Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation

The surface integral formulation is a flexible, multiscale and accurate tool to simulate light scattering on nanostructures. Its generalization to periodic arrays is introduced in this paper. The general electromagnetic scattering problem is reduced to a discretizated model using the Method of Moments on the surface of the scatterers in the unit cell. The study of the resonances of an array of bowtie antennas illustrates the main features of the method. When placed into an array, the bowtie antennas show additional resonances compared to those of an individual antenna. Using the surface integral formulation, we are able to investigate both near-field and far-field properties of these resonances, with a high level of accuracy.     

Analysis of sound absorption by periodic structures using a hybrid-boundary element/mode expansion method

An improved hybrid method is proposed for analyzing sound scattering by a periodic structure. Part of the scattered field formulated with the mode expansion method is combined with other components of the field formulated with the boundary integration method in one period of the structure. Structures treated by this method can have arbitrary periodic forms made of locally reactive boundaries and porous materials. The oblique incident absorption coefficient of the structure is obtained simply from the reflection factor calculated for each elemental wave of the scattered field. The accuracy of the method is demonstrated by the agreement between calculated and measured values of the normal absorption coefficient of some test structures.     

A general method for analyzing em wave scattering from arbitrarily shaped two-dimensional periodic surfaces

In this paper the Method of Lines (MoL) is successfully extended to solve the EM wave scattering problems of periodic surfaces with arbitrary profile. As examples, the scattering coefficients of space harmonics of corrugated and sinusoidal surfaces are calculated. The results are in good agreement with available data from Wirgin and from A.K.Jordan et al. In addition, the results of comb structure are also calculated. The flexibility and less computation of this method make it eligible for analyzing various two-dimensional periodic structures.     

Phase space structure and chaotic scattering in near-integrable systems

We investigate the bifurcation phenomena and the change in phase space structure connected with the transition from regular to chaotic scattering in classical systems with unbounded dynamics. The regular systems discussed in this paper are integrable ones in the sense of Liouville, possessing a degenerated unstable periodic orbit at infinity. By means of a McGehee transformation the degeneracy can be removed and the usual Melnikov method is applied to predict homoclinic crossings of stable and unstable manifolds for the perturbed system. The chosen examples are the perturbed radial Kepler problem and two kinetically coupled Morse oscillators with different potential parameters which model the stretching dynamics in ABC molecules. The calculated subharmonic and homoclinic Melnikov functions can be used to prove the existence of chaotic scattering and of elliptic and hyperbolic periodic orbits, to calculate the width of the main stochastic layer and of the resonances, and to predict the range of initial conditions where singularities in the scattering function are found. In the second example the value of the perturbation parameter at which channel transitions set in is calculated. The theoretical results are supplemented by numerical experiments.     

Scattering of light by a periodic structure in the presence of randomness VII: Application of statistical detection test

Detection of periodic structures, hidden in random surfaces has been addressed by us for some time and the ‘extended matched filter’ method, developed by us, has been shown to be effective in detecting the hidden periodic part from the light scattering data in circumstances where conventional data analysis methods cannot reveal the successive peaks due to scattering by the periodic part of the surface. It has been shown that if r ₀ is the coherence length of light on scattering from the rough part and Λ is the wavelength of the periodic part of the surface, the extended matched filter method can detect hidden periodic structures for (r ₀/Λ) ≥ 0.11, while conventional methods are limited to much higher values ((r ₀/Λ) ≥ 0.33). In the method developed till now, the detection of periodic structures involves the detection of the central peak, first peak and second peak in the scattered intensity of light, located at scattering wave vectors v _x = 0, Q, 2Q, respectively, where Q = 2Gp/Λ, their distinct identities being obfuscated by the fact that the peaks have width Δv _x = 2Gp/r ₀ ≫ Q. The relative magnitudes of these peaks and the consequent problems associated in identifying them is discussed. The Kolmogorov-Smirnov statistical goodness test is used to justify the identification of the peaks. This test is used to ‘reject’ or ‘not reject’ the null hypothesis which states that the successive peaks do exist. This test is repeated for various values of r ₀/Λ, which leads to the conclusion that there is really a periodic structure hidden behind the random surface.      

Determination of the phonon spectrum from coherent neutron scattering by polycrystals

As proposed by Bredovet al. [2, 3] the phonon spectrum can be obtained approximately from coherent neutron scattering by polycrystals if suitable averages over scattering angles are considered. The accuracy of this method is estimated by comparison with analytical results for simple lattice models (discussed here in particular for Aluminium). The errors are about 5% for low order moments and about 50% near van Hove singularities for “cold” neutrons (wavelength of the order of the nearest-neighbour-distance).     

相干声线跟踪理论中的周期界面迭代散射 仿真方法*

界面声反射模拟是室内复杂声学现象仿真的关键。针对传统声学仿真方法对于周期散射结构存在条件下声场仿真精度较低的问题,本文发展了一种基于迭代散射模型的室内相干声线跟踪法。此方法以经典的相干声线跟踪法为基础,将室内中常见的周期散射结构进行几何形状上的简化处理,然后依据周期散射定理给出声波在界面上的散射方向及能量,并将原始声线迭代分裂为相应的散射子声线,继续对其跟踪处理,此迭代散射模型对周期散射结构上的界面散射现象进行了准确的模拟。数值验证结果表明,本文方法可以有效地在低频段提高室内声场仿真精度,可为具有复杂散射现象的室内仿真提供新思路。     

Volume integral equation for analysis of three-dimensional scattering from dielectric frequency-selective structures

The space-domain volume integral equation method is presented for the analysis of three-dimension scattering from dielectric frequency-selective structures involving homogeneous and inhomogeneous lossy materials. The method directly solves for the electric field in order to easily enable the periodic boundary conditions in the spatial domain. The special basis and test functions are introduced to deal with the current continuity in periodic boundaries. The computation of the spatial domain periodic Green’s function (PGF) is accelerated by the modified Ewald transformation, so that a very thick periodic structure can also be analyzed efficiently and accurately. The PGF mentioned above is of free-space type and very smooth and amenable to interpolation. Thus, optimized interpolation procedures for the PGFs can be applied, resulting in a considerable reduction of matrix-filling time without any significant influence on the accuracy. A study of the scattering parameters of a multilayered dielectric periodic structure is accomplished by imposing the boundary conditions in terms of the multimode scattering matrix. Numerical examples show the reliability and accuracy of the proposed method.     

Compacton and periodic wave solutions of the non-linear dispersive Zakharov-Kuznetsov equation

In this paper, the nonlinear dispersive Zakharov-Kuznetsov equation is solved by using the sine-cosine method. As a result, compactons, periodic, and singular periodic wave solutions are found.      

A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter

We deal with nonlinear T-periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation. In the end two applications are done. The key tool for proving the main result is the Lyapunov-Schmidt reduction method applied to the T-Poincaré-Andronov mapping.     

A modified K-matrix approach to many-channel processes

The definition of a K-matrix, that does not have the direct channel singularities of the scattering amplitude, is not unique. Different choices lead to different integral equations for K with different physical content. We discuss the choice where the intermediate states have a simple off-shell behaviour for each partial wave. Crossing leads to an identification of the singularities of the K-matrix elements and to an iterative method of finding the scattering amplitude starting from a “bare” world.     

2D electrons in lateral superlattice in a strong B⊥

A periodic 1D potential acting upon the surface electrons on vicinal planes smears up the Landau levels into energy bands. The density of states singularities result in a specific line shape in cyclotron absorption and inelastic light scattering. In both cases there are two maxima caused by the density of states increasing at the band edges.     

A hybrid finite/boundary element method for periodic structures on non-periodic meshes using an interior penalty formulation for Maxwell’s equations

This paper presents a hybrid finite element/boundary element (FEBE) method for periodic structures. Periodic structures have been efficiently analyzed by solving for a single unit cell utilizing Floquet’s theorem. However, most of the previous works require periodic meshes to properly impose the boundary conditions on the outer surfaces of the unit cell. To alleviate this restriction, the interior penalty method is adopted and implemented in this work. Also, the proper treatment of the boundary element part is addressed to account for the non-conformity of the boundary element mesh. Another ingredient of this work is the use of the efficient boundary element computation, accelerated by the Ewald transformation for the calculation of the periodic Green’s function. Finally, the method is validated through examples which are discretized without the constraint of a periodic mesh.     

Tissue structure study through ultrasonic forward scattering

A procedure is demonstrated for characterization of biological tissues at small scattering angles. The power spectra of ultrasonic pulses transmitted through excised tissue samples were measured and compared to the spectra of signals transmitted through a water path. The specimens were examined in two spatial-frequency bands by acquiring data at scattering angles of 10 degrees and 20 degrees using 2.25 MHz transducers. Peaks in the measured power spectra are interpreted using two signal models. The medium is modelled either as a periodic structure producing a single spectral peak, or by two discrete targets producing a periodic modulation of the spectrum. The periodic structure model appears to be the more promising method for interpretation of forward-scattered signals. Data acquired from hyperplastic spleen and atheromatous aorta specimens both exhibited increases in pulse-tissue interaction at low spatial frequencies compared to normal specimens of those tissues. This observation is tentatively linked to increases in the size or separation of distributed scattering structures resulting from those pathologies.     

A fast direct solver for the integral equations of scattering theory on planar curves with corners

We describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners. It is rather comprehensive in that it applies to a wide variety of boundary value problems; here, we treat the Neumann and Dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids. It achieves high accuracy, is applicable to large-scale problems and, perhaps most importantly, does not require asymptotic estimates for solutions. Instead, the singularities of solutions are resolved numerically. The approach is efficient, however, only in the low- and mid-frequency regimes. Once the scatterer becomes more than several hundred wavelengths in size, the performance of the algorithm of this paper deteriorates significantly. We illustrate our method with several numerical experiments, including the solution of a Neumann problem for the Helmholtz equation given on a domain with nearly 10000 corner points.     

An exact inversion method for the determination of spin–orbit potentials from scattering data

A generalized Newton–Sabatier inversion method which permits extraction from scattering data of central and spin-orbit potentials is presented. The inversion method originally developed by Sabatier and further elaborated by Hooshyar and Richardson, has been reformulated to lead to physically reasonable solutions and to allow for its numerical implementation. Numerical problems due to the occurrence of singularities in the transformation kernel are discussed and a successful application using schematic scattering data is reported. Received: 30 April 1997 / Revised version: 20 October 1997     

Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

The famous scientist Hermann von Helmholtz was born 200 years ago. Many complex physical wave phenomena in engineering can effectively be described using one or a set of equations named after him: the Helmholtz equation. Although this has been known for a long time, from a theoretical point of view, the actual numerical implementation has often been hindered by divergence-free and/or curl-free constraints. There is further a need for a numerical method that is accurate, reliable and takes into account radiation conditions at infinity. The classical boundary element method satisfies the last condition, yet one has to deal with singularities in the implementation. We review here how a recently developed singularity-free three-dimensional boundary element framework with superior accuracy can be used to tackle such problems only using one or a few Helmholtz equations with higher order (quadratic) elements which can tackle complex curved shapes. Examples are given for acoustics (a Helmholtz resonator among others) and electromagnetic scattering.     

一维周期结构散射近-远场外推的两种方法

 首先讨论了自由空间中一维周期结构近-远场外推的Floquet模方法,即对FDTD计算所得散射近场中输出边界上散射场进行级数展开,求出各阶Floquet模的复数幅值,再求得远区场。接着介绍了求解一维周期结构远区散射场的周期Green函数方法,即根据FDTD计算所得散射近场中输出边界上散射场,求得等效面电磁流后,再借助周期Green函数进行外推得到远区场。两种方法均仅用一个周期单元内的散射近场进行外推。计算结果验证了上述两种方法外推的有效性。