Band Gaps In Periodic Structures Research Articles

Reduced-order modeling for time domain analysis of finite periodic structures with absorbing boundary conditions

A reduced-order model of finite periodic structures with absorbing boundary conditions (ABCs) and localized time-dependent excitations is proposed. 1D-periodic structures whose cells can represent any 2D or 3D arbitrary substructures are considered. The key steps for modeling an ABC in the time domain are: (i) expression of the impedance matrix with the wave finite element (WFE) method; (ii) partial rational decomposition of the impedance matrix; (iii) introduction of vectors of supplementary variables to describe the ABC in the time domain. In this paper, a model reduction approach is proposed to speed up the computation of the ABCs. The focus is on the reduction of the number of internal degrees of freedom of substructures, and the reduction of the number of degrees of freedom at the substructure interfaces. The approach allows the consideration of complex substructures with large FE models, and the computation of the time response of periodic structures at a low computational cost. Numerical experiments are carried out to highlight the relevance of the proposed approach. Specifically, the analysis of a metamaterial structure containing nonlinear resonant substructures is carried out with the proposed approach. Results show that, when compared to the linear case, band gaps in periodic structures with nonlinear substructures can be significantly improved.

Generation of Hofstadter's butterfly spectrum using circular arrays of microring resonators.

Hofstadter's butterfly spectrum, which characterizes the energy bands of electrons in a 2D lattice under a perpendicular magnetic field, has been emulated and experimentally characterized in periodic bandgap structures at microwave and acoustic frequencies. However, measurement of the complete spectrum at optical frequencies has yet to be demonstrated. Here, we propose a simple topological photonic structure based on a circular array of microrings with periodic resonant frequency detunings that can be implemented on an integrated optics platform. We show that this ring-of-rings structure exactly emulates the Harper equation and propose an experimental approach for measuring Hofstadter's butterfly spectrum at optical frequencies.

Band gap transmission in periodic bistable mechanical systems

We theoretically and numerically investigate the supratransmission phenomenon in discrete, nonlinear systems containing bistable elements. While linear waves cannot propagate within the band gaps of periodic structures, supratransmission allows large-amplitude waves to transmit energy through the band gap. For systems lacking bistability, the threshold amplitude for such energy transmission at a given frequency in the linear band gap is fixed. We show that the topological transitions due to bistability provide an avenue for switching the threshold amplitude between two well-separated values. Moreover, this versatility is achieved while leaving the linear dispersion properties of the system essentially unchanged. Interestingly, the behavior changes when an elastic chain is coupled to bistable resonators (in an extension of the well-studied linear locally resonant metamaterials). Here, we show that a fraction of the injected energy is confined near the boundary due to the resonators, providing a means of regulating the otherwise unrestrained energy flow due to supratransmission. Together, the results illustrate new means of controlling nonlinear wave propagation and energy transport in systems having multi-stable elements.

In-Plane Vibration Response of the Periodic Viaduct on Saturated Soil under Rayleigh Surface Wave

In this study, the in-plane vibration response of the periodic viaduct on saturated soil under Rayleigh surface wave is studied. The Floquet transform method is used to decompose Rayleigh surface wave into a set of spatial harmonic waves. Considering the periodic condition of the viaduct, the wave number domain dynamic response of the periodic viaduct on saturated soil subjected to Rayleigh surface wave excitation is obtained by the transfer matrix method. Then the space domain dynamic response is retrieved by means of the inverse Floquet transform. Numerical results show that when the periodic viaduct is undergoing in-plane vibration, there exist three kinds of characteristic waves corresponding to axial compression, transverse shear, and bending vibration. Furthermore, when the frequency of Rayleigh wave is within the pass band of the periodic viaduct, the disturbance propagates over a very long distance and the attenuation of the wave motion far from the source is determined by the characteristic wave with the smallest attenuation, while the vibration attenuates rapidly and propagates in a short distance when the frequency of excitation source is in the range of band gap of periodic structure.

A Systemized View of Superluminal Wave Propagation

This paper reviews earlier studies on superluminal wave propagation in anomalously dispersive media that have been carried out in the electronic, microwave, and optical regimes. Those studies are relevant to observation of modulated Gaussian pulses transmitted through various media at speeds apparently faster than c without distortion. This paper also presents the condition for superluminal propagation that is established based on the magnitude-phase relation of a causal and minimum-phase filter. Since the condition is modeled on the basis of filter theory, it is applicable to all types of media. A terahertz experiment with a periodic bandgap structure is also included to illustrate superluminal propagation.

Study of periodic band gap structure of the magnetized plasma photonic crystals

The characteristics of the periodic band gaps of the one dimension magnetized plasma photonic crystals are studied with the piecewise linear current density recursive convolution (PLCDRC) finite-differential time-domain (FDTD) method. In frequency-domain, the transmission coefficients of electromagnetic Gaussian pulses are computed, and the effects of the periodic structure constant, plasma layer thickness and parameters of plasma on the properties of periodic band gaps of magnetized photonic crystals are analyzed. The results show that the periodic band gaps depend strongly on the plasma parameters.

Resonance-induced band gaps in a periodic waveguide

The band gaps in periodic structures are usually regarded as being induced by the Bragg resonances. Until the recent years, the non-Bragg nature resonances were not taken into account in analysing and computing the band gaps, though it can exist in all kinds of waveguides with periodic structures. Here, the resonance-induced band gaps in a periodic acoustic duct are investigated extensively and a graphical method is introduced to analyse the dependence of these resonances on the duct geometry. With this method, it becomes quite easy to estimate the frequency band gaps of the waveguide and shape the band structures by choosing the proper geometric parameters. Our analysis show that the location and the width of the band gap are closely related to the wavenumber and the amplitude of the wall corrugations, and the non-Bragg resonance can result in the obvious band gap when the wall wavenumber is close to the cut-off frequency of the first mode.

Simultaneous complete elastic and electromagnetic band gaps in periodic structures

We study elastic and electromagnetic properties in periodic structures and present “deaf and blind” structures, i.e. materials having simultaneous complete phononic and photonic band gaps, that is, transverse electric (TE) and transverse magnetic (TM) electromagnetic waves, pure shear elastic waves, and mixed shear and dilatation elastic waves, cannot propagate within these structures. These composite materials can control the flow of light and sound at the same time. The existence of complete gaps for electromagnetic and elastic waves can lead to the simultaneous localization of light and sound, a novel phenomena that can have strong influence on photon–phonon interactions. We study the dependence of the simultaneous and complete gaps on material parameters to provide design guidelines on how to create these photonic–phononic crystals.

Transmission lines on periodic bandgap metamaterials: from microwaves to optics applications

The paper presents a transmission line (TL) approach developed in the field of metamaterials for planar devices and circuits. Materials under consideration are dielectric substrates, containing or not nanoscaled ferromagnetic inclusions, that can be arranged in a periodic way with specific impedances and phase velocities. They form photonic or periodic bandgap (PBG) substrates, on which planar guiding structures are deposited. These PBG substrates show some similarities with integrated Bragg reflector topologies previously investigated in the optical range. Using the TL concept, defect modes in PBG planar devices are studied. A theoretical explanation of defect mode operation, which is validated by measurement on a dielectric PBG structure, is proposed. It is then transposed to a magnetic PBG structure based on arrays of nanoscaled ferromagnetic nanowires. The effect on its frequency response of varying the defect length is discussed, in order to design planar sensors and filters. Extension of the magnetic PBG structure for applications in the optical range is finally proposed.

Silicon micromachined EBG resonator and two-pole filter with improved performance characteristics

A novel micromachined resonator at 45 GHz based on a defect in a periodic electromagnetic bandgap structure (EBG) and a two-pole Tchebyshev filter with 1.4% 0.15 dB equiripple bandwidth and 2.3 dB loss employing this resonator are presented in this letter. The periodic bandgap structure is realized on a 400 μm thick high-resistivity silicon wafer using deep reactive ion etching techniques. The resonator and filter can be accessed via coplanar waveguide feeds.

Broadband Characterization of Complex Periodic EBG Structures:An FDTD/Prony Technique Based on the Split-Field Approach

The main challenge in this paper is to provide an efficient and powerful computational engine to effectively characterize the performance of periodic electromagnetic band-gap (EBG) structures composed of complex scatterers of dielectrics and conductors of arbitrary configurations. The finite difference time domain (FDTD) full wave analysis technique with periodic boundary conditions/perfectly matched layers (PBCs/PMLs) integrated with the Prony method is applied to accurately obtain the interactions of EM waves with the periodic band-gap structures and detail their broadband behaviors. The split-field approach is used to discretize the Floquet-transformed Maxwell's equations derived in the periodic structures. The developed engine is successfully applied to characterize and create insight into the electromagnetic performances of different classes of challenging EBG structures in the areas of frequency selective surfaces (FSSs), dielectric band-gap materials, and left-handed (LH) composite meta-materials. It is demonstrated that the FDTD/Prony technique utilizing the split-field approach is a very useful method in analyzing complex periodic electromagnetic band-gap structures.

Band gaps and localization in acoustic propagation in water with air cylinders

Multiple scattering of waves leads to many peculiar phenomena such as complete band gaps in periodic structures and wave localization in disordered media. Within a band gap, excitations are evanescent; when localized, they remain confined in space until dissipated. Here, we report acoustic band gap and localization in a two-dimensional system of air cylinders in water. Exact numerical calculations reveal the unexpected result that localization is independent of the precise location or organization of the scatterers. Localization occurs within a finite region of frequencies, coincident with the complete band gap for a periodic lattice of scatterers. Inside the gap or localization regime, a previously uninvestigated collective behavior of the cylinders appears.

Acoustic band gaps and localization in water with air-cylinders

Multiple scattering of waves may lead to many peculiar phenomena such as complete band gaps in periodic structures and wave localization in disordered media. Within the gaps excitations are evanescent, when localized waves remain confined in space until dissipated. Here we show that waves are not always localized in 2-D, but when localized there is a coincidence between the gaps in periodic structures and the ranges of localized states in the case of acoustic propagation through water containing an array of air-cylinders. We report that inside the gap or localization regime, an interesting collective behavior of the cylinders appears. Exact numerical calculations reveal the unexpected result that localization is relatively independent of the precise location or organization of the scatterers.