Heddle D.W. - Electrostatic Lens Systems (2000) (1).pdf

Electrostatic Lens Systems Second Edition Electrostatic Lens Systems Second Edition D W O Heddle Emeritus Professor of Physics Royal Holloway and Bedford New College University of London Institute of Physics Publishing Bristol and Philadelphia 
c IOP Publishing Ltd 2000 All rights reserved. No...

Electrostatic Lens Systems Second Edition Electrostatic Lens Systems Second Edition D W O Heddle Emeritus Professor of Physics Royal Holloway and Bedford New College University of London Institute of Physics Publishing Bristol and Philadelphia 
c IOP Publishing Ltd 2000 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright LicensingAgency under the terms of its agreement with the Committee of Vice-Chancellors and Principals. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0697 1 Library of Congress Cataloging-in-Publication Data are available Publisher: Nicki Dennis Commissioning Editor: Jim Revill Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing Executive: Colin Fenton Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in TEX by Digital by Design, Cheltenham Printed in the UK by Bookcraft, Midsomer Norton, Somerset Contents Preface ix 1 The optics of simple lenses i 1.1 Analogies between particle and photon optics 1 1.2 The cardinal points of a lens 4 1.2.1 The construction of an image 5 1.2.2 The general ray 8 1.2.3 The nodal points 9 1.3 Matrix representation of lens parameters 10 2 The motion of charged particles in an electrostatic field 13 2.1 The equation of paraxial motion 13 2.2 Some general results 15 2.2.1 The nodal and principal points 17 2.2.2 Relativistic effects 18 3 The determination of the axial potential 20 3.1 Analytic expressions 21 3.1.1 Approximations 23 3.1.2 The value of ω 25 3.2 The boundary element method 26 3.3 The finite element method 29 3.3.1 The basic equations 30 3.4 The finite difference method: relaxation 32 3.4.1 A two dimensional example 33 3.4.2 Cylindrical symmetry 38 3.4.3 Target conditions 40 3.4.4 Precision and accuracy 43 3.5 The nine point method applied to cylindrical geometry 46 3.5.1 The nine point formula for a general point 46 3.5.2 The nine point formula for an axial point 48 v vi Contents 3.5.3 The convergence of the potential in the lens geometry 49 3.5.4 Comparisons of the five and nine point methods 50 3.6 Bessel function expansions 52 3.6.1 The variational principle 54 3.6.2 The problem of the non-linear potential in the gap 56 3.6.3 Calculation of the potential 57 4 The optics of simple lens systems 61 4.1 Aperture lenses 61 4.1.1 Three-aperture lenses 62 4.2 Cylinder lenses 64 4.3 The calculation of ray paths 65 4.3.1 Ray tracing in a known potential 65 4.3.2 Integration of the Picht equation 69 4.4 Real and asymptotic cardinal points 70 4.4.1 The determination of the cardinal points by ray tracing 71 4.5 Windows and pupils 71 4.5.1 The energy-add lens 73 4.6 Multi-element lenses 73 4.6.1 Zoom lenses 73 4.6.2 The afocal lens 75 4.6.3 The zoom afocal lens 77 5 Aberrations 80 5.1 Spherical aberration 80 5.1.1 The axial displacement 82 5.1.2 The disc of minimum confusion 83 5.2 Chromatic aberration 84 5.3 Off-axis aberrations 86 5.4 Interrelations of the spherical aberration coefficients 87 5.4.1 The principal surfaces 88 5.5 The determination of aberration coefficients 89 5.5.1 Direct ray tracing 89 5.5.2 Perturbation of the paraxial solution 92 5.6 The aberrations of retarding lenses 98 5.6.1 Spherical aberration 98 5.6.2 Fifth order effects 99 5.6.3 Chromatic aberration 102 5.6.4 Figures of merit 103 6 The LENSYS program 105 6.1 The programs on the disk 105 6.1.1 Loading and running the programs 106  Contents vii 6.2 The screen display 106 6.2.1 The menu system 106 6.3 The Set Options option 107 6.4 The Lens Data option 107 6.4.1 Error conditions 108 6.5 The Imaging option 109 6.5.1 Types of solution 109 6.5.2 Aperture lenses 111 6.5.3 Cylinder lenses 111 6.5.4 The Afocal and Varimag lenses 112 6.5.5 The special lenses 114 6.6 The Print/Save option 115 6.6.1 The Save to Files option 115 6.6.2 The printing options 116 6.6.3 Printing under Windows 117 6.7 The auxiliary programs 117 6.7.1 The RELAX51A.EXE and RELAX51P.EXE programs 117 6.7.2 The RELAXCYL.EXE program 118 Appendix: Technical aspects of LENSYS 120 A.1 The programming language 120 A.2 Initialization 120 A.2.1 The effect of processor type on the speed of the program 120 A.3 Calculation of potentials 121 A.3.1 Aperture lenses 121 A.3.2 Cylinder lenses 121 A.3.3 Details of the calculations 121 A.4 Ray tracing for focal data 124 A.4.1 Accuracy and consistency 125 A.5 Solving the imaging problem 125 A.5.1 Finding the image 125 A.5.2 Solving the lens 126 A.5.3 The afocal lens 126 A.5.4 The Varimag lens 127 References 128 Preface Interest in electrostatic lenses for the control of ion and (especially) electron beams is of long standing and has grown considerably in the past few decades. In addition, recent innovations in the production of low energy positrons have opened a whole new field of research for which electrostatic lenses are required. This book makes no attempt to be a comprehensive text on electron and ion optics, but addresses the need for data on simple lenses consisting of two or three apertures or cylinders, together with some more complex cylinder lenses which have interesting properties. There are a number of quite detailed and extensive texts, of which two recent books, by Hawkes and Kasper and by Szilagyi , contain modern treatments of the subject and are well worth consulting. The book by Harting and Read which has objectives similar to the present volume has, unfortunately, been out of print for some years. The book itself forms an introductory text on electrostatic optics and is accompanied by a disk containing a program for the IBM personal computer. This program, which is described in detail in chapter 6 and an appendix, will calculate the focal and aberration properties of a range of lenses and will also allow the study of the imaging behaviour between conjugate planes. Ray paths are traced in the paraxial approximation. A graphical display (Hercules, CGA, EGA or VGA) is required to run the program, and a maths coprocessor is very desirable. Only non-relativistic energies are considered and the electrode geometries are all very simple and easily manufactured. The text bears some resemblance to courses I have given to first year postgraduate students over a number of years. The LENSYS program, on the other hand, is the result of a pressing need for lens data in a readily usable form for research purposes, and includes a number of lenses which have been found to be particularly useful for the control of electron and positron beams. It is a pleasure to acknowledge the contribution made by two former research students, Dr Roger Cook and Dr Tony Renau, to the development of the Bessel function expansion method and its application to reveal hitherto unsuspected interrelationships between the aberration coefficients. I am particularly indebted to my colleague, Dr Susan Kay, who has worked with the LENSYS program from its ix x Preface earliest manifestation as a number of separate routines. Her suggestions of material to include or omit, and her assistance in developing a reasonably ‘user-friendly’ interface, have been invaluable. The principal changes in this new edition are in chapters 3 and 5 though chapter 6 and the appendix have also been substantially revised to reflect the changes in the LENSYS program itself. Chapter 3 now includes an account of the finite element method for the solution of the potential problem as well as the boundary element, finite difference and Bessel function expansion methods, which have all been expanded. There is a particular emphasis on the FDM, particularly in the nine point form, because it is the easiest of the methods for the reader to develop from scratch. The full equations are derived and, with material in the appendix, allow efficient, convergent programs to be written. While LENSYS remains a DOS program, the rise in the Windows operating system and the associated great increases in computer memory since the first edition have been taken into account. The LENSYS program has been expanded to include the calculation of the coefficients of chromatic aberration and the Petzval integral. The solution of the imaging problem has been extended and now, as well as the ability to find a focusing voltage appropriate to a specified object/image pair, the image position can be found for a specified set of potentials even if it lies outside the displayed screen. Data calculated during the ray tracing process may now be saved to files for further study and screen copies in colour may now be made via the Windows clipboard. D W O Heddle 6 July 2000 1 The optics of simple lenses 1.1 Analogies between particle and photon optics The actions of lenses and mirrors in controlling beams of light are matters of everyday experience and it may be helpful to look at some systems which have similar behaviour in photon and particle optics. The simplest, shown in figure 1.1, is the plane boundary separating two regions which differ in some property. In the case of photon optics the important property is the refractive index. Figure 1.1 (a) Refraction of light at a plane boundary between two media having refractive indices n1 and n2 ; and (b) deviation of a beam of charged particles at a plane boundary separating regions having potentials V1 and V2. The path of a ray of light incident non-normally onto the boundary is changed on crossing the boundary, the directions in the two regions being related by Snell’s law. The analogue in particle optics is a boundary separating two regions at different electrostatic potentials. Here we consider a particle having a velocity v1 in the first region and v2 in the second. These velocities are related to the potentials, V1 1 2 Electrostatic Lens Systems and V2 in the two regions by 1 2 mv12 + eV1 = 21 mv22 + eV2 = 0 (1.1) where e is the particle charge. This expression, which is fundamental to the simple analysis of electrostatic lenses, defines the zero of potential as that for which the particle is at rest. The potential changes abruptly at the boundary, but there is no change parallel to the boundary and so no force parallel to the boundary acts on the particle. The component of momentum parallel to the boundary is therefore unchanged and we can write mv1 sin α1 = mv2 sin α2 (1.2) where α1 and α2 are the angles between the normal to the boundary and the path of the particle in the two regions. From equations (1.1) and (1.2) we have (sin α1 )/(sin α2 ) = (V2 /V1 )1/2 which is exactly Snell’s law with V 1/2 playing the role of the refractive index. That example was rather artificial, because abrupt changes of potential do not occur in free space, but can be only approximated by the use of closely spaced grids. Our second example is more realistic and demonstrates the focusing behaviour of a lens consequent on curved boundaries. Figure 1.2(a) depicts two regions, having refractive indices n1 and n2 , separated by a spherical boundary of radius R. Rays of light incident parallel to the axis, at small distances, h2 and h1 , from the right and left are refracted at the boundary in such a way that they cross the axis at points labelled F1 and F2 respectively. These are the first and second focal points of this simple lens and the distances measured from the boundary (and here we assume that h  R so we can define the position of the boundary by the pole, O, where the boundary cuts the axis) are called the first and second focal lengths and denoted by f1 and f2 respectively. Remembering that h/R is very small, so the sines and tangents of the angles can be approximated by the angles themselves, we can write for the upper part of figure 1.2(a) n1 α1 = n2 α2 α1 = h1 /R α1 − α2 = h1 /f2 and for the lower part n 1 α4 = n2 α 3 α3 = h2 /R α3 − α4 = h2 /f1. Eliminating the angles we find 1 (n1 − n2 ) 1 1 (n2 − n1 ) 1 = =. (1.3) f1 n1 R f2 n2 R Taking the ratio of these two expressions we see that the ratio of the two focal lengths is the negative of the ratio of the refractive indices. The sign is a  The optics of simple lenses 3 (a) Figure 1.2 (a) Refraction of light at a spherical boundary between two media having refractive indices n1 and n2. The inset illustrates the sign convention. The angles α1 and α2 are positive as are the angles between the rays and the axis at the focal points. Since α3 and α4 are negative, the angle at F1 is equal to α3 − α4. (b) Paths of charged particles in a two cylinder lens. The paths become asymptotic to straight lines only away from the central region of the lens. The values of the equipotential surfaces, shown in section, are discussed in the text. consequence of defining the focal lengths as distances measured from the boundary and taking the pole to be the origin of Cartesian coordinates. The sign convention for angles is that angles measured from the axis to the ray are positive if the rotation is clockwise. These conventions lead to angles having the signs of their tangents, with the two distances involved measured from the right angle. One consequence of this choice is that the sign of the angle a ray makes with the axis is opposite to that of dr dz for that ray. For angles defined with respect to some other direction, such as a radius (as in figure 1.2(a)) or a second ray (as in figure 1.3), clockwise rotation 4 Electrostatic Lens Systems from the ray to the other direction defines the positive sense. In a number of the figures we have indicated the direction in which angles are measured according to these conventions. Figure 1.2(b) illustrates an electrostatic analogue consisting of two coaxial cylinders separated by half their common diameter for clarity and held at potentials V1 and V2. No mathematical analysis should be needed to show that, deep inside each cylinder, the potentials are very close to those of the electrodes, but within a diameter or so of the centre of the system the potential changes quite rapidly and we show a number of equipotentials to illustrate this. These are symmetric about a flat, central, equipotential which has a value of (V2 + V1 )/2 and those to the left have values V1 + (V2 − V1 )/2n where n = 2,... , 5. Notice that the paths of the particles are curved and only away from the central region of the lens do they become asymptotic to straight lines. We shall have more to say about the curvature later, but for the moment we concentrate on the asymptotes. As in figure 1.2(a), rays entering the system parallel to the axis from the right and left are deviated towards the axis, crossing it at the first and second focal points, but as the deviation is not abrupt in this case there is no physical reference point from which to measure the focal lengths. Instead we consider the asymptotic paths and measure the focal lengths from the intersections of the asymptotes to the incident and emergent rays. These intersections define, for incident rays close to the axis, the principal planes of the lens. We shall see later that, for non-paraxial rays, the principal surfaces are actually curved; nonetheless the term ‘principal plane’ is universally employed. Notice that the actual paths are not perpendicular to the equipotential surfaces and approach the asymptotes from the outside on the high potential side of the lens. It would be possible to regard the equipotentials as boundaries separating regions having some mean potential and to use Snell’s law to follow the refraction, boundary by boundary, through the lens, but this would be rather tedious and not very accurate. There are much better methods. 1.2 The cardinal points of a lens The contrast between the ray paths in the photon and particle lenses of the previous section was very marked. In the case of a photon lens of many elements, such as is common in even quite simple cameras, the ray paths, while still made up of straight line segments, may experience many changes of direction within the lens and resemble somewhat the ray paths in the particle lens. The details of the paths are of little importance as long as the asymptotic paths are well defined and the appropriate planes of intersection known, because the imaging properties of both sorts of lens can be completely described (in the paraxial approximation, at least) by the positions of the focal points and the principal points, that is, the intersections of the principal planes with the axis. These are four of the six cardinal points of the lens and their use is illustrated in figure 1.3. The lens is specified in this figure by these four points, and their separations and the distances from an arbitrary  The optics of simple lenses 5 R K O’ 2 1 r 1 F H H F O 1 2 1 2 I r 2 2 I’ K 1 p f 2 f q 1 F 2 F 1 v u Q P Figure 1.3 This diagram shows the focal and principal points of a lens and illustrates a construction, using asymptotes to the principal rays, for finding the position of the image of a given object. The angles α1 and α2 are measured from the upper ray to the lower and have opposite signs. reference plane, R, are indicated. If there is a plane of (mechanical) symmetry in the lens, this plane is commonly used as reference and the distances from this plane to the focal points are called the ‘mid-focal distances’ and usually denoted by F1 and F2. The ambiguity of the same symbols being used for these distances and for the focal points themselves is not usually a problem. Chapter 5 will contain a discussion of the aberrations of particle lenses, but until then we shall concentrate on the rather simpler situation of Gaussian imaging appropriate to paraxial rays, and in the next section shall develop some important interrelations. We have restricted the field of this book to lenses of axial symmetry, and we shall consider only rays which lie in planes containing the optic axis, that is, the meridional rays. 1.2.1 The construction of an image We consider an object, OO of height r1 at a distance p from the first focal point, u from the first principal point or P from the reference plane, and construct the image II by drawing two principal rays from O. The first is drawn parallel to the axis to represent the initial asymptote and, from its intersection with the second principal plane at K2 , a line representing the asymptote to the emergent ray is drawn to pass through the second focal point, F2. The second is drawn through the first focal point to meet the first principal plane at K1 from where it continues parallel to the axis. The intersection of these two rays defines the image point, I. 6 Electrostatic Lens Systems The image has a height r2 and is a distance q from the second focal point, v from the second principal point or Q from the reference plane. With our Cartesian sign convention p, P , f1 , F1 , u and r2 are all negative. The other distances are positive. The angles α1 and α2 have opposite signs because the rotations from one ray to the other are in different senses: we have chosen to mark α1 as the positive angle. In order to find relationships among the various distances, we consider a number of similar triangles. From triangles OF1 O and H1 F1 K1 we see that r1 /p = −r2 /f1 and, from triangles IF2 I and H2 F2 K2 , r2 /q = −r1 /f2. Triangles H1 F1 K1 and H2 F2 K2 are not similar and allow us to represent the angles as α1 = r2 /f1 and α2 = −(−r1 /f2 ). The (transverse) magnification of the lens, M, is given by the ratio r2 /r1 and the angular magnification, Mα , by α2 /α1 so we can write M = −f1 /p = −q/f2 and Mα = (−r1 /r2 )(f1 /f2 ) so MMα = −f1 /f2 = n1 /n2 = (V1 /V2 )1/2. (1.4) This last relationship, between the magnification, angular magnification and the ratio of the potentials in the object and image regions, is known as the law of Helmholtz and Lagrange and is essentially a statement that the brightness of an image cannot exceed that of the object. Using the two expressions for the magnification we obtain pq = f1 f2 , which is Newton’s relation, and if we substitute for u (=p + f1 ) and v (=q + f2 ) we obtain f1 /u + f2 /v = 1. This is not often used (except in an approximate form valid for thin lenses), but substitution for P (=p + F1 ) and Q (=q + F2 ) shows that a graph of Q against P is a rectangular hyperbola with its centre at F1 , F2. This is frequently used to present data for lenses of two elements with the voltage ratio as a parameter. Lines of constant magnification can be shown on the same axes, leading to a very simple procedure for the choice of a (two element) lens to meet a specification in terms of a voltage ratio and magnification. In the same way that it is only the ratio of the potentials which governs the behaviour of a lens, so the actual size of the lens does not matter and it is only the ratios P /D, F1 /D etc where D is some characteristic dimension of the lens, usually the diameter, which are important. It is very common indeed to find focal lengths and other distances represented with D as the unit of length. In this book we shall normally write ‘F1 ’, for example, instead of ‘F1 /D’ unless a specific point is to be made. Figure 1.4 illustrates the general form of a family of P –Q curves showing both branches of the hyperbolæ and also the real object-image region for a lens of two closely spaced coaxial cylinders. With our sign convention, the transverse and angular magnifications of a real image are negative. It is, however, common usage to speak of magnifications as though they were positive and to ignore the sign when using terms such as ‘larger’, ‘ smaller’, ‘increase’ etc. Except in critical situations we shall cite magnifications in this fashion. The optics of simple lenses 7 Figure 1.4 (a) P –Q diagram for a lens. The upper left region (P < 0, Q > 0) applies to the most common case of real objects and images. In the other regions one or both are virtual. Notice that, close to the origin, the P –Q relationship is virtually independent of the voltage ratio. (b) P –Q diagram for a two cylinder lens, with logarithmic scales to simplify the presentation of data over a wide range of distances and voltage ratios. Lines of constant magnification and of voltage ratio are shown. 8 Electrostatic Lens Systems H H 2 1 r 2 (z) F1 F2 O I r 1 (z) Figure 1.5 Asymptotes to two principal rays, r1 (z) and r2 (z), and to a general ray r(z) = −[r1 (z) + (1/M)r2 (z)]. It is easy to see that an object in the first principal plane will be imaged in the second principal plane unchanged in size and the principal planes are sometimes referred to as ‘planes of unit positive magnification’. Note that either the object or the image must be virtual in this case. We shall see later that this property of the principal planes has been exploited in particle optics. 1.2.2 The general ray While the two principal rays shown in figure 1.3 offer a simple construction of the position and size of an image, a ray from an axial object point to the corresponding image point is often of greater interest. Provided that we restrict ourselves to paraxial, Gaussian, imaging, any linear combination of two rays which themselves satisfy the focusing condition will also satisfy that condition. Figure 1.5 illustrates a construction to demonstrate this property using the asymptotes to the various rays. Any independent pair of rays may be used as the basis, but it is convenient to choose rays which are parallel to the axis in one region. We define the first ray, r1 (z), as that passing through the first focal point of the lens at an angle of +45◦. This ray is therefore parallel to the axis in the image space and a distance f1 from the axis. The second ray, r2 (z), is parallel to the axis in the object space and a distance, −f1 , from the axis. It crosses the axis at the second focal point, with a slope f1 /f2. The general ray, r(z), is then represented by r(z) = ζ r1 (z) + ξ r2 (z). The object is a distance p from the first focal point, so at the object we have r(z) = ζ (−p) + ξ(−f1 ) = 0, and we can therefore write ξ = ζ /M where M is the lens magnification between these conjugates. At the image we write  The optics of simple lenses 9 r(z) = ζ [f1 + q(f1 /f2 )/M] = 0 for all values of ζ because the term in the square brackets is zero. The single parameter, ζ, is sufficient to define a particular ray and we write
 1 r(z) = ζ r1 (z) + r2 (z). (1.5) M Figure 1.5 illustrates the construction of the asymptotes to one such general ray. We take ζ = −1, and draw the asymptote from the object point, O, until it reaches the first principal plane, H1. Then, from the second principal plane, H2 , at the same radial distance, draw the emergent asymptote to the image point, I. The slope of this asymptote is (−1/M)f1 /f2 and, as M is known from the ratio of the first focal length and the object distance, the position of the image point does not have to be known beforehand. The values used for illustration in figure 1.5 are f1 = −4 f2 = 8 p = −5 q = 6.4 giving M = −0.8. 1.2.3 The nodal points The two remaining cardinal points are the nodal points, N1 and N2. These are points of unit positive angular magnification in the sense that a ray directed towards the first emerges as though from the second with the same slope. All rays from a point on the object must pass through the corresponding point on the image and we show a particular linear combination of the two principal rays in figure 1.6. We draw a ray for which the incident asymptote crosses the first principal plane at some distance h from the axis. The emergent asymptote appears to come from a point in the second principal plane at this same distance, h, from the axis. If these asymptotes are parallel we have r1 − h r2 − h = p + f1 q + f2 which leads to f + f  1 2 h = r1. f2 − p These asymptotes cross the axis at the nodal points which are clearly separated by the same distance as the principal points. We leave it as an exercise for the reader to show that these points are at distances f2 and f1 from the focal points F1 and F2 respectively. Notice that if the refractive indices (potentials) on the two sides of the lens are the same, i.e. f1 = −f2 , then h = 0 and the asymptotes cross the axis at the principal points which, for this case, coincide with the nodal points. 10 Electrostatic Lens Systems f q r 2 1 h F N F 1 1 2 H H N 2 1 2 r 2 p f 1 Figure 1.6 Diagram to illustrate the action at the nodal points of a lens. 1.3 Matrix representation of lens parameters While the focal lengths and distances are the parameters most often presented, it is possible to express the action of a lens by a two by two matrix which acts on some initial ray asymptote for which the ray position, r, and slope, r  , at some plane are expressed as a column vector, to produce a modified asymptote at a second plane.


 r2 a11 a12 r1 =. (1.6) r2 a21 a22 r1 The planes between which the matrix acts determine the values of the matrix elements and some compromise is needed between simple expressions for the elements and a convenient choice of planes. The significance of the individual elements is clear from equation (1.6): a11 is the linear magnification between the planes and a22 the angular magnification. The other elements relate the radius at one plane and the slope at the other. The simplest matrix is one which transfers the ray from the first principal plane to the second. In this case the element a11 = 1 because the principal planes are planes of unit positive magnification, and a22 = − ff21 from equation (1.4). r2 does not depend on r1 so a12 = 0 and it is easy to show that a21 = − f12 by considering an incident asymptote parallel to the axis. The positions of the principal planes of an electrostatic lens vary with the potentials and it would be more convenient to work with a matrix operating between planes which did not move in this way. We now develop the matrix which transfers an incident asymptote at the reference plane to an emergent asymptote at the same plane. Despite the change in radial position which may occur, this is usually referred to as a bending matrix. We do this by operating on the incident column vector with three matrices. The first transfers the ray from the reference plane to the  The optics of simple lenses 11 Figure 1.7 Matrix elements for a lens of two coaxial cylinders separated by 0.1 diameters. Note that all four matrix elements are continuous as the potential ratio passes through a value of one. first principal plane, the second is the matrix acting between the principal planes and the third transfers the ray from the second principal plane to the reference plane.




 r2 1 f2 − F2 1 0 1 F1 − f1 r1 = r2 0 1 − f12 − ff21 0 1 r1
F2 F1 F2 −f1 f2 
 f2 f2 r1 =. − f12 − Ff21 r1 To use this matrix to study the formation of an image we first translate the ray from the object to the reference plane, a distance −P , then use the lens matrix to produce the emergent ray and finally transfer this ray a distance L. The product of these three matrices gives the transfer matrix

F2 F1 F2 −f1 f2 

F2 −L (F2 −L)(F1 −P )−f1 f2  1 L f2 f2 1 −P f2 f2 =. 0 1 − f12 − Ff21 0 1 − f12 P −F1 f2 The condition for imaging is that the position of the ray in the exit plane does not depend on the slope of the incident ray. This is equivalent to the element, a12 , of the transfer matrix being zero, which will be the case if L = Q. The element a11 is equal to the magnification of the lens between the conjugate planes, and a22 is the angular magnification. 12 Electrostatic Lens Systems The analysis of combinations of lenses is frequently simpler if the matrix formulation is used, because the only calculation required is a sequence of matrix multiplications. A further situation in which the matrix formulation has advantages is the analysis of very weak two element lenses. This is because the matrix elements vary smoothly as the potential ratio passes through one, in contrast to the focal lengths and distances which become infinite for this value of the ratio. Figure 1.7 shows the matrix elements for a lens of two coaxial cylinders separated by a gap of 0.1 diameters. 2 The motion of charged particles in an electrostatic field 2.1 The equation of paraxial motion Our concern is with axisymmetric lenses and so we use cylindrical polar coordinates with the z-axis as the axis of symmetry of the lens system. Laplace’s equation is ∂ 2V 1 ∂  ∂V  + r =0 (2.1) ∂z2 r ∂r ∂r where V (r, z) is the potential. Because of the rotational symmetry an expansion of the potential in even powers of r can be made ∞  V (r, z) = An (z)r 2n. n=0 The two terms of Laplace’s equation are 1 ∂  ∂V   2 ∞ r = 4n An (z)r 2n−2 r ∂r ∂r n=0 and  ∞ ∂ 2V = An (z)r 2n ∂z 2 n=0 where primes ( ) are used to denote differentiation with respect to z. The sum of the coefficients of each power of r must be zero so we have a recurrence relation An (z) An+1 (z) = − 4(n + 1)2 giving the series expansion as A0 (z)r 2 A(4) 0 (z)r 4 V (r, z) = A0 (z) − + + ···. (2.2) 2 2 2 ·4 2 2 13 14 Electrostatic Lens Systems The axial potential, V (z), is just A0 (z) and we can write the axial and radial components of the electric field as ∂V ∂V r Ez = − = −V  Er = − = V  ∂z ∂r 2 where we omit the explicit reference to z and retain terms only to second order because we are considering only paraxial rays. When we come to consider the aberrations of particle lenses we shall have to make allowance for the higher order terms. The axial velocity is so much greater than the radial in these circumstances that we can write 21 m(dz/dt)2 + eV = 0 for the total energy of the particle. The equation of radial motion is d2 r er m 2 = eEr = V  dt 2 writing e for the charge on the particle. The sign of the charge matters only insofar as the potential must always have the opposite sign and we shall soon see that the magnitude of the charge does not matter. To eliminate t from these expressions we write d2 r dz d  dz dr  = dt 2 dt dz dt dz giving  −2eV 1/2 d  −2eV 1/2 dr  er  = V m dz m dz 2m which reduces to d2 r 1 V  dr r V  + = −. (2.3) dz2 2 V dz 4 V This equation contains neither the charge of the particle nor its mass and it is therefore valid for electrons, positrons and ions of either sign. The only constraint is that the potential should have a sign opposite from that of the particle to ensure a positive total energy. In order to integrate this equation of motion we require very precise information about the axial potential, because we need to determine the second derivative. An alternative approach is to change the independent variable in such a way as to remove this term. We introduce a reduced radius, R, defined by R = rV 1/4 where V must be read as |V | for positive particles. Successive differentiation of this expression yields dR dr 1 = V 1/4 + V −3/4 V  r dz dz 4 d2 R  d2 r 1 V  dr r V  3r  V  2  = V 1/4 + + − dz 2 dz 2 2 V dz 4 V 16 V =0  The motion of charged particles in an electrostatic field 15 where equation (2.3) shows that the grouped terms sum to zero, and we are left with the very simple equation of motion d2 R 3  V  2 =− R (2.4) dz 2 16 V which is known as the Picht equation. 2.2 Some general results The independent variable in the Picht equation is the ratio of the axial potential gradient to the potential itself and it will be convenient to define a new variable, T (z) = (V  (z)/V (z)), and to note that, because this appears squared the sign does not matter. Figure 2.1(a) shows the variation of the axial potential, the potential gradient and the parameter T (z) for a lens having a gap equal to the radius with a potential ratio of 1 : 16. Figure 2.1(b) shows the lens geometry and equipotentials at values of 321 , 16 1 , 81 , 41 , 21 , 43 , 78 , 15 16 31 and 32 of the overall potential difference. The potential distribution was calculated by the nine point finite difference method of section 3.5 and the two principal rays were traced in this lens by the routines of the LENSYS2 program. It is not difficult to show that all electrostatic lenses having uniform potential regions to each side are converging and to obtain an approximate expression for their focal lengths. We consider a ray incident parallel to the axis at a reduced radius R1 and make the assumption that this reduced radius does not change in passing through the lens. Naturally, r, the true radius, will change or there would be no lens action, but the change in V will act in the opposite sense. A formal integration of the Picht equation gives ∞ ∞ 3 R  dz = R2 − R1 = − R1 T 2 dz −∞ 16 −∞ though in practice the limits of integration can be very much narrower as a consequence of the sharply peaked nature of T 2. R1 = 0 because the incident ray is parallel to the axis and so R2 has the opposite sign to R1. Writing r2 −3/4  R2 = r2 V2 1/4 + V V2 4 2 and, noting that V2 will be zero away from the lens proper and that V2 is 1/4 intrinsically positive, we see that r2 has the same sign as R2 and so for rays incident above the axis the emergent ray moves towards the axis giving a convergent lens action. Lenses for which the object and image positions lie in regions of uniform potential are known as immersion lenses. 16 Electrostatic Lens Systems (a) T V' V z/D (b) Figure 2.1 (a) The variation of V (z), V  (z) and their ratio, T (z) with axial position z/D in a two cylinder lens having a gap to diameter ratio of 0.5. Note that the maximum value of T occurs on the low potential side of the lens. The ordinate scale applies to the axial potential. (b) Ray paths for a voltage ratio of 1 : 16 in this lens showing that most of the deviation of the particle paths occurs in the region where |T | is large. We saw in section 1.2.1 that the second focal length of a lens can be written as −r1 /α2. We identify α2 with r2 and write 1 r R   V1 1/4 3  V1 1/4 ∞ V  2 =− 2 =− 2 = dz. (2.5) f2 r1 R 1 V2 16 V2 −∞ V  The motion of charged particles in an electrostatic field 17 Figure 2.2 Paths of the ray r(z) (· · · · · ·) and the reduced ray R(z) (——) in the vicinity of the nodal points. The reduced ray is bounded by the asymptotes and the curvature of its path is always towards the axis. The slopes of the asymptotes to the ray, r(z), are equal, while those of the reduced ray depend on the potential ratio. The radial scales have been chosen to make the incident asymptotes coincide. If we were to trace a ray incident parallel to the axis, but from the other side, we would obtain 1 r R   V2 1/4 3  V2 1/4 −∞  V  2 =− 1 =− 1 = dz (2.6) f1 r2 R 2 V1 16 V1 ∞ V and would find, for the ratio of the focal lengths, f   V 1/2 n1 1 1 =− =−. f2 V2 n2 The ratio is negative because the integrals in equations (2.5) and (2.6) have opposite signs and the result is what we expect from our previous general discussion. An alternative expression is sometimes quoted ∞ 1 1 V 2 = dz f2 8V2 1/2 −∞ V 3/2 but the derivation of this expression is valid only if V  does not change sign and this is never true unless the potential is constrained by grids. 2.2.1 The nodal and principal points We consider an incident ray having its asymptote directed towards the first nodal point as illustrated in figure 2.2. By a suitable choice of vertical scale the same 18 Electrostatic Lens Systems line can be used to represent the asymptotes of both the ray itself and the reduced ray. The Picht equation shows that R  ∝ −R and so the path of the reduced ray is always curved towards the axis. The reduced ray therefore crosses the axis to the left of the first nodal point; the curvature then reverses and the ray approaches its emergent asymptote from below. This asymptote passes through the second nodal point which must lie to the left of the point at which the ray crossed the axis. Note that both the ‘real’ ray, shown by the dotted line, and the reduced ray cross the axis at the same point. While the slope of the real asymptotes are equal, those of the reduced asymptotes are in the ratio of the one fourth power of the potential ratio. As figure 2.2 shows, the actual ray path is not bounded by the asymptotes. In chapter 1 we saw that the separation of the nodal points is equal to that of the principal points and our result, above, shows that the principal points of an electrostatic immersion lens are crossed, in the sense that H2 is further to the low potential side of the lens than H1. This is just the way in which the principal planes were represented in figure 1.3. 2.2.2 Relativistic effects The independence of the paths of charged particles in electrostatic fields from the charge to mass ratio is only true for velocities sufficiently small that the mass of the particle remains essentially equal to the rest mass. Certainly the lenses discussed in chapter 6 and illustrated in the program would not be suitable for use with high potentials, but it is perhaps worth noting here the ways in which relativistic effects can be taken into account. Writing the momentum of a particle as p = m0 v(1 − β 2 )1/2 (2.7) and the kinetic energy as E = m0 c2 ((1 − β 2 )−1/2 − 1) (2.8) the velocity can be expressed in terms of the momentum (using equation (2.7)) and substituted into equation (2.8) to give m0 c2 + E = c m20 c2 + p 2 which can be rearranged to give an explicit expression for the momentum in terms of the kinetic energy. Recalling that this is just equal to the negative of the charge multiplied by the potential (the zero being taken as usual for the particle to be at rest), we finally express the momentum in terms of the accelerating potential, V.  E  p 2 = 2m0 E 1 + 2m0 c2  eV  = −2m0 eV 1 − 2m0 c2 = −2m0 eVrel  The motion of charged particles in an electrostatic field 19 where
 eV Vrel =V 1− = V (1 + )) 2m0 c2 = V (1 + 0.9785 × 10−6 V ) for electrons is known as the relativistically corrected acceleration potential. Note that the product eV is always negative, and therefore ) is a positive quantity. The equations of motion can be analysed in terms of this corrected potential and the Picht equation retains much the same form. The reduced radius, R, is now 1/4 defined by R = rVrel = rV 1/4 (1 + ))1/4 and the equation itself becomes 3  V  2 R  + R × K()) = 0 16 V where 1 + 43 ) + 43 ) 2 K()) =. (1 + ))2 3 The determination of the axial potential In order to trace the path of a particle through a lens, we have to integrate the equation of motion and it is convenient and simple to work in terms of the Picht equation. We need to know the function T (z) to find the focal properties and we shall see later that we also need to know the axial derivative, T  (z), if we wish to obtain information on the aberrations. The first stage in the calculation is the determination of the axial potential, from which, with its derivative, we construct T (z). We shall consider ways of generating the potential distribution for lenses with more than two electrodes later, but in the first instance we concentrate our attention on lenses having only two electrodes. If the electrode potentials are V1 and V2 , we can express the axial potential in one of the forms V (z) = 21 (V1 + V2 ) + 21 (V2 − V1 )+(z) = V1 + (V2 − V1 ),(z) where +(z) and ,(z) are functions which range between ∓1 and between 0 and 1 respectively as z → ∓∞. There are various methods of calculating these functions. There is an exact solution for the case of two coaxial cylinders at different potentials, but with no gap between them. This is an unrealistic situation, but it has been used as an aid to the normalization of other expressions. The general form can be approximated by a number of analytic expressions chosen more for their mathematical tractability than their connection with the basic physics of the problem. Such an approach has a long history, but with the amount of computing power readily available nowadays, there is little point in using this sort of method though we give a brief account in the next section. Results of much greater accuracy can be obtained by approximations based on the laws of electrostatics. In later sections we consider three such approaches: the direct application of Coulomb’s law, numerical solutions of Laplace’s equation on discrete networks of points and the analytical solution of Laplace’s equation subject to boundary conditions which closely approximate the true situation. 20  The determination of the axial potential 21 3.1 Analytic expressions An exact analytic solution of Laplace’s equation can be found for the axisymmetric case, where the potential, V (z, r), can be written as a constant term plus the product of separate functions, Z(z) and R(r) of the axial and radial coordinates. If we differentiate this product and substitute into Laplace’s equation we obtain an equation in which the terms in z and in r may each be set equal to some parameter which we write as −k 2.
 1 d2 R 1 dR 1 d2 Z + =− = −k 2. R dr 2 r dr Z dz2 The differential equations for the two functions are then d2 Z d2 R 1 dR − k2 Z = 0 + + R = 0. dz2 d(kr) 2 kr d(kr) The solution for Z(z) is straightforward: Z(z) = Ak ekz + Bk e−kz while that for R(r) requires some consideration of the context. The equation is Bessel’s equation of zero order for which the formal solution would be R(kr) = C1 J0 (kr) + C2 N0 (kr) where J0 and N0 are Bessel functions of zero order and of the first and second kind respectively. The latter become infinite for zero argument and cannot therefore give a physically real description of the potential on the axis, so C2 = 0 and the overall solution for the potential becomes V (z, r) = (Ak ekz + Bk e−kz )J0 (kr) + V0 (3.1) where we have incorporated the coefficient C1 into Ak and Bk and V0 is a constant potential determined by the boundary conditions. Notice that the zero order Bessel function of the first kind can be expressed as ∞  (−1)i (r/2)2i r2 r4 r6 J0 (r) = =1− + − + ··· i=0 (i!)2 4 64 2304 which is exactly the form given in equation (2.2) by the Taylor expansion. The general solution for a particular lens geometry will consist of a sum of such terms with, perhaps, a constant representing the potential of an electrode. 22 Electrostatic Lens Systems If k is a real number then, in equation (3.1), Ak must be set to zero for positive values of kz and Bk must be zero for negative values. In section 3.6 we shall use these distinct forms, but for a general case we have to assume that k = jκ and is an imaginary number. Using the modified Bessel function, In (x) = j−n Jn (jx), and expanding the exponentials of equation (3.1) we write   V (z, r) = Gκ cos(κz) + Fκ sin(κz) I0 (κr) + V0. This is a solution of Laplace’s equation for an axially symmetric field and for a specific value of κ. To find the general solution we must integrate over all possible values of κ. ∞  V (z, r) = Gκ cos(κz) + Fκ sin(κz) I0 (κr) dκ + V0. −∞ We consider the case of two coaxial cylinders of radius R separated by an infinitesimal gap and take the following values of the potentials to define the boundary conditions: for z < 0 V (z, R) = V1 for z = 0 V (0, r) = (V1 + V2 )/2 for z > 0 V (z, R) = V2. The second of these conditions both determines the value of V0 and requires that Gκ should be zero to remove the cosine term. ∞ V 1 + V2 V (z, r) = Fκ sin(κz)I0 (κr) dκ +. (3.2) −∞ 2 Application of the first and third boundary conditions allows us to show, after an application of Fourier’s integral theorem, that the coefficient Fk can be expressed as (V2 − V1 ) 2πκI0 (κR) and if we substitute this into equation (3.2), rearrange the terms and note that the integrand is an even function, we find that the potential can be written as ∞ V1 + V2 V2 − V1 2 sin(κz) I0 (κr) V (z, r) = + dκ (3.3) 2 2 π 0 κ I0 (κR) which reduces to the correct boundary values for r = R as ∞ sin(κz) π π dκ = − , 0, for z < 0, 0, > 0. 0 κ 2 2  The determination of the axial potential 23 This expression is not amenable to evaluation except by numerical methods and there is only one reason for continuing the analysis. 3.1.1 Approximations It was pointed out by Gray in 1939 that the axial potential function for the two cylinder zero gap lens can be represented approximately by the function ωz +(z) = tanh (3.4) R which has the correct asymptotic behaviour. The precise shape is governed by the parameter ω. If we differentiate this expression we have d+ ω ωz ω = sech2 = at z = 0. dz R R R From equation (3.3) we can extract the true shape of the axial potential as ∞ 2 sin(κz) dκ +(z) = (3.5) πR 0 κ I0 (κ) and we can equate the differential of this at z = 0 to ω/R to find ∞ 2 dκ ω=. (3.6) π 0 I0 (κ) Cylinders with no gap between them are somewhat impracticable, but this approximate form for +(z) does give a reasonable guide to the behaviour of the lens with a small gap. Writing γ for the potential ratio V1 /V2 , the parameter, T (z), of the Picht equation can be expressed as 2ω(1 − γ ) exp 2ωz T = (3.7) (1 + exp 2ωz)(γ + exp 2ωz) which has a maximum value,  γ 1/2 − 1  Tm = −2ω γ 1/2 + 1 at zm = (1/4ω) ln γ. Analysis using this approximate potential function usually continues by replacing equation (3.7) by one of two expressions having a similar shape: Tm T = (3.8) 1 + ((z − zm )/a)2 or T = Tm sech((z − zm )/a) (3.9) 24 Electrostatic Lens Systems Figure 3.1 The differences between the potential functions, +, of equations (3.4) ( ) and (3.10) (
) and the true axial potential for a two cylinder lens having a gap of D/10. where a is found by equating the area under the curves of equations (3.7), (3.8) and (3.9). The focal lengths and distances can then be expressed as rather complicated functions of γ , a, ω and zm. For the more practical case of cylinders separated by a small, but not infinitesimal, gap, g, the function 1  cosh(ωz/R + ω g/2R)  += ln (3.10) ω g/R cosh(ωz/R − ω g/2R) has often been used. If ω = ω this reduces to tanh ωz/R as g → 0. The expression for T (z) is more complex and the approximation becomes progressively poorer as g increases. Some improvement may be made by modifying the value of ω to take account of the smaller value of the slope of the potential at z = 0, and a value of 1.67 has been used, but this is not really a good model for a wide gap lens. It was in any case common practice to use equations (3.8) or (3.9) for actual calculations. Figure 3.1 shows the differences between the values of + given by these two approximations and the true axial potential for a two cylinder lens with a gap of D/10.  The determination of the axial potential 25 Some of the other approximate forms which have been suggested are: 1 z  z/a  , =1− arccot , = 1+ π a 1 + |z/a| z 1 += exp(−π(z/a)2 ) dz a 0 where the value of a depends on the length of the gap, g. There is little point in giving further discussion of this type of approximation. 3.1.2 The value of ω Values for ω were first calculated some 60 years ago by three rather different methods which gave values of 1.318, 1.32 and 1.315. These values have appeared in many textbooks since then, including the first edition of the present work, frequently without specific attribution. A value of 1.326 227 5, obtained by a numerical integration of equation (3.6), was quoted by Verster in 1963 , but this seems to have passed unnoticed and has not been cited in a textbook. This omission might suggest that there were doubts about the accuracy of this value and a recalculation would seem appropriate. It is now quite practicable to evaluate the integral with much higher precision and accuracy, and we have used a standard series representation of I0 (κ) to determine values of the integrand from 0 to 50 at intervals of 0.001. We have then used three quadrature expressions, each at intervals of 0.01, 0.002 and 0.001, to calculate the integral. While there are differences beyond the 11th decimal place all the results are consistent with a value of ω = 1.326 227 505 1 in complete agreement with Verster, but with three more significant figures. This confirmation of Verster’s value underlines the question of why an older, approximate, value was still being cited 25 years later [1, 2]. The numerical methods described in this chapter cannot calculate the field for a true zero gap lens, but values can be found for progressively smaller gaps and may be extrapolated to zero. While it is no surprise that the Bessel function expansion method of section 3.6 gives an excellent agreement, two other methods have been applied to the problem and also agree well with our calculated value. The finite difference method of section 3.5 lends itself to a simple type of extrapolation in which both the size of the gap and the density of the finite lattice are extrapolated together. With a cylinder radius, R1 , in terms of the lattice spacing, the smallest value of the gap to diameter ratio which can be modelled is 1/R1. Figure 3.2 shows values of the field at the centre of the gap as a function of 1/R12 , which is proportional to the lattice density. A linear fit to these data extrapolates to 1.326 227 506 with a standard deviation of 5 in the last figure. This is a very satisfactory agreement. Similar agreement has been found by Read using a special version of the CPO2D program which uses the boundary element method of section 3.2. 26 Electrostatic Lens Systems Figure 3.2 Values of the field at the centre of a two cylinder lens of radius R1. Extrapolation to the zero value of 1/R12 implies zero gap length and also infinite lattice density. 3.2 The boundary element method The solution of Laplace’s equation requires the conditions to be known over a boundary which completely encloses the space. Almost without exception this calls for some approximation of the conditions in the gaps between the electrodes. The boundary element method, also known as the charge density method, does not have this constraint. It is in principle a very simple method: any electrode may be replaced by a system of charges at the electrode surface provided that the potentials they produce are everywhere the same as those produced by the electrode. In particular, the potential of the electrode itself must be reproduced by the charge distribution. The formal way to express this is n 1  σj (rj ) dSj Vk = (3.11) 4π)0 j =1 Sj |rj − rk | where Vk is the potential at a position rk due to charges qj = σj dSj at positions rj on each of n electrodes. In the present context of round lenses with axial symmetry, the problem reduces to the summation of the potentials at a point due to rings of charge on all the electrodes. The geometry of this situation is illustrated in figure 3.3 in which the potential at the ring of radius ri due to the charge on the ring of radius rj is to be found. In terms of the distances shown in this diagram the denominator of equation (3.11) is l = [ri2 + rj2 − 2ri rj cos(αi − αj ) + (zi − zj )2 ]1/2.  The determination of the axial potential 27 Figure 3.3 Diagram to indicate the coordinates appropriate to the calculation of the potential of a point due to a ring of charge. The absolute values of the angles are arbitrary because of the axial symmetry so, for convenience, let αi = π which allows us to simplify this expression as l = [(ri + rj )2 − 4ri rj sin2 (αj /2) + (zi − zj )2 ]1/2   4ri rj sin2 (αj /2 1/2 = [(ri + rj )2 + (zi − zj )2 ] 1 −. (ri + rj )2 + (zi − zj )2 The total charge on the ring, Qj , can be written in terms of a line density, λj = Qj /2π rj , and each term in the summation of equation (3.11) becomes 1 2π λj rj dαj 4π )0 0 l 1 λj rj = π)0 [(ri + rj )2 + (zi − zj )2 ]1/2 π/2  4ri rj sin2 (αj /2) −1/2 × 1− d(αj /2). 0 (ri + rj )2 + (zi + zj )2 The integral is just K(τ ), the complete elliptic integral of the first kind, π/2 (1 − τ 2 sin2 β)−1/2 dβ 0 28 Electrostatic Lens Systems with 4ri rj τ2 = (ri + rj )2 + (zi − zj )2 and so in terms of τ and Qj the potential at a point i can be written as  n Qj τ Vi = 2 ) √r r K(τ ). (3.12) j =1 4π 0 i j The practicability of solving this set of equations depends on a sensible choice of ‘ring’elements and the distribution of charge between them. The charge density on the surface of an electrode is directly proportional to the electric field normal to the surface and a preliminary estimate of this field may be made in other ways, such as by the finite difference method. The charge density near the edge of an electrode will be high and change rapidly so small rings are needed here: conversely, deep inside a cylindrical electrode the charge density will be small and a substantial length of the cylinder may be used as a single ‘ring’. This choice of elements is probably the factor which most severely limits the overall accuracy of the method. The most direct way of solving the problem is to treat equation (3.12) as a matrix equation relating the column vectors Vi and Qj with the individual elements being functions of the positions of the two rings involved, but other methods have been used. The use of discrete rings of charge leads to a non-uniformity of potential on the electrode surfaces. In place of a constant potential, the potential will vary in a ‘saw-tooth’ fashion with discontinuities at the edges of the ring elements. This effect can be minimized by making the charges on each element the same. An alternative approach would be to use a continuous charge distribution with a surface charge density described analytically. Even though the only boundary conditions which need to be applied are those on the electrode surfaces, the effects of field penetration from charges on the surface of the vacuum system enclosing the electrodes should be assessed and the electrodes designed to minimize such effects. Once the charge distribution consistent with the required electrode potentials has been found, the axial potential distribution follows readily, since the elliptic integrals are then just equal to 1. The gradient of the axial potential can also be expressed in closed form and the function T (z) of the Picht equation follows immediately. When properly optimized, the boundary element method is capable of very high accuracy. There are commercial programs available which use the boundary element method and, in some cases, a limited, trial version is available for download from the Web†. † The CPO program written by RB Consultants Ltd can be downloaded from cpo.ph.man.ac.uk and the Optics program written by Dr Li Wang can be downloaded from www.jps.net/liwang/download.htm.  The determination of the axial potential 29 Figure 3.4 The right hand half of a two cylinder lens showing a pattern of triangular elements, suitable for analysis by the finite element method. Note that the sizes of the elements are far from constant. 3.3 The finite element method Instead of treating the space within a system of electrodes as continuous, we may establish a set of discrete points at which we determine the potential by some means. The finite elements of this method are, usually, triangles and it is the potentials at the vertices of these triangles which we seek to determine. Figure 3.4 shows a system of triangular elements appropriate to the solution of the potential distribution in a two cylinder lens with thick walls. We consider the walls to be thick compared to the length of the gap for two reasons. For purposes of calculation it is necessary to specify the potential at all boundaries, and at the outer edge of the gap a linear variation with position is usually a good approximation and one which can be tested. In addition, for practical purposes, the thick walls reduce the effect of field penetration from, for example, the walls of the vacuum chamber. This factor is important whatever method is used for the calculation. Only half of the lens system is shown in figure 3.4 as the system is symmetric about the mid-plane. If we are representing a lens with potentials 1 and 0 on the left hand and right hand electrodes, we should set the potential at all points on the mid-plane to be 0.5 and at points at the outer edge of the gap to fall linearly from 0.5 to 0. The axial potential gradient has a maximum value at the centre of the lens and is also large close to the inner end of the cylinder because of the 90◦ change of the boundary equipotential. In both regions we should use elements of small size though elsewhere, particularly deep inside the cylinder, they may be much larger. 30 Electrostatic Lens Systems Figure 3.5 A representative element of the network illustrated in figure 3.4. The physical principle behind the finite element method is that the potential distribution will be such that the potential energy of the electrostatic field is a minimum and so we need to express this potential energy in terms of the potentials at the vertices of each and all of the triangular elements. 3.3.1 The basic equations We consider a typical element in figure 3.5 where we have shown both Cartesian and axisymmetric cylindrical coordinates. We shall use the latter for the lens example. We assume that the potential varies linearly with position and write V1 = V (z1 , r1 ) = a + bz1 + cr1 etc (3.13) where the coefficients can be expressed in terms of the coordinates of the vertices and their potentials. a = [V1 (z2 r3 − z3 r2 ) + V2 (z3 r1 − z1 r3 ) + V3 (z1 r2 − z2 r1 )]/2S b = [V1 (r2 − r3 ) + V2 (r3 − r1 ) + V3 (r1 − r2 )]/2S (3.14) c = [V1 (z3 − z2 ) + V2 (z1 − z3 ) + V3 (z2 − z1 )]/2S S = [z1 (r2 − r3 ) + z2 (r3 − r1 ) + z3 (r1 − r2 )]/2 where S is the area of the triangle.  The determination of the axial potential 31 The potential energy of the electrostatic field is  ∂V 2  ∂V 2  ) 0 W = + 2π r dr dz. (3.15) ∂r ∂z 2 The condition for minimum potential energy is that ∂W ∂W ∂W ∂W = = ··· = = ··· = = 0. (3.16) ∂V1 ∂V2 ∂Vk ∂VN ∂V ∂V For a single element, ∂r = c and ∂z = b so the associated potential energy can be written as )0  2  BW = b + c2 2π r dr dz 2   = πr)0 S b2 + c2. From figure 3.4 it is clear that any given node is common to a number of triangular elements and the potential at that node affects the potential energy of all of these. We should, therefore, write each term of equation (3.16) in the form ∂   k+n ∂W = BWj = 0 (3.17) ∂Vk ∂Vk j =k+1 where there are n elements having node k in common. We may move the differential operator inside the summation and write for a representative element ∂(BW )   = πr)0 b(rj +1 − rj +2 ) + c(zj +2 − zj +1 ). ∂Vj The radial coordinate r in this equation has come from the volume element in equation (3.15) and should be assigned the value at the centroid of the triangle, that is the average of the values at the vertices. If we substitute for b and c using equation (3.14) this expression reduces to the sum of the potentials at the three nodes multiplied by coefficients which involve the coordinates of these nodes and which can be calculated once for all. Equation (3.17) can then be written as  k+n ∂W = gm Vm = 0. ∂Vk m=k This is one of a set of N equations which are usually solved by matrix methods. The solution is the set of free node potentials from which equipotential plots, such as that shown in figure 3.6, may be generated. Remember that all the nodes, including those at the boundaries, must be considered in evaluating these equations. The finite element method is of limited accuracy for electrostatic lenses as the linear interpolations used in equation (3.13) militate against the determination 32 Electrostatic Lens Systems Figure 3.6 Equipotentials for the electrode and finite element system shown in figure 3.4. Notice that the equipotentials are not quite smooth; this is the result of using the linear potential variation of equation (3.13) within each triangular element. of the accurate axial potentials and gradients required for ray tracing and the calculation of aberrations. It is a much more useful tool for the analysis of magnetic field problems and has been extensively used in this area. Commercial programs based on the method exist, and the data of figures 3.4 and 3.6 were obtained using Student’s QuickField 4.1† which is a limited version of a very powerful program. We noted earlier the need to vary the size of the elements such that, in regions of high field strength, the elements were small. QuickField allows the user to determine the sizes during the problem definition phase, but other programs, such as FlexPDE‡ will generate an optimal set of elements from the boundary conditions. 3.4 The finite difference method: relaxation One of the simplest methods for calculating the potential distribution in a bounded region is to treat the space not as continuous, but as a regu- lar lattice of discrete points. We can then replace the differential form of Laplace’s equation by difference equations. We shall analyse systems of the † This program, a development of one called ELCUT, can be downloaded from www.tor.ru/quickfield. ‡ FlexPDE lite can be downloaded from www.pdesolutions.com.  The determination of the axial potential 33 Figure 3.7 Lattice of points spaced by a distance h on which the potential may be relaxed using groups of five points ( ). cylindrical symmetry appropriate to electrostatic lenses later, but first con- sider a simpler case which allows us to illustrate some general features. 3.4.1 A two dimensional example Laplace’s equation in rectangular Cartesian coordinates is ∂ 2V ∂ 2V + = 0. (3.18) ∂x 2 ∂y 2 We can represent this space by a rectangular array of points separated by distances h in both x and y directions as shown in figure 3.7, and identify five points each marked by the symbol. The point at the centre of this group is at (x, y). We can express the potential at the four outer points using a Taylor expansion of the potential at the centre point and its derivatives at that point. ∂V 1 ∂ 2V V (x − h, y) = V (x, y) − h + h2 2 ∂x 2 ∂x 1 3 ∂ 3V 1 4 ∂ 4V − h + h − ··· 6 ∂x 3 24 ∂x 4 ∂V 1 ∂ 2V V (x + h, y) = V (x, y) + h + h2 2 ∂x 2 ∂x 1 3 ∂ 3V 1 4 ∂ 4V + h + h + ··· 6 ∂x 3 24 ∂x 4 34 Electrostatic Lens Systems with two similar expressions, but involving y-derivatives, for V (x, y − h) and V (x, y + h). Adding all four equations we find V (x − h, y) + V (x + h, y) + V (x, y − h)+V (x, y + h) − 4V (x, y)  ∂ 2V ∂ 2V  = h2 + ∂x 2 ∂y 2 ignoring higher order terms beginning with 1 4 ∂ 4V ∂ 4V  + h +. 12 ∂x 4 ∂y 4 Equation (3.18) shows that the right hand side of this equation is zero. The potential at any point on the lattice can therefore be written as the mean of the potentials at the four nearest neighbour points. In order to apply this result to a real problem, fixed values of potential must be assigned to all the points on a boundary enclosing a region of our two dimensional space. For the non-boundary points it is sometimes possible to guess approximate potentials, but unless the array is quite large it is usually enough to set all the other potentials to zero. We then move systematically over the space replacing the potential at each point by the appropriate average. It is usual to start at a place on the boundary where regions at different potentials meet. As an example, suppose that the potentials at points (x, 10) and (0, y) are fixed at potential 1 and those at (0, x) and (10, y) are fixed at zero, x and y taking all values from 1 to 9; the potentials at the corners are set to 0.5, 1, 0.5, 0 in sequence, defining a square boundary. We would then begin by writing V (1, 1) = 41 [V (0, 1) + V (2, 1) + V (1, 0) + V (1, 2)] = 41 [1 + 0 + 0 + 0] = 0.25 then increase x to the end of the row and then move up one row, return to the left and so on. A single passage across the array is not going to give us the solution and we must repeat the whole cycle many times until the changes from one cycle to the next are sufficiently small. This process of repeated iteration is known as relaxation. Notice the change in notation here: distances are expressed in units of h. 3.4.1.1 Improving the speed and precision. The relaxation method outlined above will always give a solution, but the precision will depend on the scale of the array of points and the time taken to converge to an adequate degree will increase as the square of the number of points. This conflict between speed and precision can be resolved to some extent by first relaxing a rather coarse array and then inserting additional points between the array points, interpolating to find starting potentials for these new points and then continuing the relaxation process. However, a much greater increase in speed can be obtained by a procedure known The determination of the axial potential 35 Figure 3.8 The approach of the potential on the diagonal of a square lattice (51×51 points) to the final value. (a) The difference between the potential after n cycles of iteration and the final potential, illustrating the exponential approach. The upper curve and those of figure 3.8(b) are the result of five point iteration; the lower curve is the result of nine point iteration, discussed later. (b) The potential after n cycles of iteration with acceleration parameters of (from the right) 1.0, 1.4, 1.8, 1.85, 1.90. 36 Electrostatic Lens Systems as ‘successive over-relaxation’. The basis for this is that each time we calculate the potential at some point, we move it towards its final value, but by only a small amount, and indeed by an amount which is progressively smaller. Over-relaxation involves replacing the potential at a point not by the simple mean of that at its four neighbours but, recognizing that next time around it will be changed even more, we make a bigger change immediately. We write g V (x, y) = (V (x − 1, y) + V (x + 1, y) + V (x, y − 1) + V (x, y + 1)) 4 − (g − 1)V (x, y)† where V (x, y)† is the value of V (x, y) from the preceding iteration and g is called the acceleration parameter. As an example of the increased speed that can be obtained by appropriate choice of acceleration parameter, we show in figure 3.8 the behaviour of a square array similar to that of figure 3.7, but with 51 points on a side. We know from symmetry that the potential of all the points on the diagonal with x = y will become 0.5 eventually and so the sum of the potentials for the 49 points with 0 < x = y < 50 will be 24.5. Figure 3.8(a) shows the exponential approach to this final value for no acceleration, and figure 3.8(b) shows, on linear scales, the sum potential for a number of values of g. With no acceleration the sum reaches a value within 0.0001 of 24.5 only after more than 3000 cycles of iteration, while for g = 1.90 this condition is reached after 95 cycles. Notice that the final value is reached, in this case, from above. The effect of increasing g from 1 to larger values resembles very much the effect of reducing the damping of a resonant circuit and, if g is increased too much, the system becomes unstable and will oscillate for g
2; it is the first signs of this which we see for g = 1.90. The maximum safe value of g depends on the size of the array and also on its shape. This example has only 2401 free points and would be regarded as a rather small array. We commonly use arrays of up to some 500 000 points with aspect ratios of six to 20 and have found no serious problems with values as high as 1.97. The Pascal program RELAX51 used for this calculation is shown, in outline, on the following page, and a compiled version, with certain refinements, is included on the program disk. 3.4.1.2 Estimates of g. Expressions from which the value of the optimum value of the acceleration parameter may be estimated are given by Press et al and figure 3.9 is based on these expressions. It is clear that only for very small arrays will values less than 1.7 be needed.  The determination of the axial potential 37 PROGRAM relax51; VAR V:array[0..50,0..50] of double; x,y,c:integer; g,Vsum:double; PROCEDURE average; VAR x,y:integer; BEGIN for x:=1 to 49 do BEGIN for y:=1 to 49 do BEGIN V[x,y]:=g*(V[x+1,y]+V[x-1,y]+V[x,y+1]+V[x,y-1])/4 -(g-1)*V[x,y]; END; END; Vsum:=0; for x:=1 to 49 do Vsum:=Vsum+V[x,x]; writeln(c:4,Vsum:16:5); END; BEGIN REPEAT REPEAT clrscr; gotoxy(16,6); write(’ENTER a value for the acceleration parameter, g: ’); c:=wherex; writeln; gotoxy(16,7); write(’(1’,chr(243),’ g < 2), but g=0 terminates the program’); gotoxy(c,6); readln(g); IF g=0 THEN exit; UNTIL (g >= 1) and (g < 2); clrscr; c:=1; Vsum:=0; for x:=1 to 50 do for y:=0 to 49 do V[x,y]:=0; for x:=0 to 49 do V[x,50]:=1; for y:=1 to 49 do V[0,y]:=1; V[0,0]:=0.5; V[50,50]:=0.5; while abs(Vsum - 24.50) > 0.0001 do BEGIN average; c:=c+1; END; readln; UNTIL g=0; END. 38 Electrostatic Lens Systems Figure 3.9 Estimates of the acceleration parameter, g, for arrays of Y (R) × X(Z) points. The abscissa is the aspect ratio, X/Y , which allows values for a wide range to be shown compactly. 3.4.2 Cylindrical symmetry In cylindrical polar coordinates Laplace’s equation (2.1) is ∂ 2V ∂ 2V 1 ∂V + + = 0. ∂z 2 ∂r 2 r ∂r Consider a rectangular array of points separated by distances h with the polar axis forming the lower boundary. We can write for the potential close to the point (z, r) ∂V V (z, r + h) = V (z, r) + h ∂r 1 ∂ V 2 1 3 ∂ 3V 1 ∂4 + h2 2 + h + h4 4 + · · · (3.19a) 2 ∂r 6 ∂r 3 24 ∂r ∂V V (z + h, r) = V (z, r) + h ∂z 1 ∂ V 2 1 3 ∂ 3V 1 ∂4 + h2 2 + h + h4 4 + · · · (3.19b) 2 ∂z 6 ∂z 3 24 ∂z with similar expressions for the two other neighbouring points. Taking appropriate sums and differences and ignoring terms of fourth order and above we can write  The determination of the axial potential 39 ∂ 2V V (z + h, r) + V (z − h, r) − 2V (z, r) = h2 ∂z2 ∂ 2V V (z, r + h) + V (z, r − h) − 2V (z, r) = h2 2 ∂r h  h2 ∂V V (z, r + h) − V (z, r − h) =. 2r r ∂r The right hand sides of these three equations are, term by term, equal to h2 times the terms of Laplace’s equation, and so the sum of the left hand sides must be zero. We can therefore write   V (z, r) = V (z + h, r) + V (z − h, r) + V (z, r + h) + V (z, r − h) 1 4 h  + V (z, r + h) − V (z, r − h)  8r
 h = V (z + h, r) + V (z − h, r) + 1 + V (z, r + h) 2r
  h  + 1− V (z, r − h) 4 2r where the latter version is better suited to computation as the r-dependent coefficients can be calculated once for all. These expressions cannot be applied to points on the axis, as the final term would become infinite. For this case we have to consider six points: two axial points, one to either side of the target point, and four points in a plane perpendicular to the axis.   V (z, 0) = 1 6 V (z + h, 0) + V (z − h, 0) + 4V (z, h). As in the two dimensional case, the convergence of the calculation can be made significantly faster by over-relaxation. 
 g h V (z, r) = V (z + h, r) + V (z − h, r) + 1 + V (z, r + h) 4 2r   h + 1− V (z, r − h) − (g − 1)V (z, r)† 2r g V (z, 0) = (V (z + h, 0) + V (z − h, 0) + 4V (z, 1)) − (g − 1)V (z, 0)†. 6 3.4.2.1 Boundary conditions. It is a straightforward matter to apply the conditions at the electrode surfaces, but the gaps between electrodes can present problems and there are other ‘boundaries of convenience’ which can be introduced to reduce the computational space and time requirements. Considerable reductions 40 Electrostatic Lens Systems can be made by recognizing planes of symmetry in the problem. Figure 3.10 shows the electrodes of two aperture lenses, one with two apertures and the other with three. The origin of coordinates is at the centre of the lens. In the case of the two aperture lens, the symmetry is obvious and a suitable set of boundary conditions would be V = 0 on the left hand electrode, V = 0.5 on the plane of symmetry (a boundary of convenience) and a linear rise from 0 to 0.5 along the line from a to b. It is easy to test the validity of this last condition by applying it at a large radial distance and examining the change in potential distribution between the aperture plates as a function of radial position: it is usually very good provided that (D3 − D1 )/A is greater than about two. Only the mesh in the region indicat