Periodic Green's Function Research Articles

Electromagnetic scattering from perfectly conducting periodic surfaces by transforming into equivalent boundary condition

AbstractIn this study, electromagnetic scattering from perfectly conducting periodic surface have been solved by means of transformation of problem into equivalent problem, that is scattering from plane represented by high order inhomogeneous impedance boundary condition (IBC). High order impedance functions are determined by function of the roughness of the surface. Then, transformed equivalent problem is solved by means of series expansion method using Floquet modes. This transformation makes the problem simple formulation and computational effectively without involving calculation of slowly converging periodic Green's function. Results and computational times obtained by transform method and those obtained by Method of Moment (MoM) technique are compared. Good agreements are observed in results. It is also observed that transform method needs much less computational time than MoM method. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1997–2000, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23567

Accurate FEM/BEM Simulation of Wireless Passive Surface Acoustic Wave Sensors

The wireless passive surface acoustic wave (SAW) impedance-loaded sensors offer new and exciting perspectives for remote monitoring and controlling of moving parts, even in harsh environments. However, the classical simulation models such as equivalent circuit model (ECM) and coupling of modes (COM) model are not rigorous enough for these sensors since these sensors usually work in the RF band. We try to make it more accurate by using FEM/BEM (combined finite element method and boundary element method) model to determine some key parameters used in the COM model of impedance-loaded SAW sensors. In this paper, we use periodic Green function which is the basis of FEM/BEM model to determine the key parameters. The result of measurements on real SAW is nearly in accordance with that of simulations.

A higher order numerical method for 3-d double periodic electromagnetic scattering problem

We develop a method for 3D doubly periodic electromagnetic scattering. We adapt the Muller integral equation formulation of Maxwell's equations to the periodic problem, since it is a Fredholm equation of the second kind. We use Ewald splitting to efficiently calculate the periodic Green's functions. The approach is to regularize the singular Green's functions and to compute integrals with a trapezoidal sum. Through asymptotic analysis near the singular point, we are able to identify the largest part of the smoothing error and to subtract it out. The result is a method that is third order in the grid spacing size. We present results for various scatterers, including a test case for which exact solutions are known. The implemented method does indeed converge with third order accuracy. We present results for which the method successfully resolves Wood's anomaly resonances in transmission.

546 3次元Maxwell方程式直交異方性周期境界値問題における高速多重極境界要素法(OS12.境界要素法の高度化と最新応用(2),オーガナイズドセッション)

This presentation discusses an FMM (Fast Multipole Method) for orthotropic periodic boundary value problems for Maxwell's equations in 3D. We consider scattering problems where metallic or dielectric scatters are aligned periodically with different periodic lengths in two orthogonal directions. The multipole and local expansions of the periodic Green function are used for accelerating the solution of the integral equations. We derive Fourier integral expressions for the periodic Green function and its derivatives, which are essential ingredients in our formulation. The cells used in the proposed FMM are not cubic because all the cells in the FMM tree are geometrically similar to the unit cell of the periodic structure. We show modifications required to the FMM algorithm for introducing non-cubic cells. Through numerical tests we conclude that the proposed method is efficient and accurate.

Comments on "Comparative Analysis of Acceleration Techniques for 2-D and 3-D Green's Functions in Periodic Structures Along One and Two Directions"

The paper by Valerio et al. (2007), contains a lot of useful information on the problem of efficient calculation of the periodic Green's function. However, making their review of the existing methods, the authors have missed one approach, which, in our opinion, deserves attention as well. That approach was originally proposed by Ivanishin (2002) for calculating the Green's function of a rectangular waveguide. Later, it was generalized and successfully applied in (Skobelev and Mukhamedov, 1993) and (Skobelev, 2002) for solving 2D and 3D problems associated with analysis of 1D- and 2D-periodic arrays, respectively.

Time Domain Weyl's Identity and the Causality Trick Based Formulation of the Time Domain Periodic Green's Function

Time domain Floquet modes for analyzing scattering (radiation) from periodically arranged planar scatterers (sources) have been presented recently in a series of papers by Felsen and Capolino. In these papers, they have investigated phenomenological details of transient scattering (radiation) by such structures. Typically, time domain Floquet modes are derived either by Fourier transforming their frequency domain counterparts or using spatial synthesis using Poisson summation. This paper proposes an alternative means to develop similar expressions and is motivated by the possibility of easily extending the method to the analysis of periodic structures in a layered medium. As will be shown, the time domain Weyl's identity can be used to construct the periodic Green's function using both homogeneous and inhomogeneous plane waves. We will also derive expressions using the Causality Trick. This technique involves using time-symmetric (non-causal) signals; however, it yields much simpler expressions that involve only homogeneous time domain plane waves and the results are identical to those obtained using the time domain Weyl's identity for the time region of interest. Finally, we will use the Whittaker formulation, which relies on real signals, to derive time domain Floquet modes. This formulation is applicable for simultaneous excitation only.

Comparative Analysis of Acceleration Techniques for 2-D and 3-D Green's Functions in Periodic Structures Along One and Two Directions

The problem of accelerating the calculation of the periodic Green's function is addressed here for both 3-D and 2-D free-space configurations. In the 3-D case, periodicity is considered both along one axis and along two, generally skew, axes. A comprehensive review of the existing methods is first presented and some extensions are developed. The possibility of treating the case of complex phase shifts between unit cells, necessary for the study of complex modes in periodic structures, is also investigated. Comparisons among the various acceleration methods are performed, thus providing fundamental information on their actual efficiency in typical problems.

An FMM for periodic boundary value problems for cracks for Helmholtz' equation in 2D

AbstractThis paper presents an FMM (fast multipole method) for periodic boundary value problems for Helmholtz' equation in 2D. The periodic Green function is an important ingredient in our formulation, which is computed efficiently with the help of the Fourier analysis. We validate the proposed method by comparing the obtained numerical results with those computed with the conventional approach. We then apply the proposed method to the analysis of scattering problems for periodic array of cracks and plot the energy transmittance vs wave numbers. The stopband and related phenomena are observed clearly. Copyright © 2007 John Wiley & Sons, Ltd.

Optical scattering from a planar array of finite-length metallic carbon nanotubes

A model is developed for optical scattering from planar arrays of finite-length single-wall metallic carbon nanotubes. The scattered field is predicted using a periodic Green's function for the array, which includes all electromagnetic interactions, and a quantum conductance function ${\ensuremath{\sigma}}_{\mathit{cn}}(\ensuremath{\omega})$ for the carbon nanotubes. It is found that for both individual carbon nanotubes and nanotube arrays, the optical far scattered field is proportional to ${\ensuremath{\sigma}}_{\mathit{cn}}(\ensuremath{\omega})$, so that scattering characteristics are governed by effects associated with electronic transitions. This is in strong distinction to the case for far-infrared arrays, where mutual electromagnetic coupling effects were previously found to be very important for a wide range of broadside nanotube spacings. Furthermore, due to strong damping, single-wall nanotubes do not exhibit longitudinal current resonances (optical antenna effects) associated with the finite length of the tubes, which is also quite different from the far-infrared result.

Periodic Green's function for skewed 3‐D lattices using the Ewald transformation

AbstractWe apply the Ewald acceleration technique to the efficient evaluation of periodic Green's functions (GFs) for 3‐D skewed lattices like those arising in electromagnetic/photonic band‐gap (EBG/PBG) structures and metamaterials (MTMs). We develop the expression for the optimal value of an associated splitting parameter, derive the gradient of the scalar potential GFs and address the extraction of singularity for both vector/scalar potential GF and their curl/gradient. Several numerical implementation issues are also discussed leading to further enhancement in computational speed, accuracy, and numerical stability. Finally, the accuracy of the developed GFs is verified against well established algorithms. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1353–1357, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22429

Spatial Sensitivity Distribution of Surface Acoustic Wave Resonator Sensors

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The sensitivity distribution of surface acoustic wave (SAW) resonator sensors is investigated by theoretical and experimental means. It is shown that the sensitivity to mass loading varies strongly across the surface due to the confinement of acoustic energy toward the center of the device. A model is developed for this phenomenon based on the extraction of coupling of modes parameters from a rigorous boundary element method analysis based on a periodic Green's function. As SAW sensors for many applications include a layer covering the electrodes, a new technique is introduced to account for the mechanical interactions with buried electrodes. Using this technique, the sensitivity calculations are found to be in good agreement with measurements. It is also shown that while changes in other parameters influence sensitivity, it is velocity change that most strongly determines overall frequency change. </para>

Calculation of the local electric field for an infinite array of conducting nanosized objects

The electric field for an infinite array of conducting nanosized objects in two-dimensional space has been calculated. The mirror symmetry for this physical problem has been introduced. By taking into account this symmetry, we transform the original problem into an infinite two-dimensional array of nanosized objects with the same solution. The electric field equation of the model has been successfully constructed using a single-layer potential of the periodic Green function. The electric field operator has been introduced. This mathematical approach yields a solution for determining the optimum structure of nanosized electronic devices such as carbon nanotube-based field emitters.

Doubly periodic volume–surface integral equation formulation for modelling metamaterials

A generalised volume-surface integral equation is extended by way of the periodic Green's function to model arbitrarily complex designs of metamaterials consisting of high-contrast inhomogeneous anisotropic material regions as well as metallic inclusions. The unique aspect of the formulation is the integration of boundary and volume integral equations to increase modelling efficiency and capability. Specifically, the boundary integral approach with equivalent surface currents is adopted over regions consisting of piecewise homogeneous materials as well as metallic perfect electric/magnetic conductor inclusions, whereas the volume integral equation is employed only in inhomogeneous and/or anisotropic material regions. Because the periodic Green's function only needs to be evaluated for the equivalent surface currents enclosing an inhomogeneous and/or anisotropic region, matrix fill time is much less as compared to using a volume formulation. Furthermore, the incorporation of curvilinear finite elements allows for greater geometrical modelling flexibility for arbitrarily shaped high-contrast regions found in typical designs of engineered metamaterials

Numerical study on surface plasmon polariton behaviors in periodic metal-dielectric structures using a plane-wave-assisted boundary integral-equation method

A novel hybrid technique based on the boundary integral-equation method is proposed for studying the surface plasmon polariton behaviors in two-dimensional periodic structures. Considering the periodicity property of the problem, we use the plane-wave expansion concept and the periodic boundary condition instead of using the periodic Green's function. The diffraction efficiency can then be readily calculated once the equivalent electric and magnetic currents are solved that avoids invoking the numerical calculation of the radiation integral. The numerical validity is verified with the cases of highly conducting materials and practical metals. Numerical convergence can be easily achieved even in the case of a large incident angle as 80o. Based on the numerical scheme, a metal-dielectric wavy structure is designed for enhancing the transmittance of optical signal through the structure. The excitation of the coupled surface plasmon polaritons for the high transmission is demonstrated.

３次元Ｍａｘｗｅｌｌ方程式の周期境界値問題における高速多重極境界要素法

This paper presents an FMM for periodic boundary value problems for Maxwell's equations in 3D. The periodic Green function and its derivatives, which are essential to the present method, are derived with Fourier analysis. We verify the proposed method by comparing the obtained numerical results with analytic solutions for models of the multi-layered dielectric slab. We then apply the proposed method to scattering problems for periodic two dimensional arrays of dielectric spheres and compare the obtained energy transmittances with those in the previous studies. We observe good agreements.

The linear-wave response of a periodic array of floating elastic plates

The problem of an infinite periodic array of identical floating elastic plates subject to forcing from plane incident waves is considered. This study is motivated by the problem of trying to model wave propagation in the marginal ice zone, a region of ocean consisting of an arbitrary packing of floating ice sheets. It is shown that the problem considered can be formulated exactly in terms of the solution to an integral equation in a manner similar to that used for the problem of wave scattering by a single elastic floating plate, the key difference here being the use of a modified periodic Green function. The convergence of this Green function in its original form is poor, but can be accelerated by a transformation. It is shown that the results from the method satisfy energy conservation and that in the particular case of a fixed rigid rectangular plate which spans the periodicity uniformly the solution reduces to that for a two-dimensional rigid dock. Solutions for a range of elastic-plate geometries are also presented.

Electromagnetic scattering from finite-length metallic carbon nanotubes in the lower IR bands

A model is presented for electromagnetic scattering from infinite planar arrays of finite-length metallic carbon nanotubes, and isolated nanotubes, in the lower IR bands. The scattered field is predicted using a semiclassical formulation based on a periodic Green's function for the array, and a quantum conductance function for the carbon nanotubes. The finite length of the tubes is accounted for electromagnetically by imposing a boundary condition on the tube ends. Scattering characteristics are investigated for various armchair carbon nanotube array configurations, as well as for isolated nanotubes. The principle observations of this study are that longitudinal (end-to-end) coupling between carbon nanotubes is not very important, although transverse (side-to-side) coupling in an array environment shifts and broadens resonance line shapes compared to the isolated tube case. Nanotube length and radius also play critical roles in governing scattering characteristics.

Theoretical analysis of micro-machined ultrasonic transducer using a simple 1-D model

Micro-machined ultrasonic transducers (MUT) appear as an attractive alternative to standard bulk transducers mainly based on PZT ceramic actuators. However, the simulation of these new devices requires one to take correctly into account their operating conditions. Particularly, most of the MUT structures are periodic, associating a very large number of elementary actuators excited in phase. Furthermore, the development of an equivalent to the Mason model for MUTs would help in the promotion of this new kind of transducers. In this work, we propose a very simple model based on the material resistance theory to describe the flexural motion of a MUT. It is associated with a periodic Green's function development to take into account radiation in water. Basic working principles of MUT then are deduced from computing results, which coincides with already published data on that topic.

Schlömilch series and grating sums

We consider sums over the set of positive integers relevant to construction of periodic Green's functions for diffraction gratings and similar problems, and provide a general formula for a combination of Bessel functions of complex order and complex powers of distance from the origin. This general formula is investigated in a number of particular cases, and in particular we provide expressions which enable sums of functions with Neumann series to be re-expressed as combinations of hypergeometric series. We also investigate sums of Neumann functions of integer order, using analytic continuation techniques to provide formulae for their evaluation which we demonstrate are accurate and efficient in both the high and low frequency regions. We also exhibit sums which may be evaluated analytically, and recurrence formulae linking sums.

Efficient Analysis of Periodic Structures with Arbitrary Shape Using Volume-surface Integral Equation Method

A coupled volume-surface integral equation method is presented for analyzing arbitrarily shaped periodic structure of mixed dielectric and conducting objects. Free space periodic Green’s function is used in the formulation of both integral equations. In the method of moments solution to the integral equations, the target is discretized using triangular patches for conducting surfaces and tetrahedral cells for dielectric volume. Ewald’s method is used to accelerate the convergence of computing each element in the impedance matrix. Numerical results are presented to demonstrate the accuracy and efficiency of the technique. I. Introduction Electromagnetic analysis of periodic structures has a wide range of application. The spectral method of moments is a power tool in analyzing periodic structures [1]. To improve the flexibility in modeling various geometries, the Rao-Wilton-Gillison (RWG) [2] triangular discretization was reported in [3]. In this paper, the coupled volume-surface integral equation method [4] is presented for the electromagnetic wave scattering from arbitrarily shaped periodic structures composed of dielectrics and conducting objects. Free space periodic Green’s function is used in the formulation of both integral equations. The triangular patches and tetrahedral elements are used to mesh the dielectric and conducting objects, respectively. The coupled integral equations are solved by the method of moments (MoM) [5] with surface RWG and volume SWG basis functions [6] discretization. To rapidly generate the impedance elements, Ewald’s method [7] is used to speed the computation of the periodic Green function due to the fact that the periodic Green function, which is a summation of series, converges very slowly. The proposed technique can also be extensively applied to analyze the periodic structure consisted of inhomogeneous dielectric material and conducting body. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed technique for analyzing the scattering problem of arbitrarily shaped periodic structure.