Scattering of electromagnetic radiation by a two-layer nonconfocal spheroid that is small compared to the wavelength

We have considered the electrostatic problem for a two-layer nonconfocal spheroid. The approach is based on surface integral equations that are similar to equations in terms of the extended boundary condition method for wave problems. Electrostatic fields are related to scalar potentials, which are represented as expansions in terms of eigenfunctions of the Laplace equation in two spheroidal coordinate systems, while unknown expansion coefficients are determined from infinite systems of linear algebraic equations. The constructed rigorous solution to the problem coincides with the known solution in a particular case of a confocal two-layer spheroid. In addition, for the nonconfocal two-layer spheroid, we have constructed an explicit approximated solution assuming that the field in the particle core is constant. This solution coincides with the rigorous solution if the scatterer shells are confocal. The formula found for the polarizability of the two-layer nonconfocal spheroid has a very simple form compared to the previously proposed cumbersome algorithm (B. Posselt et al., Measur. Sci. Technol. 13, 256 (2002)) and is more efficient numerically.     

Wavefront reconstruction algorithm for wavefront sensing based on binary aberration modes

We propose a new algorithm for wavefront sensing based on binary intensity modulation. The algorithm is based on the fact that a wavefront can be expended with a series of orthogonal and binary functions, the Walsh series. We use a spatial light modulator(SLM) to produce different binary-intensity-modulation patterns which are the simple linear transformation of the Walsh series. The optical fields under different binary-intensity-modulation patterns are detected with a photodiode.The relationships between the incident wavefront modulated with the patterns and their optical fields are built to determinate the coefficients of the Walsh series. More detailed and strict relationship equations are established with the algorithm by adding new modulation patterns according to the properties of the Walsh functions. An exact value can be acquired by solving the equations. Finally, with the help of phase unwrapping and smoothing, the wavefront can be reconstructed. The advantage of the algorithm is providing an analytical solution for the coefficients of the Walsh series to reconstruct the wavefront. The simulation experiments are presented and the effectiveness of the algorithm is demonstrated.     

An Approach to Permittivity Determination at Microwave Frequencies Through the Eigenvalue Problem

A simple method of solving the inverse medium problem in order to determine the complex dielectric constant of a lossy dielectric material at microwave frequencies is presented. The arbitrarily shaped dielectric sample placed in a rectangular waveguide is considered. Electromagnetic fields in the waveguide are expanded in series of harmonic functions. A set of linear equations is obtained using the finite–difference technique. An additional empirical condition, which is related to the measured reflection and/or transmission coefficients and results in an overdetermined system, is added and then the system is defined by the addition of a trivial extra parameter. The sample's dielectric properties are obtained by solving the resulting eigenvalue problem. An algorithm to extract proper results from the set of eigenvalues is defined by considering the properties of numerical solution. The system is tested by using an alternative method.     

New exact solutions of Einstein's field equations: Gravitational force can also be repulsive!

This article has not been written for specialists of exact solutions of Einstein's field equations but for physicists who are interested in nontrivial information on this topic. We recall the history and some basic properties of exact solutions of Einstein's vacuum equations. We show that the field equations for stationary axisymmetric vacuum gravitational fields can be expressed by only one nonlinear differential equation for a complex function. This compact form of the field equations allows the generation of almost all stationary axisymmetric vacuum gravitational fields. We present a new stationary two-body solution of Einstein's equations as an application of this generation technique. This new solution proves the existence of a macroscopic, repulsive spin-spin interaction in general relativity. Some estimates that are related to this new two-body solution are given.     

Numerical Analysis of Electromagnetic Fields in Multiscale Model

Modeling technique for electromagnetic fields excited by antennas is an important topic in computational electromagnetics, which is concerned with the numerical solution of Maxwell's equations. In this paper, a novel hybrid technique that combines method of moments(MoM) with finite-difference time-domain(FDTD) method is presented to handle the problem. This approach employed Huygen's principle to realize the hybridization of the two classical numerical algorithms. For wideband electromagnetic data, the interpolation scheme is used in the MoM based on the dyadic Green's function. On the other hand, with the help of equivalence principle, the scattered electric and magnetic fields on the Huygen's surface calculated by MoM are taken as the sources for FDTD. Therefore, the electromagnetic fields in the environment can be obtained by employing finite-difference time-domain method. Finally, numerical results show the validity of the proposed technique by analyzing two canonical samples.     

An algorithm for the calculation of the net induction energy and induced multipole tensors in a set of charge distributions with non-linear contributions of all potentials of external fields and both permanent and induced multipoles

This algorithm for the calculation of the induced multipole tensors in a set of charge distributions includes the contributions of partial derivatives of arbitrary order of the potentials defined by both induced and permanent multipoles as well as non-uniform external fields of arbitrary strength. Specific equations are given for both systems with and without translational order. The algorithm, which is based on the Maxwell invariant form, uses direct extensions of algorithms previously developed and tested for the calculation of permanent multipole energies and induced dipole vectors (when the non-uniformity of the field is neglected). The induced tensor components are calculated iteratively. The first approximation, which gives the components as the solution of a set of simultaneous linear equations, includes all nonlinear, non-uniform contributions of permanent multipoles and external fields as well as all contributions linear in derivatives of the induced potentials. The induced tensor components are then used to calculate the net induction energy. The general relations between the polarizability tensors with respect to a centre and the moments of the polarizability densities about the centre are derived.     

Specular and diffuse reflections of rays in coupled thin plates at high frequencies

The flexural energy distribution in two right-angled point-excited thin plates at high frequencies is investigated by means of an integral energy flow approach. The fields of energy averaged over time and frequency are described by the superposition of uncorrelated cylindrical waves stemming from both boundaries and direct sources. Specular and diffuse laws are considered for the reflection and transmission of rays, giving rise to two kinds of energy equations. The diffuse law leads to a Fredholm integral equation over the boundary sources while the specular law is shown to allow an image source solution when the plates have identical propagation properties. The algorithm for computing the image position, magnitude and directivities is described. Then, some comparisons between the results from the both energy formulations and also from the statistical energy analysis and the numerical solution of the equations of motion are performed with two damped plates at high frequency. The non-diffuse pattern of the averaged flexural energy fields is well described by the energy flow approaches.     

Finite volume method for calculation of electrostatic fields in electrostatic precipitators

This paper presents a numerical method for calculation of coupled electric and space-charge density fields in electrostatic precipitators. It is based on the finite volume discretization of the solution domain by arbitrary polyhedral control volumes and employs an iterative segregated solution procedure of the resulting set of algebraic equations, amounting to a simple, accurate and efficient numerical technique. The method is tested on a number of cases for which analytical solution, numerical and/or experimental results exist. Also, shown are the results of calculation of a 3D model of electrostatic precipitator with spike discharge electrodes.     

Spatially homogeneous and inhomogeneous cosmologies with equation of statep=μ

This paper gives an algorithm for generating solutions of the Einstein field equations which have an irrotational perfect fluid, with equation of statep=, as source, and which admit a two-parameter Abelian group of local isometries. The algorithm is used to derive a variety of new and known spatially homogeneous cosmological models, both tilted and nontilted. However, since the solutions in general only admit two Killing vectors, spatially inhomogeneous models are also obtained. Finally, it is pointed out that the solution generation technique used in this paper is closely related to solution generation techniques that have been used to generate solutions of the source-free Brans-Dicke field equations, and of the Einstein field equations with a massless scalar field as source.     

Quantum Mechanics of Two-Field Parametric Frequency Converter Interacting with a Single Atom

In this communication we deal with a new model Hamiltonian representing the interaction between a two-level atom and two electromagnetic field modes in a cavity. The interaction between the modes has been taken into account and considered to be of a parametric frequency converter type. The model can be regarded as a generalization of two different systems: the Jaynes–Cummings model (atom–field interaction) and the two-mode frequency converter model (field–field interaction). Under a certain condition an exact solution for the equations of motion in the Heisenberg picture is given. The wavefunction in the Schrödinger picture is also constructed and used to discuss some statistical properties related to the model. We assume that the fields are initially in coherent states. We discuss atomic inversion, photon number distribution, squeezing and other phenomena. We show in all cases that the system is very sensitive to any variation in the mean photon numbers.     

Light scattering by axisymmetric multilayer particles

A new recursive algorithm for solving the problem of scattering a plane electromagnetic wave by axisymmetric dielectric multilayer particles is constructed. The approach that was proposed earlier and demonstrated for uniform axisymmetric particles is used. It has the following basic features: (1) the fields are represented in the form of a sum of two terms, one of which is independent of the azimuthal angle, whereas averaging of the second term over this angle gives zero; (2) the axisymmetric problem is solved by using the scalar potentials related to the azimuthal components of electromagnetic fields; and (3) the non-axisymmetric problem is solved by using the superposition of Debye potentials and vertical components of the magnetic and electric Hertz vectors. It is of principal importance for the solution proposed here that the scattering problem is formulated in the form of surface integral equations in these scalar potentials, which are represented in the form of expansions in wave spherical functions. Infinite systems of linear algebraic equations for unknown expansion coefficients are obtained, which are rather simple in structure. The reduced systems for multilayer particles have the same dimension as the systems for identical uniform particles. In the case of multilayer spherical particles, the algorithm gives an explicit solution to the problem, and the dependence on the radial spherical functions for the layers is specified in terms of the derivative of the logarithm (i.e., the ratio of the derivative to the function itself) and the ratio of the functions of neighboring layers. Numerical calculations demonstrated the high efficiency of the algorithm.     

Stationary axially symmetric electrovac solutions in the general scalar-tensor theory

A method is presented which reduces the Bergmann-Wagoner-Nordtvedt field equations for a stationary axisymmetric electrovac space-time, to the Einstein-Maxwell equations. In this formalism the solution generation technique of Singh and Rai for Brans-Dicke theory yields a particular class of solutions, for which the conformal scalar field depends upon the radial coordinate only. As an application of the method, new cylindrically symmetric and nonstatic scalar-Maxwell solutions are obtained for null and non-null electromagnetic fields.     

Possibility to break through limitation of measurement range in dual-wavelength digital holography

By using the beat frequency technique, the dual-wavelength digital holography(DWDH) can greatly increase the measurement range of the system. However, the beat frequency technique has a limitation in measurement range. The measurement range is not larger than a synthetic wavelength. Here, to break through this limitation, we propose a novel DWDH method based on the constrained underdetermined equations, which consists of three parts:(i) prove that the constrained underdetermined equation has a unique integer solution,(ii) design an algorithm to search for the unique integer solution,(iii) introduce a third wavelength into the DWDH system, and design a corresponding algorithm to enhance the anti-noise performance of DWDH. As far as we know, it is the first time that we have discovered that the problem of DWDH can belong in a problem of contained underdetermined equations, and it is also the first time that we have given the mathematical proof for breaking through the limitation of the measurement range. A series of results is shown to test the theory and the corresponding algorithms. More importantly, since the principle of proposed DWDH is based on basic mathematical principles, it can be further extended to various fields, such as dual-wavelength microwave imaging and dual-wavelength coherent diffraction imaging.     

Simultaneous measurement of D and T2 using the distant dipolar field

The presence of long-range dipolar fields in liquids is known to introduce a non-linear term in the Bloch-Torrey equations which is responsible for many interesting effects in nuclear magnetic resonance as well as in magnetic resonance imaging. We show here, for the first time, that the diffusion coefficient D and the spin-spin relaxation time T2 can be obtained simultaneously from the time evolution profile of the long-range dipolar field refocused signal. In a COSY Revamped by Z-asymmetric Echo Detection sequence, the analytical first-order approximation solution of the Bloch-Torrey equations modified to include the effect of the distant dipolar field is used to demonstrate the technique in an experiment using doped water.     

Algorithm for solving the inverse problem of ballistics in the case of an asymmetric object

An algorithm to identify the aerodynamic characteristics of an asymmetric object from its trajectory data obtained in a ballistic experiment is developed based on the technique for estimating the nonlinear system’s parameters. Using the method of successive approximations, the coefficients of the aerodynamic function polynomial representation are found that best describe measuring data. The essence of the algorithm is the solution of the direct problem of the symmetric object’s dynamics using the complete set of Euler dynamic equations. The variation of the desired parameters is statistically estimated during calculations. The algorithm allows for jointly processing data of a series of experiments with similar models. Thereby, the volume of processed data is augmented and the final result becomes more accurate.     

Recent developments in conformal invariant quantum field theory

A review of the recent results concerning the kinematics of conformal fields, the analysis of dynamical equations and dynamical derivation of the operator product expansion is given.The classification and transformational properties of fields which are transformed according to the representations of the universal covering group of the conformal group have been considered. A derivation of the partial wave expansion of Wightman functions is given. The analytical continuation to the Euclidean domain of coordinates is discussed. As shown, in the Euclidean space the partial wave expansion can be applied either to one-particle irreducible vertices or to the Green functions, depending on the dimensions of the fields.The structure of Green functions, which contain a conserved current and the energy-momentum tensor, has been studied. Their partial wave expansion has been obtained. A solution of the Ward identity has been found. Special cases are discussed.The program of the construction of exact solution of dynamical equations is discussed. It is shown, that integral dynamical equations for vertices (or Green's functions) can be diagonalized by means of the partial wave expansion. The general solution of these equations is obtained. The equations of motion for renormalized fields are considered. The way to define the product of renormalized fields at coinciding points (arising on the right-hand side) is discussed. A recipe for calculating this product is presented. It is shown, that this recipe necessarily follows from the renormalized equations.The role of bare term and of canonical commutation relations (for unrenormalized fields) is discussed in connection with the problem of the field product determination at coinciding points. As a result an exact relation between fundamental field dimensions is found for various three-linear interactions (section 16 and Appendix 6). The problem of closing the infinite system of dynamical equations is discussed.Al above said results are demonstrated using Thirring model as an example. A new approach to its solving is developed.The program od closing the infinite system of dynamical equations is discussed. The Thirring model is considered as an example. A new approach to the solution of this model is discussed.Methods are developed for the approximate calculation of dimensions and coupling constants in the 3-vertex and 5-vertex approximations. The dimensions are calculated in the γ_?³ theory in 6-dimensional space.The problem of calculating the critical indices in statistics (3-dimensional Euclidean space) is considered. The calculation of the dimension is carried out in the framework of the γ_?⁴ model. The value of the dimension and the critical indices thus obtained coincide with the experimental ones.     

A Spectral Time-Domain Method for Computational Electrodynamics

Ever since its introduction by Kane Yee over forty years ago, the finite-difference time-domain (FDTD) method has been a widely-used technique for solving the time-dependent Maxwell's equations that has also inspired many other methods. This paper presents an alternative approach to these equations in the case of spatially-varying electric permittivity and/or magnetic permeability, based on Krylov subspace spectral (KSS) methods. These methods have previously been applied to the variable-coefficient heat equation and wave equation, and have demonstrated high-order accuracy, as well as stability characteristic of implicit time-stepping schemes, even though KSS methods are explicit. KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral, rather than physical, domain. We show how they can be generalized to coupled systems of equations, such as Maxwell's equations, by choosing appropriate basis functions that, while induced by this coupling, still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields. We also discuss the application of block KSS methods to problems involving non-self-adjoint spatial differential operators, which requires a generalization of the block Lanczos algorithm of Golub and Underwood to unsymmetric matrices.     

Static and time-dependent solutions of Einstein-Maxwell-Yukawa fields

An exact solution of Einstein-Maxwell-Yukawa field equations has been obtained in a space-time with a static metric. A critical analysis reveals that the results previously obtained by Patel [9], Singh [10], and Taub [11] are particular cases of our solution. The singular behavior of the solutions has also been discussed in this paper. Further, extending the technique developed by Janis et al. [12], for static fields, to the case of nonstatic fields, an exact time-dependent axially symmetric solution of EMY fields has been obtained. Our solution in the nonstatic case is nonsingular in the sense of Bonnor [15] and presents a generalization of the results obtained by Misra [7] to the case when a zero-mass scalar field coexists with a source free electromagnetic field.     

On numerical solution of the shallow water equations with chemical reactions on icosahedral geodesic grid

The shallow water equations coupled to the set of reaction–advection–diffusion equations are discretized on a geodesic icosahedral mesh using the finite volume technique. The method of solution of this coupled system is based on the principle of semi-discretization. The algorithm is mass conserving and stable for advection with the Courant numbers up to 2.7. The important part of the methodology is the optimization of the node positions in the icosahedral grid. It is shown that a slight adjustment of the mesh is instrumental in improving the accuracy of the numerical approximation. The convergence of the approximation of the differential operators is evaluated and compared to the data published in the literature. Numerical tests performed with the shallow water solver include two advection experiments, steady and unsteady zonal balanced flow, mountain flow, and the Rossby wave. The mountain flow and the Rossby wave cases are used to test the transport properties of the method in the case of both passive and reactive scalar fields. The investigation of essential numerical characteristics of the method is concluded by the simulation of an unstable zonal jet. The numerical simulation is performed using the set of shallow water equations without dissipation as well as with the viscosity term added to the momentum equation. Results show that the behavior of the model is consistent with both the literature published on the subject and the general empirical evidence.     

Three-dimensional vectorial analysis of waveguide structures with the method of lines

The method of lines (MoL) a special eigenmode algorithm has been proven as an efficient tool for the analysis of waveguide structures in optics and microwaves. The electric and magnetic fields in the cross-section and their derivatives with respect to the cross-section coordinates are discretized with finite differences (FD) while analytic expressions are used in the direction of propagation. The numerical effort for analyzing three-dimensional structures with a two-dimensional discretization can be very high, particularly if vectorial characteristics have to be taken into account. In this paper we introduce a reduction of the eigenmode system to keep the effort moderate. Only a certain number of eigenmodes is determined with the Arnoldi algorithm. We will show then how the electric field distribution of the eigenmodes can be computed from the magnetic field and vice versa. To match the fields at the interfaces we introduce left eigenvectors which are the inverse of the field distributions. The formulas were applied to the analysis of a polarization converter consisting of a periodical perturbation of a waveguide structure. A rotation angle greater than 80° was determined.