Hybrid computer aided design of thick electrostatic electron lenses

The synthesis of space charge free electron lenses can be approached by first assuming solutions for Laplace's partial differential equation in terms of cylindrical harmonics. The hybrid computer can quickly generate these assumed solutions and their gradient by working with the differential equations obtained from product separation of variables. The gradient is used in the integration for the trajectory to determine if the lens is useful. If so, either equipotential surfaces or the potentials along a desired surface are generated with the computer to guide the physical realization of the lens. Certain cases where space charge cannot be neglected are solved by truncation of rectilinear electron flow. Poisson's equation is solved in the computer to serve as a Cauchy boundary condition to Laplace's equation outside the electron beam. The hybrid computing technique and the experimental verification of some of the results in cathode-ray tube electron lenses are presented.     

Formation of arbitrary axisymmetric relativistic beams

A new form of the solution to the exactly formulated problem of formation of arbitrary axisym-metric relativistic beams is obtained. This form includes the calculation of forming electrodes and the external magnetic field continuously matched with the field of the beam on the beam’s boundary. The problems of nonrelativistic cylindrical and conical beams, butt-end electrodes for a cylindrical diode, and toroidal Brillouin formations based on nonrelativistic conical and relativistic cylinder flows are considered.     

On the asymptotic theory of narrow electron beams

Solutions to equations of the paraxial theory of spatial relativistic beams with an inhomogeneous density distribution over the cross section are presented. Equations for quasi-plane paraxial flows are formulated. Analytic solutions are obtained on the basis of the formal coincidence of the relationship on a current tube in the geometrized theory and the equation of the asymptotic theory of strip beams. For these solutions, basic current tubes are determined by the trajectories of a plane periodic electrostatic flow described by the exact solution to the beam equations.     

Application of extended boundary conditions and the Haar wavelets in the analysis of wave scattering by thin screens

A recent approach to solution of 2D scattering problems for electromagnetic waves scattered by thin screens is analyzed. With the use of examples of scattering by a strip and an unclosed cylindrical surface, it is shown that the proposed approach has no advantages in terms of the efficiency of numerical solution over a well-known approach based on exact integral equations for currents that have singular kernels and that are solved with the Krylov-Bogoliubov method.     

Vector Variational Formulation of Electromagnetic Fields in Anisotropic Media

Maxwell's equations can be cast into a basic differential operator equation, the curlcurl equation, which lends itself easily to variational treatment. Various forms of this equation are associated with problems of practical importance. The formulation includes the treatment of loss-free anisotropic media. The boundary conditions associated with electromagnetic-field problems are treated in detail and the uniqueness of the solution is discussed. A functional is derived for the curlcurl equation in Cartesian and cylindrical coordinates.     

Integral equation method in problems of electromagnetic-wave reflection from inhomogeneous interfaces between media

Problems on reflection of a plane electromagnetic wave from various irregular interfaces between media are studied by the integral equation method in the cases of two- and three-dimensional incident electromagnetic field. The reflecting surfaces are meant as periodic transparent interfaces between two media and plane boundaries with locally inhomogeneous and transparent sections. The boundary value problems for the system of Maxwell’s equations in an infinite domain with an irregular boundary are reduced to Fredholm or singular integral equations, depending on the problem considered. Numerical algorithms for solving such integral equations are developed. Results of calculation of currents induced on inhomogeneities and characteristics of the electric field in the far zone are presented.Problems on reflection of a plane electromagnetic wave from various irregular interfaces between media are studied by the integral equation method in the cases of two- and three-dimensional incident electromagnetic field. The reflecting surfaces are meant as periodic transparent interfaces between two media and plane boundaries with locally inhomogeneous and transparent sections. The boundary value problems for the system of Maxwell’s equations in an infinite domain with an irregular boundary are reduced to Fredholm or singular integral equations, depending on the problem considered. Numerical algorithms for solving such integral equations are developed. Results of calculation of currents induced on inhomogeneities and characteristics of the electric field in the far zone are presented.     

A three-dimensional iterative scheme for an electromagneticcapacitive applicator

An efficient iterative method for solving quasi-static electromagnetic field problems is presented. A relaxation function is introduced in the quasi-static field equations. Then, the resulting equations can be solved by iteration. The method is similar to the one of solving a Laplace equation by computing the stationary state of a diffusion equation. Next, for a radially layered configuration the numerical results are compared with the results from an existing integral equation method. Subsequently, for a realistic three-dimensional model of a human knee numerical results are arrived at.     

部分相干修正贝塞尔-高斯光束通过硬边光阑ABCD光学系统的传输特性

对部分相干修正贝塞尔-高斯光束通过有硬边光阑的近轴ABCD光学系统的传输进行了研究，采用将硬边光阑窗口函数展开为有限个复高斯函数之和的方法，推导出了部分相干修正贝塞尔-高斯光束的近似解析传输公式，作为特例，给出了在无光阑情况下的传输公式。以光束在有光阑情况下自由空间的传输为例作了数值计算，其结果与直接对柯林斯(Collins)公式作数值积分的结果进行了比较，表明用Wen的方法便于进行物理分析，可节约大量机时，并对解析公式的计算误差和适用范围作了讨论。     

Self-consistent solution of the diffusion and current spreading problems in oxide stripe lasers using integral equations: An application to triple lasers

The problem of current spreading and diffusion in oxide stripe lasers leads to two coupled boundary value problems. This paper presents an efficient method for the simultaneous solution of these two problems based on the conversion of the two-dimensional Laplace equation representing the current spreading into integral equations by means of a contour integral. The power of the method is illustrated by its application to a coupled triple-stripe laser. Highlights of the numerical method are discussed.     

An analysis of the scalar Helmholtz equation using the integralequation method

The scalar Helmholtz equations are investigated by using the integral equation method (IEM). In the IEM analysis, the fundamental solution of the Laplace equation is used as a weighting function. Two IEM formulations are obtained: one is a standard formulation and the other is obtained from an elimination of the unknown boundary value. The accuracy and computational time of the IEM are compared with those of the finite element method in two dimensional scalar Helmholtz problems. The analysis of a resonant cavity is reduced to a simple eigenvalue problem. Resonant frequencies of the IEM agree well with those of the finite difference method. Usefulness of the IEM is confirmed through the analyses of the scalar Helmholtz equations     

Reproducibility of Laplacian Wall Thickness Measurements of the Gallbladder with Varying CT Slice Thickness

In measuring changes of gallbladder wall thickness using CT, robustness to differences in acquisition protocols including slice thickness can be important. We have developed an automated technique based on Laplace’s equation to measure the gallbladder wall thickness using computer tomography (CT). The purpose of this work is to investigate the usefulness of the Laplacian technique in obtaining gallbladder wall thickness measurements that are reproducible with variations in CT slice thickness. This study included 2D (2D) and 3D (3D) wall thickness measurements using Laplace’s equation. Ten subjects who had 5 mm (thick) and 2.5 mm (thin) reconstruction (from a single set of raw data) through the abdomen were randomly selected from a research database. Their volumetric CT images were acquired using a multidetector GE MEDICAL SYSTEMS–LightSpeed 16 scanner at 120 KVP, ~250 mAs, with standard filter reconstruction algorithm and manually segmented on all CT cross sections by a radiologist. The inner and outer boundaries of the gallbladder wall were obtained from the segmentation. The thickness of the wall was quantified by computing the distance between the boundaries for each scan and over the entire volume using Laplace’s equation from mathematical physics. The distance between the surfaces is found by computing normalized gradients that form a vector field. The vector fields represent tangent vectors along field lines connecting both boundaries. The Laplacian technique was compared with the conventional Euclidean distance transformation (EDT) technique using coefficient of variation. EDT results in an Euclidean distance mapping between the two extracted surfaces. Both techniques were compared in 2D and 3D. For the 2D and 3D wall thickness measurements, a mean difference of 0.35 and 0.25 mm between thin and thick reconstruction was found respectively using Laplace’s equation. EDT resulted in a higher mean difference for both 2D and 3D. In addition, a significant difference in thickness between the Laplacian technique and EDT techniques (p?<?0.001) were obtained. The Laplacian measurement of gallbladder wall showed significantly lower variation compared to EDT on different CT slice thickness for both 2D and 3D techniques. Hence, proving to be an important technique for obtaining reproducible wall thickness measurements of the gallbladder using CT.     

Non-local singularities as the point-source images in thegeneralized method of D. A. Grave for solving wave-scattering problems.(On the possible origin of a legend about optical properties ofwerewolves and ghosts)

The method of D. A. Grave (1885) in the theory of two-dimensional boundary-value problems for the Laplace equation in domains with smooth analytical (algebraic) boundaries represents the solution as an infinite superposition of static fields of external sources with alternating signs, located along an infinite curve orthogonal to the boundary. This method is a generalization of the classical method of images for the simplest boundaries (half-plane, flat strip, circular, and rectangular cylinders). All of the images are local singularities of the pole type of the solution continuation outside the boundary. The type of solution of D. A. Grave may be obtained for convex algebraic boundaries as a result of a certain regular procedure. In the case of the Helmholtz equation, a generalization of this method for curved boundaries is also possible, but leads to more complicated non-local singularities. Besides the poles, it also contains a series of weak singularities, distributed along an infinite set of segments     

Butt-end forming electrodes for dense electron beams and axisymmetric Brillouin formations

Exact solutions are constructed for the butt-end region of electron beams in the form of finite-length cylinders; truncated cones; cylindrical and conical rings; toroidal, spherical, and ellipsoidal Brillouin formations; a cylindrical magnetron in the subcritical mode; and a cylindrical diode. Problems for finite-length cylinders with elliptic and circular cross sections are investigated. The cylinders are considered in the presence of a uniform longitudinal magnetic field for the case when the longitudinal velocity in a flow exhibits a cycloidal variation and the Cauchy conditions are exactly fulfilled on the flow’s boundary. Relativistic and nonrelativistic flows are analyzed. In the latter case, both the electrostatic and magnetostatic problems are solved. Solution of the magnetostatic problem provides for the continuous transition from the eigenfield to the external magnetic field on a beam’s boundary.     

Scattering from dielectric structures above impedance surfaces andresistive sheets

Interest in understanding of electromagnetic interaction with rough surfaces has prompted the study of scattering from typical dielectric humps over impedance surfaces. It is shown that the Green's function of the problem for a resistive sheet resembles that of the impedance surface. Hence both problems are considered here. A numerical solution for the scattered field of a two-dimensional dielectric object, possibly inhomogeneous, with arbitrary cross section above the impedance surface or resistive sheet is sought. First the Green's function of the problem is derived based on the exact image theory. This form of the Green's function is amenable to numerical computation. Then the induced polarization currents are calculated by casting the integral equations into a matrix equation via the method of moments. Numerical problems in calculation of the Green's function when both source and observation points are close to the surface are discussed. Comparison of numerical results with a perturbation solution shows excellent agreement between the two methods     

Thermal velocity effects in magnetically confined beams

The spreading of magnetically confined electron beams caused by thermal velocities has been investigated theoretically and experimentally. An analysis of the spreading of confined beams of various geometries (strip, rectangular, and cylindrical) is presented. The thermal spreading of a confined cylindrical beam was measured at the anode of a parallel-flow Pierce gun. A transparent fluorescent screen was used for the anode. The spot size at the anode was measured as a function of magnetic field and an attempt made to relate the results to the theory.     

Kinetic description of cylindrical plasma structures

A kinetic model describing the nonlinear dynamics of multicomponent charged plasma in a cylindrical geometry is developed. The kinetic model is applied in the linear approximation to derive the general equation for the potential of the electric field in relativistic plasma. With allowance for electrostatic and azimuthal magnetic eigenfields, the equation is generalized to the case of arbitrary radial irregularities of the density and of the longitudinal and angular velocities of plasma components. A few electrostatic irregularities of axially symmetric electron beams are considered, and expressions for increments are specified. The nonlinear problem of stationary injection into a cylindrical drift half-space is solved for a relativistic partially neutralized electron beam. The dynamics that characterizes the formation of the spatially steady regime and the evolution of the beam’s equilibrium configurations is studied. It is shown that, under certain conditions, the laminar motion of electrons becomes turbulent; i.e., different transverse layers of the beam are mixed.     

Toroidal Resonators and Waveguides of Arbitrary Cross Section

After introducing a new method to solve Maxwell's equations using a complex electromagnetic field vector F, a rotational coordinate system xi, Theta, psi is introduced. In this coordinate system, the field vector components F/sub xi/, F/sub Theta/ may be expressed by F/sub psi/. This component can be obtained from a two-dimensional Hehlmholtz equation. Specifying xi, Theta by cylindrical coordinates r, z the complex Maxwell equation curl F= gamma F is solved for the axisymmetric case (/spl part///spl part/psi = 0) and for the nonsymmetric case. The differential equations for magnetic field lines are solved and surfaces on which the normal component of B and the tangential components of E vanish are recognized as metallic walls of toroidal resonators of various arbitrary cross sections. In the Appendix the results of the new method are compared with well known results for circular cylindrical waveguides.     

Analysis of hard surfaces of cylindrical structures of arbitrarily shaped cross section using asymptotic boundary conditions

The surface equivalence principle is used to formulate the problem of infinitely long cylindrical structures of arbitrary cross section and loaded with metallic strips. The strips are tilted by an angle with respect to the cross section plane. Special cases are considered when the strips are directed along or transverse to the cylinder axis. The excitation is an oblique plane wave incident with arbitrary polarization. The asymptotic strip boundary condition (ASBC) is used to simplify the problem. The surface integral equations are solved using the method of moments. The numerical solution is verified with the series solutions of the circular cylinders loaded with helical strips. Several applications of hard surfaces are presented for structures loaded with tilted strips such as the struts loaded with tilted strips to reduce the equivalent blockage width of the struts and make it polarization independent.     

Finite element implementation of Maxwell's equations for image reconstruction in electrical impedance tomography

Traditionally, image reconstruction in electrical impedance tomography (EIT) has been based on Laplace's equation. However, at high frequencies the coupling between electric and magnetic fields requires solution of the full Maxwell equations. In this paper, a formulation is presented in terms of the Maxwell equations expressed in scalar and vector potentials. The approach leads to boundary conditions that naturally align with the quantities measured by EIT instrumentation. A two-dimensional implementation for image reconstruction from EIT data is realized. The effect of frequency on the field distribution is illustrated using the high-frequency model and is compared with Laplace solutions. Numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation. The results show that scalar/vector potential reconstruction produces images which are essentially indistinguishable from a Laplace algorithm for frequencies below 1 MHz but superior at frequencies reaching 10 MHz.     

Plane wave diffraction by a slit in a perfectly conducting plane and a parallel complementary strip

The diffraction of plane electromagnetic waves by the configuration formed by a slit in a perfectly conducting plane and a parallel complementary strip is investigated. The related boundary-value problem is formulated into a modified matrix Wiener-Hopf equation. The factorization of the kernel matrix is accomplished through Abrahams’ method and the modified matrix Wiener-Hopf equation is first reduced to a pair of coupled Fredholm integral equations of the second kind and then solved by iterations. Several numerical results illustrating the effects of various parameters such as the spacing between the slit and the strip and their width on the diffraction phenomena are presented.