Normalization corrections to perturbation theory based on a matrix method

We present the derivation of the normalization constant for the perturbation matrix method recently proposed. The method is tested on the problem of a binary waveguide array for which an exact and an approximate solution are known. In our analysis, we show that to third order the normalized matrix method approximate solution gives results coinciding with the exact known solution.     

Singular perturbation theory and its application to the computation of electromagnetic fields

In the past decade the theory of singular perturbations has been well-established. The techniques developed to solve singular perturbation problems were succesfully applied to practical problems. In this paper we will give a brief survey of singular perturbation theory and discuss a possible application to the computation of electromagnetic fields.     

Scattering of the electromagnetic waves from a rough surface

The electromagnetic field scattered by a rough surface of a semi-infinite body is computed up to the second order of a perturbation scheme with the surface roughness as a perturbation parameter. The calculations are based on the equation of motion of the polarization within the Lorentz–Drude (plasma) model of polarizable, non-magnetic, homogeneous matter. The surface roughness contributes both to the main (specularly) reflected and refracted fields and diffuse scattering, or gives rise to secondary (second-order) diffraction peaks for a regular grating. The calculations are performed both for the s- and p-waves. Two-dimensional modes, resonant at certain frequencies, are identified, confined to and propagating only on the surface, as a consequence of the surface roughness.     

Multi-level biharmonic and bi-Helmholtz interpolation with application to the boundary element method

The scattered data interpolation problem is investigated. Instead of the direct use of radial basis functions, the interpolation function is sought as a solution of a higher order partial differential equation supplied with the interpolation equations as special boundary conditions. In this paper the methods based on the biharmonic and the bi-Helmholtz equations are analysed. The interpolation problem is reformulated in variational forms. Existence and uniqueness theorems are proved in Sobolev spaces. The approximation properties of this interpolation are also investigated. A representation theorem is proved which shows the similarity to the method of radial basis functions based on the fundamental solution of the applied partial differential operator. To solve the appearing biharmonic/bi-Helmholtz equation, a multi-level method is presented which is based on a quadtree/octtree cell system generated by the interpolation points. It is shown that the overall computational cost of the presented method is much less than that of the traditional method of radial basis functions. The method makes it possible to avoid the solution of large, fully populated and often ill-conditioned systems of linear equations. Finally, some applications to solving partial differential equations are outlined. The biharmonic/bi-Helmholtz interpolation technique immediately defines a grid-free method, but can be combined with the boundary element method as well. A possible application in the dual reciprocity method is also presented.     

Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates

A method other than the extended-boundary-condition method (EBCM) to compute the T matrix for electromagnetic scattering is presented. The separation-of-variables method (SVM) is used to solve the electromagnetic scattering problem for a spheroidal particle and to derive its T matrix in spheroidal coordinates. A transformation is developed for transforming the T matrix in spheroidal coordinates into the corresponding T matrix in spherical coordinates. The T matrix so obtained can be used for analytical calculation of the optical properties of ensembles of randomly oriented spheroids of arbitrary shape by use of an existing method to average over orientational angles. The optical properties obtained with the SVM and the EBCM are compared for different test cases. For mildly aspherical particles the two methods yield indistinguishable results. Small differences appear for highly aspherical particles. The new approach can be used to compute optical properties for arbitrary values of the aspect ratio. To test the accuracy of the expansion coefficients of the spheroidal functions for arbitrary arguments, a new testing method based on the completeness relation of the spheroidal functions is developed.     

The perturbation method and the extended finite element method. An application to fracture mechanics problems

The extended finite element method has been successful in the numerical simulation of fracture mechanics problems. With this methodology, different to the conventional finite element method, discretization of the domain with a mesh adapted to the geometry of the discontinuity is not required. On the other hand, in traditional fracture mechanics all variables have been considered to be deterministic (uniquely defined by a given numerical value). However, the uncertainty associated with these variables (external loads, geometry and material properties, among others) it is well known. This paper presents a novel application of the perturbation method along with the extended finite element method to treat these uncertainties. The methodology has been implemented in a commercial software and results are compared with those obtained by means of a Monte Carlo simulation.     

Boundary element method calculation for coherent electromagnetic scattering from two and three dielectric spheres

A quadratic, isoparametric boundary element formulation has been used to calculate the multiple scattering of electromagnetic waves from systems of two and three dielectric spheres. Extinction efficiency results for the scattering of a plane wave are presented for variations of the separation of the two spheres in three kinds of orientations of the system with respect to the incident wave. These have been verified against analytical calculations based on Mie's theory and calculations by other authors. The results demonstrate a large side scattering resonance (the so-called specular resonance). Agreement between the results establishes the boundary element method as a very powerful tool for solving multiple scattering problems because the method applies to arbitrarily shaped objects having a homogeneous dielectric constant in any configuration. To illustrate the versatility of the method, scattering from three spheres is calculated.     

An application of modified reductive perturbation method to long water waves

In this work, we extended the application of “the modified reductive perturbation method” to long water waves and obtained the governing equations as the KdV hierarchy. Seeking a localized travelling wave solutions to these evolution equations we determined the scale parameter c₁so as to remove the possible secularities that might occur. The present method is seen to be fairly simple as compared to the renormalization method [Kodama, Y., & Taniuti, T. (1977). Higher order approximation in reductive perturbation method 1. Weakly dispersive system. Journal of Physics Society of Japan, 45, 298–310] and the multiple scale expansion method [Kraenkel, R. A., Manna, M. A., & Pereira, J. G. (1995). The Korteweg–deVries hierarchy and long water waves. Journal of Mathematics Physics, 36, 307–320].     

A systematic method for the targeted discovery of chemical attribution signatures: application to isopropyl bicyclophosphate production

Potential attribution signatures for the synthesis of a highly toxic bicyclophosphate, 4-isopropyl-2,6,7-trioxa-1-phosphabicyclo[2.2.2]octane 1-oxide (Isopropyl Bicyclophosphate or IPBCP) were discovered using a trilateral synthetic, analytical, and statistical approach. Initially, five synthetic routes were confirmed to successfully produce IPBCP using a range of reaction solvents, reactant ratios, and reaction temperatures. Experimental design principles were subsequently used to guide a formal study specifically aimed at discovering attribution signatures that could be used to differentiate forensic samples. A comparison of three-dimensional scatter plots comprised of the detected ions, their relative retention times (RRTs) and intensities (from LC-MS analyses) identified: (1) signatures that were unique to a synthetic route; (2) signatures associated with a combination of synthetic route and reaction solvent; (3) signatures related to reaction solvent, and (4) signatures associated with reagent source. Top level analysis revealed that the majority of the signatures are related to the synthetic route or a combination of the synthetic route and reaction solvent. Deeper analysis utilizing high resolution mass spectrometry (HRMS) and MS(n) revealed that most of the signatures stem from impurities in the reagents or byproducts formed from incomplete reactions between the reagents used in a given synthetic route. Finally, a subsequent validation study was performed to assess the presence and absence of the key route dependent signatures.     

Scattering of electromagnetic radiation by a metal nanoparticle

The scattering cross section of electromagnetic radiation by a small spherical metal particle has been calculated in the framework of the standard kinetic theory in a dipole approximation. The calculation has been performed for relatively small (～10 nm) particles, which allows the skin effect to be ignored. A mechanism of mixed specular-diffuse reflection of conduction electrons from the particle surface is considered. It is established that, at certain angles of scattering, the mechanism of magnetic-dipole scattering becomes dominating. The influence of kinetic effects on the differential scattering cross section is analyzed.     

Application of a perturbation method to the solution of eddycurrent testing problems

This paper presents an analytical solution for the change of the impedance of a double conductor line due to eddy currents induced in presence of an infinitely long horizontal cylindrical flaw with vertically symmetric cross-section. The solution, found by a perturbation method, is applicable in the case the conductivities of the flaw and of the surrounding material do not differ by much. Numerical results for flaws of circular and elliptic cross-sections are presented     

A perturbation method for non-linear dispersive waves with an application to water waves

Summary A multiple scale perturbation method is developed to obtain asymptotic evolution equations for slowly varying wave train solutions to non-linear dispersive wave problems. The method appears to give results which are a generalization of Whitham's theory on one hand and a generalization of the ray theory on the other hand. First an application is given to a non-linear Klein-Gordon equation, then the method is applied to two-dimensional water waves on water of finite depth (Stokes waves).     

Scattering by simple and nonsimple shapes by the combined method of ray tracing and diffraction: application to circular cylinders

The combined method of ray tracing and diffraction (CMRD) is an efficient and accurate technique for computing the scattered field in focal regions of optical systems. Here we extend the CMRD concept so it can be used to compute fields scattered by objects of simple as well as nonsimple shapes. To that end we replace the scattering object by an equivalent, planar phase object; use ray tracing to determine its location, aperture area, amplitude distribution, and phase distribution; and use standard Kirchhoff diffraction theory to compute the field scattered by the equivalent phase object. To illustrate the practical use of the CMRD we apply it to a two-dimensional problem in which a plane or cylindrical wave is normally incident upon a circular cylinder. For this application we determine the range of validity of the CMRD by comparing its results for the scattered field with those obtained by use of an exact eigenfunction expansion.     

微穿孔板传递矩阵计算方法的改进及实验

传统的声电类比法对双层微穿孔板吸声结构进行计算时,忽略了空腔声质量的影响,对于空腔距离比较短,频率比较低时是适用的,但是当空腔距离比较大或频率比较高时,则存在误差。用传递矩阵法对微穿孔板吸声结构进行分析：解决了空腔声阻抗的近似计算带来的误差;对于微孔部分传递矩阵中的声阻抗计算仍然采用马大猷教授的理论,不需要引入修正参数δx。通过上述工作,进行了微穿孔板吸声结构的吸声系数的计算和相应的实验验证,理论计算结果与实验数据吻合良好。     

Discrete dipole moment method for calculation of the T matrix for nonspherical particles

A computational method, based on a moment solution to the discrete dipole approximation (DDA) interaction equations, is proposed for calculation of the T matrix of arbitrary-shaped particles. It is shown that the method will automatically provide the conservation-of-energy and origin-invariance properties required of the T matrix. Furthermore, the method is significantly faster than a T-matrix calculation by direct inversion of the DDA equations. Because the method retains the dipole lattice representation of the particle, it can be applied with relative ease to particles with irregular shapes-although in the same respect it will not automatically simplify for axisymmetric particles. Calculations of scattering matrix distributions, in fixed and random orientations, are made for tetrahedron, cylindrical, and prolate spheroid particle shapes and compared with DDA and extended boundary condition method results.     

Scattering of electromagnetic plane wave from a perfect electric conducting strip located in topological insulator medium

In this article, scattering from a perfectly electric conducting strip located in an infinitely extended topological insulator (IT) medium is investigated using the Kobayashi potential method. For solving mixed boundary value problem, KP method is a fast and semi-analytical technique. In this method, edge conditions and boundary conditions are incorporated simultaneously. The scattered field is calculated from the strip geometry embedded in IT medium. The far zone scattered field is investigated with respect to different parameters of the geometry, i.e. topological parameter of the IT medium, size of the strip etc. It has been observed that the scattered field can be enhanced by increasing topological parameter of the IT medium. On the other hand, size of strip, permittivity of IT medium may also be used to enhance the main lobe of the scattering width/scattered field.     

Analytic calculation of the interaction between a pulsed electromagnetic field and a two-electrode horn antenna

An analytic solution of a system of differential equations derived earlier is obtained. Expressions are given for the current in the load as a function of the orientation of Poynting's vector with respect to the axis of the antenna, and the trasfer, characteristic of the transducer.     

Scattering of s-polarized electromagnetic plane waves from a film with a shallow random rough surface on a perfect conductor

Scattering of s-polarized electromagnetic planes waves from a film, with a shallow random rough one-dimensional surface, bounded by vacuum and a perfect conductor is calculated. An integral equation that relates the amplitude of the scattered field to the incident wave is found by use of the Rayleigh hypothesis. The integral equation is solved numerically and by use of the perturbation theory, up to the fourth order in the surface profile function. In the angular dependence of the incoherent part of the differential reflection coefficient, the backscattering peak and two additional satellite peaks are observed, owing to two guided waves supported by the film. Analysis of the perturbation solution reveals that the background scattering exhibits minima and maxima as functions of the thickness. By studying the behavior of the scattering as a function of the absorption index of the film, it is shown that the amplitudes of the peaks are low when k ~ 10(-2) and high when k ~ 10(-4).     

Scattering matrix approach to the resonant states and Q values of microdisk lasing cavities

We develop a scattering matrix approach for the numerical calculation of resonant states and Q values of a nonideal optical disk cavity with an arbitrary shape and with an arbitrary varying refraction index. The developed method is applied to study the effect of surface roughness and inhomogeneity of the refraction index on Q values of microdisk cavities for lasing applications. We demonstrate that even small surface roughness (deltar < or approximately equal to lambda/50) can lead to a drastic degradation of high-Q cavity modes by many orders of magnitude. The results of the numerical simulation are analyzed and explained in terms of wave reflection at a curved dielectric interface, combined with an examination of Poincaré surfaces of section and of Husimi distributions.