Two-dimensional Periodic Lattice Research Articles

Sub-THz and THz Cherenkov radiation source with two-dimensional periodic surface lattice and multistage depressed collector

We present the theory, concept and design of an efficient, megawatt coherent Cherenkov radiation source based on a two-dimensional periodic surface lattice (2D-PSL) cavity combined with a novel energy recovery system for the generation of highly efficient (> 50%) single-frequency radiation. We demonstrate the scalability of the transverse dimension of the 2D-PSL cavity of the Cherenkov source and thus the potential for efficient, continuous-wave, high-power (> 1 MW) operation; fundamental to the eventual realization of clean, fusion energy. These new sources, with the capacity to operate in the 0.1-10THz range, hold strong promise to address the long-standing “Terahertz gap”. By combining a Cherenkov oscillator driven by a non-gyrating beam with an innovative four-stage depressed collector energy recovery system, the overall device efficiency can be increased to be competitive with gyrotrons in the requirements for heating and current drive in fusion plasma. In these Cherenkov devices, the frequency independence of the magnetic guide field enables advantageous frequency scaling without deployment constraints, making them especially attractive for high-impact applications in fusion science, turbulence diagnostics, non-destructive testing and biochemical spectroscopy. The novel energy recovery techniques presented in this paper have broad applicability to many electron-beam driven devices, bringing revolutionary potential to future THz source technologies.

Numerical study of dispersion characteristics of cylindrical spiral lines and periodic gratings based on them

Numerical studies of the dispersion characteristics of cylindrical spiral lines have been carried out. One-dimensional and two-dimensional periodic lattices formed by such lines are also considered. The eigenvalue method was used in combination with periodic boundary conditions. The dispersion characteristics are calculated in the range of parameters for which the conditions of applicability of the known approximate numerical-analytical methods for analyzing spiral lines are not met. The dispersion characteristics of the first two eigen-waves in one-dimensional and two-dimensional periodic lattices of spirals are obtained for different values of phase shifts between lattice periods. It is shown that the main type of wave in them is a wave slowed down relative to the axes of the lines with the direction of rotation of the field vector determined by the direction of winding of the spiral.

High-performance plasmonic metasurface sensor by triangular nano-structures

The increasing need to sense different materials has caused sensors and efforts to improve their performance to receive a lot of attention in the optics field. In this report, we present a high-performance plasmonic metasurface sensor by the DGTD method that includes a two-dimensional periodic lattice of nano-triangles on a glass film and a TiN mirror. The TiN prevents light transmission from the resonators, causing a deeper reflection dip. Two resonances are observed, which offer a maximum sensitivity of about 417 nm/RIU and 735 nm/RIU, correspondingly. In this structure, augmenting the nano-triangles enhances hot spots, improving spectral response and sensitivity. Moreover, the structure maintains an almost unchanged spectral response for small angles. Overall, this sensor can be suitable for various applications such as biosensing.

Terahertz frequency selective surfaces using heterostructures based on two-dimensional diffraction grating of single-walled carbon nanotubes

ABSTRACT For single-walled carbon nanotubes (SWCNTs) with a length of 1–50 µm, the surface plasmon-polariton (SPP) resonance is within the terahertz frequency range; therefore, SWCNT lattices can be used to design frequency-selective surface (FSS). A numerical model of electromagnetic wave diffraction on a two-dimensional periodic SWCNT lattice can be described by an integro-differential equation of the second-order with respect to the surface current along SWCNT. The equation can be solved by the Bubnov–Galerkin method. Frequency dependence of reflecting and transmitting electromagnetic waves for FSSs near the SPP resonance is studied numerically. It is shown that the resonances are within the lower-frequency part of the terahertz range. We also estimate the relaxation frequency of an individual SWCNT and demonstrate the applicability of the Kubo formula for graphene conductivity to array of strips similar in size to SWCNTs under consideration.

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.

Efficient, 0.35-THz Overmoded Oscillator Based on a Two-Dimensional Periodic Surface Lattice

We present the theory, design, and numerical modeling of a cylindrical, two-dimensional periodic surface lattice (2D-PSL) intended for use as the interaction region of an electron beam-driven, pulsed source. The production of 1.95-MW peak, pulsed 0.35-THz radiation with an electronic efficiency of 24% is reported. Mode selection in the oversized cavity, where the diameter D is almost <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3.5\times $ </tex-math></inline-formula> larger than the operating wavelength <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> , is achieved by coupling volume and surface fields to form a coupled cavity eigenmode. We demonstrate the advantages (including enhanced output power and improved spectral purity) of using a 2D-PSL over a simpler 1-D structure. The cylindrical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${D}/\lambda \sim {3.5} 2\text{D}$ </tex-math></inline-formula> -PSL demonstrates the “proof-of-principle” high-order mode coupling with the potential to increase <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${D}/\lambda $ </tex-math></inline-formula> to values of 20 or more for the realization of CW 2D-PSL sources or very powerful pulsed sources. The theory is applicable over a broad frequency range.

Random Graphene Metasurfaces: Diffraction Theory and Giant Broadband Absorptivity

We introduce a diffraction-theory-inspired analytical description of the metasurface comprising an array of graphene subwavelength hemispheres. Our theory describes light interaction with the random metasurface, in which the periodicity is broken by accidentally damaged meta-atoms in the nodes of a two-dimensional periodic lattice. Both numerical modeling and experiments show that such a nanometer-thin metasurface possesses giant broadband absorption in the terahertz spectral range that remains intact, even when a substantial portion of meta-atoms, i.e., graphene hemispheres, is damaged. Moreover, defective fabrication of graphene free-standing metasurface may enhance the absorptive properties.

Wave propagation and directionality in two-dimensional periodic lattices considering shear deformations

The effects of shear deformation on analysis of the wave propagation in periodic lattices are often assumed negligible. However, this assumption is not always true, especially for the lattices made of beams with smaller aspect ratios. Therefore, in the present paper, the effect of shear deformation on wave propagation in periodic lattices with different topologies is studied and their wave attenuation and directionality performances are compared. Current experimental limitations make the researchers focus more on the wave propagation in the direction perpendicular to the plane of periodicity in micro/nanoscale lattice materials while for their macro/mesoscale counterparts, in-plane modes can also be analyzed as well as the out-of-plane ones. Four well-known topologies of hexagonal, triangular, square, and Kagomé are considered in the current paper and their wave propagation is investigated both in the plane of periodicity and in the out-of-plane direction. The finite element method is used to formulate the governing equations and Bloch’s theorem is used to solve the dispersion relations. To investigate the effect of shear deformation, both the Timoshenko and Euler-Bernoulli beam theories are implemented. The results indicate that including shear deformation in wave propagation has a softening effect on the band diagrams of wave propagation and moves the dispersion branches to lower frequencies. It can also reveal some bandgaps that are not predicted without considering the shear deformation.

Plasmonic crescent nanoarray-based surface lattice resonance sensor with a high figure of merit.

Due to the natural accumulation of radiation losses arising from the localization and random arrangement of nanoparticles, the figure of merit (FOM) of localized surface plasmon resonance (LSPR) sensors is usually very low (the value is usually less than 5 RIU-1). However, radiation losses of individual particles will be offset by adjusting the phase of the scattered field which is dependent on the structure parameters of arrays. Based on this, a two-dimensional periodic crescent nanoarray-based surface lattice resonance (SLR) sensor with a high FOM is proposed in this work. Some significant results have been obtained by mode field analysis and adjustment of structural parameters. On the one hand, the line-shape of the SLR spectrum is divided into a Fano-like line and a separate line. And the former usually has an FOM of 101 magnitude while the latter has an FOM of 103 magnitude. On the other hand, the relative size of the excitation wavelengths between SLR and LSPR is also vital. The FOM is higher but resonance depth decreases faster when the relative size increases. In this work, a full width at half-maximum (FWHM) of less than 0.5 nm and FOM of more than 1000 RIU-1 (the quality factor is more than 3000) are achieved by the proposed crescent nanoarrays. In addition, this structure demonstrates that plasmonic nanoarray-based SLR has enormous potential in trace substance detection.

Macroscopically degenerate localized zero-energy states of quasicrystalline bilayer systems in the strong coupling limit

When two identical two-dimensional (2D) periodic lattices are stacked in parallel after rotating one layer by a certain angle relative to the other layer, the resulting bilayer system can lose lattice periodicity completely and become a 2D quasicrystal. Twisted bilayer graphene with 30-degree rotation is a representative example. We show that such quasicrystalline bilayer systems generally develop macroscopically degenerate localized zero-energy states (ZESs) in strong coupling limit where the interlayer couplings are overwhelmingly larger than the intralayer couplings. The emergent chiral symmetry in strong coupling limit and aperiodicity of bilayer quasicrystals guarantee the existence of the ZESs. The macroscopically degenerate ZESs are analogous to the flat bands of periodic systems, in that both are composed of localized eigenstates, which give divergent density of states. For monolayers, we consider the triangular, square, and honeycomb lattices, comprised of homogenous tiling of three possible planar regular polygons: the equilateral triangle, square, and regular hexagon. We construct a compact theoretical framework, which we call the quasiband model, that describes the low energy properties of bilayer quasicrystals and counts the number of ZESs using a subset of Bloch states of monolayers. We also propose a simple geometric scheme in real space which can show the spatial localization of ZESs and count their number. Our work clearly demonstrates that bilayer quasicrystals in strong coupling limit are an ideal playground to study the intriguing interplay of flat band physics and the aperiodicity of quasicrystals.

Synchronization transition in the two-dimensional Kuramoto model with dichotomous noise.

We numerically study the celebrated Kuramoto model of identical oscillators arranged on the sites of a two-dimensional periodic square lattice and subject to nearest-neighbor interactions and dichotomous noise. In the nonequilibrium stationary state attained after a long time, the model exhibits a Berezinskii-Kosterlitz-Thouless (BKT)-like transition between a phase at a low noise amplitude characterized by quasi long-range order (critically ordered phase) and an algebraic decay of correlations and a phase at a high noise amplitude that is characterized by complete disorder and an exponential decay of correlations. The interplay between the noise amplitude and the noise-correlation time is investigated, and the complete, nonequilibrium stationary-state phase diagram of the model is obtained. We further study the dynamics of a single topological defect for various amplitudes and correlation time of the noise. Our analysis reveals that a finite correlation time promotes vortex excitations, thereby lowering the critical noise amplitude of the transition with an increase in correlation time. In the suitable limit, the resulting phase diagram allows one to estimate the critical temperature of the equilibrium BKT transition, which is consistent with that obtained from the study of the dynamics in the Gaussian white noise limit.

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.

Phase synchronization in the two-dimensional Kuramoto model: Vortices and duality.

We study a system of Kuramoto oscillators arranged on a two-dimensional periodic lattice where the oscillators interact with their nearest neighbors, and all oscillators have the same natural frequency. The initial phases of the oscillators are chosen to be distributed uniformly between (-π,π]. During the relaxation process to the final stationary phase, we observe different features in the phase field of the oscillators: initially, the state is randomly oriented, then clusters form. As time evolves, the size of the clusters increases and vortices that constitute topological defects in the phase field form in the system. These defects, being topological, annihilate in pairs; i.e., a given defect annihilates if it encounters another defect with opposite polarity. Finally, the system ends up either in a completely phase synchronized state in case of complete annihilation or a metastable phase locked state characterized by presence of vortices and antivortices. The basin volumes of the two scenarios are estimated. Finally, we carry out a duality transformation similar to that carried out for the XY model of planar spins on the Hamiltonian version of the Kuramoto model to expose the underlying vortex structure.

A simple real-space scheme for periodic Dirac operators

We address in this work the question of the discretization of two-dimensional periodic Dirac Hamiltonians. Standard finite differences methods on rectangular grids are plagued with the so-called Fermion doubling problem, which creates spurious unphysical modes. The classical way around the difficulty used in the physics community is to work in the Fourier space, with the inconvenience of having to compute the Fourier decomposition of the coefficients in the Hamiltonian and related convolutions. We propose in this work a simple real-space method immune to the Fermion doubling problem and applicable to all two-dimensional periodic lattices. The method is based on spectral differentiation techniques. We apply our numerical scheme to the study of flat bands in graphene subject to periodic magnetic fields and in twisted bilayer graphene.

Spectral singularities and asymmetric light scattering in PT-symmetric 2D nanoantenna arrays.

The intriguing physics of non-Hermitian systems satisfying parity-time (PT) symmetry has spurred a surge of both theoretical and experimental research in interleaved gain-loss systems for novel photonic devices. In this work, we investigate vertically stacked GaInP PT-symmetric nanodisk resonators arranged in two-dimensional periodic lattice using full-wave numerical simulations and scattering matrix theory. The proposed dielectric metasurface supports lasing spectral singularities with asymmetric reflection and highly anisotropic far-field scattering patterns. It offers a much broader design parameter space to control wavelength, scattering direction, and efficiency of optical emission when compared to the predominantly one-dimentional (1D) or quasi-1D structures studied so far. The proposed system with Q-factor >105 serves as a powerful platform for enhanced light-matter interaction by enabling extensive control of asymmetric light scattering, amplification, and unprecedented localization of electromagnetic fields.

Topology and Toughening of Sparse Elastic Networks.

The toughening of sparse elastic networks, such as hydrogels, foams, or meshes against fracture is one of the most important problems in materials science. However, the principles of toughening have not yet been established despite urgent engineering requirements and several efforts made by materials scientists. Here we address the above-mentioned problem by focusing on the topology of a network. We perform fracture experiments for two-dimensional periodic lattices fabricated from rubber strings and connecters with well-defined topological structures. We find that systematic increase in the largest coordination number while maintaining the average coordination number (=4) as constant leads to significant improvement in toughness. We reproduce the observed toughening behavior through numerical simulations and confirm that the stress concentration in the vicinity of a crack tip can be controlled by the topology of the network. This provides a new strategy for creating tough sparse elastic networks, especially hydrogels.

Numerical investigations and experimental measurements on the structural dynamic behaviour of quasi-periodic meta-materials

Abstract The periodic structures have various applications in vibroacoustic engineering fields since they introduce frequency band effects due to the periodic discontinuities in the geometrical or material configurations: this can lead to increased performances. This paper is focused on the analysis of quasi-periodic structures: instead of using strictly repeated patterns, a certain degree of irregularity is introduced. Quasi-periodic lattices are defined as assemblies of two different elements in two directions. The assembly follows a Thue-Morse Morphism sequence which results in asymmetry in both directions. Numerical studies and experimental measurements on two-dimensional periodic and quasi-periodic lattices are thus performed. First validations are carried out by comparing the quasi-periodic lattice modelled by using finite element model with a prototype manufactured by laser machine. The wave characteristics in quasi-periodic lattice introduce elements of novelty for designing wider frequency stop bands in low frequency ranges.

Wave Manipulation of Two-Dimensional Periodic Lattice by Parametric Excitation

Abstract Phononic crystals composed of delicately designed periodic units are used to control spatial and spectral properties of acoustic or elastic waves. The ability to manipulate transmitting waves in a real-time dynamic manner provides a new concept in programable phononic crystals and metamaterials. In this study, the mechanical waves and bandgaps in a two-dimensional spring-mass array loaded by high-frequency parametric excitation have been investigated by both analytical and numerical methods. It is found that the high-frequency parametric excitation provides an equivalent additional stiffness which leads to low-frequency bandgaps. By tuning the parametric excitation, the versatility of such a dynamic modulating technique has been presented. The waveguide structure has also been designed and studied by non-uniformly distributed parametric excitations.

Electromagnetic Crystal with Capacitive Cylinders

The article investigates an electromagnetic crystal in the form of metal cylinders with capacitive gaps in the nodes of a rectangular two-dimensional periodic lattice between metal screens forming a flat waveguide. Using a standard electrodynamic modeling system, the authors studied the diffraction of a plane wave at the boundary of an infinite layer for one coordinate and a finite layer for another coordinate, and determined the band structure of the electromagnetic crystal. The behavior of the pass and stop bands was studied as a function of the crystal parameters. An additional passband was discovered, the occurrence of which is associated with excitation of a higher-type wave of a plane waveguide. The effect of series resonance in a capacitive cylinder on the damping of a wave in the stop band is considered. The possibility of an approximate description of an electromagnetic crystal in the stop band is investigated. A number of models are proposed, obtained via numerical solution of the problem of oblique incidence of a plane wave on the crystal boundary.

Bandgaps in two-dimensional high-contrast periodic elastic beam lattice materials

We consider elastic waves in a two-dimensional periodic lattice network of Timoshenko-type beams. We show that for general configurations involving certain highly-contrasting components a high-contrast modification of the homogenization theory is capable of accounting for bandgaps, explicitly relating those to low resonant frequencies of the “soft” components. An explicit example of a square-periodic network of beams with a single isolated resonant beam within a periodicity cell is considered in detail.