, introducing a time delay of 2 picoseconds (ps) between the two cross-polarized laser pulses resulted in the division of the initially cross-oriented 2D-LIPSS into square-shaped structures, particularly along the spatially overlapped region of the laser beams. The emergence of triangle- and hexagonal-shaped two-dimensional laser-induced periodic surface structures (2D-LIPSS) is analyzed within the framework of a self-organization model, particularly in the context of solidifying the molten layer under cold temperature conditions on the tungsten surface. We examine the formation mechanism of LIPSS, attributing it to a combination of the self-organization/hydrodynamic mechanism initially, which transitions to the electromagnetic mechanism as the effective number of pulses increases.

Research on out-of-plane vibration band gap calculation of two-dimensional periodic grillage structures based on spectral element method

The vibration of ship and marine structures directly impacts their structural performance and noise levels. Therefore, the vibration band gap characteristics of two-dimensional periodic grillage structures are investigated based on spectral element method in this article. Firstly, the governing motion equation of out-of-plane vibration of periodic grid structures is established in light of the Bloch theorem, and the shear spring oscillator system is introduced to simulate the first-order vertical shear vibration of bottom plate. As a result, the periodic oscillator coupling grid structure model can be proposed. Based on the periodic grid structures, the influence of material distribution on vibration band gap is obtained. Meanwhile, the dispersion relations and vibration transmission of periodic coupling model are calculated and the coupling effect of local resonance system on the band gap characteristics is analyzed. At last, a general approach for equating periodic grillage unit to numerical model is outlined and summarized on basis of the similarity of dynamic properties. The effectiveness and accuracy of the numerical method in out-of-plane vibration band gap analysis of periodic grillage structures are validated through the computation and model test.

Computation of the dynamic response of finite two-dimensional periodic structures

Computation of the dynamic response of finite two-dimensional periodic structures

Active Regulation of Elastic Waves in a Type of Two-Dimensional Periodic Structures With Piezoelectric Springs

Abstract Wave propagations exhibit direction and frequency selectivity in two-dimensional (2D) periodic structures, which provides possibilities to regulate wave dispersion and bandgap properties. Most of current researches focus on regulations of 1D waves, and there are few works about active regulations of 2D waves, especially in the structures with strong nonlinearities that have remarkable influences on dispersions. In this work, two types of 2D periodic nonlinear lattice structures with piezoelectric springs, which include a monatomic and a diatomic structure, are designed to implement controllable dispersion and propagation direction of 2D waves. Considering the strong nonlinearities caused by the cubic spring, dynamic models of the wave propagations in the two kinds of periodic structures are established, and an improved incremental harmonic balance (IHB) method is developed to implement efficient and accurate calculations of the 2D wave propagation. Influences of active and structural parameters on dispersion and bandgap properties are comprehensively studied, and the regulation ability of the piezoelectric springs is demonstrated where the proportional voltage constant is the active control parameter with particle displacements as the feedback. Results also show that a piezoelectric modulated bandgap and a critical wave vector region are created by positive and negative proportional constants, respectively, which indicate that the structures can be used to filter a wide range of low-frequency long-wavelength noises and waves at particular directions. The properties predicted by the improved IHB method are verified by numerical experiments.

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.

Formation of two-dimensional laser-induced periodic surface structures on titanium by GHz burst mode femtosecond laser pulses

GHz burst mode femtosecond (fs) laser pulses, which consist of a series of pulse trains with ultra-fast intervals of several hundred picoseconds, have offered distinct features for material processing compared to conventional irradiation of laser pulses (single-pulse mode). We apply GHz burst mode processing to fabricate laser-induced periodic surface structures (LIPSS) on the material surfaces. In our previous work for silicon (Si), we have found that GHz burst mode can create unique two-dimensional (2D) LIPSS composed of both parallel and perpendicular to the laser polarization direction. We proposed that the formation of 2D-LIPSS is attributed to the synergetic contributions of electromagnetic and hydrodynamic mechanisms. To further investigate more detailed formation mechanisms and explore practical applications, we employ titanium (Ti), whose properties are significantly different from Si. We demonstrate that GHz burst mode fs laser pulses (central wavelength: 1,030 nm, intra-pulse width: 230 fs, intra-pulse repetition rate (an intra-pulse interval): 4.88 GHz (205 ps) and burst pulse repetition rate: 10 kHz) can also fabricate 2D-LIPSS on Ti surfaces. We attribute the dominant formation mechanism of 2D-LIPSS to the generation of hot spots with highly enhanced electric fields due to transient change of material properties during GHz burst pulse irradiation. Based on this speculation, properly tailoring the shapes of the burst pulse with an optimum intra-pulse number enables the creation of well-defined 2D-LIPSS. Furthermore, essentially homogeneous 2D-LIPSS can be formed in a large area by laser scanning of a focused fs laser beam with a stage scanning speed of 5 mm/s.

Laser-driven self-organized evolution of 1D- and 2D-Nanostructures from metal thin-films on silicon: Influence of alloying and oxidation

One- and two-dimensional laser-induced periodic surface structure (LIPSS) formation of various metal (Ag, Cu, Fe, Zn, Ti) thin films on silicon wafers is achieved by irradiation with circularly polarized 532 nm nanosecond laser pulses. The occurrence and the evolution of pseudo-1D linear periodic, 2D honeycomb-like structures, and 2D periodic nanoparticle arrays were mapped for each element in dependence on the energy applied, i.e. the effective pulse number, and the metal thin film thickness. Samples series with increasing pulse counts monitor, like snapshots, the progressive pattern evolution. Because the metal thin films are of nanometer thickness only, the co-structuring of the silicon substrate must be taken into account in all cases. Even more complex are LIPSS formations on samples carrying two stacked thin films. The LIPSS formation process of thin films becomes rather complex as soon as co-structuring of the substrate, oxidation of the material, and alloy formation with substrate material or a potential second thin film layer are considered. All these effects affect the morphology of the formed LIPSS patterns.

Demystifying data evaluation in the measurement of periodic structures

Periodic structures are often found in various areas of nanoscience and nanotechnology with many of them being used for metrological purposes either to calibrate instruments, or forming the basis of measuring devices such as encoders. Evaluating the period of one or two-dimensional periodic structures from topography measurements, e.g. performed using scanning probe microscopy, can be achieved using different methodologies with many grating evaluation methods having been proposed in the past and applied to a handful of examples. The optimum methodology for determining the grating period/pitch is not immediately obvious. This paper reports the results of extensive large-scale simulations and analysis to evaluate the performance of both direct and Fourier space data processing methods. Many thousands of simulations have been performed on a variety of different gratings under different measurement conditions and including the simulation of defects encountered in real life situations. The paper concludes with a summary of the merits and disadvantages of the methods together with practical recommendations for the measurements of periodic structures and for developing algorithms for processing them.

Two-dimensional laser-induced periodic surface structures formed on crystalline silicon by GHz burst mode femtosecond laser pulses

Femtosecond laser pulses with GHz burst mode that consist of a series of trains of ultrashort laser pulses with a pulse interval of several hundred picoseconds offer distinct features in material processing that cannot be obtained by the conventional irradiation scheme of femtosecond laser pulses (single-pulse mode). However, most studies using the GHz burst mode femtosecond laser pulses focus on ablation of materials to achieve high-efficiency and high-quality material removal. In this study, we explore the ability of the GHz burst mode femtosecond laser processing to form laser-induced periodic surface structures (LIPSS) on silicon. It is well known that the direction of LIPSS formed by the single-pulse mode with linearly polarized laser pulses is typically perpendicular to the laser polarization direction. In contrast, we find that the GHz burst mode femtosecond laser (wavelength: 1030 nm, intra-pulse duration: 220 fs, intra-pulse interval time (intra-pulse repetition rate): 205 ps (4.88 GHz), burst pulse repetition rate: 200 kHz) creates unique two-dimensional (2D) LIPSS. We regard the formation mechanism of 2D LIPSS as the synergetic contribution of the electromagnetic mechanism and the hydrodynamic mechanism. Specifically, generation of hot spots with highly enhanced electric fields by the localized surface plasmon resonance of subsequent pulses in the bursts within the nanogrooves of one-dimensional LIPSS formed by the preceding pulses creates 2D LIPSS. Additionally, hydrodynamic instability including convection flow determines the final structure of 2D LIPSS.

PEEC Model Based on a Novel Quasi-Static Green's Function for Two-Dimensional Periodic Structures

A quasi-static periodic Green's function (PGF) is proposed for modeling and designing metasurfaces in the form of two-dimensional (2D) periodic structures. By introducing a novel quasi-static approximation on the full-wave PGF in the spectrum domain, the quasi-static PGF is derived that can retain the contribution from propagating and evanescent modes below resonant frequency by a second-order polynomial of frequency. Unlike full-wave PGF, the proposed quasi-static PGF polynomial coefficients are frequency-invariant. Consequently, it can save the modeling time significantly by calculating the coefficients only once for a frequency band of interest. Moreover, a quasi-static PEEC model is developed from the proposed quasi-static PGF. It circumvents the breakdown problem around near-zero frequencies since the singularity in PGF is separated in the quasi-static PGF and eliminated analytically in model development. Therefore, both the time- and frequency-domain analysis can be conducted easily using a SPICE-like solver on the PEEC model. Two examples are given, one of which validates the accuracy of the proposed quasi-static PGF in a 2D periodic unit cell working at 0–12 GHz; the other demonstrates the superior performance in terms of model efficiency and stability by a Jerusalem-cross frequency selective surface (FSS) working at 0–20 GHz. The numerical results show that the proposed method is accurate and efficient in a wide band for metasurfaces made of two-dimensional periodic structures.

All-metal wakefield accelerator: Two-dimensional periodic surface lattices for charged particle acceleration

Interest in ``all-metal'' surface periodic structures to control charged particle beam dynamics and properties, and to generate coherent radiation in the THz frequency range has recently risen exponentially. While such structures have the benefits of metal, they are capable of providing similar properties to a dispersive dielectric. All-metal structures allow good temperature and space charge management. Transportation of high energy charge particle beams in the vicinity of the structure without risking material degradation makes them very attractive for particle acceleration and terahertz radiation generation applications. Here, the results of theoretical studies (via numerical modeling) of the electron beam dynamics and wakefield excitation and evolution along the cylindrical two-dimensional periodic surface structure will be presented, and the properties of the wakefields generated will be discussed. The wakefield acceleration of the witness bunch in the potential reaching $1\text{ }\text{ }\mathrm{GV}/\mathrm{m}$ will be demonstrated. The challenges will be identified and discussed, and the concept of the cascade accelerator based on such all-metal wakefield accelerator (AMWA) structures will be presented.

Magnetic metasurfaces with metallic inclusions

Purpose of this paper is the development and creation of the magnetic metasurfaces with metallic inclusions operating both in the microwave and terahertz frequency ranges. Methods. The Maxwell’s equations and the expressions for the effective medium parameters are used to build the analytical models of the magnetic metasurfaces based on either a ferromagnetic (FM) or antiferromagnetic (AFM) dielectric matrix, containing a two-dimensional periodic structure of thin metal (non-magnetic) wires surrounded by insulators. Numerical simulation of such structures operating in the microwave range is carried out using the MaxLLG software package. The magnetron sputtering, liquid etching, optical lithography, and lift-off photolithography are used to create bicomponent magnetic metasurfaces, consisting of two magnetic materials with very different values of magnetization. The study of linear and nonlinear characteristics of the bicomponent magnetic metasurfaces is carried out using the methods of microwave and Brillouin spectroscopy. Results. Based on the developed analytical model of the magnetic metasurface with metallic (nonmagnetic) inclusions it is shown that the FM metasurface possesses properties of a left-handed medium in a microwave range and the AFM metasurface possesses similar properties in a terahertz range. In the last case, the material parameters of the AFM metasurface are twice negative in two frequency bands. For the magnetic metasurfaces with metallic magnetic inclusions, the formation of absorption bands in the spectrum of a traveling magnetostatic surface spin wave due to the resonant properties of the inclusions has been established. In the nonlinear regime, the effect of nonreciprocal parametric three-wave resonance was obtained. Conclusion. The results presented in the paper demonstrate a number of physical phenomena that are observed only in the magnetic metasurfaces with metallic (nonmagnetic and magnetic) inclusions.

Vibration band gap characteristics of two-dimensional functionally graded grids using the spectral element method

In this article, the band gap characteristics of two-dimensional periodic grid structures composed of functionally graded materials are studied. According to the first-order shear deformation theory and the spectral element method, the motion equation of the functionally graded beam is established at first, and then the spectral stiffness matrix of the lattice structure is assembled through the coordinate transformation matrix. With the aid of the Bloch theorem, the in-plane wave dispersion relation of periodic grid structures composed of two functionally graded materials is analyzed. The availability and accuracy of the band gap calculation method are verified by structural vibration transmission results based on the finite element method. In addition, the influence of the material parameters of the structure on the dispersion relation of the in-plane wave is obtained. The results extend the range of research on two-dimensional grids which can be applied to the design of filters.

One-step curing process of conductive paste based on a UV pulse laser for a frequency selective surface.

A frequency selective surface (FSS) is a two-dimensional periodic array structure, in which a common structure is composed of conductive paste and metal film. An ultraviolet (UV) pulse-laser-induced curing of conductive paste towards FSS was used to simultaneously realize the efficient curing of conductive paste and the removal of surplus materials. Through simple defocus control of the UV pulse laser when irradiating the workpiece, this technology was capable of reducing the energy density and increasing the irradiation range, thus preventing the conductive paste from being ablated and achieving efficient curing. In this context, the curing process of UV curable conductive paste was systematically studied through the Ohmic resistance and bonding force of the cured conductive paste. The curing effects of various conductive paste thicknesses and different laser scanning times at a certain laser scanning speed were also analyzed. The results showed that, after a controllable defocusing treatment, the UV pulse laser could effectively solidify the conductive paste and realize the electrical connection of materials on both sides of the pattern. The peel strength of the conductive paste was greater than 7.1 N/cm. However, the pulse laser curing method needs a longer curing time when compared with the continuous UV curing method.

Nonlinear Bragg scattering of surface waves over a two-dimensional periodic structure

Bragg scattering of nonlinear surface waves over a wavy bottom is studied using two-dimensional fully nonlinear numerical wave tanks (NWTs). In particular, we consider cases of high nonlinearity which lead to complex wave generation and transformations, hence possible multiple Bragg resonances. The performance of the NWTs is well verified by benchmarking experiments. Classic Bragg resonances associated with second-order triad interactions among two surface (linear incident and reflected waves) and one bottom wave components (class I), and third-order quartet interactions among three surface (linear incident and reflected waves, and second-order reflected/transmitted waves) and one bottom wave components (class III) are observed. In addition, class I Bragg resonance occurring for the second-order (rather than linear) transmitted waves, and Bragg resonance arising from quintet interactions among three surface and two bottom wave components, are newly captured. The latter is denoted class IV Bragg resonance which magnifies bottom nonlinearity. It is also found that wave reflection and transmission at class III Bragg resonance have a quadratic rather than a linear relation with the bottom slope if the bottom size increases to a certain level. The surface wave and bottom nonlinearities are found to play opposite roles in shifting the Bragg resonance conditions. Finally, the results indicate that Bragg resonances are responsible for the phenomena of beating and parasitic beating, leading to a significantly large local free surface motion in front of the depth transition.

A macroscopic model for immiscible two-phase flow in porous media

This work provides the derivation of a closed macroscopic model for immiscible two-phase, incompressible, Newtonian and isothermal creeping steady flow in a rigid and homogeneous porous medium without considering three-phase contact. The mass and momentum upscaled equations are obtained from the pore-scale Stokes equations, adopting a two-domain approach where the two fluid phases are separated by an interface. The average mass equations result from using the classical volume averaging method. A Green's formula and the adjoint Green's function velocity pair problems are used to obtain the pore-scale velocity solutions that are averaged to obtain the upscaled momentum balance equations. The macroscopic model is based on the assumptions of scale separation and the existence of a periodic representative elementary volume allowing a local description as usually postulated for upscaling. The macroscopic momentum equation in each phase includes the generalized Darcy-like dominant and viscous coupling terms and, importantly, an additional compensation term that accounts for surface tension effects to momentum transfer that is, otherwise, incompletely captured by the Darcy terms. This interfacial term, as well as the dominant and viscous coupling permeability tensors, can be predicted from the solutions of two associated closure problems that coincide with those reported in the literature. The relevance of the compensation term and the upscaled model validity are assessed by comparisons with direct numerical simulations in a model two-dimensional periodic structure. Upscaled model predictions are found to be in excellent agreement with direct numerical simulations.

Self-organization of stearic acid salts on the hemispherical surface of the aqueous subphase allows functionalization of matrix-assisted laser desorption/ionization mass spectrometry target plates for on-plate immobilized metal affinity chromatography enrichment

In this work we propose an approach for formation of periodic two-dimensional structures on a solid substrate via self-assembly of stearic acid molecules on an aqueous subphase drop containing metal ions. It was shown that when a saturated solution of stearic acid in n-hexane was applied to the hemispherical surface of an aqueous media containing barium ions, barium stearate monolayer was formed, which spontaneously collapsed during movement from the drop surface to the substrate. By means of infrared and Raman spectroscopy, light and scanning electron microscopy, atomic force microscopy and matrix-assisted laser desorption/ionization mass spectrometry it was proved that a multi-structure of collapsed metal stearate monolayers with a developed surface was formed on the substrate surface. Thin films formed by a single application of stearic acid solution in n-hexane to a drop of aqueous lanthanum nitrate solution were used for selective enrichment of human butyrylcholinesterase adducts with diisopropyl fluorophosphate.

Design and evaluation of three–dimensional axisymmetric mechanical metamaterial exhibiting negative Poisson's ratio

Mechanical metamaterials are generally two-dimensional periodic structures or three-dimensional cellular structures that exhibit mechanical properties beyond the ordinary. Due to size and boundary effects, three-dimensional mechanical metamaterials typically display anisotropic changes even if they are isotropic in construction and composition. However, in this study, the comprehensive design and fabrication of a three-dimensional axisymmetric auxetic structure that exhibits uniform and axisymmetric transverse deformation under longitudinal compression loading are proposed. Extending the concept of two-dimensional periodically perforated auxetic sheet structures to the third dimension, the design of the metamaterial was generated by revolving a two-dimensional parabolic curve along the axis of rotation and subsequently perforating the structure periodically with elliptical voids varying in size longitudinally along the curvature of the structure in order to promote the exhibition of isotropic negative Poisson's ratio. Furthermore, this study elucidates the significance of the perforations by comparing the metamaterial structure to a so-called plain structure.

A band-pass frequency selective surface with polarization rotation

A band-pass frequency selective surface (FSS) with polarization rotation property is proposed. The proposed polarization rotating FSS (PR-FSS) is a two-dimensional periodic structure, its unit cell is an antenna-filter-antenna (AFA) module, and the polarization directions of the upper and lower antennas in each AFA module are orthogonal to each other, so the PR-FSS can achieve frequency selection and 90 degrees polarization rotation at the same time. The numerical simulation demonstrate that the anticipated frequency selection and polarization rotation are realized by the PR-FSS in the frequency band from 8.84 GHz to 10.30 GHz with a relative bandwidth of 15.26%, and the maximum insertion loss in the pass band is only 0.17 dB. Finally, one effective experiment validation is carried out, a reasonable agreement is observed between the experimental and simulated results except for a slight deviation caused by fabrication error and measurement tolerance.