Showing papers on "Elliptic coordinate system published in 2005"

Helmholtz–Gauss waves

Abstract: A detailed study of the propagation of an arbitrary nondiffracting beam whose disturbance in the plane z=0 is modulated by a Gaussian envelope is presented. We call such a field a Helmholtz–Gauss (HzG) beam. A simple closed-form expression for the paraxial propagation of the HzG beams is written as the product of three factors: a complex amplitude depending on the z coordinate only, a Gaussian beam, and a complex scaled version of the transverse shape of the nondiffracting beam. The general expression for the angular spectrum of the HzG beams is also derived. We introduce for the first time closed-form expressions for the Mathieu–Gauss beams in elliptic coordinates and for the parabolic Gauss beams in parabolic coordinates. The properties of the considered beams are studied both analytically and numerically.

Integrable Hamiltonian systems with vector potentials

Abstract: We investigate integrable two-dimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In the case of a linear second invariant, we provide some examples of weakly integrable systems. In the case of a quadratic second invariant, we recover the classical strongly integrable systems in Cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates.

A finite element in elliptic coordinates with application to membrane vibration

Abstract: A nine-node Lagrangian finite element is derived and implemented in elliptic coordinates. The element is used to formulate an analysis for the free vibration of an elliptical membrane. Frequencies and mode shapes are computed for vibration of both solid and annular elliptic membranes. The annular membrane is defined as a conformal ellipse. Both free and fixed interior boundary conditions are analyzed. Results for frequencies and mode shapes are presented in tabular and graphic format, respectively.

Ince-Gaussian series representation of the two-dimensional fractional Fourier transform

Abstract: We introduce the Ince–Gaussian series representation of the two-dimensional fractional Fourier transform in elliptical coordinates. A physical interpretation is provided in terms of field propagation in quadratic graded-index media whose eigenmodes in elliptical coordinates are derived for the first time to our knowledge. The kernel of the new series representation is expressed in terms of Ince–Gaussian functions. The equivalence among the Hermite–Gaussian, Laguerre–Gaussian, and Ince–Gaussian series representations is verified by establishing the relation among the three definitions.

Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature

Abstract: We characterize and completely describe some types of separable potentials in twodimensional spaces, S 2 [�1]�2 , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere S 2 , hyperbolic plane H 2 , AntiDeSitter / DeSitter two-dimensional spaces AdS 1+1 / dS 1+1 as well as for their flat analogues E 2 and M 1+1 . This is achieved through an approach of Cayley-Klein type with two parameters, �1 and �2, to encompass all curvatures and signature types. We discuss six coordinate systems allowing separation of the Hamilton-Jacobi equation for natural Hamiltonians in S 2�1]�2 and relate them by a formal triality transformation, which seems to be a clue to introduce general “elliptic coordinates” for any CK space concisely. As an application we give, in any S 2 [�1]�2 , the explicit expressions for the Fradkin tensor and for the Runge-Lenz vector, i.e., the constants of motion for the harmonic oscillator and Kepler potential on any S 2

DOE-generated laser beams with given orbital angular moment: application for micromanipulation

Abstract: A countable set of linearly independent solutions of the paraxial wave (Schroedinger-type) equation is derived and given the name hyper-geometric modes. These solutions describe pure optical vortices that can be generated when a spiral phase plate is illuminated with a plane wave. The distinction between these modes and the familiar paraxial modes is that in propagation the radius of the former increases as a square root of distance and the phase velocity is the same for all modes. In the present work experimental results on trapping and rotation of 5-10 micron-sized biological objects (yeast cells) and polysty rene beads of diameter 5 P m using various laser beams are discussed. Keywords: diffractive optical elements, pure op tical vortices, orbital angular moment, hyper-geometric modes, optical microparticle manipulation 1. INTRODUCTION The higher-order Bessel and Laguerre-Gaussian (LG) modes contain optical vortices providing screw character and presence of orbital angular moment. A microparticle, trapped in such a beam, receives a rotary movement. The new types of laser beams having orbital angular moment - optical vortices "imbedded" in a plane or a Gaussian beam, are considered. After passing some distance, such fields get rather stable configuration, reminding of LG modes, and are distributed under the similar law. In optics, the Hermite-Gauss (HG) and LG modes, which are partial solutions of the paraxial wave equation (PWE) or Schroedinger equation in the Cartesian or cylindrical coordinates, have long been in wide use [1]. They represent the transverse modes of stable laser resonators. Such modes preserve their structure (cross-section intensity distribution), changing only the scale along the propagation axis. Because these modes form an orthogonal basis it is possible to use their linear combinations for constr ucting other solutions of the PWE. In the cylindrical coordinates, the PWE has other modal solutions that, similar to the HG and LG modes, preserve their structure, changing only in scale. These are referred to as paraxial diffracted Bessel modes [2] and should be distinguished from the paraxial diffraction-free Bessel beams [3 ], which will be reffered to as the Durnin-Bessel modes, to distinguish them from the diffracting Bessel modes. As di stinct from the Gaussian mode s, both Bessel modes possess the infinite energy (their intensity being finite at every space point). The effective diameter of the diffracted Bessel beam increases linearly along the optical axis with increa sing distance from the initial plane. The Durnin-Bessel (DB) beam have a constant diameter. Recently introduced [4-8] new modal solutions of the PWE have been studied theoretically [4-7] and experimentally [8]. These are the Ince-Gaussian modes derived as a solution of the PWE in the elliptic coordinates. In these coordinates, the PWE is solved via separation of variab les, with the solution found as a product of the Gaussian function by the Ince polynomials. Note that the Ince pol ynomials are properly a solution of the Whitteker-Hill equation [2]. The Ince-Gaussian (IG) modes represent an orthogonal basis that generalizes the HG and LG modes. When the elliptic coordinates change to cylindrical (the ellipses change to the circumferences) the IG modes change to the LG modes. With the ellipse eccentric ity tending to infinity (the ellipse changing to a line segment), the IG modes change to the HG modes.

Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

Abstract: We describe Jacobi’s method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrodinger operator with potential on a Riemannian space is second-order superintegrable if there are 2n − 1 (classically) functionally independent second-order symmetry operators. (The 2n − 1 is the maximum possible number of such symmetries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general ...

Numerical Study of Bidimensional Steady Natural Convection in a Space Annulus Between Two Elliptic Confocal Ducts Influence of the Internal Eccentricity

Abstract: The authors express the Boussinesq equations of the laminar thermal and natural convection, in the case of permanent and bidimensional jlow, in an annular space between two confocal elliptic cylinders. A new calculation code using the finite volumes with the primitive functions (velocity-pressure formulation) and the elliptic coordinates system is proposed. The Prandtl number isfixed at 0.7 (case of the air) with varying the Rayleigh number. The efJect of the geometry of the interior (...)

Subgroup Type Coordinates and the Separationof Variables in Hamilton-Jacobi and Schrödinger Equations

Abstract: Separable coordinate systems are introduced in complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also graphically)

A river-flow model based on a simplified boundary-fitted coordinate

Abstract: 河川流計算で用いる水平座標系として, デカルト座標系や直交曲線座標系のような数値モデル上の簡便さを有しつつ, 一般座標系と同程度の計算精度を兼ね備えた簡易境界適合座標系 (水平σ座標系) を新たに提案した. この水平σ座標系とは, 鉛直座標系として気象・海洋計算に用いられる一種の境界適合座標系であるσ座標系を, 河川流計算における水平座標系に応用するものである. 水平σ座標系に基づく河川流モデルの基本的な有効性を調べるために, 本モデルを直線開水路流れや実河川流場へ適用し, 他の水平座標系と計算精度や計算負荷を比較・検討した. その結果, 水平σ座標系は一般座標系と比べて計算精度に大差なく, その上, デカルト座標系と同程度の低計算負荷を保つことが示された.

Quasi-periodic solutions for an extension of AKNS hierarchy and their reductions

Abstract: Quasi-periodic solutions of an extension of the AKNS hierarchy are derived. Based on finite-order expansion of the Lax matrix, the elliptic coordinates are introduced, from which the equations are separated into solvable ordinary differential equations. Then various flows are straightened out through the Abel–Jacobi coordinates. By the standard Jacobi inversion treatment, explicit quasi-periodic solutions of the evolution equations are constructed in terms of the Riemann theta functions. Furthermore, the solutions of a new generalized nonlinear Schrodinger equation, which are the reductions of the above system, are deduced.

A special case of spatial averaging of the diffusion coefficient

Abstract: Steady-state mass transfer through a heterogeneous layer formed by coordinate planes of an orthogonal coordinate system is considered under the assumption that the boundary conditions at both layer boundaries are independent of coordinates. It is established that, if the coordinate dependence of the diffusion coefficient is described by the product of functions depending on different coordinates, then the surfaces of equal concentration (isohalines) coincide with the coordinate planes. It is shown that, in this case, the diffusion problem has an exact solution and the averaging of the diffusion coefficient is performed by elementary methods.

Coordinate System for 3-D Model Used in Robotic End-Effector

Abstract: This paper revises the concept of coordinate system on new 3-D model used in robotic End-Effector. A very brief design of this model has been mentioned and shown. This model is used to calculate the unknown applied force, but here its kinematics configuration is explained through its coordinate system. First local coordinate system for each individual leg is drawn. Then these local coordinates are mapped into global coordinate system. These global coordinates are shaped into unit vectors by concept of rotation matrix using certain trigonometric identities. This paper refreshes the geometrical orientation of coordinate system through configuration to recognize the operation of robotic model.

Adaptive Coordinate Transformation Methods

Abstract: A series of adaptive corrdinate transformation methods are proposed in which the grid angles are preserved approximately, and the material interface is kept to be Lagrangian description and a minimum difference (in the least-squares sense) between the mesh velocity and the fluid velocity is achieved. The new coordinate system is adapted to important features of flow fields.

Analytical approach for resolving stress states around elliptical cavities udc 624.191.8(045)=20

Abstract: The determination of stress states around cavities in the stressed elastic body, regardless of cavity shapes, that may be spherical, cylindrical, elliptical etc. in its analytical approach has to be based on selection of a stress function that will satisfy biharmonic equation ∇ 2 ∇ 2 Ψ = 0, under given boundary conditions. This paper is concerned with formulation and solution of the cited differential equation using elliptical coordinates in conformity with the cavity shape of oblong ellipsoid (1). It is therefore considered that the formulation of the stress tensor will be done in conformity to the cited coordinates. The paper describes basic statements and definitions in connection to harmonic functions used for determination of stress states around cavities formed in the stressed homogeneous space. The particular attention has been paid to the use of Legendres functions, with definitions and derivation of recurrent formulas, that have been used for determination of stress states around an oblong ellipsoidal cavity, (1). The paper also includes the description of procedures used in forming series based on Legendres functions of the first order.

Eigenmodesofatwo-dimensional elliptical laserresonator

Abstract: karkoV. Ha Abstract -Itiswell-known, that theFresnel -Kirchhoff diff-raction theory imposes boundary conditions onafield strength andanormal derivative simultaneously. Itisalso knownthat these requirements contradict eachother. A. Sommerfeld hasshown that forelimination ofthese conflicts ofthediffiraction theory, itisnecessary andenough to select aGreen function sothat thefunction orits normal derivative onaboundary surface areequal tozero. Butsuch Green function isknown, however, forpractically unique caseofadiffraction onaplane screen. According tothe Sommerfeld, this Green function isformed byapoint source andits mirror image inaplane screen. Ifthepoint source andits image areantiphased, theGreen function isidentically equal tozeroonall surface ofthescreen. Ifthe point source andits image arecophased, thenormal derivative oftheGreen function isequal tozeroonthescreen. Theconflicts oftheFresnel -Kirchhoff theory areeliminated ineach casc. Inthework, wepresent oneofpossible versions ofaGreen function forinterior field ofcylindrical elliptic cavities orscreens. Atthesolution ofdiffraction problems insuchfields forsimplification oftheequations everyone usually transfers Cartesian coordinates to elliptical coordinates. Theinterrelations between coordinates arefeatured bymultivalued functions, therefore are available realfield oftheexistence ofcoordinates andset"dummy" fields overlapped onrealfield. This circumstance allows todesign foranelliptic cavity aGreen function satisfying totheSommerfeld conditions. For this purpose, inaninterior point ofreal field ofacavity weputthepoint source featured bytheHankel function HI10 (kr) ofthefirst kindandofthezeroorder, andintherelevant conjugate points "dummy" field weputthepoint source featured bytheliankel function H20(kr) ofthesecond kind andofthezeroorder. TheGreen function is formed asthetotal sumoftheaforesaid cophased point sources.We notethat argument rfortheHlankel function in "dummy" field isnecessary totake bynegative. Itiseasy toverify that such Green function isequal tozeroandits normal derivative isnotequal tozero onaboundary surface ofanelliptic cavity. Anumerous investigations during 40years haveledtoagoodunderstanding oftheeigen- modestructure andlosses both unstable (1-3) andstable laser resonators (4-6). However, satisfac- tory analytical solution formarginally unstable resonators islacking; therefore, itispossible that suchresonators donothavetheapplied utility. Inthis paper, wepresent approximate analytical expression foreigenmodes oftwo-dimensional elliptic emptyresonator withsmall instability which canbeutilised inmetrology forevaluation oftheFresnel phase shift atreproduction ofthe unit ofalength -meter. Foranalysing ofmarginally unstable resonator, weusethemethod ofintegral equations in two-dimensional elliptical coordinates. Suchanapproach hasfollowing advantages: -theelliptic mirror surface hastheleast deviation fromspherical form; -theelliptic mirror surface coincides acoordinate surface andintegral equations havethe mostsimple fonn, thereby; -inelliptical coordinates, wecananalyse both unstable (prolate ellipse) andstable (oblate ellipse) resonators bythesamewayanditisconvenient for thedifferent resonator comparison; -themethod ofintegral equations isgood forusing ofthediverse simplification; -thewaveequation isseparated inelliptical coordinates andwecanusesimple one- dimensional integral equations; -intwo-dimensional coordinates, wecanusethescalar theory ofdiffraction; -intwo-dimensional coordinates, weemploy thepowerful system ofcomplex functions. Weintroduce thesystem ofprolate elliptical coordinates associated withCartesian coordi- nates byformulas oftransformations

Eigenmodes of a two-dimensional elliptical laser resonator

Abstract: It is well-known that the Fresnel - Kirchhoff diffraction theory imposes boundary conditions on a field strength and a normal derivative simultaneously. It is also known that these requirements contradict each other. A. Sommerfeld has shown that for elimination of these conflicts of the diffraction theory, it is necessary and enough to select a Green function so that the function or its normal derivative on a boundary surface are equal to zero. But such Green function is known, however, for practically unique case of diffraction on a plane screen. According to the Sommerfeld, this Green function is formed by a point source and its mirror image in a plane screen. If the point source and its image are antiphased, the Green function is identically equal to zero on all surface of the screen. If the point source and its image are cophased, the normal derivative of the Green function is equal to zero on the screen. The conflicts of the Fresnel - Kirchhoff theory are eliminated in each case. In the work, we present one of possible versions of a Green function for interior field of cylindrical elliptic cavities or screens. At the solution of diffraction problems in such fields for simplification of the equations everyone usually transfers Cartesian coordinates to elliptical coordinates. The interrelations between coordinates are featured by multivalued functions, therefore are available real field of the existence of coordinates and set "dummy" fields overlapped on real field. This circumstance allows to design for an elliptic cavity a Green function satisfying to the Sommerfeld conditions. For this purpose, in an interior point of real field of a cavity we put the point source featured by the Hankel function H10 (kr) of the first kind and of the zero order, and in the relevant conjugate points "dummy" field we put the point source featured by the Hankel function H20 (kr) of the second kind and of the zero order. The Green function is formed as the total sum of the aforesaid cophased point sources. We note that argument r for the Hankel function in "dummy" field is necessary to take by negative. It is easy to verify that such Green function is equal to zero and its normal derivative is not equal to zero on a boundary surface of an elliptic cavity.

Assembly of Element Matrices and Vectors, and Derivation of System Equations

Abstract: The element matrices and vectors are assembled to obtain the characteristic equations of the entire system of elements. Coordinate transformation is the prerequisite to the assembly of matrices and vectors. The coordinate transformation is necessary when the field variable is a vector quantity such as displacement and velocity. Sometimes, the element matrices and vectors are computed in local coordinate systems suitably oriented for minimizing the computational effort. The local coordinate system may be different for different elements. When a local coordinate system is used, the directions of the nodal degrees of freedom will also be taken in a convenient manner. In such a case, before the element equations can be assembled, it is necessary to transform the element matrices and vectors derived in local coordinate systems so that all the elemental equations are referred to a common global coordinate system. Once the element characteristics, namely, the element matrices and element vectors, are found in a common global coordinate system, the next step is to construct the overall or system equations. The procedure for constructing the system equations from the element characteristics is the same regardless of the type of problem and the number and type of elements used. The assembly procedure is further implemented by computer.

Analytical approach for resolving stress states around elliptical cavities

Abstract: The determination of stress states around cavities in the stressed elastic body, regardless of cavity shapes, that may be spherical, cylindrical elliptical etc. in its analytical approach has to be based on selection of a stress function that will satisfy biharmonic equation, under given boundary conditions. This paper is concerned with formulation and solution of the cited differential equation using elliptical coordinates in conformity with the cavity shape of oblong ellipsoid [1]. It is therefore considered that the formulation of the stress tensor will be done in conformity to the cited coordinates. The paper describes basic statements and definitions in connection to harmonic functions used for determination of stress states around cavities formed in the stressed homogeneous space. The particular attention has been paid to the use of Legendre`s functions, with definitions and derivation of recurrent formulas, that have been used for determination of stress states around an oblong ellipsoidal cavity, [1]. The paper also includes the description of procedures used in forming series based on Legendre`s functions of the first order.

Scalar representation of paraxial and nonparaxial laser beams

Abstract: The development of technology of small dimensions requires a different treatment of electromagnetic beams with transverse dimensions of the order of the wavelength. These are the nonparaxial beams either in two or three spatial dimensions. Based on the Helmholtz equation, a theory of nonparaxial beam propagation in two and three dimensions is developed by the use of the Mathieu and oblate spheroidal wave functions, respectively. Mathieu wave functions are the solutions of the Helmholtz equation in planar elliptic coordinates that is a special case of the prolate spheroidal geometry. So we may simply refer to the solutions, either in two or three dimensions, as spheroidal wave functions. Besides the order mode, the spheroidal wave functions are characterized by a parameter that will be referred to as the spheroidal parameter. Divergence of the beam is characterized by choosing the numeric value of this spheroidal parameter, having a perfect control on the nonparaxial properties of the beam under study. When the spheroidal parameter is above a given threshold, the well known paraxial Laguerre-Gauss and Hermite-Gauss beams are recovered, in their respective dimensions. In other words, the spheroidal wave functions represent a unified theory that can describe electromagnetic beams in the nonparaxial regime as well as in the paraxial one.