Finite Periodic Structures Research Articles

Fraunhofer diffraction of electromagnetic radiation by finite periodic structures with regular or irregular overall shapes.

Based on an essentially different theoretical foundation than common classical diffraction theories that remain in extensive use, this paper discusses from a fresh perspective the theoretical interpretation and prediction of the far-field diffraction of a plane monochromatic wave by a finite periodic array (PA) of identical obstacles. The theoretical treatment rests on the PA extension of the rigorous generalized multiparticle Mie solution (GMM). The truncated periodic structures may have an irregular overall shape with an arbitrary spatial orientation with respect to the incident beam. It is shown that the overall shape and intrinsic geometrical structure of a finite PA play a decisive role in giving rise to an associated far-field diffraction pattern. It is also shown that, when the physical dimensions of individual component units are much smaller than the incident wavelength, the extracted diffraction pattern of a densely packed PA of such small volumes in forward directions exhibits the distinct features predicted from classical diffraction theories for an aperture with the same shape as the overall finite PA. Several typical examples are presented, including two complementary arrays used in the specific discussion concerning Babinet's principle. There are brief preliminary discussions on some fundamental concepts in connection with the involved theoretical basis and on potential further development and application of the present GMM-PA approach.

Loop coupled resonator optical waveguides.

We propose a novel coupled resonator optical waveguide (CROW) structure that is made up of a waveguide loop. We theoretically investigate the forbidden band and conduction band conditions in an infinite periodic lattice. We also discuss the reflection- and transmission- spectra, group delay in finite periodic structures. Light has a larger group delay at the band edge in a periodic structure. The flat band pass filter and flat-top group delay can be realized in a non-periodic structure. Scattering matrix method is used to calculate the effects of waveguide loss on the optical characteristics of these structures. We also introduce a tunable coupling loop waveguide to compensate for the fabrication variations since the coupling coefficient of the directional coupler in the loop waveguide is a critical factor in determining the characteristics of a loop CROW. The loop CROW structure is suitable for a wide range of applications such as band pass filters, high Q microcavity, and optical buffers and so on.

Numerical and experimental investigation of stop-bands in finite and infinite periodic one-dimensional structures

Adding periodicity to structures leads to wavemode interaction, which generates pass- and stop-bands. The frequencies at which stop-bands occur are related to the periodic nature of the structure. Thus structural periodicity can be shaped in order to design vibro-acoustic filters for reducing vibration and noise transmission. The aim of this paper is to investigate, numerically and experimentally, stop-bands in periodic one-dimensional structures. Two methods for predicting stop-bands are described: the first method applies to infinite periodic structures using a wave approach; the second method deals with the evaluation of a vibration level difference (VLD) in a finite periodic structure embedded within an infinite one-dimensional waveguide. This VLD is defined to predict the performance in terms of noise and vibration insulation of periodic cells embedded in an otherwise uniform structure. Numerical examples are presented, and results are discussed and validated experimentally. Very good agreement between the numerical and experimental models in terms of stop-bands is shown. In particular, the results show that the stop-bands obtained using a wave approach (applied to a single cell of the structure) predict those obtained from the VLD of the corresponding finite periodic structure.

A memory efficient discontinuous galerkin finite-element time-domain scheme for simulations of finite periodic structures

A discontinuous Galerkin finite-element time-domain (DG-FETD) scheme with nonconforming interface meshes is proposed for transient simulations of finite periodic structures. A finite periodic structure is divided into an array of identical building blocks. Each building block is decomposed into a volume part and six surface parts, and they are computed and stored separately. By doing this one building block can represent the whole discretized structure, and the repeat of meshes and system matrices can be exploited to the maximum extent. This DG-FETD scheme costs much less memory compared to conventional methods in simulating finite periodic structures. © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:1929–1933, 2014

A Fast Analysis of Scattering From Large-Scale Finite Periodic Microstrip Patch Arrays Arranged on a Non-Orthogonal Lattice Using Sub-Entire Domain Basis Functions

A sub-entire domain (SED) basis function method, which was first introduced for modeling large-scale finite periodic PEC structures in free space, has been extended for fast characterization of electromagnetic scattering from an electrically large planar finite periodic microstrip patch array. The microstrip array may have a nonrectangular layout and non-orthogonal lattice configurations (e.g., hexagons or quadrangles). Based on the mixed potential integral equation, and utilizing the proposed SED basis function algorithm, the original large-scale finite periodic array of microstrip patches can be efficiently simulated by decomposing it into two problems with matrix equations of small dimensions. The first is to construct the SED basis functions for the corresponding microstrip arrays with orthogonal/non-orthogonal lattices. Three kinds of the SED basis functions are constructed, including those related to the edge patch elements, the interior patch elements, and the corner patch elements. The second is to solve the system equation with significantly reduced problem dimension as compared to the original larger problem. Based on the obtained SED basis functions, the reduced matrix equation of small size can be generated by the Galerkin procedure, and solved by use of the LU (lower-upper) decomposition-based direct solver, which results in a fast solution. The accuracy and efficiency of the developed algorithms are demonstrated by numerical tests that include the scattering from several large-scale finite periodic arrays of microstrip patches with rectangular, non-orthogonal lattices.

Dynamic characteristics of a periodic rib-skin structure

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.

Dynamic analysis of finite periodic nanoplate structures with various boundaries

Dynamic responses of the finite periodic nanoplate structures with various boundary conditions are researched by using the wave method. The dynamics model of the nanoplate structure is established based on the nonlocal Mindlin plate theory. From the Fourier transform, the vibration responses of the single layer graphene sheets under various boundary conditions are obtained by the wave method. The dynamic responses of the finite periodic nanoplate can be determined by the continuous and boundary conditions. In the numerical calculation, the natural frequencies of single layer graphene sheets computed by the presented method agree well with those of molecular dynamics (MD) simulation, which verifies the reliability of the present method. It can be found that the dynamic responses of the finite periodic nanoplates are much smaller than those of the uniform nanoplates in some frequencies which are called the band gap frequencies. It is significant to design the periodic array structure for the vibration suppression of the nanoplates. Moreover, the influences of the nanoplate width and thickness and boundary conditions on the dynamic response of the finite periodic nanoplate structures are also studied.

On the complexity of aperiodic Fourier modal methods for finite periodic structures

The Fourier modal method (FMM) is based on Fourier expansions of the electromagnetic field and is inherently built for infinitely periodic structures. When the infinite periodicity assumption is not realistic, the finiteness of the structure has to be incorporated into the model. In this paper we discuss the recent extensions of the FMM for finite periodic structures and analyze their complexity both with respect to the main discretization parameter Nˆ as well as with respect to the number of periods R. We show that among the three FMM-based approaches able to represent finiteness, the aperiodic Fourier modal method with alternative discretization has the lowest computational cost given by either O(Nˆ3log2R) or O(Nˆ2R) depending on the values of Nˆ and R. This result demonstrates that the method is highly suited for rigorous modeling of scattering from large periodic structures. For instance, for Nˆ=100 and R<1000 the complexity of the aperiodic Fourier modal method with alternative discretization is comparable to the complexity of the standard FMM.

Efficient integral equation-based analysis of finite periodic structures in the optical frequency range

The optical response of dense finite arrays of nanoparticles can be efficiently analyzed with the help of macro basis functions obtained by employing the array scanning method. This is demonstrated by analyzing optical collimation in arrays of silver nanorods. The accuracy of the solution obtained with the proposed method has been validated by comparison with solutions obtained employing the Krylov subspace iterative method. The relative error in the electric field distribution on an observation plane above the finite array is of the order of -25 dB, while the number of unknowns is reduced by a factor of 32.

Transfer-matrix approach for finite-difference time-domain simulation of periodic structures

Optical properties of periodic structures can be calculated using the transfer-matrix approach, which establishes a relation between amplitudes of the wave incident on a structure with transmitted or reflected waves. The transfer matrix can be used to obtain transmittance and reflectance spectra of finite periodic structures as well as eigenmodes of infinite structures. Traditionally, calculation of the transfer matrix is performed in the frequency domain and involves linear algebra. In this work, we present a technique for calculation of the transfer matrix using the finite-difference time-domain (FDTD) method and show the way of its implementation in FDTD code. To illustrate the performance of our technique we calculate the transmittance spectra for opal photonic crystal slabs consisting of multiple layers of spherical scatterers. Our technique can be used for photonic band structure calculations. It can also be combined with existing FDTD methods for the analysis of periodic structures at an oblique incidence, as well as for modeling point sources in a periodic environment.

Spectral Methods and Domain Decomposition for Nanophotonic Applications

Nanophotonic applications often involve large-scale problems with excessive demand on computational resources. We develop a domain decomposition method (DDM) to reduce computer memory and central processing unit (CPU) time requirements by combining the spectral element method (SEM) and the spectral integral method (SIM) for large-scale finite periodic structures. The interior scattering subdomains within each period are modeled by the SEM while the exterior scattering problem is modeled by the SIM. The interactions between neighboring subdomains are modeled by the frequency-domain version of the Riemann solver. Numerical convergence of the Riemann solver is fast and weakly dependent on the size of the system. Two sets of examples demonstrate the typical nanophotonic applications: The first periodic system is a vertical coupling waveguide based on a photonic crystal slab which opens a way to construct and simulate optical circuits. The second periodic system is a finite-sized metamaterial with an effective negative refractive index, whose edge effects are visualized and analyzed.

Numerical Design of Matching Structures for One-Dimensional Finite Superlattices

This paper describes how to design matching structures to improve the frequency characteristics of one-dimensional finite periodic structures. In particular, it deals with one-dimensional finite superlattices. A downhill simplex method is used to determine some of the structural parameters of the matching structure. Numerical examples show that this method is effective in improving the frequency characteristics of finite superlattices. key words: superlattice, downhill simplex method, electron band pass filter, equivalent circuit

Flexural wave propagation in beams with periodically attached vibration absorbers: Band-gap behavior and band formation mechanisms

This paper is concerned with flexural wave propagation and vibration transmission in beams with periodically attached vibration absorbers. Such periodic systems feature unique wave filtering characteristics that can find applications in the control of wave propagation in flexural beam structures. The study is performed by using an exact analytical approach based on a combination of the spectral element method and periodic structure theory. Both infinite and finite periodic structures are considered. An explicit expression is provided for the calculation of propagation constants and thus the complex band structures, and it is further developed to examine the effects of various system parameters on the band-gap behavior, including the position, width and wave attenuation performance of all the band gaps. The band formation mechanisms of such periodic systems are explained via both derivations and physical models, yielding explicit equations to enable the prediction of all the band edge frequencies in an exact manner without the need to calculate propagation constants. Based on these equations, explicit formulas are further derived to determine the conditions for the transition and near-coupling between local resonance and Bragg scattering, each being a unique band-gap opening mechanism.

Hybrid PMM-MoM Method for the Analysis of Finite Periodic Structures

In this paper, a hybrid method (hybrid PMM-MoM method) is presented for the effective and accurate analysis of finite periodic structures. This method divides a finite periodic structure into two parts. The inner part of an approximate infinite periodic structure is analyzed by periodic method of moment (PMM); the outer part is then analyzed by method of moments (MoM). For the finite periodic structures, the accuracy of the new method is much better than that of the pure PMM, and is almost the same as that of pure MoM. Because pure PMM uses the periodic boundary conditions, it takes much less memory resources and computation time. For hybrid PMM-MoM method, because the inner part is calculated by PMM, the calculation work concentrates on the outer part. Consequently, compared with the exact MoM, the new method saves much more memory resources and computation time, which provides a drastic reduction of unknowns.

Hybrid PMM-MOM Method for the Analysis Of Finite Periodic Structures

In this paper, a hybrid method (hybrid PMM-MOM method) is presented for the effective and accurate analysis of finite periodic structures. This method divides the finite periodic structure into two parts. The inner part of an approximate infinite periodic structure is analyzed by periodic method of moment (PMM), and the outer part is then analyzed by method of moments (MOM). For the finite periodic structures, the accuracy of the new method is much better than that of the pure PMM and almost the same as that of pure MOM. Because pure PMM uses periodic boundary conditions, it takes much less memory resources and computation time. For hybrid PMM-MOM method, because the inner part is calculated by PMM, the calculation work concentrates on the outer part. Consequently, compared with the exact MOM, the new method saves much more memory resources and computation time, which provides a drastic reduction of unknowns.

Millimeter-wave phase resonances in compound reflection gratings with subwavelength grooves

Experimental evidence of phase resonances in a dual-period reflection structure comprising three subwavelength grooves in each period is provided in the millimeter-wave regime. We have analyzed and measured the response of these structures and show that phase resonances are characterized by a minimum in the reflected response, as predicted by numerical calculations. It is also shown that under oblique incidence these structures exhibit additional phase resonances not present for normal illumination because of the potentially permitted odd field distribution. A satisfactory agreement between the experimental and numerical reflectance curves is obtained. These results confirm the recent theoretical predictions of phase resonances in reflection gratings in the millimeter-wave regime, and encourage research in this subject due to the multiple potential applications, such as frequency selective surfaces, backscattering reduction and complex-surface-wave-based sensing. In addition, it is underlined here that the response becomes much more complex than the mere infinite analysis when one considers finite periodic structures as in the real experiment.

Analysis of finite-sized guided-mode resonant gratings using the fast multipole boundary element method

Guided-mode resonant grating filters are dispersive devices that utilize resonance anomalies. When they are modeled as finite imperfect periodic structures, it is time-consuming to calculate their optical properties precisely, and huge computational memories are needed to accommodate the large analytical domain. This limits the numerical performance, and so existing reports, which use the conventional boundary element method, refer to structures with less than 100 periods. This paper shows that general optical properties and the impact of production error distributions can be calculated for guided-mode resonant grating filters with several hundred periods using the fast multipole boundary element method.

Multiobjective topology optimization for finite periodic structures

Many engineering structures consist of specially-fabricated identical components, thus their topology optimizations with multiobjectives are of particular importance. This paper presents a unified optimization algorithm for multifunctional 3D finite periodic structures, in which the topological sensitivities at the corresponding locations of different components are regulated to maintain the structural periodicity. To simultaneously address the stiffness and conductivity criteria, a weighted average method is employed to derive Pareto front. The examples show that the optimal objective functions could be compromised when the total number of periodic components increases. The influence of thermoelastic coupling on optimal topologies and objectives is also investigated.

Space-FFT-accelerated marching-on-in-degree methods for finite periodic structures

A fast, yet unconditionally stable, solution of time-domain electric field integral equations (TD EFIE) pertinent to the scattering analysis of uniformly meshed and/or periodic conducting structures is introduced. A one-dimensional discrete fast Fourier transform (FFT)-based algorithm is proffered to expedite the calculation of the recursive spatial convolution products of the Toeplitz–block–Toeplitz retarded interaction matrices in a new marching-without-time-variable scheme. Additional saving owing to the system periodicity is concatenated with the Toeplitz properties due to the uniform discretization in multi-level sense. The total computational cost and storage requirements of the proposed method scale as O(Nt2Nslog Ns) and O(Nt Ns), respectively, as opposed to O(Nt2Ns2) and O(NtNs2) for classical marching-on-in-order methods, where Nt and Ns are the number of temporal and spatial unknowns, respectively. Simulation results for arrays of plate-like and cylindrical scatterers demonstrate the accuracy and efficiency of the technique.

Determination of ultrasonic vibration modes of a graded solid cylinder using a modified wave‐expansion technique.

Research on wave propagation in cylinders—hollow, solid, or graded—has been driven by its potential use in the inspection of long‐span engineering structures. Its practical application largely depends on the understanding of modal vibration inside the structure. Typically, the plane wave approach is used to model phase velocity, group velocity and dispersion curves in periodic systems. This proves cumbersome when used to model finite periodic structures, semifinite plates, or concentric cylinders. Here, a simpler approach to determine the vibration modes and dispersion curves of a solid concentric cylinder using a modified wave‐expansion approach, used mostly in phononic crystals research, is proposed. The advantage of this technique is the possible application on a wide variety of cylindrical geometries and graded structures with various transverse sections. The results of the proposed technique are compared with those obtained by the resonance scattering theory for the simple case of a solid cylinder with a single core having different mechanical properties. [Work partially sponsored by SEP‐CONACYT through project # 58951.