Symmetric Electrostatic Lenses Research Articles

High precision electrostatic potential calculations for cylindrically symmetric lenses

A method is developed for a potential calculation within cylindrically symmetric electrostatic lenses using mesh relaxation techniques, and it is capable of considerably higher accuracies than currently available. The method involves (i) creating very high order algorithms (orders of 6, 8, and 10) for determining the potentials at points in the net using surrounding point values, (ii) eliminating the effect of the large errors caused by singular points, and (iii) reducing gradients in the high gradient regions of the geometry, thereby allowing the algorithms used in these regions to achieve greater precisions-(ii) and (iii) achieved by the use of telescopic multiregions. In addition, an algorithm for points one unit from a metal surface is developed, allowing general mesh point algorithms to be used in these situations, thereby taking advantage of the enhanced precision of the latter. A maximum error function dependent on a sixth order gradient of the potential is defined. With this the single point algorithmic errors are able to be viewed over the entire net. Finally, it is demonstrated that by utilizing the above concepts and procedures, the potential of a point in a reasonably high gradient region of a test geometry can realize a precision of less than 10(-10).

Lie algebraic analysis for the nonlinear transport of intense pulsed beams in electrostatics lenses

The Lie algebraic method is applied to the analysis of the nonlinear transport of an intense pulsed beam in cylindrically symmetrical electrostatic lenses and particle orbits in a six-dimensional phase space (x, px, y, py, τ, pτ) are obtained in the second order approximation. They can also be acquired in the third or higher order approximation if needed. In the analysis, we divide the electrostatic lenses into several segments. Each segment is considered as a uniform accelerating field and each dividing point is treated as a thin lens. The particle distribution in a three-dimensional ellipsoid is of Gaussian type.

Focusing ground-state atoms with an electrostatic field

A general formula for the trajectory of atoms in an arbitrary potential with axial symmetry is derived. We apply this formula to show that an axially symmetric electrostatic imaging lens for ground-state neutral atoms is possible. Because of its simple construction and capability of focusing atoms of any species, such a lens system can be used in a variety of applications.

Accuracy of electrostatic lens computations with FOFEM

The increased speed of personal computers enables fast computation of rotationally symmetric electrostatic lenses with the first-order finite element method in meshes with a large number of mesh points. In order to produce an estimate of accuracy of the computed potential, we propose a simple procedure based on doubling the number of mesh points in each coordinate. In this way, we can produce for the lower-density mesh at each point the information about error of the potential and visualize the sources of the computation errors.

High-order maps with acceleration for optimization of electrostatic and radio-frequency ion-optical elements

A method has been developed to calculate accurate high-order ion-optical maps for electrostatic and radio-frequency accelerating elements. The method has been incorporated into the arbitrary-order map-based beam optics code COSY Infinity. The treatment is restricted to the case of negligible magnetic fields, as is typical of heavy-ion accelerating cavities, and does not include space charge. For validation purposes, the beam dynamics calculated for these elements is compared against ray tracing for typical beam and cavity parameters. Different from the ray-tracing approach, parameter changes of individual components typically require only recalculation of the maps of the particular components and not the entire system, and thus the method is particularly suitable for optimization. The approach developed for accurate analytical representation of the on- and off-axis electric fields of cylindrically symmetric electrostatic lenses and radio-frequency cavities is described. Some of the many possible applications for using accurate high-order map representations of Einzel lenses, electrostatic accelerating gaps, and radio-frequency accelerating structures are discussed.

Studies on the asymptotic cardinal elements and aberration coefficients of symmetric unipotential electrostatic lenses

Summary Numerical and theoretical evidence is provided to establish the number of independent coefficients needed to characterize fully the magnification-dependence of the spherical aberration coefficient of symmetric unipotential electrostatic lenses. Claims that the number is smaller than believed are shown to be unfounded.

Analytical and numerical analysis of third-order geometric and first-order chromatic aberrations in a bi-potential electrostatic lens

Glaser and Robl (Z. Angew. Math. Phys. 2 (1951) 444) first proposed a theoretical model for a bi-potential electrostatic lens, whose axial potential distribution satisfies the law of electrostatics and whose paraxial trajectory equation is soluble. In this paper, the Gaussian optical property in the bi-potential electrostatic lens has been thoroughly studied. Meanwhile, based on electron optical canonical aberration theory, analytical formulas of third-order geometrical and first-order chromatic aberration coefficients have first been derived and their computational results obtained for this bi-potential electrostatic lens. It is to be emphasized that this theoretical study can be used to estimate third-order geometric and first-order chromatic aberrations and to provide a theoretical criterion for numerical computation in a rotationally symmetric electrostatic immersion lens.

Synthesis of electrostatic focusing and deflection systems

In this article, we present synthesis of an ion beam column as a single entity, based on combined focusing and deflection (CFD). We present seven cases as examples of designing CFD systems by synthesis. Four of those are examples resulting from the a priori given multielectrode approach using a multipurpose multipole system with nine or five units. In these cases, the units have short cylindrical rings or disks cut into segments with geometrically octupole symmetry. In one example we are using the cubic spline method. The other two cases are examples obtained by using both methods in the case of a separate group of axially symmetric electrostatic lenses and electrostatic postlens deflectors. By using a sequential optimization technique combined with the gradient method and the Hooke–Jeeve’s method for a multipurpose multipole system with nine units, our calculations show that one can produce a beam spot radius less than 7 nm and current density of 614 A/cm2, scanned over a 2-mm-square field for a field ionization source. We also present comparison of our CFD system with other published data. The results display substantial improvements, especially for a field ionization source with smaller energy spread ΔU=1 eV and higher acceptance half-angle γ0=2.3 mrad. A sensitivity analysis is given, and the optimum working distances of a given CFD system are presented for the minimum beam spot radius.

Sensitivity of the focusing property of optimized electrostatic lenses

The optical properties of a given axially symmetric electrostatic multielectrode lens were examined by changing its external parameters (source parameters, object‐side and image‐side potentials, and working distances, etc.) and optimizing the electrode potentials of the lens for each considered value of an external parameter. Our main result is the discovery of the tendencies how the optimal total beam radius depends on the external parameters, i.e., the sensitivity of the optimum related to the change of the external parameters. We established that the external parameters can be distinguished clearly from each other on the basis of the scale of the sensitivity of the optimum related to them.

A Numerical Design Study of Auto-Convergent Electron Gun for Color Cathode Ray Tube

An auto-convergent electron beam gun system for a cathode ray tube (CRT) is proposed, designed and numerically simulated. The system can resolve the beam distortion caused by an aberration of symmetric electrostatic lenses and a higher order field of the deflection yoke, and produce a small beam spot diameter and convergence on the screen. The system consists of two kinds of lenses. One kind consists of three pre-forming lenses for the three beams and the other kind involves the assemblage of two quadrupole lenses, which are common to all beams. After the optimization of the shape of the electrodes and applied voltage on the electrodes, the performance of the lens system is verified by a ray tracing of the electron beams. From the calculation, the spot diameters of 0.3 mm×0.45 mm for the central beam and 0.4 mm×0.45 mm for the outer beam on the screen center are obtained.

Numerical computation of higher-order chromatic aberrations for rotationally symmetric electrostatic lenses

In the present paper third order chromatic aberrations have been computed for an immersion lens and an einzel lens based on canonical aberration theory. The electron trajectories with up to third order chromatic aberrations have been compared with the real trajectories. They are in good agreement, and the computational errors are believed to be about 0.2%. Therefore, the canonical theory of higher order chromatic aberration has been verified. As a practical application, for a typical electrostatic lens, the first, third order chromatic aberrations and the third order geometric aberrations, which are functions of the object side beam half-angle, have been presented in detail. It is expected that the results are useful for designing and optimizing a low chromatic aberration electrostatic lens.

Application of luminous ion beams in direct ion optical characteristics measurements

A novel direct technique for measuring the ion optical characteristics of systems used for the focusing, selection, transportation, energy analysis and mass-separation of ion beams is proposed. This method is based on detecting the light (350–600 nm) emitted because of decay ( τ≈10 −5s) of the metastable excited ions produced by electron-atom collisions. We found that the beam trajectories formed by such ions are visible in free space as well as in electric or magnetic fields under UHV conditions. Some widely employed ion optical elements have been investigated by using the proposed method. For example, our measurements of the focal length, the location of the principal plane, and the aberrations of well-known symmetrical electrostatic lenses are in agreement with computer calculations. This technique may be useful for ion optical characteristics studies and practical designs of ion beam equipment.

Ion Trajectory Analysis Program (ITAP)

ITAP is a 2 1 2 -dimensional FORTRAN code developed for the first-order design of charged-particle transport systems. The Ion Trajectory Analysis Program (ITAP) utilizes the paraxial-ray equation with no space charge to determine image size and divergence along the beam line. A discretized transfer-matrix technique is used to model particle transport through drift spaces, symmetrical electrostatic lenses, quadrupoles, wein filters, sector mass separators, deflection plates, and solenoids. Dispersion effects are also included for the prisms. ITAP contains an iterative design option which can determine the excitations, thicknesses, or gaps required in the model to produce either a desired size or divergence at the pseudo-image. Multiple elements may be designed in sequence permitting complete model optimization. Output consists of trajectory data as well as summary tables which describe the focal properties of each ion-optical element. Plots of the trajectories, divergences, and specific transverse planes are also produced.

Transfer matrices of electrostatic lenses and ion mirrors both consisting of rotationally symmetric electrodes

Abstract To determine the behaviour of time-of-flight mass analyzers containing rotationally symmetric electrostatic lenses and mirrors, transfer matrices up to third and fifth order have been determined. For this purpose the distribution of the electrostatic potential is calculated from the geometry and the potentials of the electrodes and a reference particle is traced through this distribution using differential algebra. By varying some parameters specific elements of the transfer matrix or of combinations thereof can be minimized by numerical multiparameter fits thus obtaining for instance energy-independent transversal and/or longitudinal focusing to first and/or second order.

Achromatic quadrupole focusing systems for use with liquid metal ion sources

We discuss the use of achromatic quadrupole lenses to overcome the limitation due to the energy spread of ions from liquid metal ion sources as they are currently focused with axially symmetric electrostatic lenses. Achromatic quadrupoles require both magnetic and electrostatic focusing elements. Each lens acts to focus in one plane and defocus in the other so that multiplet focusing systems are required to form an image of the source. Lens acceptance angles are limited by the bore of the electrostatic elements which in turn is limited by the necessary magnetic gradients required of the magnetic elements. The coefficients relating image distance to lens excitation of some of the elements of the quadrupole multiplets are found to be much larger than those in an axially symmetric electrostatic system. These factors are discussed in terms of the stability requirements of lens power supplies for a particular achromatic triplet system which is undergoing initial test.

Anticipated performance of achromatic quadrupole focusing systems when used with liquid metal ion sources

Current focusing systems for producing finely focused ion beams from liquid metal ion sources (LMIS) using axially symmetric electrostatic lenses are limited by chromatic aberration due to the relatively high (∼10 eV) energy spread inherent in LMIS. The highest current densities achievable are in the range of 1–5 A/cm2 at energies above 20 keV. In this paper, we discuss the use of achromatic quadrupole lenses to overcome the limitation due to energy spread. These lenses require both magnetic and electrostatic focusing elements. Lens acceptance angles are limited by the electrode positions and by the necessary magnetic gradients. The coefficients relating image distance to lens excitation of some of the elements in the quadrupole multiplet are found to be much larger than in electrostatic systems. These factors combine to require excitation stabilities as high as 1 part per million in order to achieve current densities more than 10 A/cm2, although such performance can be expected at beam energies of 5 keV or below.

Electrostatic lens potentials with small relativistic spherical aberration

We discuss a computerized systematic investigation to find axially symmetric electrostatic lens potentials with acceptable first-order properties and small spherical aberrations. The calculation of the spherical aberration is based on the direct solution of the relativistic trajectory equation in paraxial and nonparaxial approximation. The potentials are characterized by their properties in infinite magnification mode. Conditions for the computations and constraints to satisfy practical requirements are prescribed. A successful axial potential distribution and its relativistic spherical aberration coefficient, the spherical disk radius, and some first-order optical parameters, are also presented in the paper as an example.

A novel approach to the synthesis of electrostatic lenses with minimized aberrations

A novel approach to the synthesis of axially symmetric electrostatic lenses with minimum aberrations is presented. The method uses computer optimization to determine the electrode potentials of a multielectrode lens. The method can be used for designing lenses for many different applications. As a preliminary result, an example of a lens with very small spherical aberration is given. The synthesis of electrostatic lenses can be accomplished automatically with an experimental setup controlled by an online computer.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A systematic analysis of symmetric three-electrode electrostatic lenses

A thorough investigation of a class of axially symmetric three-electrode einzel lenses is given. The study is based on the analysis of axial potential distributions. Two different central/outer voltage ratios are examined. The spherical and chromatic aberration coefficients of a large class of lens models are calculated and plotted. For both voltage ratios general conclusions are drawn for the axial potential functions and electrode shapes that have small aberration coefficients. An electrostatic lens with small spherical aberration is given. General criteria are established for the design of symmetric three-electrode lenses.

A systematic analysis of two-electrode electrostatic lenses

A thorough investigation of axially symmetric two-electrode electrostatic lenses is given. The study is based on the analysis of axial potential distributions, according to the principle of synthesis. The aberration coefficients of a large class of axial potential distributions are examined and plotted. General conclusions are drawn for the axial potential functions and electrode shapes that have small aberration coefficients. The electrode shapes of an optimized lens with small spherical aberration are given. General criteria are established for the design of two-electrode lenses.