Imaging of periodic dielectrics

We consider imaging of periodic penetrable structures from measurements of scattered electromagnetic waves. The importance of this problem stems from the decreasing size of periodic structures in photonic devices, together with an increasing demand in fast non-destructive testing. This demand makes qualitative inverse scattering techniques particularly attractive since they do not use time consuming optimization techniques for reconstruction but rather directly transform measured data into a picture of the scattering object. We present the Factorization method as an algorithm for imaging of a special class of periodic dielectric structures known as diffraction gratings. Our sampling method computes a picture of the shape of the periodic structure from measured near-field data in a rapid way. We provide numerical examples illustrating this imaging technique.     

Casimir effect for periodic systems separated by a vacuum gap

We describe the general method for calculating the Casimir energy of two diffraction gratings periodic in one spatial direction and separated by a vacuum gap. We express the Casimir energy in terms of the Rayleigh coefficients.     

Modelling of relief of phase reflection diffraction grating

This article describes theoretical and experimental results on phase reflection diffraction gratings. Based on Fourier optics the mathematical formulation describing diffraction dispersion of light from a relief grating of the trapezium profile is derived. We propose a method that lets us estimate the grating’s geometric parameters in a versatile modelling system. We have designed an original programme that estimates diffraction intensities and calculates diffraction efficiency. The estimated intensities are used to reconstruct the grating’s geometrical properties using our mathematical model. The precision of the method is evaluated as the deviation of obtained results from microscopy data.     

Volume integral equations for scattering from anisotropic diffraction gratings

We analyze electromagnetic scattering of transverse magnetic polarized waves from a diffraction grating consisting of a periodic, anisotropic, and possibly negative index dielectric material. Such scattering problems are important for the modelization of, for example, light propagation in nano‐optical components and metamaterials. The periodic scattering problem can be reformulated as a strongly singular volume integral equation, a technique that attracts continuous interest in the engineering community but has rarely received rigorous theoretic treatment. In this paper, we prove new (generalized) Gårding inequalities in weighted and unweighted Sobolev spaces for the strongly singular integral equation. These inequalities also hold for materials for which the real part of the material parameter takes negative values inside the diffraction grating, independently of the value of the imaginary part. Copyright © 2012 John Wiley & Sons, Ltd.     

Conical diffraction by periodic structures: Variation of interfaces and gradient formulas

This paper studies the dependence of solutions to conical diffraction problems upon geometric parameters of non–smooth profiles and interfaces between different materials of diffractive gratings. This problem arises in the design of those optical devices to diffract time–harmonic oblique incident plane waves to a specified far–field pattern. We prove the stability of solutions and give analytic formulas for the derivatives of reflection and transmission coefficients with respect to Lipschitz perturbations of interfaces. These derivatives are expressible as contour integrals involving the direct and adjoint solutions of conical diffraction problems.     

Thermoelasticity of a regularly inhomogeneous curved layer with wavy surfaces

A curved inhomogeneous anisotropic layer of variable thickness is considered that has wavy surfaces. It is assumed that the elastic, thermo-physical characteristics of the layer material and the shape of its upper and lower surfaces are periodic in structure with a single periodicity cell (PC). The period of the structure is here comparable in magnitude with the layer thickness, which is assumed to be much less than the other linear dimensions of the body and the radius of curvature of its middle surface.On the basis of a general scheme for taking the average of processes in periodic media /1, 2/, a method is developed which enables a transition to be made from a spatial quasistatic thermoelasticity problem to a system of thermoelasticity equations for an average shell whose effective and thermophysical coefficients are determined from the solution of local problems in a PC. Results obtained for the static theory of elasticity in /3/ are used. The heat conduction problem is averaged to determine the temperature components occurring in the equation of motion.The model constructed enables thermoelastic strains, stresses and the temperature distribution to be obtained in shells and plates of composite or porous materials with a different kind of reinforcement of the periodic structure (waffle, ribbed, corrugated) in reinforced and grid-like shells and plates. In the limiting case of “smooth” surfaces and a homogeneous material, the thermoelasticity equations are obtained for shallow anisotropic shells and the heat conduction equations of anisotropic shells assuming a linear temperature distribution law over the thickness.     

On Approximation Orders of Functions of Several Variables in the Lorentz Space

We consider the anisotropic Lorentz space of periodic functions. Sufficient conditions are proved for a function to belong to the anisotropic Lorentz space. Estimates for the order of approximation by trigonometric polynomials of the Nikol’skii–Besov class in the anisotropic Lorentz space are established.     

On orders of approximation of function classes in Lorentz spaces with anisotropic norm

In this paper, we study the anisotropic Lorentz space of periodic functions. We establish a sharp estimate of the order of approximation for the Besov class by trigonometric polynomials in Lorentz spaces with anisotropic norm.     

The Two-dimensional Electromagnetic Scattering from Periodic Chiral Structures and Its Finite Element Approximation

In this paper, we consider the electromagnetic scattering from periodic chiral structures. The structure is periodic in one direction and invariant in another direction. The electromagnetic fields in the chiral medium are governed by the Maxwell equations together with the Drude-Born-Fedorov equations. We simplify the problem to a two-dimensional scattering problem and we show that for all but possibly a discrete set of wave numbers, there is a unique quasi-periodic weak solution to the diffraction problem. The diffraction problem can be solved by finite element method. We also establish uniform error estimates for the finite element method and the error estimates when the truncation of the nonlocal transparent boundary operators takes place.     

Equilibria for anisotropic surface energies with wetting and line tension

We study the stability of surfaces trapped between two parallel planes with free boundary on these planes. The energy functional consists of anisotropic surface energy, wetting energy, and line tension. Equilibrium surfaces are surfaces with constant anisotropic mean curvature. We study the case where the Wulff shape is of “product form”, that is, its horizontal sections are all homothetic and have a certain symmetry. Such an anisotropic surface energy is a natural generalization of the area of the surface. In particular, we study the stability of parts of anisotropic Delaunay surfaces which arise as equilibrium surfaces. They are surfaces of the same product form of the Wulff shape. We show that, for these surfaces, the stability analysis can be reduced to the case where the surface is axially symmetric and the functional is replaced by an appropriate axially symmetric one. Moreover, we obtain necessary and sufficient conditions for the stability of anisotropic sessile drops.     

在面斜对称载荷下无限各向异性介质内的周期裂纹

本文试图藉助复变函数方法求解在面斜对称载荷下无限各向异性弹性介质的周期裂纹问题.这问题将化为欲确定满足某种边值条件的两个复变函数.文中假定应力、位移与边值条件是周期的,而且假定应力在无穷远处有界.这里的解答已表示为封闭形式.     

Wave Properties of Double One-Dimensional Periodic Sheet Grating

We show that the double one-dimensional periodic sheet gratings always have waveguide properties for acoustic waves. In general, there are two types of pass bands: i.e., the connected sets of frequencies for which there exist harmonic acoustic traveling waves propagating in the direction of periodicity and localized in the neighborhood of the grating. Using numerical-analytical methods, we describe the dispersion relations for these waves, pass bands, and their dependence on the geometric parameters of the problem. The phenomenon is discovered of bifurcation of waveguide frequencies with respect to the parameter of the distance between the gratings that decreases from infinity. Some estimates are obtained for the parameters of frequency splitting or fusion in dependence on the distance between the simple blade gratings forming the double grating. We show that near a double sheet grating there always exist standing waves (in-phase oscillations in the neighboring fundamental cells of the group of translations) localized near the grating. By numerical-analytical methods, the dependences of the standing wave frequencies on the geometric parameters of the grating are determined. The mechanics is described of traveling and standing waves localized in the neighborhood of the double gratings.     

First kind symmetric periodic solutions and their stability for the Kepler problem and anisotropic Kepler problem plus generalized anisotropic perturbation

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As an application of this general situation, we discuss the existence of symmetric periodic solutions for the anisotropic Kepler problem plus a generalized anisotropic perturbation, (shortly, p-AKPQ problem) and for the Kepler problem plus a generalized anisotropic perturbation (shortly, p-KPQ problem), as continuation of circular orbits of the two-dimensional Kepler problem. To get this objective, we consider different types of perturbations and then we apply our main result.     

Correlation Function of the Two-Dimensional Ising Model on a Finite Lattice: II

We calculate the pair correlation function and the magnetic susceptibility in the anisotropic Ising model on the lattice with one infinite and one finite dimension with periodic boundary conditions imposed along the second dimension. Using the exact expressions for lattice form factors, we propose formulas for arbitrary spin matrix elements, thus providing a possibility to calculate all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. We analyze passing to the scaling limit.     

A linear sampling method for inverse problems of diffraction gratings of mixed type

This paper is concerned with the direct and inverse problem of scattering of a time‐harmonic wave by a Lipschitz diffraction grating of mixed type. The scattering problem is modeled by the mixed boundary value problem for the Helmholtz equation in the unbounded half‐plane domain above a periodic Lipschitz surface on which a mixed Dirichlet and impedance boundary condition is imposed. We first establish the well‐posedness of the direct problem, employing the variational method, and then extend Isakov's method to prove uniqueness in determining the Lipschitz diffraction grating profile by using point sources lying above the structure. Finally, we develop a periodic version of the linear sampling method to reconstruct the diffraction grating. In this case, the far field equation defined on the unit circle is replaced by a near field equation defined on a line above the surface, which is a linear integral equation of the first kind. Numerical results are also presented to illustrate the efficiency of the method in the case when the height of the unknown grating profile is not very large and the noise level of the near field measurements is not very high. Copyright © 2012 John Wiley & Sons, Ltd.     

强非线性广义 Boussinesq 方程孤波解的波形分析及求解

上海理工大学理学院\quad 上海 200093该文建立了强非线性广义 Boussinesq 方程的耗散项、波速、渐进值与波形函数的导数之间的关系.利用适当变换和待定假设方法,作者求出了上述广义 Boussinesq 方程的扭状或钟状孤波解,还求出了以前文献中未曾提到过的余弦函数的周期波解.进一步给出了波速对波形影响的结论,即:``好'广义 Boussinesq 方程的行波当波速由小变大时,波形由钟状孤波变成余弦函数周期波解;``坏'广义 Boussinesq 方程的行波当波速由小变大时,波形由余弦函数周期波解变成钟状孤波.     

FFT-based homogenization on periodic anisotropic translation invariant spaces

In this paper we derive a discretisation of the equation of quasi-static elasticity in homogenization in form of a variational formulation and the so-called Lippmann–Schwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the Lippmann–Schwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both de la Vallée Poussin means and Box splines illustrate the flexibility of this framework.     

Anisotropic interpolation error estimates via orthogonal expansions

We prove anisotropic interpolation error estimates for quadrilateral and hexahedral elements with all possible shape function spaces, which cover the intermediate families, tensor product families and serendipity families. Moreover, we show that the anisotropic interpolation error estimates hold for derivatives of any order. This goal is accomplished by investigating an interpolation defined via orthogonal expansions.     

Nitsche-mortaring for singularly perturbed convection–diffusion problems

In the present paper we analyse a finite element method for a singularly perturbed convection–diffusion problem with exponential boundary layers. Using a mortaring technique we combine an anisotropic triangulation of the layer region (into rectangles) with a shape regular one of the remainder of the domain. This results in a possibly non-matching (and hybrid), but layer adapted mesh of Shishkin type. We study the error of the method allowing different asymptotic behaviour of the triangulations and prove uniform convergence and a supercloseness property of the method. Numerical results supporting our analysis are presented.