Interactions beyond nearest neighbors in a periodic structure: Force analysis

Evanescent Bloch waves and complex band structure in magnetoelectroelastic phononic crystals

In the history of antenna development, periodic structures have been extensively studied and applied. Typically, through the dispersion analysis, the propagation and radiation performances of periodic structures can be predicted. There is no doubt that dispersion relation plays a vital role in periodic structures based antenna designs.
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\nIn this thesis, a systematic study on dispersion engineering of periodic structures and its applications to antenna designs is presented. The unifying goal of this study is to change the antenna properties by manipulating the dispersion relations to meet various design requirements and address the bottleneck problems. In addition, several novel antenna designs with excellent performances are proposed and presented in this thesis.
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\nFirstly, a novel one dimensional multiple periodic (MP) structure is proposed. Different from the conventional periodic structures that have single periodicity, the proposed MP structure increases the periodicity by enclosing different unit cells (UCs) into each periodic element. Through the dispersion analysis, it is found that some new features can be offered by the MP structures. With the increase of the periodicity, the separation distance between the space harmonics are reduced and several new stopbands are opened up. The former feature will lead to excitations of multiple space harmonic modes in a certain frequency range. These new features are proved to be universal for both dispersive and non-dispersive materials. Regarding to each material, the analyses are performed from media and lumped circuit perspectives. The MP structures provide room to manipulate the dispersions. By proper adjusting the geometric parameters, the dispersion relations can be engineered. Furthermore, a general dispersion relation and a general Bragg condition for MP structures are also derived.
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\nSecondly, the proposed theory of MP structures is extended to two new designs of supercell (SC) based dual-beam leaky-wave antennas. By proper engineering the dispersion relations, two space harmonic modes (m= -1 and m= -2) are excited to form two radiation beams. The two designs improve the link quality of the communication system and address the problems that are dominated by the intrinsic characteristics of the links, such as the multipath effect and the mutual interference. The first antenna is based on the microstrip lines and operates in the microwave region. It validates the feasibility of the proposed theory. On the other hand, another antenna is realized by the dielectric grating and works in the millimeter-wave (MMW) region (60-GHz). It employs a symmetrically configured SC and have better radiation performance. Both two designs can realize beam steering in the same clockwise or anti-clockwise direction against the frequency, which is rarely discussed in the previous literatures. They are of very simple structures that are easy for manufactures.
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\nThirdly, a new collimated surface-wave (SW) excited high impedance surface (HIS) leaky-wave antenna is proposed. This antenna realizes a transformation from TM0 SW mode to m= -1 leaky-wave mode. A novel substrate-integrated waveguide (SIW) based planar parabolic reflector is designed as the SW launcher (SWL). Different from the previous literatures that the SWs propagate in a cylindric wave fashion, the proposed SWL can efficiently excite collimated SWs. On the other hand, the dispersion relation is engineered by introducing an additional secondary slit on the HIS to solve the bottleneck problem, open-stopband effect. Through proper design, the open-stopband effect is suppressed successfully in the design and implementation.
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\nFinally, two antenna designs that can operate in the near-field region by using the inductive coupling principle are proposed for radio frequency identification (RFID) applications. The first antenna works in dual frequency bands (HF/UHF bands) that have nearly 70 times frequency difference. The design is challenged by the impaired UHF bandwidth due to existence of the spiral coil that operates in the HF band. A novel meander lines based diagonal symmetric structure is designed to support multiple resonance modes to expand the impedance bandwidth under the large inductive circumstance of the coil. On the other hand, the second design starts from a segmented loop antenna that behaves as a magnetic dipole and has better performance than coil. In order to meet the requirement of compactness of the RFID tag antenna, the dispersion relation is engineered by adopting multiple identical radiators. The zeroth-order resonance frequency is reduced with same dimension by this method. As a result, the physical dimension of the antenna can be further reduced at the desired frequency. Through the dispersion engineering, the antenna miniaturization is realized.
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\nIn summary, this thesis contributes to 1) the development of novel periodic structures and their relevant theory for the new features, 2) addressing practical issues by engineering the dispersion relations and 3) providing a new procedure of using the dispersion engineering for antenna designs.

In solid-state physics, including photonics and wherever periodic lattice structures occur, it is essential to establish the fundamental features associated with wave propagation through the lattice: This is achieved using Bloch waves, the reciprocal lattice, and the reduction, using periodicity, to consider the irreducible Brillouin zone. A general approach, although widely accepted as not being perfectly legitimate, is to plot the dispersion relations around the edges of the Brillouin zone. We show definitively that this can be dangerous and that an important mode of practical significance is missed if this is done in too cavalier a fashion: This missing mode is illustrated for the design of endoscopes based on spring-mass (discrete) periodic structures and photonic crystals.

Periodic rib-skin structure is composed of an array of identical elements connected in parallel and each element is a stiffened structure consisting of skins and ribs which are modeled as elastic beams and rigid bodies respectively. The periodic structure is designed to gain excellent longitudinal and flexural vibration band gaps in which corresponding waves cannot propagate in the structure freely. Firstly, longitudinal and flexural vibration mobility matrices of the skins and ribs are derived to obtain mobility equations of single-floor and multi-floor periodic elements. Then, propagation constants of the infinite rib-skin structure are calculated and an in-depth numerical study is performed to examine the relationships between the wave propagation constants and dynamic response of semi-infinite and finite periodic structures. Furthermore, flexural vibration reduction performances of the periodic rib-skin structure below 3000 Hz are discussed in detail. Finally, an experimental setup for measuring the acceleration frequency response of a two-floor periodic rib-skin structure is carried out to verify the numerical solution. Both predicted and measured results have demonstrated that longitudinal and flexural band gaps exist in the periodic rib-skin structure. The flexural vibration transmitted through the structure is substantially suppressed in a wide frequency range below 3000 Hz.

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.

Peierls transition that modifies electronic band structure has attracted intensive attention in solid state physics. In the present work, we report that a photonic analog of Peierls transition has been observed in a 1-D triangular metal diaphragm array, where the photonic bandgap structures have been designed at will by adjusting periodically metal diaphragm positions. It is shown by the numerical analysis that the transmission and radiation effect of the present periodic metal structure designed through the Peierls transition rule exhibits the behavior significantly different from an original periodic structure with each unit cell containing a metal diaphragm. The near- and far-field measurement results are in good agreement with our theoretical simulation. The present effect of photonic Peierls transition can serve as a working mechanism for designing new types of guided wave devices. It can be seen that the photonic Peierls transition would be one of the simplest ways for modifying the transport characteristics of electromagnetic waves in periodic structures.

Conventional passive periodic structures exhibit unique dynamic characteristics that make them act as mechanical filters for wave propagation. As a result, waves can propagate along the periodic structures only within specific frequency bands called the “Pass Bands” and wave propagation is completely blocked within other frequency bands called the “Stop Bands.” In this paper, the emphasis is placed on actively controlling the spectral width and location of the pass and stop bands in two-dimensional periodic struc tures depending on the direction of wave propagation. In this paper, an idealized periodic structure is considered which consists of a two-dimensional array of masses that are coupled by sets of active and passive springs with the active springs distributed in a periodic manner. The unique filtering characteristics of the active periodic structure are demonstrated by evaluating the phase constant surfaces and directivity plots, and verified by computing the response to point harmoni c loading and modal density. The presented examples demonstrate the feasibility of tuning the spectral width and location of the pass and stop bands according to the nature of the external excitation. This uniqu e directional behavior makes the applica tion of the two-dimensional periodic structures as directional mechanical filters potentially attractive for various applications such as panels with active periodic stiffeners for effective vibration isolation and acoustic stealth. .Keywords : 2-D Periodic structures, Active Periodic Structures, Propagation Surfaces, Directivity plots, Directional mechanical filters.

A system of equations is formulated to describe the propagation of surface acoustic waves in a piezoelectric substrate whose surface has a periodic structure formed by transverse grooves. Dependences of the reflection coefficient and the wave velocity on the geometry of the periodic structure (the width and the depth of the grooves) are obtained in a wide range of variation of these parameters for five different orientations in various piezoelectric crystals (quartz, lithium niobate, lithium tantalate, and langasite).

The wave propagations in the periodic structure composed of auxetic star-shaped honeycombs are investigated. The matrix of the periodic structure is filled with polymer. The effective material properties of the unit cell of the periodic structure such as the Poisson’s ratio, the Young’s modulus, and the shear modulus are varied with different filling materials. The finite element method is used to solve the wave propagation within the auxetic star-shaped honeycombs. The dispersion analysis of the band structures and iso-frequency contour are presented. The auxeticity is found to have a significant effect on the wave propagation in the honeycomb structure, and the collimation of the auxetic star-shaped honeycomb structure is also studied. Effects of different types of filling on the wave propagation are also investigated. Among three filling types, the outer-filled honeycomb has the best self-collimation effect. The self-collimation of the polymer-filled auxetic honeycomb structure presented in this study can be applied to various acoustic devices.

Cellular structures like honeycombs or reticulated micro-frames are widely used in sandwich construction because of their superior structural static and dynamic properties. Aim of this study is to evaluate the dynamic behavior of bi-dimensional cellular structures, with focus on the effect of the geometry of the unit cell composing the solid on the dynamics of the propagation of elastic waves within the structure. The characteristics of wave propagation for the considered class of cellular solids are analyzed through the Finite Element model of the unit cell and the application of the theory of periodic structures. This combined analysis yields the phase constant surfaces, which define the directions of wave propagation in the plane of the structure for assigned frequency values. The analysis of iso-frequency contour lines in the phase constant surfaces allows predicting the location and extension of angular ranges, and therefore regions within the structures, where waves do not propagate. The performance of honeycomb grids of regular hexagonal topology is compared with that of grids of various geometries, with emphasis on configurations featuring a negative Poisson's ratio behavior. The harmonic response of the considered structures at specified frequencies confirms the predictions from the analysis of the phase constant surfaces and demonstrates the strongly spatial dependent characteristics of periodic cellular structures. The presented numerical results indicate the potentials of the phase constant surfaces as tools for the evaluation of the wave propagation characteristics of this class of two-dimensional periodic structures. Optimal design configurations can be identified in order to achieve desired transmissibility levels in specified directions and to obtain efficient vibration isolation capabilities. The findings from the presented investigations and the described analysis methodology will provide invaluable guidelines for the prototyping of future concepts of honeycombs or cellular structures with enhanced vibro-acoustics performance.

In order to design reflectarray antennas within reasonable CPU times, fast and accurate numerical tools for the analysis of periodic multilayered structures are required. In this paper the method of moments in the spectral domain (MoM-SD) based on multilayered Green's functions (MGF) is applied to the analysis of periodic structures containing multilayered stacked patches in the unit cell. These multiresonant cells are potential elements for the design of reflectarray antennas. In the paper we show that the use of basis functions with edge singularities in the approximation of the current density on the patches leads to important computer memory and CPU time savings in the analysis of the periodic structures. Also, a rational fitting technique is introduced which makes it possible to obtain closed-form expressions for the scattering matrix of the periodic structures in terms of the dimension of the patches used to adjust the phase of the reflectarray elements.

Differential transmission lines (DTL) built as periodic structure (PS) can be composed of identical unit cells (UC). The PS can be used either for matching circuits to each other and or for realizing small resonators serving in filters or oscillators. In this paper, we propose new, integrated DTLs with possible characteristic impedances beyond that of classical TLs and - even more importantly - at much smaller size. The reduction factor of approximately 22 can be achieved. This result provides a real alternative for DTL applications in MMIC design below 10GHz

Multi-scale wave propagation modelling for two-dimensional periodic textile composites

Band structure analysis is central to understanding wave propagation in periodic media; however, it becomes challenging in open systems owing to energy leakage. Photonic crystal (PhC) slabs exemplify such systems, featuring periodicity in thex-yplane and finite extent in thez-direction, and supporting diverse guided-mode resonances whose interactions give rise to phenomena such as bound states in the continuum (BICs), exceptional points (EPs), and circular polarisation states. Although numerical simulations can reveal these effects, effective non-Hermitian Hamiltonians are often employed to elucidate the underlying physical mechanisms. This approach, however, relies on manually selected resonant modes and may suffer from basis incompleteness. Here, a systematic first-principles approach is presented to derive the complex band structure. The minimal channels in the scattering matrix, either open or closed, are determined by the number of propagating bulk Bloch waves. The interactions between these waves fully reveal the complex band structure. For instance, two Bloch waves predict the leading-order imaginary frequencyω''and identify accidental BICs, each associated with a dual Fabry-Pérot mode, whereas three waves reveal robust Friedrich-Wintgen and symmetry-protected BICs together with the associated linewidth behaviours. Orthogonally polarised waves are further incorporated to characterise far-field polarisation and EPs. When extended to a two-dimensional periodic structure, this framework accurately predictsω'', encompasses all known BICs, and tracks their evolution with system parameters. Overall, this first-principles approach provides a unified foundation for studying complex band structure and facilitates the exploration of light confinement in periodic media.

Wave localization in randomly disordered periodic piezoelectric rods