Selected Problems of Computational Charged Particle Optics 978-0-12-374717-4

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EDITOR-IN-CHIEF

PETER W. HAWKES CEMES-CNRS Toulouse, France

HONORARY ASSOCIATE EDITORS

TOM MULVEY BENJAMIN KAZAN

Academic Press is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 84 Theobald’s Road, London WC1X 8RR, UK 30 Corporate Drive, Suite 400, Burlington, MA01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA First edition 2009 Copyright # 2009 Elsevier Inc. 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 written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/ permissions, and selecting Obtaining permission to use Elsevier material. Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-374717-4 ISSN: 1076-5670 For information on all Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 09 10 10 9 8 7 6 5 4 3 2 1

PREFACE

Charged-particle optics enables us to study an extensive family of devices. At one extreme we have electron beams of low current density traversing static electron lenses, while at the other there are broad beams in which the current density may be very high and the optic axis curved. D. Greenfield and M. Monastyrskiy have assembled here many of the mathematical tools needed to study such situations; there is more emphasis than in most of the related books on the charged-particle optics of systems in which the current density is significant and on time-dependent focusing. In these respects, the present text complements those intended for students of electron optics. The contents of the individual chapters are presented in the authors’ Foreword and not repeated here. I shall, however, just draw attention to Chapter 5, in which the approach to the study of aberrations developed by the authors, the tau-variation technique, is presented at length. I am very pleased to include this useful text in these Advances, where so many articles on related topics have appeared over the years. I have no doubt that many readers will profit from the explanations set out so clearly here. Peter W. Hawkes

ix

FOREWORD Il n’existe pas de sciences applique´es mais seulement des applications de la science.There are no such things as applied sciences, only applications of science. Louis Pasteur, 1872

Charged particle optics, as one of the most intellectually saturated branches of science, has borrowed its theoretical and numerical methods from classical and celestial mechanics, light optics, mathematical physics, and perturbation theory. Contemporary charged particle optics represents a peculiar ‘‘alloy’’ that includes charged particle optics itself as a study of the regularities of motion of charged particle bunches in electromagnetic fields and principles of image formation, specialized numerical methods, and problem-oriented programming. The penetration of modern computer technologies in charged particle optics cannot be overestimated: many problems that had been inaccessible for the founders of charged particle optics even in their dreams represent a common routine for today’s researchers. On the other hand, it is important for researchers to resist the temptation of entrusting the intellectual work to the computer, thus confining themselves to the most simple, general-purpose, numerical tools. The simplicity of the basic equations of charged particle optics is rather deceptive: a good example is the firstkind Fredholm integral equation for the surface charge density, which trivially follows from the Coulomb law. The long-term intensive work of pure mathematicians and experts in numerical analysis and the extensive experience of practical calculations were needed to understand how to make the solution of this equation accurate and stable, and thus acceptable for charged particle optics applications. This monograph is an attempt by the authors to summarize the knowledge and experience they have acquired during their many years of work in computational charged particle optics. Our main message for readers is that, even in our era of powerful computers and the multitude of general-purpose and problem-oriented programs, the asymptotic analysis based on the perturbation theory still remains one of the most effective tools to penetrate deeply into the essence of the problem in question. Undoubtedly, such an approach assumes that the researcher possesses some general mathematical culture and a certain knowledge of numerical analysis and clearly realizes the illusiveness of gaining the reliable and comprehensive solution to any problem of charged particle optics by means of a ‘‘frontal’’ computer attack. Speaking figuratively, the use of xi

xii

Foreword

perturbation theory allows the researcher to consider a computational procedure not as a ‘‘black box’’ but as an open Chinese fan, with all variety of its hues and details. This is especially important in imaging charged particle optics, when the quality of an electron or ion image is characterized not by the charged particle trajectories themselves but by the points of their intersection. The first, and perhaps the most important, step in the application of perturbation theory is to answer the question ‘‘what is big and what is small?’’ or, in other words, to pick out a set of small parameters as being peculiar to the problem in question. This is not always simple, and physical intuition supported by the properly arranged numerical experiments is commonly of profound help. Do such small parameters always exist? The authors think that a good answer to this (in definite sense philosophical) question was given by a friend of one of the authors, Prof. A.V. Guglielmi, who once said that ‘‘a system with no small parameters simply cannot work’’. Maybe this statement sounds too generally but it seems to be close to the truth. This monograph consists of nine chapters and seven appendices. Chapters 1 through 4 are dedicated to different aspects of electrostatic and magnetostatic field calculation based on the first-kind Fredholm integral equation method, the finite element method, and their generalizations. The main emphasis of Chapter 1 is the problem of boundary triangulation and construction of a special basis of functions that allows effective approximation of the surface charge density on the triangulated boundary in the three-dimensional case. We use locally analytical finite elements, either flat or curved. The proposed basis contains both regular functions represented by local polynomials and the functions with prescribed singularities determined by the asymptotic behavior of the surface charge density in the vicinity of the edges and ribs of the boundary. Those singularities are studied in Chapter 2 for different cases of conducting and dielectric boundaries. In Chapter 3 perturbation theory and the integral equations method are employed for the numerical evaluation of potential perturbations caused by small deviations of electrode geometry and voltages from a nominal state. The contribution of fringe effects, which comprise a special class of the so-called locally strong perturbations, is considered as well. The two- and three-dimensional linear and nonlinear magnetostatic problems are considered in Chapter 4 on the basis of the finite-element and boundary-element methods, and their combinations. Chapter 5 presents a versatile version of aberration theory based on the tau-variation method in tensor form, which allows unified construction of the aberration expansions in any stationary or nonstationary electromagnetic field for both narrow and wide charged particle beams. Applications of the tau-variation method to the multiple principal

Foreword

xiii

trajectory approach, tolerance analysis, tracking technique, and charged particle scattering are discussed. The problem of simulating space charge effects is considered in Chapter 6. The central focus of this chapter is the technique of tree-type preordering, which is used for Coulomb field calculation in short bunches of charged particles. The possibility of exclusion of the external field in space charge problems is discussed as a generalization of the numerical approach suggested for charged particle scattering simulation in Chapter 5. Chapter 7 is devoted to the most general regularities of electron image formation. The basic notions of isoplanatic system, spatiotemporal spread function, modulation and phase transfer functions, spatial and temporal resolution are studied using the strict concepts of linear system theory. Those notions are applied in Chapter 8 to study electron image formation in static and time-analyzing image tubes with axial symmetry. Spatial and temporal focusing of photoelectron bunches in timedependent electric fields is the subject of Chapter 9. The practical goal connected with the theoretical and experimental results presented in this chapter was to create a photoelectron gun capable of compressing the photoelectron bunches in time-dependent electric fields. Chapter 9 may serve as a pithy example of application of almost all numerical methods and approaches presented in the monograph. The appendices contain auxiliary information, mainly of mathematical nature, that may be helpful for better understanding of the basic text. This monograph by no means claims to be a regular textbook on computational charged particle optics. It represents only the authors’ viewpoint on the problems presented. Interested readers can find more details regarding those problems in the original papers and monographs listed in the references. We are most grateful to everyone who has promoted the work on this monograph. Special thanks are extended to our colleagues – experimentalists whose permanent collaboration has always been exceedingly stimulating and encouraging.

SOME BASIC NOTATION Vectors in three-dimensional Euclidean space are denoted by bold capital characters such as R, P, Q. We usually do not distinguish the points and their radius-vectors. The vector components and Cartesian coordinates of the points are denoted by lower-case characters such as x, y, z. If necessary, the components or coordinates are supplied with a subindex indicating a vector or a point (for example, xQ ). In Chapter 5 we use the tensor notations xi for Cartesian coordinates, with superscript indexes taking the values 1, 2 and 3, so that x1
x,

xiv

Foreword

x2
y, x3
z. The components of the vectors are likewise given superscript indexes; for example, Ei and Bi are the components of electric and magnetic fields, correspondingly. Summation over the repeated dummy indexes is assumed by default. The two-dimensional vector composed of the two first Cartesian coordinates ispdenoted ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as r ¼ fx; yg, and its absolute value is denoted as r ¼ jrj ¼ x2 þ y2 . The cylindrical coordinate system thus appears as ðz; r; cÞ. Derivatives with respect to time are usually denoted by dot, and the derivatives with respect to the arc-length by prime. The normal derivative of a scalar field F on a surface is denoted as @n F , or @n F when the normal derivatives on the two sides of the surface are different. The variation of a functional g, function G, and parameter q are denoted as dg, dG, and dq, respectively. The Dirac delta function has special notation dD . The scalar product in three-dimensional or two-dimensional vector space is denoted by angle brackets whereas the scalar product in functional space is shown by double angle brackets . The operation of vector multiplication is denoted by the sign . In some cases, we use parenthetical notations ðjÞ for aberration coefficients. For example, ðrjx0 Þ, ðTje0 x0 Þ denote, correspondingly, the partial derivatives @r=@x0 and @ 2 T=@e0 @x0 . The corresponding binominal coefficients are included, for example, ðTjx0 x0 Þ
ð1=2Þ@ 2 T=@x20 , ðTjx0 x0 x0 Þ
ð1=6Þ@ 3 T=@x30 . The particle charge is denoted by q and the particle mass by m. The centimeter-gram-second (CGS) system of units is used unless otherwise stated. For the particle energy we usually use the ‘‘voltage’’ units assuming the energy divided by the particle charge.

FUTURE CONTRIBUTIONS

S. Ando Gradient operators and edge and corner detection V. Argyriou and M. Petrou (vol. 156) Photometric stereo: an overview K. Asakura Energy-filtering x-ray PEEM W. Bacsa Optical interference near surfaces, sub-wavelength microscopy and spectroscopic sensors C. Beeli Structure and microscopy of quasicrystals C. Bobisch and R. Mo¨ller Ballistic electron microscopy G. Borgefors Distance transforms Z. Bouchal Non-diffracting optical beams F. Brackx N. de Schepper and F. Sommen (vol. 156) The Fourier transform in Clifford analysis A. Buchau Boundary element or integral equation methods for static and time-dependent problems B. Buchberger Gro¨bner bases T. Cremer Neutron microscopy N. de Jonge (vol. 156) Electron emission from carbon nanotubes P. Dombi Ultra-fast monoenergetic electron sources

xv

xvi

Future Contributions

A. N. Evans Area morphology scale-spaces for colour images A. X. Falca˜o The image foresting transform R. G. Forbes Liquid metal ion sources B. J. Ford Physics and the pioneers of microscopy C. Fredembach Eigenregions for image classification J. Giesen Z. Baranczuk K. Simon and P. Zolliker Gamut mapping J. Gilles Noisy image decomposition ¨ lzha¨user A. Go Recent advances in electron holography with point sources M. Haschke Micro-XRF excitation in the scanning electron microscope L. Hermi M. A. Khabou and M. B. H. Rhouma Shape recognition based on eigenvalues of the Laplacian M. I. Herrera The development of electron microscopy in Spain J. Isenberg Imaging IR-techniques for the characterization of solar cells K. Ishizuka Contrast transfer and crystal images A. Jacobo Intracavity type II second-harmonic generation for image processing L. Kipp Photon sieves G. Ko¨gel Positron microscopy

Future Contributions

T. Kohashi Spin-polarized scanning electron microscopy R. Leitgeb Fourier domain and time domain optical coherence tomography B. Lencova´ Modern developments in electron optical calculations H. Lichte New developments in electron holography M. Mankos High-throughput LEEM M. Matsuya Calculation of aberration coefficients using Lie algebra S. McVitie Microscopy of magnetic specimens I. Moreno Soriano and C. Ferreira Fractional Fourier transforms and geometrical optics M. A. O’Keefe Electron image simulation D. Oulton and H. Owens Colorimetric imaging N. Papamarkos and A. Kesidis The inverse Hough transform K. S. Pedersen A. Lee and M. Nielsen The scale-space properties of natural images E. Rau Energy analysers for electron microscopes E. Recami and M. Zamboni-Rached (vol. 156) Superluminal solutions to wave equations R. Shimizu T. Ikuta and Y. Takai Defocus image modulation processing in real time S. Shirai CRT gun design methods

xvii

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Future Contributions

A. S. Skapin The use of optical and scanning electron microscopy in the study of ancient pigments T. Soma Focus-deflection systems and their applications P. Sussner and M. E. Valle Fuzzy morphological associative memories S. Svensson The reverse fuzzy distance transform and its applications I. Talmon Study of complex fluids by transmission electron microscopy M. E. Testorf and M. Fiddy Imaging from scattered electromagnetic fields, investigations into an unsolved problem N. M. Towghi Ip norm optimal filters E. Twerdowski Defocused acoustic transmission microscopy Y. Uchikawa Electron gun optics K. Vaeth and G. Rajeswaran Organic light-emitting arrays M. van Droogenbroeck and M. Buckley Anchors in mathematical morphology V. Velisavljevic and M. Vetterli Space-frequence quantization using directionlets M. H. F. Wilkinson and G. Ouzounis Second generation connectivity and attribute filters D. Yang Time lenses M. Yavor (vol. 157) Optics of charged particle analysers P. Ye Harmonic holography

CHAPTER

1 Integral Equations Method in Electrostatics

Contents

1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11.

Statement of the Problem Boundary Surface Approximation Surface Charge Density Approximation Interface Boundary Conditions for Dielectric Materials Reducing the Integral Equations to the FiniteDimensional Linear Equations System Accuracy Benchmarks for Numerical Solution of 3D Electrostatic Problems More Complicated Examples of 3D Field Simulation Planar and Axial Symmetries Calculation of Potential and its Derivatives Near the Boundary Acceleration of Field Calculation: Finite-Difference Meshes and Calculation Domain Decomposition Microscopic and Averaged Fields of Periodic Structures

2 6 8 12 14 16 20 22 26 29 32

Simulation of electric and magnetic fields in which electron and ion beams are focused and transported is the first, and perhaps, the most complicated, part of numerical modeling and computer-aided design in charged particle optics. The obvious fact that only very few geometrical constructions allow exact field representation makes the numerical methods for field calculation of profound practical importance. This chapter considers only electrostatic systems; Chapter 4 covers the magnetic case. It should be clearly understood that we have not endeavored to provide an exhaustive survey of numerical methods for solving the field problem. Advances in Imaging and Electron Physics, Volume 155 ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)00801-X

2009 Elsevier Inc. All rights reserved.

#

1

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Integral Equations Method in Electrostatics

Moreover, the content of this chapter does not include all the aspects of computational electrostatics. The well-known monograph by Hawkes and Kasper (1989) provides interested readers with details not included herein. The emphasis of this chapter is on the integral equations approach, or the boundary element method (BEM), which, in our opinion, is the most suitable for imaging charged particle optics applications. This is why original papers are mentioned and quoted only to the extent necessary to discuss the questions being considered. Section 1.1 gives the mathematical statement of the general threedimensional (3D) electrostatic problem as the Dirichlet problem for the Laplace-Poisson equation, introduces the basic definitions, and outlines the principles of the BEM. The details of the electrode surfaces representation and the charge density approximation are presented in Sections 1.2 and 1.3, correspondingly. Extension of the BEM to the case of dielectric materials is considered in Section 1.4. Some essential aspects of numerical solution of the integral equations are given in Section 1.5. A number of test problems and examples of electrostatic simulation are presented in Sections 1.6 and 1.7. In Section 1.8 we derive appropriate Green functions for the cases of planar and axial symmetries. Some aspects of field calculation near the electrode boundaries are considered in Section 1.9. Section 1.10 is devoted to techniques to make the BEM more efficient. Finally, a special type of electrostatic problem, namely, simulation of the electric field in the vicinity of periodic structures such as electronoptical grids, is considered in Section 1.11.

1.1. STATEMENT OF THE PROBLEM Electrostatic field modeling consists in numerical solving the Dirichlet boundary-value problem for the Poisson equation D’ðPÞ ¼ 4prðPÞ;
ðQÞ; ’j@O ¼ ’

P2O

Q 2 @O

ð1:1Þ ð1:2Þ

with respect to the potential ’(P) as a function of the point radius-vector P. The domain O may be unrestricted, and the boundary @O may consist of a finite number of connected components @Oi ; i ¼ 1; . . . ; n. When the domain O is unrestricted, the boundary-value problem (1.1), (1.2) should be supplemented by the condition at infinity, which may be imposed in the form
 1 ; ð1:3Þ ’ðPÞ ¼ ’1 ðPÞ þ O jPj

Integral Equations Method in Electrostatics

3

with a smooth potential distribution ’1(P) satisfying the Laplace equation. Normally one can put ’1(P)  0; however, it is of interest in some applications to know to what extent a system of electrodes (for instance, a fine-structure grid) affects the external electrostatic field. The condition (1.3) should be modified in the case of planar systems, with their electrodes formally extending to infinity (we touch on this question in Section 1.7). The function r(P) represents the space charge density distribution in the domain O. As soon as we have postponed the discussion of space charge simulation until Chapter 6, this section considers only the spacecharge-free case r = 0. The traditional axial and planar symmetries gradually give up their exclusive positions in charged particle optics. The analysis and computeraided design of many essentially 3D optical elements, such as deflectors, quadrupole lenses, dispersive elements, and ion traps, require numerical solution of Laplace and Poisson equations in 3D domain of rather complicated configuration. The algorithms to solve this problem may be divided into three broad classes: (1) the algorithms based on the finite difference method (FDM), (2) the algorithms based on the finite volumes method (FVM), and (3) those using the method of integral equations or, in other words, the BEM. As seen in the analysis made by Cubric et al. (1999), each approach has its specific advantages and faults. The FDM is relatively easy to implement, which is why a great variety of 3D computer programs based on this approach exist. The most well-known of them are probably the programs by Munro (1990), Rouse (1994), and the program package SIMION (Dahl, Delmore, Appelhans, 1990). Conversely, the FDM requires exceedingly bulky meshes of up to several million nodes to gain acceptable calculation accuracy, which leads to considerable time and memory consumption while the calculation accuracy still remains rather moderate. The use of finite-volume elements, with their shapes adjusted to the electrode configuration (in contrast to the FDM, which uses the regular meshes), may provide substantially better accuracy. Nevertheless, it is not easy to elaborate a finite-volume meshes construction algorithm versatile enough for application in precise simulations of charged particle optics. The use of curvilinear finite-difference meshes is an alternative approach (see, for instance, Yamamoto and Nagami, 1977). The field distributions calculated with FVM suffer from the lack of smoothness even more than the FDM solutions – the higher derivatives of the field can barely be retrieved with proper accuracy. This drawback is especially crucial when high-order aberration coefficients are needed. The main advantages of BEM, the basis of which forms the integral equations technique, follow from the fact that, in contrast to FDM and FVM, that approach does not require digitization of the entire calculation domain but only of the electrode surfaces. First, this implies that the

4

Integral Equations Method in Electrostatics

number of mesh nodes and, correspondingly, the rank of the system of linear equations to be solved may be significantly reduced. Second, it is quite realistic to match an automatically generated 2D finite-element mesh with the electrode shape to gain better approximation accuracy. Finally, the kernels of integral representations for potential functions in BEM allow explicit analytical differentiation, which essentially improves the accuracy of calculating the higher derivatives. These features make BEM most effective for 3D electrostatic field simulation in high-accuracy problems of imaging charged particle optics. Whenever a conducting electrode is immersed into an electric field, an electric charge density s(Q) is induced on its surface. The corresponding electric potential is given by the Coulomb integral ð ð1:4Þ ’ðPÞ ¼ GðP; QÞ sðQÞ dSQ þ ’1 ðPÞ; Q 2 G; P 2 O G

with the Green function GðP; QÞ ¼ 1= j P Q j, and dSQ being the area element on G ¼ @O. Hereafter, the index Q indicates that integration is performed with respect to the point Q running across the boundary G ¼ @O. The Green function G satisfies the equation DGðP; QÞ ¼

4pdD ðP

QÞ

ð1:5Þ

with the Dirac delta-function dD in the right-hand side. The potential defined by Eq. (1.4) automatically satisfies the Laplace equation everywhere except the electrode surfaces bearing the charge, and meets the condition (1.3) in infinity. Thus, the problem of calculating the electric potential in the 3D domain O is reduced to the problem of finding a 2D surface charge density distribution s to satisfy the Dirichlet condition (1.2) on the boundary G. In doing so, we arrive at the first-kind Fredholm integral equation ð sðQÞ
ðPÞ ’1 ðPÞ; P; Q 2 G dSQ ¼ ’ ð1:6Þ jP Qj G

on G. It is easy to verify that the kernel G is integrable; nonetheless, the correctness of the integral equation (1.6) deserves special analysis. To briefly illustrate this, let us consider, according to Tikhonov and Arsenin (1979), the 1D first-kind integral equation ðb A½yŠ  Kðx; x0 Þyðx0 Þdx0 ¼ f ðxÞ a

ð1:7Þ

Integral Equations Method in Electrostatics

5

with some integrable kernel Kðx; x0 Þ. It is easy to see that the solution y0 ðxÞ to this equation, as understood in the classical sense y0 ¼ A 1 ½ f0 Š, where A 1 is the inverse operator to A, is generally not stable with respect to small variations of the right-hand side f (or, in other words, Eq. (1.7) is illconditioned) if the kernel K is smooth enough. Indeed, the function yðxÞ ¼ y0 ðxÞ þ N sin ox is the solution to Eq. (1.7) with the perturbed right-hand side ðb f ðxÞ ¼ f0 ðxÞ þ N Kðx; x0 Þ sinox0 dx0 : a

ð1:8Þ

Obviously, for any arbitrary given number N, the discrepancy of the right-hand side in the L2-norm, 8 2 32 91=2 > > =

> ; : a

a

can be made arbitrarily small at o large enough whereas the corresponding discrepancy between the solutions in the C-norm ky

y0 kC½a;bŠ ¼ max j N sinox j¼ N x2½a;bŠ

ð1:10Þ

remains arbitrarily large. The same deduction as to ill-definition of Eq. (1.7) can be easily derived if the discrepancy between the solutions y(x) and y0(x) is estimated in the L2-norm. At the same time, if the kernel K contains a delta-function singularity, so that Kðx; x0 Þ ¼ 6 0 and a regular residual K1 ðx; x0 Þ, C1 dðx x0 Þ þ K1 ðx; x0 Þ with C1 ¼ we immediately come to the well-defined second-kind Fredholm integral Ð equation C1 yðxÞ þ K1 ðx; x0 Þ yðx0 Þdx0 ¼ f ðxÞ. It is very important that even some essentially weaker integrable singularities of the kernel Kðx; x0 Þ in the coincidence limit x0 ! x may improve the conditionality of the first-kind integral equation (1.7) within a restricted set of possible solutions. For instance, as was shown by Voronin and Tsetsoho (1981), the 1D first-kind Fredholm integral equation with logarithmic singularity in the kernel is well-conditioned in the class of functions obeying the Ho¨lder condition. This remarkable theorem has not only laid a reliable theoretical ground for solving the first-kind Fredholm integral equation numerically but also pointed out the important fact that the more thoroughly all the boundary properties and singularities are preliminary taken into account (both in boundary surface approximation itself and in numerical representation for the surface charge density s(Q)), the higher stability and accuracy can be expected from BEM.

6

Integral Equations Method in Electrostatics

1.2. BOUNDARY SURFACE APPROXIMATION In some cases the electrode surfaces may be described analytically using a set of simple shapes (geometrical primitives). For instance, in the software packages developed by Harting and Read (1976) and Murata, Ohye, and Shimoyama (1996), each of the electrodes is represented by one or several simple geometric primitives such as rectangles; ovals; spherical and elliptical segments; cylindrical, conical, and toroidal surfaces; and so on. This approach, effective for comparatively simple numerical models, often fails in the description of a somewhat complicated boundary by joining or subtraction of a number of overlapping primitives. The main point here is that the intersection line of even the simplest surfaces is not easy to describe analytically, while the Voronin-Tsetsoho’ theorem implies that the accurate representation of the electrode ribs formed by such intersections is of crucial importance from the viewpoint of computational accuracy and stability. For these reasons we do not follow the traditional approach consisting in the global representation of entire electrode surfaces by analytical expressions. In contrast, in our approach the surfaces are subdivided into a large number of elements, each of them defined by a polynomial with specific coefficients. The simplest way to construct such a locally analytical representation is to split the surfaces into a set of the triangles touching each other along their common edges. A local coordinate system is introduced within each of the triangular elements, as shown in the Figure 1a. The local coordinates u and v run within the range restricted by the inequalities u  0;

(a)

v  0;

ð1:11Þ

(b)

v P3

u þ v  1:

P6

u=0 v=1

V u=0 v=1

P3 u = 0, v = 0,5

P5 u P2 u = 1, v=0

P1 u=U v=0

FIGURE 1

P1 u=0 v=0

u = 0,5 v = 0,5

P2 u = 0,5 v=0

P4 u = 1, v=0

u

Local parameterization of flat (a) and curved (b) surface elements.

Integral Equations Method in Electrostatics

7

Any point Q inside a flat triangular element can be expressed as a convex linear combination ð1Þ

ð1Þ

ð1Þ

Qðu; vÞ ¼ C1 ðu; vÞP1 þ C2 ðu; vÞP2 þ C3 ðu; vÞP3

ð1:12Þ

of its corners P1, P2, P3 with the coefficients ð1Þ

C1 ¼ 1

u

ð1Þ

ð1Þ

ð1:13Þ

v; C2 ¼ u; C3 ¼ v: ð1Þ

The superscript index (1) indicates that the functions Ck , k = 1, 2, 3 are the first-order (linear) polynomials with respect to the local coordinates u, v. Although representation of the curved surfaces requires the elements of higher approximation order, the triangular topology may be retained as shown in Figure 1b. For the second-order elements, we get Qðu; vÞ ¼

6 X m¼1

ð2Þ Cm ðu; vÞPm ;

ð1:14Þ

with ð2Þ

C1 ¼ ð1 ð2Þ

u

vÞð1

2u

2vÞ

ð2Þ

C4 ¼ uð2u ð2Þ

C2 ¼ 4uð1

u

vÞ

C5 ¼ 4uv

C3 ¼ 4vð1

u

vÞ

C6 ¼ vð2v

ð2Þ

1Þ

ð2Þ

1Þ

ð1:15Þ

Here the three points P1, P4, and P6 are the curved triangle’s corners, and the other three points – P2, P3, and P5 – belong to the curved triangle’s edges. The local surface approximation of higher orders may be constructed in a similar manner, and the elements of different approximation orders may be simultaneously used to describe the geometry with proper accuracy. Local surface approximation substantially facilitates the 3D geometry input, which is rather important from the viewpoint of the software versatility. First, a flat shape selected from a number of standard ones or drawn with a 2D graphic editor is converted into a 3D object by means of the extrusion procedure shown in Figure 2. The resultant shapes are automatically triangulated. Second, proper mutual positions and orientations (in terms of the Euler angles) are given to the newly constructed 3D objects. If an electrode under construction should possess a more complicated shape than could be obtained with the use of simple extrusion, this electrode can be built up by the special intersection procedure to be applied to several simpler shapes (Figure 3). The intersection line is found automatically to preserve proper triangulation. Finally, a special assembly operation ensures removal of the surfaces that have become redundant after the shape intersection.

8

Integral Equations Method in Electrostatics

FIGURE 2 Extrusion of 3D shapes from a 2D cross-section. (a) cylindrical, (b) conical, (c) toroidal.

FIGURE 3 Intersection of the shapes. (a) two overlapping simple shapes; (b) extra subdivision of the boundary elements near the intersection line (dashed); (c) the product of removing some redundant surfaces (assembly operation).

1.3. SURFACE CHARGE DENSITY APPROXIMATION To numerically solve the integral equation (1.6), we construct an approximation for the surface charge density s(Q) in the linear finite-dimensional space L of the piecewise-smooth scalar functions defined on G. The basis functions (or shape functions) of this space should meet two requirements. First, these functions should be continuous wherever G is smooth – this is to guarantee the continuity of the field strength in the vicinity of G (from

9

Integral Equations Method in Electrostatics

each side of the surface separately). Second, the shape functions should tend to infinity in the vicinity of the sharp ribs of the surface, with the singularity order matching that of the solution s(Q) to be approximated. We further consider the sets of shape functions represented by first-, second-, and third-order polynomials. The higher the order of the surface charge density approximation, the more accurate computation may be performed. However, as soon as the computation time and memory requirements substantially increase with the approximation order M, we do not recommend the use of M > 3. Each of the shape functions of the space L is associated with a certain point of the surface, referred to as a nodal point. The shape function is nonzero only within the elements containing this point. The location of the corresponding nodes in terms of the local parameters (u, v) is shown in Figure 4 for different approximation orders M = 1, 2, 3. The nodal points located on the element sides and at the element corners are assumed to belong to two or more adjacent elements simultaneously. Any shape function of the Mth order associated with the kth node is unambiguously defined by the following requirement: such function should be equal to 1 at the kth node and zero at the others. The relations (1.13) and (1.15) define three shape functions of the first order and six shape functions of the second order, respectively. The set of ten polynomials for M = 3 is as follows: ð3Þ

C1 ¼ ð1

u

vÞð1

1:5u

1:5vÞð1

ð3Þ C2 ¼ 9uð1 1:5u 1:5vÞð1 ð3Þ C3 ¼ 9vð1 1:5u 1:5vÞð1 ð3Þ C4 ¼ 4:5uð3u 1Þð1 u vÞ ð3Þ C5 ¼ 27uvð1 u vÞ

v 3

u

vÞ

u

vÞ

3u

ð3Þ

C6 ¼ 4:5vð3v

3vÞ

1Þð1

ð3Þ C7 ¼ uð3u 1Þð1:5u ð3Þ C8 ¼ 4:5uvð3u 1Þ ð3Þ C9 ¼ 4:5uvð3v 1Þ ð3Þ C10 ¼ vð3v 1Þð1:5v

u

vÞ

1Þ

1Þ

ð1:16Þ

v

v

10

6

M=3

M=2

M=1

9 6

5 3

3 1

2

1 u

2

4

1 u

5

8 2

4

7 u

FIGURE 4 The nodal points for the shape functions of the first-, second-, and thirdapproximation orders.

10

Integral Equations Method in Electrostatics

In accordance with the requirements previously mentioned, the shape functions are continuous provided that the approximation order M is the same across the entire electrode. Figure 5(a) provides an example of the second-order shape function associated with a nodal point located on the element side. Now we proceed to constructing the shape functions with singularities in the vicinity of ribs and edges of the electrode surface. Toward this end, the set of regular polynomials should be complemented with the singular functions of two types: the first-type singular functions (which turn to infinity along one side of the given element), and the second-type singular functions (which turn to infinity just at one of the element corners). The first-type singular shape functions appear as ð1Þ

C0
v

ð1Þ

C1
ð1

ð1Þ

C2
ð1

C0
u g ; C1
v g ; C2
u g ;

ð1Þ

g

ð1Þ

u

vÞ

g

ð1Þ

u

vÞ

g

1 ð0 < g  Þ 2 ð1:17Þ

and the second-type singular functions are ð1Þ

C0
ðu þ vÞ ð1Þ

C1
ð1

ð1Þ C2
ð1

g

uÞ

g

vÞ

g

ð0 < g < 1Þ

ð1:18Þ

The singular shape functions of higher orders can be obtained in similar manner: A regular function is multiplied by a power-like singular

FIGURE 5

Examples of the second-order shape functions: regular (a) and singular (b).

Integral Equations Method in Electrostatics

11

term. Every shape function associated with a nodal point located on a sharp rib is constructed of the first-type and second-type singular functions (Figure 5b), and every corner node is associated with a second-kind singular function. Accurate evaluation of the singularity index g is of profound importance because it fully determines the asymptotic behavior of the surface charge density distribution s(Q) in the vicinity of the electrode ribs and vertexes. The index g depends on the local properties of the electrode geometry. Under the assumption that two faces of the surface form the angle 0  a < p and the electric potential on both of them is continuous, the asymptotics of the surface charge density near the rib generally appears as sðQÞ  h g , where g ¼ ðp aÞ=ð2p aÞ and h is the distance between the point Q and the rib (Figure 6). In the case of a solitary surface edge, a = 0 and g = 1/2, respectively. In more complicated cases of 3D corners or if the dielectric materials are to be taken into account, the problem of singularity index evaluation deserves special consideration (this is the main topic of Chapter 2). Using the shape functions we have constructed, we now are able to approximate the electric charge density distribution on the electrode surfaces with the sum X si
Ci ðQÞ: ð1:19Þ sðQÞ ¼ i

Q h

a

FIGURE 6

Evaluating the surface charge singularity index on the electrode ribs.

12

Integral Equations Method in Electrostatics

The summation here is made over the full set of the shape functions, and the coefficients sk represent either the charge density values at the corresponding mesh nodes on smooth surfaces or the coefficients before the singular terms on the ribs and corners. With regard to the fact that any shape function is nonzero only within the elements containing the respective node, the calculation of the field value at any fixed point R requires calculation of ðM þ 1ÞðM þ 2Þ=2 integrals like ð1 1ð u 0

0

  ðMÞ @Q @Q Ck ðu; vÞ  dv du
  j R Qðu; vÞ j @u @v 

ð1:20Þ

ðMÞ

over each boundary element, with Ck denoting either a regular polynomial or a function with an integrable power-like singularity.

1.4. INTERFACE BOUNDARY CONDITIONS FOR DIELECTRIC MATERIALS Now consider the case when the space between the electrodes is filled with the dielectric material having piecewise constant permittivity. The Laplace equation is valid within each of the regions with constant permittivity, which offers the possibility of applying the BEM. However, in this case the surface charge density should be introduced not only on the electrode surface G ¼ @O but also on the set G0 of the interfaces separating different dielectric media (Figure 7).

Γ

Γ′ e2 Q

n

θ

e1

Γ

P Rc

FIGURE 7

Definition of the dielectric interface G0 and the normal vector n on G0 .

13

Integral Equations Method in Electrostatics

The coupling condition on the interface G0 between the two media with the permittivity values e1 and e2 reads e1 @ n ’ ¼ e2 @ þ n ’:

ð1:21Þ

The symbols @ n ’ and @ þ n ’ designate the normal potential derivatives calculated on both sides of the interface G0 . The direction of the normal vector n on G0 is shown in Figure 7. With regard to the surface charge density on the dielectric interfaces, integration in Eq. (1.4) should be extended to the full set of the surfaces involved ð GðR; QÞsðQÞdSQ þ ’1 ðRÞ R 2 O: ð1:22Þ ’ðRÞ ¼ S 0 G

G

Correspondingly, the integral equation (1.6) takes the form ð sðQÞ
ðPÞ ’1 ðPÞ; P 2 G dSQ ¼ ’ S 0jP Qj G

ð1:23Þ

G

for P, Q 2 G. Recall that the normal component of the electric field experiences the 4ps discontinuity on the interfaces separating the dielectric media and may be expressed as ð ’ðPÞ ¼ @ n GðP; QÞ sðQÞ dSQ þ @ n ’1 ðPÞ  2psðPÞ P 2 G0 : ð1:24Þ @ n S 0 G

G

The integral on the right-hand side of Eq. (1.24) contains the kernel @ n GðP; QÞ ¼

< n; P Q > ; j P Q j3

P; Q 2 G [ G0 ;

ð1:25Þ

which also reveals a singularity in the coincidence limit Q ! P, staying nonetheless integrable at any regular point of the interface surface. Indeed, the difference P Q becomes orthogonal to the normal n if the point Q approaches P, which makes the scalar product in Eq. (1.25) vanish faster than the distance between these two points. The asymptotic behavior of the kernel @ n G at Q ! P is @ n GðP; QÞ 

sinðy=2Þ 1 1 ; ¼ 2 jP Qj 2Rc j P Q j

ð1:26Þ

where Rc is the sectional curvature and y is the angle between the vectors Q and P as shown in Figure 7. It follows from Eq. (1.26) that the singularity completely vanishes at flat interfaces with Rc ¼ 1. Introducing the integral operator

14

Integral Equations Method in Electrostatics

Dn ½sŠðPÞ ¼ G

ð

S

@ n GðP; QÞsðQÞ dSQ ;

P 2 G0

ð1:27Þ

G0

and substituting Eq. (1.24) into the coupling condition expressed by Eq. (1.2), we come to the second-kind Fredholm integral equation 2psðPÞ ¼

e2 e1 fDn ½sŠðPÞ þ @ n ’1 ðPÞg P 2 G0 ; e1 þ e2

ð1:28Þ

which should be solved together with Eq. (1.23) to simultaneously determine the distribution of s on the surfaces G and G0 .

1.5. REDUCING THE INTEGRAL EQUATIONS TO THE FINITEDIMENSIONAL LINEAR EQUATIONS SYSTEM The finite-dimensional approximation (1.19) allows reducing both the integral equation (1.6) and the system of integral equations (1.23) and (1.28) describing the surface charge density in the presence of dielectrics to a discrete form. Consider the potential discrepancy function X ð Ci ðQÞ
ðPÞ; P 2 G dSQ þ ’1 ðPÞ ’ ð1:29Þ si dG ðPÞ ¼ jP Qj i G

defined on the electrode surfaces G. The coefficients si are to be determined as a result of minimizing the discrepancy norm kdG ðPÞk that can be defined in different ways. The simplest way is to define the discrepancy norm as the maximal absolute value of dG ðPi Þ over all the nodal points Pi on G: kdGkC ¼ maxj dG ðPi Þj. Since the number of unknown coefficients si i and the number of nodal points Pi are equal, we can claim that the exact equation kdGkC ¼ 0 is satisfied, which means that the discrepancy is strictly zero at all nodes. This method is referred to as the method of nodal collocation. It leads to the system of linear equations X Gij sj þ Fi ¼ 0 ð1:30Þ j

with the coefficients ð Cj ðQÞ dSQ ; Gij ¼ j Pi Q j

Fi ¼ ’1 ðPi Þ


ðPi Þ: ’

ð1:31Þ

G

The matrix elements Gij are calculated by integration over only those elements that contain the nodal point Pi. Some details of numerical

Integral Equations Method in Electrostatics

15

integration of regular and singular functions over triangular elements are considered in Appendices 1 and 2. In the presence of dielectrics, the discrepancy function on the interface G0 between the dielectric media is given by another formula ( ) e2 e1 X 0 dG ðPÞ ¼ 2ps ðPÞ si Dn ½sŠðPÞ þ @ n ’1 ðPÞ ; P 2 G0 : ð1:32Þ e1 þ e2 i This case is slightly more complicated because, first, the normal vector is poorly defined at the nodal points located on the ribs, and second, it is not clear which discrepancy formula, (1.27) or (1.29), should be used in the area where the dielectric interface meets the conductive surface. The method of weighted discrepancies helps to overcome these difficulties. Basically, we use the Galerkin method with the weight functions as the same regular shape functions Ci ðPÞ. Instead of the ‘‘uniform’’ collocation condition kdGkC ¼ 0, we arrive at the integral condition ð Ci ðPÞdG ðPÞ dSP ¼ 0: ð1:33Þ S 0 G

G

In doing so, we may again calculate the coefficients ð ð ð Ci ðPÞCj ðQÞ Gij ¼ dSP dSQ þ 2p Ci ðQÞCj ðQÞ dSQ jP Qj S G

G

e2 e1 e1 þ e2

Fi ¼

ðh G

’1 ðPÞ

G0

G0

ð

G

0

dSP

G

ð

S

@ n GðP; QÞCi ðQÞCi ðQÞ dSQ ;

G

ð1:34Þ

0

i
ðPÞ Ci ðPÞ dSP ’

e2 e1 e1 þ e2

ð

G

0

@ n ’1 ðPÞCi ðPÞ dSP

ð1:35Þ

for the linear equations system to determine s at the points of collocation. Numerical computation of the coefficients (1.34) requires double integrating over the triangular elements and is more time-consuming than computation of the coefficients (1.31) in the frame of the nodal collocation approach. However, the extra complexity is compensated by the fact that calculation of the Coulomb integrals on the element sides and corners may be avoided with the use of appropriate Gauss quadrature integration formulas. Another advantage in the dielectric-free case is that the matrix Gij becomes symmetrical. It is also noteworthy that the computation is highly facilitated if the system of electrodes allows reflection symmetry, with the boundary potential distribution (and the external field ’1, if any exists) also being

16

Integral Equations Method in Electrostatics

symmetrical or antisymmetrical. Such symmetry (or antisymmetry) is inherited by the surface charge density distribution, and each symmetry plane reduces the required number of boundary elements by the factor 2. The gain in the computational time for matrix elements computation and number of arithmetic operations needed to solve the system of linear equations (1.30) is fourfold and twofold, correspondingly.

1.6. ACCURACY BENCHMARKS FOR NUMERICAL SOLUTION OF 3D ELECTROSTATIC PROBLEMS The concern of this section is a number of simple electrostatic problems with known analytical solutions to illustrate the accuracy of the 3D boundary-element algorithm based on the principles previously described. The first example is a sole charged sphere with unit radius
¼ 1 (Figure 8). The well-known exact soluand the boundary potential ’ tion appears as  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ; r1 ð1:36Þ ; ðr ¼ x2 þ y2 þ z2 Þ: ’ðrÞ ¼ 1 r ; r>1 This model problem, despite its simplicity, has its own specific feature that is important for independent testing of various aspects of BEM algorithms: the charge density distribution is uniform; therefore the polynomial approximation considered above is exact. Thus, the numerical solution discrepancy is due only to the error of the approximate representation of the spherical surface with a set of curved boundary elements. Table I shows two numerical solutions for different mesh densities in comparison with the exact solution (1.36). The true decimal digits are shown as bold. The numerically calculated potential distribution inside the sphere (the first and the second rows in Table I) has five correct decimal digits (a)

(b)

z

y

x

FIGURE 8 (a) the spherical surface composed of 512 elements, and (b) the lines of equal potential in radial cross-section.

17

Integral Equations Method in Electrostatics

TABLE I Comparison Between Exact and Numerical Solutions for the Uniformly Charged Sole Sphere Point coordinates x

y

z

Exact solution

0 0 0 0 4

0 0.5 0 0 3

0 0 2 5 0

1 1 0.5 0.2 0.2

Numerical solution 128 elements

512 elements

0.999982 0.999982 0.4998 0.19993 0.199927

0.999987 0.999988 0.499983 0.199993 0.199993

Z Z

Y

X

X

FIGURE 9

The conducting disk subdivided into 354 elements.

for both mesh densities, which indicates, in fact, the numerical accuracy for Coulomb integrals evaluation. The accuracy outside the sphere is improved by the order of magnitude with the subdivision element number increased by factor 4, which is a direct consequence of better approximation of the spherical surface by the polynomials. A conducting disk of unit radius and unit potential is the subject of concern in the next example (Figure 9). The disk surface is subdivided into 354 elements; the peripheral ones are approximated with the secondorder accuracy. The surface charge density is no longer uniform in this example; moreover, it has a singularity with the singularity index g = 1/2 on the disk edge. The numerical results for potential distribution are compared in Table II with the ones obtained from the exact representation ’ðx; y; zÞ ¼

2 2 arcsin hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i1=2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 p þ x2 þ y 2 x2 þ y 2 þ 1 þ z 2

1

2

þ z2

i1=2

ð1:37Þ

18

Integral Equations Method in Electrostatics

TABLE II Comparison Between Analytical and Numerical Results for Conducting Disk Point coordinates x

y

z

Exact value

Numerical results

1.01 1.1 2 1 1 0 0 1

0 0 0 0 0 0 0 1

0 0 0 0.1 0.5 1 2 0

0.9103411 0.7264447 0.3333333 0.8003974 0.5703524 0.5 0.2951672 0.5

0.909 0.7263 0.33331 0.8003 0.57033 0.49998 0.29515 0.49994

given, for example, in the well-known monograph by Landau and Lifshitz (1984). It can be seen that, far from the charge density singularity, numerical calculation gives five true decimal digits. Of course, the relative accuracy near the disk edge is somewhat lower, but still remains at the level of 10 3 and 10 4 at the distance 0.01 and 0.1 from the disk edge, respectively. Such accuracy can hardly be achieved unless the surface charge singularity is not explicitly taken into account. The third model example deals with a unit-side cube shown in Figure 10. The singularity index at the cube ribs ða ¼ p=2Þ is g ¼ 1=3. Unlike the previous example, the boundary singularities are not only the cube ribs but also the eight three-faced rectilinear corners for which the singularity index is estimated as g  0:5459 (the general technique to obtain this estimation is described in Chapter 2). The numerical results presented in the Table III show that the relative error of potential calculation is 10 5 at the cube center, 10 3 at the cube ribs, and decreases down to 10 2 in the close vicinity of the cube corners. These examples show that the relative error as small as 10 5 has been obtained uniformly in the internal part of the calculation domains using 3D field calculation algorithms with a rather moderate number (a few hundreds) of boundary elements. The accuracy proves to be essentially lower in the close vicinity of the singular points and lines of the boundary 10 3. Two but still remains on the reasonable level of about 10 2 remarks should be made in this regard. First, the accuracy level gained in the first two examples could have been by at least two orders of magnitude higher (with the same number of finite elements) had we taken into account axial symmetry peculiar to these model problems. Second, the important question is what role the special boundary singularities treatment plays in the field calculation based on the integral

19

Integral Equations Method in Electrostatics

z 1.0

j

0.8

y

0.6 0.4 0.2 0.0

1.0

−0.5

0 x

0.5

1.0

x

FIGURE 10 Potential distribution along one of the charged cube’ ribs (potential vs. x-coordinate at y ¼ z ¼ 0:5). TABLE III Comparison Between Exact and Numerical Solutions for the Unit-side Conductive Cube Point coordinates x

y

z

Exact solution

Numerical solution

0 0.4 0.45 0.49 0.499

0 0.5 0.5 0.5 0.5

0 0.5 0.5 0.5 0.5

1 1 1 1 1

0.999990 0.9996 0.99986 1.0013 1.0048

equation technique. (We should note that until now some authors believe there is no need to care about such ‘‘trifles’’ as boundary singularities simply because the charged particles commonly ‘‘do not fly in immediate vicinity of the boundary’’). The answer to this question has been given by the Voronin-Tsetsoho theorem cited previously. From the computational viewpoint, this theorem states that the class of functions in which the stability of numerical solution to the first-kind Fredholm equation can be guaranteed is the class of rather smooth functions with uniformly restricted derivatives. This implies that, if we have not put ‘‘by hand’’ the correct singularity index value into the shape functions for surface charge density approximation, we are trying to determine a numerical solution in an essentially broader class of functions than that set by the Voronin-Tsetsoho theorem. In practice, this would result in calculation instability and low accuracy not only in the boundary vicinity but everywhere in the calculation domain.

20

Integral Equations Method in Electrostatics

1.7. MORE COMPLICATED EXAMPLES OF 3D FIELD SIMULATION This section presents (with minimal comments) some more complicated examples of 3D electrostatic field simulation taken from practice. The examples include two different quadrupole systems with side apertures (Figures 11 and 12), an ion injector consisting of four plates (Figure 13), a rectangular grid cell (Figure 14), and a three-channel electron lens (Figure 15).

(a)

(b)

x 1.0

z j y

2

1 0.5 0.0 ?100

3 ?50

0 x

50

100

FIGURE 11 (a) the quadrupole electrode system 1 with four cylindrical electrodes. The potential distribution along the x-direction in different z cross-sections is shown in (b): 1, in the middle plane (z = 0); 2, at the cylinders fringe; 3, between the cylinders and the end caps.

FIGURE 12

The equipotential lines in the quadrupole system 2.

21

Integral Equations Method in Electrostatics

(a)

(b)

2 kV 1 kV 0 kV -1 kV

FIGURE 13 The orthogonal ion injector with electrode voltages forming arithmetic progression (a). Generated quasi-homogenous electric field (b).

FIGURE 14 (a)

A grid cell. (b)

50

0

-50

-50

0

50

FIGURE 15 (a) three-channel electrostatic lens for color cathode ray tube. (b) potential distribution (the voltage of the middle electrode is unit; two other electrodes are grounded).

22

Integral Equations Method in Electrostatics

1.8. PLANAR AND AXIAL SYMMETRIES In the particular cases of planar or axial symmetry (when neither geometry nor the boundary conditions depend on one of the coordinates, say, Cartesian coordinate z or azimuth angle y), the 3D electrostatic problem may be obviously reduced to a 2D one. The corresponding Green function can be obtained by integrating the 3D Green function 1= j P Q j over the ‘‘unused’’ coordinate. Thus, for the planar symmetry case we get 8 > < 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðxP ; yP ; xQ ; yQ Þ ¼ > 2 : ðxP xQ Þ þ ðyP yQ Þ2 þ ðzP zQ Þ2 1 1 ð

9 > = 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi dzQ : ; 1 þ z2Q >

ð1:38Þ

The second term here is introduced to regularize the integral, otherwise diverging logarithmically. We can do so because this term does not depend on the coordinates ðxP ; yP ; zP Þ and thus does not contribute to the field strength vector. Integrating Eq. (1.38) over the finite interval L < zP < L) gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L þ zP þ D2 þ ðL þ zP Þ2 L zP þ D2 þ ðL zP Þ2 GL ¼ 2 ln D þ ln pffiffiffiffiffiffiffiffiffiffiffiffiffi2  L þ 1 þ L2 D¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxP xQ Þ2 þ ðyP yQ Þ2

ð1:39Þ

The second term vanishes in the limit L ! 1, and finally we obtain the 2D Green function G ¼ 2lnD for the planar case, which satisfies the equation DG ¼

@2G @2G þ ¼ 4pdD ðxP @x2P @y2P

xQ ÞdD ðyP

yQ Þ:

ð1:40Þ

It should be noted that the condition (1.3) at infinity cannot be met with the logarithmic Green function. This fact is closely connected with the infinite extent of the planar electrodes in the 3D space and is mathematically a consequence of the regularization applied. The function ’1 loses its meaning of the unperturbed external field and may be set to zero. Thus, we restrict our consideration to the domains with finite section in the ðxP ; yP Þ plane. In the planar case, the Green function may be defined with the accuracy up to an additive constant that obviously does not violate Eq. (1.40) and is negligible in infinity in comparison with the logarithmic term. In other words, we can calibrate the Green function with an arbitrary scaling

Integral Equations Method in Electrostatics

23

parameter D0 by putting G ¼ 2lnðD=D0 Þ. Denoting the 2D xy-crosssection of the domain O as Oxy and assuming that its boundary G ¼ @Oxy is parameterized by means of the arc-length s, so that G ¼ fxP ; yP: xP ¼ xðsÞ; yP ¼ yðsÞ; s 2 ½a; bŠg, we can attach the 1D form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ½xðsÞ xðs0 ފ2 þ ½yðsÞ yðs0 ފ2
ðsÞ 2 ln sðs0 Þ ds0 ¼ ’ ð1:41Þ D0 G

to the first-kind Fredholm integral equation for the charge density distribution s(s) on G. The charge density distribution s(s) that can be found from Eq. (1.41) depends on the scaling parameter D0. Nevertheless, as can be easily seen, the potential distribution qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ½xðsÞ xðs0 ފ2 þ ½yðsÞ yðs0 ފ2 Fðx; yÞ ¼ 2 ln sðsÞ ds ðx; yÞ 2 Oxy D0 G

ð1:42Þ in the cross-section Oxy does not depend on the choice of D0, which is in full agreement with the uniqueness theorem for solution of the Dirichlet problem inside any finite 2D domain. However, the numerical conditionality of the integral equation (1.41) at some particular D0 values may become poor. Consider two examples to illustrate this point. In the first whereas in the second example, the domain Oxy is a circle of unit-radius, p ffiffiffi one it represents a square with the side length 2. In both cases, the
ðsÞ is assumed constant: ’
ðsÞ ¼ ’
0. boundary potential ’ Let us see the total charge QS induced on the boundary. The first example allows a simple exact solution since the charge density s(s) is constant on the circumference. Eq. (1.41) degenerates into the simplest algebraic equation QS 2p

2ðp 0

ln

ð1 þ cos fÞ2 þ sin2 f
0; df ¼ ’ D20

ð1:43Þ


0 =lnD20 . As shown in Figure 16a, the solution exists for which gives QS ¼ ’ all D0 6¼ 1. The second example is more complicated, and we have solved numerically Eq. (1.42) for different D0. The total charge QS versus the ’ 0 could be termed parameter D0 is shown in Figure 16b. The ratio QS =
‘‘capacitance’’ if it were independent on D0. The total charge also becomes indefinite at D0 ¼ D0  0:83, which means that the first-kind Fredholm integral operator in the left-hand side of Eq. (1.41) is irreversible at D0 ¼ D0 . To avoid such a situation, the value of the scaling parameter

24

Integral Equations Method in Electrostatics

(a) 2.0

2.0 2

−1.0

0.0 D*0≈ 0.83

0.0

QΣ

1.0 D*0= 1.0

QΣ

1.0

2O

(b)

O f

D

−1.0 −2.0

−2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 D0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 D0

FIGURE 16 The total charge QS on the unit-radius circumference (a) and the unit-side square (b) as functions of the scaling parameter D0.

D0 should be fixed far enough from the ‘‘dangerous’’ point, say somewhere in the vicinity of 2D0 . The computation practice shows that it is reasonable to set D0 equal to the diameter of the circle circumscribed around the domain Oxy (D0 ¼ 2 in our examples). If this is the case, the kernel is non-negative for all possible pairs of points running across the boundary, and the Eq. (1.41) never degenerates. The case of dielectric materials forming a planar boundary can be considered similar to the 3D case. If we introduce the surface charge density s0 on the interfaces G0 separating the dielectric materials with different permittivity, we can evaluate normal components of the nearinterface field strength as follows : ð  @ n GðxðsÞ; yðsÞ; xðs0 Þ; yðs0 ÞÞ sðs0 Þ ds0  2psðsÞ; @ n ’ðsÞ ¼ S 0 G

@nG ¼

2

G

½xðsÞ

½xðsÞ

xðs0 ފnx þ ½yðsÞ xðs0 ފ2 þ ½yðsÞ

yðs0 ފny

yðs0 ފ2

:

ð1:44Þ

Here nx, ny are components of the normal vector. The coupling condition at the point fxðsÞ; yðsÞg 2 G0 takes the form ð ½xðsÞ xðs0 ފ nx ðs0 Þ þ ½yðsÞ yðs0 ފ ny ðs0 Þ 0 e1 e2 psðsÞ ¼ sðs Þ ds: ð1:45Þ e1 þ e2 S ½xðsÞ xðs0 ފ2 þ ½yðsÞ yðs0 ފ2 G

G0

The coincidence-limit singularity in the kernel is locally integrable at the regular boundary points, and the main part of the kernel is lnðDÞ=Rc , with Rc denoting the local curvature radius. In the case of axial symmetry, the geometry, boundary potential, and, as a sequence, the surface charge density are independent of the azimuth angle y over which the Green

Integral Equations Method in Electrostatics

25

function can be integrated. In this manner we obtain the axisymmetric Green function 0

0

Gðz; r; z ; r Þ ¼

2ðp 0

r0 dy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ2 þ ½r r0 cosyŠ2 þ r0 2 ðsÞsin2 y

4r0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ2 þ ½r þ r0 Š2

p=2 ð 0

dc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 k cos2 c

ð1:46Þ

where we have made the variable change y ¼ 2c and introduced the parameter k¼

4rr0 ðz

z 0 Þ2

þ ðr þ r0 Þ2

:

ð1:47Þ

Unlike the case of planar symmetry, the Green function (1.46) vanishes in infinity and no regularization is needed. The integral K0 ðkÞ ¼

p=2 ð 0

dc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k cos2 c

ð1:48Þ

is referred to as the complete elliptic integral. Its argument k runs within the interval [0, 1] and reaches its maximum in the coincidence limit z ! z0 , r ! r0 , where the function K0 also reveals logarithmic singularity. If the boundary surface possesses either planar or axial symmetry, the charge density s is a function of one argument only – the arc-length parameter s. This allows us to use the smooth high-order finite-dimensional approximations, which improve field calculation accuracy, especially in the vicinity of electrodes. One of the possible approaches is the use of cubic splines. Let us partition the boundary of an electrode (or a dielectric interface) into N intervals, and within each of the intervals st  s  stþ1 ; t ¼ 1; . . . ; N represent the unknown surface charge density with a set of linear combinations of four shape functions 0 1 0 1 s s s s t tþ1 A þ stþ1 c0 @ A sðsÞ ¼ st c0 @ stþ1 st stþ1 st 0 1 0 1 s s s s t A tþ1 A þ ðstþ1 st Þs0t c1 @ ðstþ1 st Þ s0tþ1 c1 @ stþ1 st stþ1 st st ¼ sðst Þ;

s0t ¼ s0 ðst Þ;

t ¼ 1; . . . ; N þ 1:

ð1:49Þ

26

Integral Equations Method in Electrostatics

The shape functions are defined as the third-order polynomials xÞ2 ;

c0 ðxÞ ¼ ð1 þ 2xÞð1

c1 ðxÞ ¼ xð1

xÞ2 ;

ð1:50Þ

which meet the boundary conditions c00 ð0Þ ¼ 0

c0 ð0Þ ¼ 1;

c01 ð0Þ ¼ 1 :

c1 ð0Þ ¼ 0;

c00;1 ð1Þ ¼ 0

c0;1 ð1Þ ¼ 0;

ð1:51Þ

to ensure the continuity of both the charge density distribution and its first derivative at the collocation points. The number of the unknown coefficients st and s0t is 2(N + 1), whereas the collocation condition at all the nodal points s1 . . . sNþ1 gives N + 1 linear equations. The remaining degrees of freedom can be used to make the second derivative of the function s(s) also continuous if we put the ‘‘left’’ second derivative s00 ðst

0Þ ¼ 6

st

1

ðst

st st 1 Þ

þ 2

þ 4s0t st 1

ð1:52Þ

4s0t þ 2s0tþ1 stþ1 st

ð1:53Þ

2s0t st

1

equal to the ‘‘right’’ one s00 ðst þ 0Þ ¼ 6

stþ1 ðstþ1

st st Þ2

at N 1 internal nodes. The two extra conditions may be used to specify the function s(s) behavior at the endpoints. In axisymmetric problems, it is reasonable to impose the symmetry condition s01 ¼ 0 or s0Nþ1 ¼ 0 at the endpoints belonging to the symmetry axis. In other cases, the free-end condition s00 = 0 is commonly used.

1.9. CALCULATION OF POTENTIAL AND ITS DERIVATIVES NEAR THE BOUNDARY In some practical cases, the charged particle trajectories pass very close to the boundary electrodes or even start from them. For example, this happens when some of the electrodes serve as cathodes, screens, narrow diaphragms, or fine-structure grids. In many charged particle problems requiring the use of aberration theory, it is very important to know electric field components and their higher derivatives (the field tensor) in the vicinity of the boundary electrodes. The direct subsequent differentiation of the Green function, being so effective far from electrodes, strengthens the coincidence-limit singularity and thus makes direct calculation of the field tensor in the electrodes’ vicinity too unstable. An efficient method based on integral equation technique was suggested by Monastyrskiy, Ivanov, Kulikov, and Ignat’ev (1983), but this method was specially

Integral Equations Method in Electrostatics

27

oriented to precise computation of axial potential distribution and its derivatives (up to the fourth-order inclusive) near the axis in the axisymmetric case. The method considered in this section is much more general and not restricted by the use of integral equation technique only. First consider the planar case, in which any equipotential surface S can be given by its generatrix G ¼ fxðsÞ; yðsÞg parameterized with the arclength s. Assume that the surface charge distribution s(s) has already been determined from solving the corresponding integral equation. Since the Green function differentiation turns its logarithmic singularity into stronger ones, it is not easy to directly calculate the electric field components near the charged surface. Nevertheless, the normal derivative of the potential exactly on the surface S appears simpler to calculate because this derivative is given by Eq. (1.44), in which the kernel @ nG has just a weak singularity and s is given by a smooth spline. Let us assume that the on-surface normal derivative on the boundary side of interest, say EðsÞ ¼ @ þ n ’, is a smooth function of the arc-length parameter s. We now can determine the electric field near the surface as a solution of the Cauchy problem for the Laplace equation. In doing so, we should remember that, in general, such a solution is ill-conditioned and is reliable only in a rather thin band-shaped region near the surface S, but this is what is needed here. Let us introduce the local curvilinear coordinates system (s, h) in the vicinity of the surface S. Here s is the arc-length corresponding to the orthogonal projection of a point onto the surface S and h is the distance of the same point to the surface S (Figure 17). The transformation to the Cartesian coordinates ðs; hÞ ! ðx; yÞ appears as x ¼ xðsÞ þ y0 ðsÞh;

y ¼ yðsÞ

x0 ðsÞh:

s

y

h

x

FIGURE 17

The near-surface curvilinear coordinates system (s, h).

ð1:54Þ

28

Integral Equations Method in Electrostatics

The curvilinear coordinate system (s,h) is orthogonal, and the square of the infinitesimal length element is dl2 ¼ ð1 þ k0 ðsÞhÞ2 ds2 þ dh2 ;

ð1:55Þ

where k0 ðsÞ ¼ y00 ðsÞx0 ðsÞ x00 ðsÞy0 ðsÞ is the local curvature coefficient. The Laplace operator transformed to the curvilinear coordinates (s, h) reads as D¼

@2 1 @2 k0 @ þ þ @h2 ð1 þ k0 hÞ2 @s2 1 þ k0 h @h

0

@ ; @s ð1 þ k0 hÞ k0 h

3

ð1:56Þ

0

where k0 ðsÞ ¼ y000 ðsÞx0 ðsÞ ðsÞy0 ðsÞ. We construct the solution to the Laplace equation D’ ¼ 0 in the local coordinate system (s,h) in the form of the Taylor expansion
ðsÞ þ EðsÞh þ ’ðs; hÞ ¼ ’

FðsÞ 2 GðsÞ 3 HðsÞ 4 h þ h þ h þ ...: 2 6 24

ð1:57Þ


is constant on S. By Let us first assume that the boundary potential ’ substituting the expansion (1.57) into the Laplace equation D’ ¼ 0 and equating to zero the terms with different powers of h independently, we unambiguously determine the coefficients F ¼ k0 E G ¼ 2k20 E E00 00 0 H ¼ ð 6k30 þ k0 ÞE þ 4k0 E0 þ 6k0 E00 :

ð1:58Þ

The generalization of Eq. (1.58) to the case of the non-constant bound
ðsÞ (for example, if the surface S represents not a conductor ary potential ’ but a dielectric media interface) can be easily obtained in the form 0


F ¼ F ’ 0 00 0

þ k0 ’ G ¼ G þ 3k0 ’ 00
ðIVÞ 11 k20 ’
H ¼ H þ ’

0


7 k0 k0 ’

0

ð1:59Þ

To consider the axisymmetric case, we need to formally substitute 0 00 00 x ! z, y ! r and transform the Laplace equation D’ ¼ ’z þ ’r þ ’r =r ¼ 0 to the curvilinear coordinates s, h. In doing so, we obtain the transformed equation 0 1 @ 2 ’ @ k0 k1 A @’ þ D’ ¼ 2 þ 1 þ k0 h 1 þ k1 h @h @h 0 1 0 0 2 1 @ ’ 1 k0 h @’ @ k1 h þ k2 A þ ¼0 þ @s ð1 þ k0 hÞ2 @s2 ð1 þ k0 hÞ2 1 þ k1 h 1 þ k0 h ð1:60Þ

29

Integral Equations Method in Electrostatics

containing two extra curvature coefficients k1 ðsÞ ¼ z0 ðsÞ=r and k2 ðsÞ ¼ r0 ðsÞ=r. The coefficients of the expansion (1.57) take the form F¼

ðk0 þ k1 ÞE

G ¼ 2ðk20 þ k0 k1 þ k21 ÞE k2 E0 E00  0 00 0 00 H¼ 6ðk0 þ k1 Þðk20 þ k21 Þ þ ðk0 þ k2 k0 Þ þ ðk1 þ k2 k1 Þ E 0

þ 4k0 E0 þ ð6k0 þ 2k1 ÞðE00 þ k2 E0 Þ:

ð1:61Þ

Thus, the method described here allows direct representation of the high-order derivatives of ’ on the surface S (up to fourth order inclusively) in terms of the normal derivative @ þ n ’, which is comparatively easy to calculate with high accuracy using Eq. (1.44), and the differential properties of the surface S itself. Simple but rather bulky formulas converting the field tensor to the Cartesian coordinates may be derived immediately from the coordinate transformation (1.54).

1.10. ACCELERATION OF FIELD CALCULATION: FINITEDIFFERENCE MESHES AND CALCULATION DOMAIN DECOMPOSITION The essential drawback of the BEM is slow ‘‘access’’ to the electric field after the surface charge distribution has been found. Indeed, the calculation of potential and its derivatives at each point of the calculation domain requires the charge surface density multiplied by the Green function to be integrated over the entire boundary. This is reasonable if the importance of high accuracy provided by BEM essentially overweighs that of computational time minimization. However, a somewhat different type of problem may need a large number of trajectories to be traced directly, with no aberrational expansion required. In this case, the solution obtained with BEM may be interpolated on a regular quadrilateral mesh constructed in the region of interest. If the potential calculation at all the mesh nodes is still too time-consuming, it is reasonable to precalculate the potential on a set of peripheral nodes surrounding the inner part of the calculation domain and then use the FDM to restore the potential at the inner nodes. The alternating directions sweep method appears to be most suitable to solve the Laplace equation on a quadrilateral mesh. We provide some finite-difference schemes that allow implementation of the method for different symmetry types. The 7-point finite-difference operator for 3D Laplace operator reads D’ 

’i

1; j; k

þ ’iþ1; j; k þ ’i; j

1; k

þ ’i; jþ1; k þ ’i; j; k h2

1

þ ’i; j; kþ1

6’i; j; k

:

ð1:62Þ

30

Integral Equations Method in Electrostatics

The 5-point operator DðplÞ ’ 

’i

1; j

þ ’iþ1; j þ ’i; j

h2

1

þ ’i; jþ1

4’i; j

ð1:63Þ

:

can be applied in the planar case. The situation is somewhat complicated in the axisymmetric case because the axisymmetric Laplace operator DðaxÞ ’ ¼ Dzr ’ þ ’r =r degenerates on the symmetry axis. The consistent finite-difference approximation may be constructed from the 3D operator (1.62) if we set k = 0 and, using the axial symmetry (Figure 18), redefine the potential at the nodes with k ¼ 1 as qffiffiffiffiffiffiffiffiffiffiffiffi ð1:64Þ ’i;j ¼ ’i; j þ ð 1 þ j2 jÞð’i; jþ1 ’i; j Þ: In doing so, we obtain the approximation

DðaxÞ ’ 

’i

1; j

þ ’iþ1; j þ ’i; j

h2

1

þ ’i; jþ1

4’i; j

þ2

pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ j2 h2

j

ð’i; jþ1

’i; j Þ; ð1:65Þ

which preserves good accuracy on the axis. Setting these operators equal to zero gives the system of linear equations to be solved. When the FDM is used in electrostatics, setting the boundary conditions on the electrodes is problematic if the electrodes do not coincide with the mesh lines. In our case, however, the FDM is used only to accelerate the access to the results that have been preliminary derived with the BEM. Thus, we have the right to set the boundary condition y

j *0

j *1

j0

j1

j *0

FIGURE 18

j *1

j *2

j2

j *2

j *3

j *4

j3

j4

j *3

x

j *4

The 5-point finite-difference operator for the axisymmetric Laplace problem.

Integral Equations Method in Electrostatics

31

not strictly on the electrodes themselves but at the mesh nodes located close to the electrodes as shown in Figure 19. This implies that the electrodes appear to be ‘‘isolated’’ from the rest volume by the set of the near-boundary nodes at which the potential is already known and which, therefore, should be excluded from the iterative solution of the Laplace equation. We outline here one more approach to the problem of accelerating the field calculation. It consists in decomposition of the computational domain into two or more subdomains (see, for example, Murata, Ohye, and Shimoyama, 2004). The field inside each subdomain is calculated independently by constructing the separate surface charge density distributions (Figure 20). Obviously, in this case the interface surface separating any two of the subdomains would carry two different surface charge distributions, each representing the field inside one of the subdomains only. Accordingly, the additional coupling condition ð ð \ ð1:66Þ GðR; QÞs1 ðQÞdSQ ¼ GðR; QÞs2 ðQÞdSQ ; R 2 G1 G2 G1

G2

should be imposed to make the potential continuous on the interface. The method of computational domain decomposition allows acceleration of both the surface charge density and potential calculations. The first relates to the fact that the matrix elements Gij are nonzero only if both nodes Pi and Pj, belong to the boundary of the same subdomain. This implies that the matrix kGk of the linear equations system may be

FIGURE 19 Constructing the FDM mesh and separating the near-electrode cells. The electric potential at the grey nodes is precalculated with BEM. The potential at the remaining (white) nodes is determined with the alternating directions sweep procedure.

32

Integral Equations Method in Electrostatics

Γ1

Γ2

s2 Ω1

s1

FIGURE 20

Ω2

The method of computational domain decomposition.

represented as a combination of the overlapping nonzero square submatrixes, which, in turn, accelerates the solution of the system of linear equations, especially if the Gauss exclusion procedure is used. Despite the increased total number of nodes, this approach may be effective for electrostatic systems with weak interpart connection. With the surface charge distribution precalculated, the potential calculation at any particular point is obviously faster because we do not need to integrate over the entire boundary but only over the boundary of the subdomain in which the point of interest is located.

1.11. MICROSCOPIC AND AVERAGED FIELDS OF PERIODIC STRUCTURES The fine-structure grids are widely used in electron/ion-optical devices to separate regions with very different electric field strengths. An ideal grid should be completely transparent for the particles and should not scatter them. At the same time, the grid behavior in regard to electric field formation should be close to that of a solid electrode. Obviously, the fine-structure grids used in real devices deviate from the ideal ones in both respects. First, only a fraction of the particles can penetrate through the cell apertures while the other ones are captured by the grid wires. Second, the charged particles are affected by the microscopic electric field generated by the grid cells, and such field scatters the particles in random directions. Finally, the ‘‘sagging’’ of the electric field through the grid cells makes the effective (averaged) electric potential of the grid differ from the potential of its conductive parts. These effects may seriously influence device operation, which is why the fine-structure grid simulation represents an important part of computational charged particle optics.

Integral Equations Method in Electrostatics

33

As shown by Williams, Read, and Bowring (1995), and Read et al. (1999), one possible approach to numerical simulation of the microscopic electric field concentrated in close vicinity of the fine-structure grid surface is direct 3D modeling with BEM. With this approach, the FDM acceleration described in Section 1.9 may help to trace a large number of trajectories and determine both the transparency coefficient and the scattering function of a grid cell. Our aim here is to show that taking into account some features specific to the problem in question may advance the calculations. The mathematical model of a fine-structure grid represents an infinite periodical array of cells. The direct use of the classic 3D BEM immediately restricts the case to only a few adjacent grid cells involved in the calculation and complimented with two additional finite-size electrodes generating the electric field on both sides of the grid. The number of cells being taken into account and the location of the auxiliary electrodes are the computational parameters whose influence on the reliability and accuracy is a subject of empirical verification in each particular case. To overcome those difficulties, we suggest a consistent and rather general modification of BEM that is specially oriented to accurate simulation of periodic structures. This approach may be useful for simulation of not only fine-structure electrostatic grids but also some composite materials and metamaterials (Lagarkov et al., 1992). Following the basic premises of BEM, we are looking for the surface charge density distribution on the conducting parts of the grid. Since the neighboring cells experience almost the same external field, we presume that the surface charge density is a periodic function. We now need to construct a special periodic Green function to take into account the translational symmetry defined by the two translation vectors A and B as shown in Figure 21. For simplicity, we assume that these vectors are orthogonal. Readers could easily conduct similar consideration for a more general case. At first sight, the Green function may be defined as the double sum ~ ðgÞ ðP; QÞ ¼ G

þ1 X

i; j¼ 1

jP

1 ; ðQ þ iA þ jBÞ j

ð1:67Þ

which is spread over the Coulomb field of the charges located at the point Q and all points obtained by means of translating the point Q by any integer number of grid periods (Figure 21). However, it is clear that the sum in Eq. (1.67) diverges and thus cannot be used for calculations. This difficulty is quite similar to that already encountered in Section 1.7 when reducing the general 3D BEM to the case of planar symmetry; in both cases the electrodes formally extend to infinity. In Section 1.7, a special regularizing term was introduced into the integrand, which made the

34

Integral Equations Method in Electrostatics

z

P

Q B

y

A

x

FIGURE 21 Constructing the periodic Green function by summing the Coulomb contributions of a point array.

integral for the planar Green function converge. A similar approach can be applied in the case of periodic structures, but let us first formulate the requirements for the Green function to be constructed. First, the Green function should be periodic GðgÞ ðP; QÞ ¼ GðgÞ ðP þ A; QÞ ¼ GðgÞ ðP þ B; QÞ:

ð1:68Þ

Second, it should satisfy the equation DGðgÞ ¼ 4p

1 X

i; j¼ 1

dD ðP

Q

iA

jBÞ:

ð1:69Þ

Finally, the third requirement should be set to specify the proper asymptotic behavior of the Green function in infinity. From the physical viewpoint, the divergence of the sum in Eq. (1.67) reflects the inapplicability of the conventional condition ’ðRÞ ¼ Oð1= j R jÞ at R ! 1. Instead, the correct asymptotics from both sides of the grid should resemble the electric field of a uniformly charged sheet  E1 z; z ! 1 ðgÞ ; ð1:70Þ G  E2 z; z ! þ1 where E1 and E2 are the asymptotic field strengths in infinity. It is easily seen that all the formulated requirements are satisfied by the ‘‘renormalized’’ Green function

35

Integral Equations Method in Electrostatics

1 QÞ ¼ GðgÞ ðx; y; zÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 þ z2 2 X 6 1 þ lim 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M!1 2 0 1 and e2 ¼ 1. This assumption corresponds to the case of one dielectric wedge with the apex angle b, and the rest of the space within the angle 2p  a  b empty (vacuum). It is also convenient to introduce the normalized variables b ¼ b=ð2p  aÞ, x ¼ 2  ð2  a=pÞl and rewrite Eq. (2.23) in the form sin½pð2  xÞ þ

e1 sin½pð2  xÞð1  2b Þ ¼ 0: eþ1

ð2:24Þ

The value b 2 ½0; 1 is the fraction of the space between the conductive faces occupied by dielectric. The meaning of the parameter x becomes clear if we express the singularity index g in terms of x and a: g ¼ gðaÞ ¼

px  a : 2p  a

ð2:25Þ

This formula generalizes Eq. (2.7) for the presence of dielectrics, with x playing the role of a correcting coefficient. In terms of the new variables introduced, the numerical solution to Eq. (2.23) is displayed graphically in Figure 26. The solution shows that two cases exist in which the correction coefficient x is equal to unit, so that the singularity index appears to be the same as in the case with no dielectric material in the vicinity of the conductive wedge (b ¼ 0). The first case is when the dielectric fully occupies the external gap between the conductive faces (b ¼ 1), and the second case is when it occupies exactly one half of this gap (b ¼ 1=2). Other apex angles of the dielectric wedge may result either in weaker (b < 1=2) or stronger (b > 1=2) charge density singularity compared with that of the corresponding solitary conductive wedge. It is noteworthy that the asymptotics of xðb ; eÞ in the limit of very high permittivity e,   2  1=ð1  b Þ; b < 1=3   xðb Þ ¼ lim xðb ; eÞ ¼ ; ð2:26Þ e!1 b  1=3 2  1=ð2b Þ; proves to be nonsmooth at b ¼ 1=3.

46

Surface Charge Singularities Near Irregular Surface Points

e=• 1.4

e = 50

e = 10 e =5 x = 2- (2 - a/p) l

1.2

e =3 e =2 e = 1.5 e =1

1.0

0.8

a b e >1

0.6 0.0

0.2

0.4

0.6

0.8

1.0

b * = b/(2p-a)

FIGURE 26 The singularity index calculation in the presence of conductive and dielectric wedges.

The particular case of a dielectric wedge touching a regular (a ¼ p) conductive surface with one of its faces merits special attention because it is frequently encountered in electrostatics. As shown in Figure 27, the singularity exists (e.g., the singularity g is positive) only for b > 0:5, which corresponds to the obtuse apex angles p=2 < b < p occupied by a dielectric.

2.4. THE CASE OF PURE DIELECTRIC VERTEX The important case of several dielectric interfaces meeting at one point, with no conductive surfaces present, needs special consideration. The angle a should be put to zero and the boundary condition (2.11) should be replaced by the periodicity conditions Yð0Þ ¼ Yð2pÞ, Zð0Þ ¼ Zð2pÞ. The system of linear equations              Yð0Þ   Yð0Þ   Yð2pÞ  ð2:27Þ   ¼ M1;N ðlÞ   ¼   Zð0Þ   Zð0Þ   Zð2pÞ 

allows a non-trivial solution if and only if the matrix M1;N ðlÞ is degenerated. Thus, the minimal positive root l0 of the equation       ð2:28Þ detMðlÞ  E ¼ 0;   with E being the unit matrix, is to be found in this case.

Surface Charge Singularities Near Irregular Surface Points

47

0.5 g

e >1 e = 50.0

b

0.4

e = 10.0

0.3

e = 5.0 0.2

e = 3.0 e = 2.0

0.1

e = 1.5

0.0 90⬚

FIGURE 27 surface.

120⬚

b

150⬚

180⬚

The singularity index when a dielectric wedge touches a regular conductive

The exact solution to Eq. (2.28) is known only for a few particular cases. As an example, we quote here, after some simplifying transformations, the singularity index expression g¼1

2 jw1 w2 j l0 ¼ arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 1 w 1 w2

ð2:29Þ

found by Veselov and Platonov (1985) for three dielectric wedges, the faces of which intersect at right angles as shown in Figure 28. Here w1 ; w2 designate the reduced permittivity ratios e1 e3 e2 e3 w1 ¼ ; w2 ¼ ð2:30Þ e1 þ e3 e2 þ e3 taking values in the interval ð 1; 1Þ. Figure 28 shows only one half of the square ð 1; 1Þ  ð1; 1Þ: Eq. (2.29) is obviously invariant with respect to the substitution w1 $ w2 . The singularity index takes values within the range 0  g < 1=2. Consider another important case of the dielectric wedge with permittivity e > 1, surrounded by vacuum. After simple transformations, we come to the problem of finding the minimal root l0 to the transcendental equation ðe  1Þsin½lðp  bÞ ¼ ðe þ 1Þsin½pð1  lÞ:

ð2:31Þ

48

Surface Charge Singularities Near Irregular Surface Points

1.0 0.8 e1

e2 0.6 e3

0.2 0

g=

0.

0.0 5

0 0.

10

0.

-0.2

15

0.

20

0.

(e2 – e3)/(e2 + e3)

0.4

-0.4

25

0.

0 .3

0

-0.6

35

0.

40

0.

-0.8

45

0.

-1.0

-0.8

-0.6

-0.4

-0.2 0.0 0.2 (e1 – e3)/(e1 + e3)

0.4

0.6

0.8

-1.0 1.0

FIGURE 28 The singularity index versus the reduced permittivity ratios in the case of three dielectric wedges, the faces of which intersect at right angles.

As above, no exact solution to this equation exists for all possible values of the apex angle b. Therefore, here we restrict ourselves to the numerical results shown in Figure 29 and investigate some limiting cases. As in the case above, the singularity index lies in the range 0  g < 1=2. The asymptotic behavior may be easily analyzed in the limits of high permittivity (e >> 1) and small apex angle (b 0 fixed, the singularity index g is approaching gc with e increasing, but the convergence in Eq. (2.32) is not uniform with respect to b in the vicinity of b ¼ 0. The coefficient before e1 in Eq. (2.32 becomes arbitrarily large at small b, which makes this asymptotic wrong in the vicinity of b ¼ 0. It can be

49

Surface Charge Singularities Near Irregular Surface Points

g

e> 1

0.5 b e = 50.0

Conductor

0.4 1/3 0.3

e = 10.0 e = 5.0

0.2

e = 3.0 e = 2.0

0.1

0.0

e = 1.5

0⬚

45⬚

90⬚

b

135⬚

180⬚

FIGURE 29 The singularity index for the dielectric wedge with different apex angles and permittivity values.

easily shown that the correct asymptotic for gðb; eÞ at e approaching unit and b being small is gðb; eÞ ¼

1

e 2p

b þ Oðb2 Þ:

ð2:33Þ

These peculiarities are evident from the numerical solution presented in Figure 29, and now let us briefly discuss the physical reasons for them. A dielectric with high permittivity ejects the electric field from its volume to the neighboring regions with lower permittivity. This means that a dielectric wedge with e 1 becomes almost equipotential, which explains the singularity index behavior in the limit of large e and not too small b. On the other hand, there is principle difference between the dielectric and conductor behavior in the limit b ! 0. Indeed, an infinitely thin conductive wedge disturbs the potential distribution around it, whereas the similar dielectric wedge does not. Accordingly, the singularity index for conductive wedge tends to its maximal value gmax ¼ 1=2 at b ! 0, whereas that for dielectric wedge vanishes. Note that such consideration does not work in the case of multiple dielectric or conductive wedges with a common vertex. Those wedges also are practically equipotential in the limit e 1, but their potentials may tend to different constants.

50

Surface Charge Singularities Near Irregular Surface Points

2.5. UPPER BOUNDS FOR THE SINGULARITY INDEX IN THE 2D CASE The field energy and, consequently, the total amount of the surface electric charge concentrated in any vicinity of a singular boundary point are finite. This physical requirement immediately imposes the upper limitation on the singularity index value as principally possible for 2D wedges: only with g < 1 is the surface charge density locally integrable. From a mathematical standpoint, the same limitation follows from the fact that the minimal eigenvalue in the Sturm-Liouville spectral problem (2.10)-(2.12), corresponding to a nontrivial (e.g., nonconstant) eigenfunction, is strictly positive. In this section we obtain more accurate upper estimations; that is we answer the question as follows: What maximal singularity index may exist at the singular point of conductive and/or dielectric boundary under the restriction 1  eðyÞ  emax

ð2:34Þ

laid on the dielectric permittivity? We first derive those estimations for the conductive wedge. In the simplest case, with no dielectric present (eðyÞ 1), Eq. (2.7) gives the obvious answer to the above question: the maximal singularity index g ¼ 1=2 is attained at the edge of a solitary infinitely thin conductive plate with a ¼ 0. The case of dielectric materials surrounding the conductive wedge is more complicated and leads to the nonclassical variational problem of constructing the lower estimations for the eigenvalue l0 in the Sturm-Liouville problem (2.10)-(2.12) under the additional restriction (2.34). It should be emphasized that we do not assume the permittivity distribution eðyÞ to be piecewise constant but only piecewise continuous. Let us make the linear transformation t ¼ ly of the independent variable y in Eq. (2.15). Following the terminology of optimal control theory, we call the new variable t and the function uðtÞ ¼ eðt=lÞ as ‘‘time’’ and ‘‘control’’, respectively. From the standpoint of optimal control theory, our problem represents a typical problem of optimal quick-action, which can be formulated as follows. The problem of optimal quick-action. Among all the piecewise continuous controls uðtÞ obeying the constraint 1  uðtÞ  eM , it is necessary to find an optimal control u0 ðtÞ that brings the phase point fYðtÞ; ZðtÞg according to the equations dY 1 Z; ¼ dt u0 ðtÞ

dZ ¼ u0 ðtÞY dt

ð2:35Þ

from the initial state fYð0Þ ¼ 0; Zð0Þ 6¼ 0g to the terminal state fYðTÞ ¼ 0; ZðTÞ 6¼ 0g for the minimal time T.

Surface Charge Singularities Near Irregular Surface Points

51

Obviously, due to the correlation l0 ¼ T=ð2p aÞ, the solution to the quick-action problem formulated above gives the lower estimation for l0. The problem can be solved with the Pontryagin maximum principle which, with some insignificant simplifications specific to the particular problem in question, is formulated below. Readers can find a more general formulation of this outstanding theorem, along with many mathematical details relevant to optimal control problems, in the specialized monographs by Girsanov (1970) and Bryson and Yu-Chi Ho (1969). Suppose Y 0 ðtÞ; Z0 ðtÞ; u0 ðtÞ; T is a solution of the optimal control problem in question. Then there necessarily exists a nontrivial solution c1 ðtÞ; c2 ðtÞ of the system dc1 ¼ u0 ðtÞc2 ; dt

dc2 ¼ dt

1 c ; u0 ðtÞ 1

ð2:36Þ

being conjugated in the Hamiltonian sense to the system (2.35), so that the Hamiltonian function 1 HðY; Z; u; c1 ; c2 Þ ¼ Zc1 u

uYc2

1

ð2:37Þ

attains its maximum over the segment 1  u  emax on the optimal solution u0 ¼ u0 ðtÞ. This means that HðY 0 ðtÞ; Z 0 ðtÞ; u0 ðtÞ; c1 ðtÞ; c2 ðtÞÞ

¼ max HðY 0 ðtÞ; Z0 ðtÞ; u; c1 ðtÞ; c2 ðtÞÞ ¼ 0 1ueM

ð2:38Þ

at any time moment 0  t  T. As soon as both the left and right endpoints of the component ZðtÞ are free (nonfixed), we additionally obtain the boundary conditions for the conjugated function c2 ðtÞ: c2 ð0Þ ¼ c2 ðTÞ ¼ 0. Thus, we now need to (1) construct the function u0 ðY; Z; c1 ; c2 Þ that brings maximum to the Hamiltonian function (2.38) with respect to u over the segment 1  u  eM at Y; Z; c1 ; c2 being fixed, (2) put this function into the differential equations (2.35) and (2.36) instead of the unknown function u0 ðtÞ, and (3) solve the boundary-value problem for these equations with the boundary conditions Yð0Þ ¼ YðTÞ ¼ 0; c2 ð0Þ ¼ c2 ðTÞ ¼ 0 and additional condition H ¼ 0. The latter condition is essential because the time moment T is not fixed but, on the contrary, is yet to be determined. Our particular problem allows this procedure to be easy implemented because of very simple structure of the Hamiltonian function (2.38). Indeed, depending on the particular correlations among Y; Z; c1 ; c2 , the maximum of H over the segment 1  u  eM can, in principle, be reached pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðZc1 Þ=ðYc2 Þ. at one of the three points: u0 ¼ 1, u0 ¼ emax , and u0 ¼ Detailed analysis of the boundary-value problem for the system (2.35), (2.36) shows that the third possibility never occurs, and the unique

52

Surface Charge Singularities Near Irregular Surface Points

optimal control u0 ðtÞ is indeed a piecewise constant function with a number of abrupt transitions (‘‘switchings’’) from 1 to emax and back. In terms of the function eðyÞ the final result reads 8 y  y0 =4 < 1; ð2:39Þ eðyÞ ¼ emax ; y0 =4 < y < 3y0 =4; y0 ¼ 2p  a: : 1; y  3y0 =4

The functions Y0 , Z0 are shown in Figure 30. The singularity index g is maximal if the dielectric wedge is located symmetrically with respect to the conductive wedge and occupies exactly one half of the outside space. The minimal transition time in our quick-action problem is 1=2 T 0 ¼ 4 arctanðemax Þ and, correspondingly,

 T0 1 emax  1 gmax ¼ 1  ; ð2:40Þ ¼ p  a þ 2 arcsin 2p  a 2p  a emax þ 1 which shows that the maximal singularity index increases from its dielectric-free value (2.7) up to 1 with emax increasing from 1 to infinity. The case of pure dielectric vertex can be studied similarly; the only difference is that the periodicity condition Zð0Þ ¼ Zð2pÞ should also be imposed. It is noteworthy that any arbitrary rotation of the dielectric media as a whole does not affect the singularity index. To avoid this ambiguity, we fix the angular coordinate origin y ¼ t ¼ 0 to make it

1.0 e = emax

e =1 0.5

e =1

Y0

q = 2p-a

a

Z0 p-a/2

-0.5

q=0

0.0

-1.0 0.0

1.0 t

2.0 T0

FIGURE 30 The solutions of the equations (2.35) under the restriction 1  e  4 outside the conductive wedge

53

Surface Charge Singularities Near Irregular Surface Points

coincide with one of zero points of the function Y (as shown further, YðyÞ turns into zero at least twice within the interval 0  y < 2p). Thus, zero boundary conditions for YðtÞ will be preserved. The periodicity condition for the function Z implies the periodicity condition for the conjugated function c2 ðtÞ: c2 ð0Þ ¼ c2 ðTÞ, whereas the boundary values of c1 stay free. The optimal solution of the corresponding quick-action problem resembles that for the problem considered above with the difference that the required periodicity can be reached only at the second zero T2 ¼ 2T1 of the Y function. The maximum singularity index appears as gmax ¼ 1

T2 2 emax 1 ¼ arcsin 2p p emax þ 1

ð2:41Þ

and takes values in the range 0  gmax < 1. It approaches 1 with emax increasing to infinity. The dielectric permittivity distribution, at which the maximum singularity index is reached, reads  emax; p=4 < y < 3p=4 or 5p=4 < y < 7p=4 ; ð2:42Þ eðyÞ ¼ 1; otherwise which describes two symmetric dielectric wedges whose four faces meet each other at a common point at right angles. The corresponding solutions are shown in Figure 31.

2.6. VARIATIONAL APPROACH TO THE SPECTRAL PROBLEM Consider an alternative, variational approach to the Sturm-Liouville spectral problem (2.10), in which the coupling conditions (2.12) appear quite naturally. The approach is based on the fact that the eigenfunction Y0 minimizes the positively defined energy functional W½Y ¼

2pa ð

0

eY 2 dy

ð2:43Þ

eY2 dy ¼ 1

ð2:44Þ

0

under the normalization condition Q½Y ¼

2pa ð 0

and the boundary conditions Yð0Þ ¼ Yð2p  aÞ ¼ 0. Indeed, the Euler variational equation with the Lagrange coefficient l20 (which is exactly the minimal eigenvalue to be found) reads

54

Surface Charge Singularities Near Irregular Surface Points

1.0 e = emax

e =1 0.5

e =1

e = emax

e =1

Y0

0.0 T1 q = 2p

p/2 q =0

-0.5 Z0 p/2

-1.0 0.0

1.0

2.0

3.0

t

T2

FIGURE 31 Solutions of Eq. (2.35) with the restriction 1  e  4 in the case of purely dielectric distribution.



d W½Y 

l20 ðQ½Y

 1Þ ¼  2

2pa ð



0

 ðeY0 Þ0 þ l20 eY dYdy

X þ0 þ 2 ¼0 ½eY0 dYyykk 0 k

ð2:45Þ

The integral part of the variation immediately yields Eq. (2.10) in the region of eðyÞ smoothness, whereas the sum gives the set of coupling conditions (2.12) on the dielectric interfaces. In the case of purely dielectric vertexes, the variational method is slightly more complicated. The spectral problem with periodical boundary conditions Yð0Þ ¼ Yð2pÞ allows the eigenvalue l0 ¼ 0 and the eigenfunction Y0 ¼ const: The corresponding radial function R0 ðrÞ also appears constant; therefore, the first term in the expansion (2.9) does not induce any surface charge density but reflects the fact that electric potential is not fixed by any Dirichlet condition and thus may contain any additive constant. The next eigenvalue is positive and gives the main part of the surface charge singularity. To define it unambiguously, we should impose the additional orthogonality condition L½Y1  ¼

2ðp

eY1 dy ¼ 0:

ð2:46Þ

0

After partitioning the interval and approximating the unknown function Y, we can find numerically the minimum of the functional (2.43)

Surface Charge Singularities Near Irregular Surface Points

55

under the conditions (2.44) and, if necessary, (2.46), which immediately gives the first positive eigenvalue. The variational approach seems more complicated than the transfer matrix method; nevertheless this approach is much more general. For example, it allows nonconstant dielectric permittivity distribution and guarantees convergence to the solution, whereas the algorithms oriented to numerical solving of the transcendental equation (2.21) may oversee the root needed. The most important advantage of the variational approach as applied to the surface charge singularities problem is that it allows direct generalization to the 3D case.

2.7. THREE-DIMENSIONAL CORNERS This section considers the problem of surface charge density singularities at conducting and dielectric corners in three-dimensional case. The knowledge of asymptotic behavior of the surface charge density in the singular point vicinity on the boundary is very important for construction of accurate 3D field calculation algorithms. The most general and sophisticated results that form theoretical basis to this problem were obtained by the renowned mathematicians Kondrat’ev (1967) and Fichera (1975). We follow the approach suggested by Fichera, based on the idea of local separating the radial and angular variables in the corner vicinity and evaluating the singularity index g in terms of the solution of an auxiliary Beltrami-Laplace spectral problem containing angular coordinates only. Below we present the general idea of Fichera’s method as applied to the singular points of conductive surface, and then extend his idea to the case of dielectrics. Consider a conical conductive surface with the corner at the point O (Figure 32). We assume that this surface is formed by a family of straight generatrixes given in terms of the angular spherical coordinates y; f by the pair of continuous functions fy0 ðsÞ; f0 ðsÞg, which depend on the parameter s running within the interval 0  s  smax : Under the periodic condition y0 ðsmax Þ ¼ y0 ð0Þ; f0 ðsmax Þ ¼ f0 ð0Þ, functions fy0 ðsÞ; f0 ðsÞg describe a close curve on the coordinate sphere of unit radius. Let us assume this curve has no self-intersections and thus divides the sphere into two spherical segments, one of them (Sþ ) being external and another one (S ) internal with respect to the conductive cone. Without loss of generality, we may put electric potential on the conductive surface equal to zero. The Laplace equation in terms of the variables r; y; f takes the form D’

1 @2 1 ðr’Þ þ 2 L½’ ¼ 0; r @r2 r

in which L denotes the Beltrami-Laplace operator

ð2:47Þ

56

Surface Charge Singularities Near Irregular Surface Points

S S+ q

R= 1

O

f

S+

h

S-

q 0(s) f0(s)

FIGURE 32 The external segment of the unit-radius sphere with the spherical coordinates system.

L¼

1 @ @ 1 @2 siny þ siny @y @y sin2 y @f2

ð2:48Þ

as the ‘‘angular’’ part of the Laplace operator on the unit-radius sphere. The boundary condition on the corner’ surface takes the form ’ðr; y0 ðsÞ; f0 ðsÞÞ ¼ 0

for

0  s  smax

and

r  0:

ð2:49Þ

By analogy with Eq. (2.9), let us represent the electric potential ’ in the vertex vicinity in the form of expansion ’¼

1 X

m¼0

Rm ðrÞYm ðy; fÞ;

ð2:50Þ

in which the radial variable r and angular variables y; f are separated. This immediately implies that the functions Ym ðy; fÞ are the eigenfunctions of the operator L on the spherical segment Sþ being external to the corner: L½Ym  þ Lm Ym ¼ 0:

ð2:51Þ

According to Eq. (2.49), the eigenfunctions Ym should vanish on the boundary @Sþ , so that Ym ðy0 ðsÞ; f0 ðsÞ ¼ 0; 0  s  smax : Since the operator L is self-conjugated and positively defined, all the eigenvalues Lm are real

Surface Charge Singularities Near Irregular Surface Points

57

and positive. We shall assume them numbered in the ascending order 0 < L0 < L1 . . .. Substituting Eq. (2.50) into Eq. (2.47), we come to the set of ordinary differential equations for the radial functions Rm ðrÞ @ 2 ðrRm Þ @r2

Lm Rm ¼0 r

ð2:52Þ

with zero boundary condition Rm ð0Þ ¼ 0: It is readily seen that the linearly independent solutions of Eq. (2.52) can be represented by a sum of two power-like functions RðrÞ ¼ C1 rpþ þ C2 rp with the exponents  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ 1  1 þ 4Lm =2: Obviously, the zero boundary condition at r ¼ 0 could be satisfied only if C2 ¼ 0. The main contribution to the sum (2.50) nearby the tip is given by the term ’  rpþ ðL0 Þ Y0 ðy; fÞ

ð2:53Þ

with minimal power of the radial coordinate, which correspond to the minimal eigenvalue l0 of the Beltrami-Laplace operator L. The surface charge density is proportional to the normal electric field component on the surface; therefore, we obtain  1 @’  y ¼ y0 ðsÞ ð2:54Þ  CðsÞrg : sðr; sÞ ffi r @h  f ¼ f0 ðsÞ Here

g¼

3

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4L0 2

ð2:55Þ

is the singularity index, h is the normal vector to the boundary @Sþ in the space of angular variables, and function C(s) is the normal derivative of the first eigenfunction, CðsÞ ¼

@Y0 : @h

ð2:56Þ

Evaluation of the main term in the surface charge density asymptotics near a conductive 3D corner is thus reduced to the calculation of the minimum eigenvalue for the Beltrami-Laplace operator on the spherical segment external to the corner. Using the projective transformation from the pole y ¼ 0 to the plane y ¼ p=2, as shown in Figure 33, we can reduce the Beltrami-Laplace spectral problem to the spectral problem ð1 þ joj2 Þ2 Dx Yk ðoÞ þ Lk Yk ðoÞ ¼ 0

ð2:57Þ

in the complex plane o ¼ x þ i. Such interpretation may be useful if we find a proper conformal transformation to simplify the computational domain in the complex plane o.

58

Surface Charge Singularities Near Irregular Surface Points

h

x

w

FIGURE 33 Reduction of the Beltrami-Laplace spectral problem to the spectral problem in the complex plane.

2.8. VARIATIONAL METHOD IN THE CASE OF DIELECTRICS We can directly generalize the variational method to the cases of conductive corners with dielectrics or pure dielectric corners if the permittivity function eðy; fÞ depends on angular variables only. By analogy with Eq. (2.10), we can rewrite the Eq. (2.51) in the form Le ½Ym  þ Lm eðy; fÞYm ¼ 0;

ð2:58Þ

where     1 @ @Y 1 @ @Y þ eðy; fÞ siny eðy; fÞ Le ½Y ¼ siny @y @y @f sin2 y @f

ð2:59Þ

is the Beltrami-Laplace operator in the case under consideration. As in the 2D case, the permittivity function eðy; fÞ may be discontinuous, but the normal component of the vector   @Y @Y ð2:60Þ ;e Z¼ e @y @f should still stay continuous on the lines of eðy; fÞ discontinuities. If we introduce two functionals  2 # ð " 2 @Y 1 @Y We ½Y ¼ e þ siny dy d’ ð2:61Þ 2 @y sin y @’ S

and

ð Ne ½Y ¼ eY2 siny dy d’; S

ð2:62Þ

Surface Charge Singularities Near Irregular Surface Points

59

we can determine the minimal eigenvalue L0 as the solution of the variational problem L0 ¼ minY We ½Y under the condition Ne ½Y ¼ 1. For a pure dielectric vertex, we must find the first strictly positive eigenvalue L1 since L0 appears to be zero and Y0 constant. The extra orthogonality condition ð eY siny dy d’ ¼ 0 ð2:63Þ S

should be added in this case.

2.9. REDUCTION TO THE 2D CASE From the methodological viewpoint, it is useful to consider the reduction of the general approach suggested by Fichera to the case of a two-faced edge with the angle a between the faces. In accordance with Figure 34a, let us place the origin of the spherical coordinate system at an arbitrary point of the edge, choose the axis y ¼ 0 orientation along the edge, and consider the outer segment of the unit sphere located in the coordinate range defined by the inequalities jfj  p  a=2, 0  y  p. It can be easily shown that the first nontrivial eigenfunction of the Beltrami-Laplace operator, vanishing on the boundary of the spherical segment S, is p : ð2:64Þ Y0 ðy; fÞ ¼ sinl y coslf; l ¼ 2p  a Accordingly, the eigenvalue we are seeking for is L0 ¼ lðl þ 1Þ, and Eq. (2.55) provides the singularity index

(a)

q

(b)

f

a a

FIGURE 34

(a) wedge with flat faces; (b) wedge with curved faces.

60

Surface Charge Singularities Near Irregular Surface Points

g¼

3

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4lðl þ 1Þ ¼1 2

l¼

p a ; 2p a

ð2:65Þ

which, as could be expected, coincides with Eq. (2.7) obtained earlier with the conformal mapping method. Our approach here is much more general and may be used to consider the nonflat edge case. The Fichera method obviously can be extended to the case of the corner with curved boundaries, provided that the faces of the corner represent smooth surfaces with limiting tangential planes at the tip. This fact provides us with a ‘‘recipe’’ to calculate the singularity index in the vicinity of any two-faced conductive wedge, even if the wedge is curved. To do so, one should cut the wedge with a plane being orthogonal to the edge made by the faces comprising the wedge, as shown in Figure 34b. The value of the crosssectional angle a then should be substituted into Eq. (2.65) to calculate the singularity index, which, obviously, may depend on the location of the particular point on the edge.

2.10. ON-RIB SINGULARITIES NEAR THREE-DIMENSIONAL CORNER This section considers the case when the spherical segment boundary @S defined by the pair of functions fy0 ðsÞ; f0 ðsÞg is nonsmooth. This happens, for example, if a multifaced corner is constructed of three of more flat angles meeting at one point. We refer to the lines of faces intersection as ribs. From our 2D analysis, we know that each rib carries the surface charge  singularity sðPÞ  hg , where h is the distance from a given point P to the rib, and the index g is calculated from Eq. (2.65) as a function of the wedge angle a. Thus, each rib of the multifaced corner introduces its own singularity into the total surface charge singularity at the tip where all the corner ribs meet. For 3D convex corners, the on-rib singularities are weaker than the corner singularity considered in Section 2.7; however, those singularities may dominate in the case of concave corners. Mathematically, the on-rib singularities near a 3D corner become apparent in the function C(s) (2.56). To construct the asymptotics of C(s) near the point at which a rib (or its limiting tangent) crosses the unit sphere, we can use the ‘‘tilted’’ spherical coordinates fy ; f g with the axis y ¼ 0 directed along the rib tangent, as shown in Figure 35. Now the variables y ; f can be separated by representing Y0 in the form X Y0 ðy ; f Þ ¼ Y0 ðyðy Þ; fðf ÞÞ ¼ Yk ðy ÞCk ðf Þ; ð2:66Þ k



where the functions Ck ðf Þ, Yk ðyÞ satisfy the equations

Surface Charge Singularities Near Irregular Surface Points

q∗

61

f∗

t h n(s)

b a

FIGURE 35

The estimation of the on-rib singularities.

d2 Ck þ l2k Ck ¼ 0; df2   1 @ l2k  @Yk  siny Yk þ L0 Yk ¼ 0;    @y siny @y sin2 y

ð2:67Þ ð2:68Þ

accordingly. With regard to the finiteness condition for Yk ðyÞ on the axis y ¼ 0, the asymptotic solution to Eq. (2.68) at small y takes the form   2

 L0 y  lk þ Oðy4 Þ ðsiny Þlk : ð2:69Þ Yk ðyÞ ¼ C0 1 þ lk þ 1 4 Since (by analogy with the 2D wedge case), the term with the lowest power of y in Eq. (2.69) is of our only interest, the asymptotics for Y0 appears as Y0  y C0 ðf Þ, with l0 ¼ p=ð2p  aÞ being the smallest nondegenerated eigenvalue of the problem (2.67) with zero boundary conditions Ck ð0Þ ¼ Ck ð2p  aÞ ¼ 0. Accordingly, the surface charge density asymptotics in the vicinity of a multifaced corner takes the form
Y   sðr; sÞ  jnðsÞ  tt jgt rg ; ð2:70Þ t

where the index g is given by Eq. (2.55), tt is the unit vector directed from the corner apex along the t-th rib, and the indexes gt are given by the formula p  at ; ð2:71Þ gt ¼ 2p  at with at being the rib apex angle. In the rib vicinity, the module jnðsÞ  tt j approximates the angular distance between the point fy0 ðsÞ; f0 ðsÞg and

62

Surface Charge Singularities Near Irregular Surface Points

the rib. The appearance of Eq. (2.71) coincides with that of Eq. (2.7) for singularity index of the 2D wedge, which makes the on-rib surface charge asymptotics (2.7) to continuously ‘‘turn over’’ into the 3D corner asymptotics when a point is approaching the corner along the rib. Another form of the singularity coefficients representation was provided by Read (2004). He distinguished two types of surface charge density dependences on the distance from the corner: one is taken along the line being equidistant from the ribs (the dashed line ‘‘a’’ in Figure 35) and another is taken on one of the ribs - that is, along the line separated from the rib by a constant distance ht ¼ rjnðsÞ tt j (the dashed line ‘‘b’’). Both dependences may be fitted with the power functions but with different exponents. In the first case, the exponent appears as sa  rg ,  whereas in the second case it is sb  rðggt Þ .

2.11. THE CASES ALLOWING SEPARATION OF VARIABLES We now consider two important examples, in which separation of variables allows reduction of the 2D spectral problems (2.51), (2.52) to 1D spectral problem. The first example is the conical tip. Restricting ourselves to the axisymmetric cone with the apex angle a, let us choose the spherical coordinates system in such a way that the axis y ¼ 0 coincides with the symmetry axis of the cone. The outer spherical segment lies within the coordinate range 0  y  p  a=2. Let us construct the first nontrivial eigenfunction in the form Y0 ¼ f ðcosyÞ, where f ðxÞ is the solution to the spectral problem

  df d  þ l0 f ¼ 0 1  x2 ð2:72Þ dx dx

with the boundary conditions f ½cosða=2Þ ¼ 0 and f ð1Þ < 1. Numerical solution of the spectral problem in question is given in Figure 36 in terms of the function gðaÞ. The asymptotic behavior of the curve gðaÞ in the ‘‘needle’’ limit a 1 is in perfect agreement with the asymptotic formula g  1  ½2 lnð2=aÞ1 given in the monograph by Landau and Lifshitz (1984a). The second example is the corner of a flat conducting plate. To solve this problem, Morrison and Lewis (1976) used special ‘‘conical’’ coordinates r; y; f that are interconnected with the Cartesian coordinates x; y; z by the transformation formulas pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ r cosf sin2 y þ k2 cos2 y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ¼ r cosy 1  k2 cos2 f

z ¼ r siny cosf:

ð2:73Þ

63

Surface Charge Singularities Near Irregular Surface Points

g 0.9 a 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

FIGURE 36

0⬚

45⬚

90⬚

135⬚

a

180⬚

The singularity index for a conical tip. y q=0

a

f=p

q=f

x

q=p

FIGURE 37 The conical coordinates introduced by Morrison and Lewis (1976) for separation of variables in the surface charge singularity problem for a pointed flat conducting plate.

By choosing the parameter value k ¼ sinðjp aj=2Þ, we make the angle a coincide with the coordinate plane f ¼ 0 as shown in Figure 37. After the variable separation Y0 ðf; yÞ ¼ LðfÞNðyÞ in terms of the conical coordinates (2.73), we derive a system of two equations ð1

k2 cos2 fÞL00 þ k2 sinf cosf L0 þ ½k2 L0 sin2 f þ dL ¼ 0

ð2:74Þ

ðsin2 y þ k2 cos2 yÞN 00 þ ð1  k2 Þsiny cosy N 0 þ ½ð1  k2 ÞL0 sin2 y  dN ¼ 0

ð2:75Þ

64

Surface Charge Singularities Near Irregular Surface Points

1.0 g a

0.9

0.8

0.7

0.6

0.5 0⬚

45⬚

90⬚

135⬚

a

180⬚

FIGURE 38 The charge singularity index for a flat conducting corner (calculated according to Morrison and Lewis, 1976)

for the functions LðfÞ and Nðy). These equations are interconnected only by the separation constant d, which should be found together with the eigenvalue L0. The numerically obtained correlation between L0 and k was recalculated in terms of the function gðaÞ and plotted in Figure 38 for the apex angle interval 0 < a  p. In the case a > p (the ‘‘dovetail’’ shape), the singularity index becomes lower than 1=2, and the on-rib singularity starts to dominate. The asymptotic formula for the ‘‘needle’’ limit g  1 ½2 lnð8=aÞ1 found by Noble (1959) differs from that for the conical tip with the same apex angle only by the constant value in the argument of the logarithm. This formula is in good agreement with the curve gðaÞ at small a.

2.12. NUMERICAL SOLUTION OF THE BELTRAMI-LAPLACE SPECTRAL PROBLEM In general, the minimal positive eigenvalue in the Beltrami-Laplace spectral problem on the spherical segment should be found numerically. In this section, a specially oriented numerical scheme is constructed for this purpose.

Surface Charge Singularities Near Irregular Surface Points

(a)

(b)

65

(c)

FIGURE 39 Approximating the unit sphere with the inscribed polyhedrons containing 22n+3 triangular faces (a, n = 0; b, n = 1; c, n = 2).

The spherical coordinate system, in which the operator (2.48) has been originally expressed, is rather inconvenient because it degenerates at the poles. Moreover, it is well known that a regular coordinate map covering the entire spherical surface does not exist. For these reasons, we will use local coordinates and approximate the spherical surface with a number of small triangular elements, as shown in Figure 39. The regular procedure to construct the mesh is as follows. The regular octahedral with eight equilateral faces shown in Figure 39a gives the first approximation. Then each of the twelve ribs is divided into two equal parts, and the middle points are shifted outward to the center onto the circumscribed sphere. By connecting these points with new ribs, we obtain the 32-faced polyhedron (Figure 39b). The procedure should be repeated n times to obtain the 22nþ3 -faced polyhedron approximating the unit sphere with sufficient accuracy. Let us parameterize each of the triangular faces of the polyhedron with the local coordinates 0 < u < 1, 0 < v < 1 u and approximate the eigenfunction Y on each of them with a linear form Y ¼ #1 ð1

u

vÞ þ #2 u þ #3 v:

ð2:76Þ

Here the coefficients #k ðk ¼ 1; 2; 3Þ represent the eigenfunction values at the vertexes of the given triangular face being simultaneously some three of the polyhedron vertexes marked as 1,2, and 3 in Figure 40. The functionals (2.61) and (2.62) can be now approximated as W

1 X V2 ð#2 e 2

N

#1 Þ2 þ U2 ð#3

#1 Þ2 2 < U; V > ð#2 jU  Vj

#1 Þð#3

1 X ejU  Vjð#21 þ #22 þ #23 þ #1 #2 þ #2 #3 þ #1 #3 Þ 12

#1 Þ ð2:77Þ

ð2:78Þ

66

Surface Charge Singularities Near Irregular Surface Points

S 1

V

3

U

S+

S+

2

FIGURE 40

Local coordinates on a polyhedron face with vertexes 1,2, 3.

where the vectors U and V represent the oriented sides of one of the polyhedron faces, the permittivity e is assumed constant within the face, and the summation is spread over all the faces. On choosing the initial approximation for the nodal values f#i g, we proceed to the numerical minimization of the functional (2.77) under the normalization condition Nð#0 ; #1 . . .Þ ¼ 1. The boundary conditions are approximated by putting the coefficients #i zero at the nodes located outside and on the boundary of the spherical segment Sþ on which the spectral problem is being solved. Technically, the quadratic form minimization may be performed with the conjugate gradients method. We have tested this numerical scheme using the example of the twofaced wedge with a ¼ p=2 considered as a 3D corner. The exact solution g ¼ 1=3 is being approached with the number of mesh elements increasing (Figure 41). The numerical error 10 2 is reached at n ¼ 4 (2048 elements), which is commonly sufficient for practical use. The accuracy can be improved if we consider the asymptotic behavior of the numerical error and extrapolate the numerical results to the case of infinite number of the polyhedron faces. Two reasons explain why the numerically determined Beltrami-Laplace operator eigenvalues are larger than the exact ones: (1) the area of the inscribed polyhedron surface is always smaller than the entire spherical surface area, and (2) the set of the test eigenfunctions is restricted to the piecewise linear ones. Accordingly, the numerical values gðnÞ are less than the exact value g0 , and we can use the empiric formula gðnÞ g0 ¼ e anþb for the discrepancy extrapolation. The coefficients a and b can be determined from numerical experiments using three different levels of mesh refinement. In doing so, we derive the extrapolation formula g0 

gðn 1Þ gðnþ1Þ ðgðnÞ Þ2 : gðn 1Þ þ gðnþ1Þ 2gðnÞ

ð2:80Þ

Surface Charge Singularities Near Irregular Surface Points

67

1.0

g–1/3 32 elements

10-1

128 512

10-2

2048 8192 10-3 32768 131072

10-4 0

1

2

3

4

5

6

7

n

FIGURE 41 The accuracy of the numerical solution of the Beltrami-Laplace spectral problem a for two-faced corner with right apex angle. The difference between the numerical and exact solutions is plotted versus the mesh refining level.

Having picked gð5Þ , gð6Þ , and gð7Þ from our numerical experiment, we obtain from the extrapolation equation (2.80) the estimation g0  0:333308 with the error jg0 1=3j being as small as 2:5  105 .

2.13. CUBICAL AND PRISM CORNERS Aside from its importance from the practical viewpoint, the problem of surface charge behavior in the vicinity of the cubic corner formed by three right ‘‘conducting’’ angles has an interesting historical background. As claimed by Fichera (1975), there are certain evidences by G. Weber and G. Kirchhoff (see Dirichlet, 1897) and K. Jacobi mentioned by Ko¨nigsberger (1904) that Dirichlet had solved the problem on distribution of electric charge over the conductive rectangular parallelepiped in terms of simple, double, and triple quadratures. Unfortunately, the Dirichlet solution has been lost, if it existed, and now we only must be content with approximate analytical estimations and numerical results. Using sophisticated mathematical technique and some very simple numerical methods, Fichera (1975) has found rather narrow bilateral estimation 0:5355 < g < 0:5665 for the singularity index of the cubic corner. Read (2004) has recently published more accurate numerical result obtained by using the BEM with the mesh density progressively

68

Surface Charge Singularities Near Irregular Surface Points

g 0.70

a

0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0⬚

FIGURE 42

45⬚

90⬚

135⬚

a

180⬚

The singularity indexes for the prism corner.

increasing near the cube ribs and corners. By fitting the numerically calculated surface charge density with the power-like functions dependent on the distance from the cube vertex, he derived the estimation g ¼ 0:540  0:004. The details of the computational technique used by Read in those experiments were described in his earlier paper (1997). The numerical value for the singularity index g found for the ‘‘cube problem’’ by the authors of this monograph with the use of the Fichera theorem and the interpolation formula (2.80) is g ¼ 0:54585  104 . We have applied the numerical scheme discussed in the Section 2.12 to analyze more general case of the prism corners composed of two right and one arbitrary angles. As shown in Figure 42, in this case the singularity index lies between 0.7034 (flat right angle, a ¼ 0) and 0.3333. . . (two-faced wedge, a ¼ p).

CHAPTER

3 Geometry Perturbations

Contents

3.1. Integral Variational Equations and Conjugate Integral Equation for the Green Function 3.2. 3D Perturbations in Axisymmetric Systems 3.3. Examples of 3D Perturbations in Axisymmetric Systems 3.4. 3D Perturbations in Planar Systems 3.5. Locally Strong 3D Perturbations in Axisymmetric Systems 3.6. 3D Fringe Fields in Planar Systems

71 78 84 93 100 103

This chapter addresses the problem of evaluating potential perturbations resulting from small deviations of the geometry and voltages of the electrodes that form the electron-optical system’s boundary (hereafter termed the boundary variations) from some ideal state, which is called nominal. This problem is the starting point and main issue for many charged particle optics problems, including mechanical tolerances computation, fringe effects evaluation, electron-optical system optimization, and so on. Indeed, it is important in practice to know which level of accuracy of mechanical parts manufacture and feeding voltages stability is needed to meet technical requirements on a particular device. If the nominal geometry is axisymmetric or planar, we necessarily face a 3D boundary-value problem that is very close to the nominal 2D one. The same situation occurs if we need to estimate the contribution of the fringe fields to the potential distribution in a planar system. In all these cases, the direct use of 3D algorithms is not the best approach. Much more accurate and reliable results can be obtained by using perturbation theory, which allows

Advances in Imaging and Electron Physics, Volume 155 ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)00803-3

2009 Elsevier Inc. All rights reserved.

#

69

70

Geometry Perturbations

construction of explicit correlations between the small boundary variations and the perturbation of potential distribution in the domain of interest. Inasmuch as any numerical optimization procedure assumes step-bystep changing of the structural parameters involved, numerical optimization in charged particle optics also requires the knowledge of the extent to which the small variations of electrode geometry and voltages may affect the electron-optical characteristics in question. In 1876 Bruns put forward a method to calculate the gravitational field perturbation induced by the deviation of the Earth’s surface shape from the ideal sphere (Bruns, 1876, Idelson, 1936). Bertein (1947, 1948) applied the Bruns approach to the problems of electrostatics and electron optics for the first time. Since then the Bruns-Bertein method was used for evaluation of tolerances in charged particle optics by many authors, including Sturrock (1951), Glaser and Schiske (1953), Der-Schwarz and Kulikov (1962), Janse (1971), Munro (1988), and others. The basis of the method is replacing the real boundary perturbation by some equivalent potential perturbation defined on the nominal boundary in terms of the unperturbed potential gradient. More exactly, if ’0 is unperturbed potential distribution, dR is boundary deformation vector, and d’ is the distribution of the potential perturbation in space, we easily derive the Dirichlet problem for Laplace equation D½d’ ¼ 0, d’j@O0 ¼  < dR; r’0 > with the boundary condition on the unperturbed boundary @O0 (Figure 43).

δr

x

ϕ ϕ0

ϕ

x

FIGURE 43 The Bruns-Bertein method. Dotted line denotes the nominal boundary, solid line shows the perturbed boundary.

71

Geometry Perturbations

Strictly speaking, the Bruns-Bertein method can be applied only to the domains with smooth boundaries on which the gradient r’0 is regular and finite. This condition is certainly violated if the boundary possesses some singular points or lines (edges, ribs, and corners) which are always present in any, pithy enough, practical problem. It is well known that in such cases the gradient r’0 tends to infinity near the boundary singularities, which renders the calculation process less accurate and essentially unstable. Another difficulty arises if both sides of the infinitely thin charged surface are immersed in the optically active part of the field – in this case, the double layer potential is needed to calculate potential perturbations with the integral equations method. This chapter develops another, more versatile and numerically efficient perturbation approach based on direct variational analysis of the first-kind Fredholm integral equation. This approach may be considered as direct extension of the boundary element technique described in Chapter 1. In Section 3.1, Fedorenko’s variational scheme (outlined in Appendix 3) is used for varying the Dirichlet problem for Laplace equation in the general 3D case. The integral variational equations obtained in Section 3.1 are then applied in Section 3.2 to the case of small 3D perturbations of axisymmetric nominal boundary. Some relevant examples are considered in Section 3.3. Section 3.4 is devoted to the small 3D perturbations of nominal planar boundary. Finally, the Sections 3.5 and 3.6 describe the numerical approaches for accurate evaluation of potential perturbations caused by the locally strong 3D boundary perturbations of axisymmetric and planar nominal boundaries.

3.1. INTEGRAL VARIATIONAL EQUATIONS AND CONJUGATE INTEGRAL EQUATION FOR THE GREEN FUNCTION Consider the Dirichlet problem for Laplace equation D’ðRÞ ¼ 0; R 2 O0 ;

 0 ðQÞ; Q 2 @O0 ’j@O0 ¼ ’

ð3:1Þ

in the domain O0 with the boundary @O0 (Figure 44). As in Chapter 1, the domain O0 may be unrestricted, and the boundary @O0 may consist of a finite number of the connected components @O0i ; i ¼ 1; . . .; n. If the domain O0 is unrestricted, the boundary-value problem (3.1) should be supplemented by the condition in infinity. In this chapter we assume that ’ðRÞ ! 0 at R ! 1. Let us assume that the boundary @O0 can be described by the vectorfunction Q0 ðu; vÞ belonging to the space C10 ðDÞ of the piecewise continuously differentiable functions defined in a 2D domain D of the parameters fu; vg. We also assume that the boundary potential distribution

72

Geometry Perturbations

Q0 dQ ⭸Ω

Q

P

⭸ Ω0

δP P0

FIGURE 44 boundary.

The definition of geometric and potential variations on the unperturbed

 0 ðQ0 ðu; vÞÞ belongs to the space L1 ðDÞ of restricted measurable functions ’  0 ðu; vÞ. in the domain D. For simplicity, we denote such distributions as ’ Following the general principle of variational analysis, let us ‘‘immerse’’ the unperturbed
boundary-value  problem (3.1) defined by the  0 ðu; vÞ into the set of the perturbed pair of functions U0 ¼ Q0 ðu; vÞ; ’ boundary-value problems D’ðRÞ ¼ 0; R 2 O;

 ðQÞ; Q 2 @O ’j@O ¼ ’

ð3:2Þ

 ðQÞ defined with the perturbed boundary O and the boundary potential ’  ðu; vÞg. The corresponding geoby the pair of functions U ¼ fQðu; vÞ; ’ metric and potential boundary variations dQðu; vÞ ¼ Qðu; vÞ  Q0 ðu; vÞ;

 ðu; vÞ  ’  0 ðu; vÞ d ’ ðu; vÞ ¼ ’

ð3:3Þ

are assumed to be small in C10 ðDÞ, which means that those variations are small, along with their partial derivatives with respect to the parameters u, v, everywhere in the domain D. It is important that both the perturbed and nonperturbed boundary-value problems (3.1), (3.2) are parameterized in the same fixed domain D. We refer to the variation or derivative of any function ‘‘attached’’ to the same point (u,v) of the domain D as Lagrangian variation or Lagrangian derivative. Geometrically, the boundary variations we have introduced may be treated as a displacement of the point Q0 2 @O0 by the small vector dQ0 dependent on the position of the point Q0 on @O0, as shown in Figure 44. The geometric and potential variations on the unperturbed boundary may be schematically represented by the following diagram  0 ðQ0 Þ Q0 ! ’ # #  ðQÞ ¼ ’  0 ðQ0 Þ þ d ’ ðQ0 Þ Q!’

ð3:4Þ

Geometry Perturbations

73

If we fix any point R within the common interior of the perturbed and unperturbed domains O and O0, we may consider the mapping  ðu; vÞg ! ’ðRÞ as a nonlinear functional determined on U ¼ fQðu; vÞ; ’ the product of spaces ℜ ¼ C10 ðDÞ  L1 ðDÞ. Similar to Chapter 1, let us represent the functional ’ðRÞ in the form of the ordinary boundary-layer potential ð s ðQÞ ðR 2 O; Q 2 @OÞ: ð3:5Þ dSQ ’ðRÞ ¼ jRQj @O

Here j R  Q j is the distance between the points R 2 O; Q 2 @O, dSQ is elementary area on @O, and s ðQÞ ¼ s ðu; vÞ is the locally integrable surface charge density on @O, obeying the first-kind Fredholm integral equation ð s ðQÞ  ðPÞ dSQ ¼ ’ ðP; Q 2 @OÞ: ð3:6Þ jPQj @O

The lower index in the surface area element dSQ indicates that integration is made with the point Q running over the boundary @O. Generally, as soon as the surface element dSQ is not invariant with respect to the boundary variations, it is convenient to use the invariant measure dm ¼ du dv in the domain D instead. As known from differential geometry, dSQ ¼ JðQÞdm

ð3:7Þ

where JðQÞ ¼j Qu  Qv j¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < Qu ; Qu >< Qv ; Qv >  < Qu ; Qv >2

ð3:8Þ

is area of the parallelogram built on the basis vectors Qu ¼ @Q=@u and Qv ¼ @Q=@v. Before we proceed to varying the integral equation (3.6), one methodological remark aimed at simplifying further calculations is necessary. Basically, we can include the Jacobian JðQÞ in the Green function and ~ introduce the modified Green function GðP; QÞ ¼ JðQÞ= j P  Q j. However, this approach is too complicated compared with the use of the standard Green function GðP; QÞ ¼ 1= j P  Q j in Chapter 1. For the sake of mathematical simplicity, it will be much better if we redefine the charge density distribution s ðQÞ by making the substitution sðQÞ ¼ s ðQÞ

JðQÞ J 0 ðQÞ

ð3:9Þ

with J 0 ðQÞ; JðQÞ respectively, the Jacobian on the unperturbed and perturbed boundary. The Jacobian J 0 ðQÞ is nonzero if the unperturbed boundary parameterization is regular. If regular parameterization cannot be

74

Geometry Perturbations

constructed for the entire boundary, we may cover the boundary @O0 with several coordinate maps and make the substitution (3.9) upon each of the maps separately. It should be emphasized that the renormalized density depends on the choice of the boundary parameterization Qðu; vÞ and, therefore, does not represent any physical surface charge distribution. To obtain the latter, the difference in the surface element areas resulting from the perturbation should be taken into consideration. By using the redefined surface charge density s(Q), we can turn in Eqs. (3.5), (3.6) from the integration over the perturbed boundary @O to the integration over the unperturbed boundary @O0: ð sðQÞ ’ðRÞ ¼ dS0 ðR 2 O; Q 2 @OÞ ð3:10Þ jRQj Q @O0

ð

@O0

sðQÞ  ðPÞ dS0 ¼ ’ jPQj Q

ðP; Q 2 @OÞ:

ð3:11Þ

 ðu; vÞg as a control U If we now consider the pair U ¼ fQðu; vÞ; ’ belonging to the space of controls fUg ¼ ℜ ¼ C10 ðDÞ  L1 ðDÞ, and the surface charge density s(Q) as an element of the phase space fXg ¼ L1 ðDÞ, we are prepared to apply to our case Fedorenko’s variational scheme considered in Appendix 3. This scheme allow us to establish linear correlations between the boundary variations we have introduced and the potential perturbation at any point R in the domain O0. Indeed, the functional F0 ½U of Appendix 3 takes the form ð ~ F0 ½U ¼ F½XðUÞ; U ¼ ’ðRÞ ¼ sðQÞGðR; QÞdS0Q ðR 2 O; Q 2 @OÞ @O0

ð3:12Þ ~ represents the unique solution in which the ‘‘phase coordinate’’ s ¼ XðUÞ of the integral equation ð  ðPÞ ¼ 0 sðQÞGðP; QÞdS0Q  ’ ðP; Q 2 @OÞ: ð3:13Þ R½X; U ¼ @O0

By varying both the functional (3.12) and the equality-type operator constraint (3.13) on the unperturbed boundary @O0 at U ¼ U0 ¼ ~ 0 Þ ¼ s0 ðu; vÞ, we obtain  0 ðu; vÞg, XðU fQ0 ðu; vÞ; ’ ð ð d’ðRÞ ¼ dF0 ½dU ¼ dsðQÞG0 ðR; QÞdS0Q þ s0 ðQÞdQ GðR; QÞdS0Q @O0

@O0

ðR 2 O0 ; Q 2 @O0 Þ

ð3:14Þ

Geometry Perturbations

ð

dR½dX; dU ¼

dsðQÞG0 ðP; QÞdS0Q þ

@O0

ð

75

s0 ðQÞdGðP; QÞdS0Q  d ’ ðPÞ ¼ 0

@O0

ðP; Q 2 @O0 Þ

ð3:15Þ

Recall that the Lagrangian variation dsðQÞ should be calculated at the fixed parameters u, v. Both the partial variation of the kernel dQ GðP; QÞ with respect to the point Q and the full variation of the kernel dGðP; QÞ can be expressed in terms of the linear forms dQ GðR; QÞ ¼< dGðP; QÞ ¼
; dQ >¼  @Q j R  Q j3

ðR 2 O0 ; Q 2 @O0 Þ

@GðP; QÞ @GðP; QÞ < dP  dQ; P  Q > ; dP > þ < ; dQ >¼  @P @Q j P  Q j3 ðP; Q 2 @O0 Þ

ð3:16Þ

which linearly depend on the boundary geometry variations. It can be seen from Eq. (3.14) that the variational derivative FX of Appendix 3 in the case under consideration takes the form FX ¼ G0 ðR; QÞ

ðR 2 O0 ; Q 2 @O0 Þ;

ð3:17Þ

while the linear operators RX ½dX; RU ½dU defined in Appendix 3 appear as ð RX ½dX ¼ dsðQÞG0 ðP; QÞdS0Q ; RU ½dU ¼

0 @O ð

 ðPÞ s0 ðQÞdGðP; QÞdS0Q  ’

ðP; Q 2 @O0 Þ:

ð3:18Þ

0

@O

The conjugate operator RX can be easily calculated according to its definition. Indeed, for any locally integrable function CðPÞ defined on @O0, we have ð ð dsðQÞG0 ðP; QÞdS0Q CðPÞdS0P >¼ @O0

¼

ð

@O0

dsðQÞdS0Q

ð

@O0

@O0

CðPÞG0 ðP; QÞdS0P

ðP; Q 2 @O0 Þ:

ð3:19Þ

76

Geometry Perturbations

This means that the conjugated operator RX is ð  RX ½C ¼ CðPÞGðP; QÞdS0P ðP; Q 2 @O0 Þ @O

ð3:20Þ

0

Reminder: everywhere in this monograph, the double brackets denote the scalar product in functional space. Similarly, >¼> ð ð 0 0 0 CðPÞs ðQÞdGðP; QÞ dSP dSQ  CðPÞd ’ ðPÞ dS0P ¼ @O0 @O0

ð3:21Þ

@O0

ðP; Q 2 @O0 Þ Thus, according to Appendix 3, the variation d’ðRÞ takes the form ð ð 0 d’ðRÞ ¼ CðP; RÞd ’ ðPÞdSP  CðP; RÞs0 ðQÞdGðP; QÞdS0P dS0Q @O0

þ

ð

s0 ðQÞdQ GðR; QÞdS0Q

@O0 @O0

ðR 2 O0 ; P; Q 2 @O0 Þ:

ð3:22Þ

0

@O

The first term in Eq. (3.22) linearly depends on the boundary potential variation d ’ ðPÞ, whereas both the second and third terms are linear forms of the geometry variations dQðu; vÞ. The conjugate function CðP; RÞ obeys the first-kind Fredholm integral equation ð CðP; RÞG0 ðP; QÞdS0P ¼ G0 ðR; QÞ ðR 2 O0 ; P; Q 2 @O0 Þ; ð3:23Þ @O0

which parametrically depends on the point R 2 O0 . This is why we have included the point R in the arguments of the conjugate function C in Eq. (3.22)). Importantly, that the conjugate function CðP; RÞ is completely determined by the geometrical structure of the unperturbed boundary @O0 and does not depend on the boundary variations. Let us consider some properties of the conjugate function. First, it can be easily seen that CðP; RÞ is harmonious in O0 with respect to R due to the harmonicity of the right-hand side of the Eq. (3.23) with respect to R in O0 . If R 2 @O0 , the conjugate equation (3.23) possesses the obvious solution CðP; RÞ ¼ dD ðP; RÞ, which means that CðP; RÞ ! dD ðP; RÞ in the

Geometry Perturbations

77

weak sense when the point R, remaining in O0 , is approaching the unperturbed boundary. Therefore the function CðP; RÞ represents the generalized solution of the boundary-value problem DR CðP; RÞ ¼ 0 Cj@O0 ¼ dD ðP; RÞ

ðR 2 O0 ; P 2 @O0 Þ ðP; R 2 @O0 Þ;

ð3:24Þ

which parametrically depends on the point P 2 @O0 . It means that CðP; RÞ as a function of R 2 O0 can be interpreted as a Green function generated by the delta-shaped boundary potential distribution concentrated at the point P 2 @O0 : Now consider the case that the nominal boundary @O0 is not varied (dQðu; vÞ ¼ 0). Inasmuch as the potential ’ðRÞ linearly depends on the  ðPÞ, the variation symbols in Eq. (3.22) boundary potential distribution ’ can be omitted and we arrive at the integral correlation ð CðP; RÞ ’ ðPÞdS0P ðR 2 O0 ; P 2 @O0 Þ; ð3:25Þ ’ðRÞ ¼ @O0

which allows us to determine the potential distribution inside the domain O0 , respondent to any boundary potential distribution on @O0 , by means of simple integration. The Eq. (3.25) not only reflects the superposition principle, which is well known in electrostatics, but also clarifies the physical meaning of the conjugate function as the limiting case of the so-called unit potential distribution. Indeed, let g be a measurable set on the boundary @O0 containing the point P 2 @O0 , with mðgÞ be its area. By definition, the unit potential function associated with the set g is the potential distribution ð ðR 2 O0 ; P 2 g  @O0 Þ ð3:26Þ ’ðgÞ ðRÞ ¼ CðP; RÞdS0P g

induced in the domain O0 by the set g. Being harmonious in O0 , this function obeys the ‘‘unit’’ boundary condition  1; P 2 g ðgÞ ðgÞ ð3:27Þ ’ ðPÞj@O0 ¼ w ðPÞ ¼ 0; P 2 @O0 ng Uniform subtending of the set g to the point P 2 g gives the limiting correlation @’ðgÞ ðRÞ g)P @mðgÞ

CðP; RÞ ¼ lim

ðR 2 O0 ; P 2 @O0 Þ;

ð3:28Þ

78

Geometry Perturbations

j− = 1

g

Ω0

⭸Ω0 \g

j− = 0

FIGURE 45

The unit function ’ðgÞ ðPÞ in a circular domain O0.

which may be treated as another, independent and more physical, definition of the conjugate Green function CðP; RÞ. According to Eq. (3.28), CðP; RÞ represents the density of the unit potential function ’ðgÞ ðRÞ with respect to the measure mðgÞ. Figure 45 shows an example of the unit potential distribution (3.26) in a circular domain. It is particularly remarkable that, according to Eq. (3.22), the knowledge of the function CðP; RÞ allows representation of potential perturbation in the form of a linear functional for any boundary variation, including the geometric one. This function can be numerically obtained as a solution of a set of the boundary-value Dirichlet problems related to the unperturbed boundary @O0. Undoubtedly, knowledge of the conjugate function CðP; RÞ is of profound importance in optimization and synthesis problems. As to the problems of tolerance computation, if such information seems to be superfluous, it is possible to restrict the calculation to numerical solution of the variational equation (3.15) with respect to the charge density variation dsQ respondent to a particular boundary variation dU ¼ fdQðu; vÞ; d ’ ðu; vÞg and then calculating the corresponding potential perturbation according to Eq. (3.14). The next two sections consider this possibility in more detail for the cases of axial and planar symmetry.

3.2. 3D PERTURBATIONS IN AXISYMMETRIC SYSTEMS Let assume that the domain O0 possesses axial symmetry with respect to the z-axis (Figure 46) and its boundary @O0 allows the piecewise smooth parametric representation

79

Geometry Perturbations

y Γ0

0

⭸Ω

Q(s,q) q

ex

er

z

q

x

ez ey

FIGURE 46

Boundary variations in the axisymmetric case.

Qðy; sÞ ¼ z0 ðsÞez þ r0 ðsÞðex cos y þ ey sinyÞ:

ð3:29Þ

Hereafter in this section, the curvilinear coordinates fu; vg on @O0 are the azimuth Sangle y 2 ½0; 2p and the arc-length s 2 ½0; smax  defined on the sum G0 ¼ ni¼1 G0i of the generatrixes G0i that correspond to the smooth connected axisymmetric components @O0i ; i ¼ 1. . .n comprising the boundary @O0 ¼ G0  ½0; 2p (Figure 46).  0 ðsÞ define the unperturbed boundary potential Let the function ’ 0 0 ! ’  as the on G . Consider the boundary variation z0 ! z, r0 ! r, ’ mapping zðs; yÞ ¼ z0 ðsÞ þ dzðs; yÞ rðs; yÞ ¼ r0 ðsÞ þ drðs; yÞ

ð0  s  smax ; 0  y  2pÞ

 ðs; yÞ ¼ ’  0 ðsÞ þ d ’ ðs; yÞ ’

ð3:30Þ

defined on the unperturbed boundary. Using Eq. (3.8), it is easy to verify that the area elements on the unperturbed and perturbed boundaries are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u " 2  2 #   u dr @z dr dz dr dz 2 0 0 t 2 þ  þ dS ¼ r dsdy; dS ¼ r ds dy; ds @s ds dy dy ds

and, accordingly,

dJ dr dr0 ddr dz0 ddz þ : ¼ þ J 0 r0 ds ds ds ds

ð3:31Þ

The variational functions dzðs; yÞ, drðs; yÞ, d ’ ðs; yÞ are assumed to be small on @O0 , together with their piecewise continuous derivatives. Inasmuch these variational functions are periodic with respect to y, they can be expanded into the Fourier series

80

Geometry Perturbations

dzðs; yÞ ¼ zð0Þ ðsÞd q þ

N  X zn;1 ðsÞcosðnyÞ þ zn;2 ðsÞsinðnyÞ d q n¼1

N  X drðs; yÞ ¼ r0 ðsÞd q þ rn;1 ðsÞcosðnyÞ þ rn;2 ðsÞsinðnyÞ d q n¼1 N  X

d ’ ðs; yÞ ¼ c0 ðsÞd q þ

n¼1

cn;1 ðsÞcosðn yÞ þ cn; 2 ðsÞ sinðnyÞ d q:

ð3:32Þ

The coefficients of these expansions are restricted functions of their arguments, and dq is variation of the structural parameter q. The zero harmonics z0 , r0 , and c0 correspond to the boundary variations preserving axial symmetry. Those harmonics are commonly used in optimization problems. The other coefficients describe a wide class of geometric and potential boundary variations that slightly perturb axial symmetry of the nominal problem. Let us consider some practically important 3D geometric variations being in use in mechanical tolerances computations, namely shift, axis misalignment, and ellipticity (Figure 47).

(a)

(b)

x

x

dq w

dq z*

z

z y y

(c) x

y

A(s) z y

w(s) x

B(s)

FIGURE 47 The definition of the boundary variations slightly perturbing axial symmetry. (a) shift; (b) axis misalignment; (c) ellipticity.

81

Geometry Perturbations

Shift. This deformation represents a shift of some connected boundary component along the given unit vector t ¼ ðcosa; cosb; cosgÞ by a small value dq. The nonzero coefficients in Eq. (3.32) appear as r1;1 ¼ cos a; r1;2 ¼ cos b; z0 ¼ cos g

ð3:33Þ

Axis misalignment. This deformation represents a turn of the symmetry axis Oz by a small angle dq around the point ð0; 0; z Þ in the plane making the angle O with the axis Ox. The nonzero coefficients in Eq. (3.32) are r1;1 ¼ ðz0  z Þ cos 2o; z1;1 ¼ r0 cos o;

r1;2 ¼ ðz0  z Þ sin 2o

z1;2 ¼ r0 sin o:

ð3:34Þ

Ellipticity. This deformation transforms a circle with radius r0 ðsÞ in the cross-section perpendicular to the axis Oz into the ellipse with the semiaxes AðsÞ ¼ r0 ðsÞ½1 þ xðsÞdq, BðsÞ ¼ r0 ðsÞ½1  xðsÞdq. The function xðsÞ is an arbitrary smooth function of the parameter s. The ellipticity tilt angle o may also smoothly depend on s. The nonzero coefficients in Eq. (3.32) appear as r2;1 ¼ r0 ðsÞxðsÞ cos 2o; r2;2 ¼ r0 ðsÞxðsÞ sin 2o:

ð3:35Þ

Our nearest goal now is to concretize the variational integral equation (3.15) as applied to the boundary variations (3.32). In the cylindrical coordinates, the Green function takes the form GðP; QÞ ¼

1 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

 jPQj z z þ r2 þ r2  2r r cos y  y P

Q

P

Q

P Q

P

Q

ð3:36Þ

and may be expanded into the Fourier series GðP; QÞ ¼

N g0 ðzP ; rP ; zQ ; rQ Þ 1 X gn ðzP ; rP ; zQ ; rQ Þ cos nðyP  yQ Þ ð3:37Þ þ 2p p n¼1

with the coefficients expressed through the elliptic integrals 4 g ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðzP  zQ Þ2 þ ðrP  rQ Þ2 n

kPQ ¼

p=2 ð 0

cosð2noÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi do 1  kPQ cos2 o

4rP rQ ðzP  zQ Þ2 þ ðrP  rQ Þ2

;

n ¼ 0; 1; :::

ð3:38Þ

ð3:39Þ

82

Geometry Perturbations

We now make one more agreement on the S notations used. Let f ðQÞ; gðQÞ be arbitrary functions defined in O @O. We will denote f ðsQ ; yQ Þ the contraction f ðQÞjQ2@O ¼ f ðQðs; yÞÞ of the function f ðQÞ onto the perturbed boundary @O expressed in terms of the Lagrangian coordinates s; y. Similarly, we will denote gðsQ Þ the contraction gðQÞjQ2G0 ¼ gðQðsÞÞ of the function gðQÞ onto the generatrix G0 parameterized by the arc length s. At times, when it does not result in a misunderstanding, we will omit the index Q in f ðsQ ; yQ Þ, gðsQ Þ. The same agreement will be applied to the functions dependent on two different points. For example, we will denote hðsP ; sQ Þ the contraction hðP; QÞjP;Q2G0 ¼ hðPðsÞ; QðsÞÞ of the function hðP; QÞ onto the generatrix G0 parameterized by the arc length s. Finally, the notation hðP; sQ Þ means that the ‘‘one-point’’ contraction hðP; QÞjQ2G0 ¼ hðP; QðsÞÞ is considered. In accordance with Eq. (3.37), we construct the perturbed surface charge density sðs; yÞ as a sum of cylindrical harmonics sðs; yÞ ¼ s0 ðsÞ þ dsðs; yÞ; dsðs; yÞ ¼ C0 ðsÞdq þ

N  X n¼1

Cn;1 ðsÞ cosðnyÞ þ Cn;2 ðsÞ sinðnyÞ d q;

ð3:40Þ

in which the first term s0 is the unperturbed surface charge density satisfying the first-kind Fredholm integral equation ð  0 ðsP Þ ðP; QÞ 2 G0 ; s0 ðsQ Þg0 ðsP ; sQ Þr0 ðsQ ÞdsQ ¼ ’ ð3:41Þ G0

which follows from Eq. (3.11) if we take the element area in the form dS0Q ¼ r0 ðsQ ÞdyQ ds and carry out the integration over the azimuth angle yQ . According to our agreement, we have denoted g0 ðsP ; sQ ) the contraction g0 ðzP ; rP ; zp ; rQ ÞjG0 of the zero harmonic g0 of the Green function GðP; QÞ onto the generatrix G0 . The equation for s0 was already obtained in Chapter 1 with the Green function G ¼ g0 r0 : We now may construct the Fourier expansion of the full Lagrangian variation dGðP; QÞ immediately from Eq. (3.37) by differentiating the kernel GðP; QÞ with respect to zP ; rP ; zQ ; rQ on the unperturbed boundary @O0: 2 3 1 4@g0 @g0 @g0 @g0 d zP þ d zQ þ d rP þ d rQ 5 dGðP; QÞ ¼ @zQ @rP @rQ 2p @zP 2 3 N n n n n 1X @g @g @g @g 4 þ d zP þ d rP þ d zQ þ d rQ 5cos nðyP  yQ Þ: ð3:42Þ @rP @zQ @rQ p n¼1 @zP

Geometry Perturbations

83

Substituting Eqs. (3.37), (3.40), and (3.42) into the variational equation (3.15), integrating over yQ , and, finally, equating to zero each of the yP -harmonics separately, we obtain the set of integral equations ð Cn;k ðsQ Þgn ðsP ; sQ Þr0 ðsQ ÞdsQ G0

n;k

¼ c ðsP Þ 

ð

G0

h i n;k s0 ðsQ Þ Gn;k ðs ; s Þ þ G ðs ; s Þ r0 ðsQ ÞdsQ P Q P Q P Q

ð3:43Þ

ðP; QÞ 2 G0 : Here Gn;k Q ðsP ; sQ Þ ¼

@gn @gn ðsP ; sQ Þzn;k ðsQ Þ þ ðsP ; sQ Þrn;k ðsQ Þ @zQ @rQ

ð3:44Þ

Gn;k P ðsP ; sQ Þ ¼

@g0 @g0 ðsP ; sQ Þzn;k ðsQ Þ þ ðsP ; sQ Þrn;k ðsQ Þ: @zP @rP

ð3:45Þ

In Eqs. (3.43)–(3.45) n ¼ 0. . .N; k ¼ 1; 2, and index k should be omitted for n ¼ 0. The structure of Eqs. (3.44) and (3.45) is very simple: their righthand sides are partial derivatives of the Green function harmonics multiplied by appropriate harmonics of the boundary variations. Here we do not discuss the aspects of the numerical solution of the integral equations (3.43), referring readers to the original paper by Monastyrskiy and Kolesnikov (1983) for necessary details. We only note that the kernels of the integrals in the right-hand side of Eq. (3.43) have similar integrable logarithmic singularities in the coincidence limit Q ! P. This implies that the right-hand side of Eq. (3.43) remain finite for any piecewise smooth boundary variation, and, therefore, these equations actually represent the first-kind Fredholm integral equations being wellconditioned in the same class of functions as the unperturbed first-kind Fredholm integral equation (3.41). Numerical solution of Eq. (3.43) gives the surface charge perturbation harmonics C0 and Cn;k ðn ¼ 1. . .N; k ¼ 1; 2Þ. According to Eq. (3.14), the potential perturbation at any point Rðz; r; yÞ 2 O in the first-order approximation with respect to the small parameter dq appears as the Fourier expansion d’ðRÞ ¼ D0 ðz; rÞd q þ

N  X Dn;1 ðz; rÞ cos ny þ Dn;2 ðz; rÞ sin ny d q; ð3:46Þ n¼1

84

Geometry Perturbations

the coefficients of which are determined by the relations ðj k 0 Dn;k ðz; rÞ ¼ Cn;k ðsQ Þgn ðz; r; sQ Þ þ s0 ðsQ ÞGn;k Q ðz; r; sQ Þ r ðsQ ÞdsQ G0

Gn;k Q ðz; r; sQ Þ ¼

@gn @gn ðz; r; sQ Þzn;k ðsQ Þ þ ðz; r; sQ Þrn;k ðsQ Þ: @zQ @rQ

ð3:47Þ

ðR 2 O; Q 2 G0 Þ: Similar to Eqs. (3.43)-(3.45), here n ¼ 0. . .N; k ¼ 1; 2, and the index k should be omitted for n ¼ 0. Generalization to the case of several structural parameters fqk g is trivial in the framework of the first-order perturbation theory. The total potential perturbation is, obviously, a sum of the potential perturbation resulting from each separate geometric or potential boundary variation. As to the substitution (3.9) we have made to simplify the calculations, we have to recall that the charge density distribution given by Eq. (3.40) does not coincide with the physical charge density distribution s since the perturbation (3.30) does not preserve the elementary area on the perturbed boundary. Using Eqs. (3.9) and (3.31), we can easily recalculate the perturbation ds into the perturbation ds :   dr0 ddr @z0 ddz 0 dr  0 dJ ds ¼ ds  s 0 ¼ ds  s þ : ð3:48Þ þ ds ds @s ds J r0

3.3. EXAMPLES OF 3D PERTURBATIONS IN AXISYMMETRIC SYSTEMS This section considers some model problems to compare the results obtained by means of the integral variational technique with the known exact solutions, and we discuss one practical example. The properly selected model problems, despite their apparent simplicity, allow verification of the validity of any theoretical or numerical approach and represent good tests for adjusting the relevant software and choosing optimal values for the numerical parameters involved. By way of the unperturbed boundary, we consider here a sole infinitely thin conductive disk with the radius Rd and the symmetry axis Oz , as shown  0 on the disk on the left side of Figure 48. If the boundary potential ’ surface is unit, the electric potential the disk generates in space is known to be [see Eq. (1.35) in Chapter 1] ’0 ðz; rÞ ¼

2 2Rd arcsin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ðr þ Rd Þ2 þ z2 þ ðr  Rd Þ2 þ z2

ð3:49Þ

85

Geometry Perturbations

(b) (a)

x

z Rd

2 2

y

1 0x

1 z

0

–1

–1

FIGURE 48 (a) the sole conductive disk; (b) the unperturbed potential distribution generated from the disk of unit radius ðRd ¼ 1Þ.

This distribution is displayed on the right-hand side of Figure 48 as a function of the coordinates x; z. It is convenient to choose parameterization of the unperturbed generatrix G0 in the form r0 ðsÞ ¼ s; z0 ðsÞ ¼ 0. The corresponding unperturbed surface charge density appears as 0 1 @’ 1 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð3:50Þ s ðsÞ ¼  2p @z z ! þ0 p2 R2  s2 d r ¼ Rd

It is easy to show that Eq. (3.50) satisfies the unperturbed integral equation (3.41). For simplicity, let us put the unperturbed disk radius R0d equal to unit. Accordingly, the parameter s will run within the segment ½0; 1.

3.3.1. Varying the Disk Radius In this example we consider the disk radius perturbation Rd ¼ 1 þ dq as a particular case of the general rescaling geometric perturbation, which can be described by the coefficients z0 ¼ z 0  z  ;

r0 ¼ r0 ;

ð3:51Þ

with all other coefficients in Eq. (3.32) equal to zero. In our particular case, the rescaling center is located on the main optical axis at z ¼ 0. The integral equation (3.43) takes the form ð1 0

ð1

C ðsQ Þg ðsP ; sQ ÞR ðsQ ÞdsQ ¼  s0 ðsQ ÞFscale ðsP ; sQ Þr0 ðsQ ÞdsQ : 0

0

0

0

ð3:52Þ

86

Geometry Perturbations

Here Fscale denotes the sum G0Q þ G0P in the right part of integral equation (3.43) at n ¼ 0. With regard to the properties of elliptic integrals, it can be easily shown that Fscale ðsP ; sQ Þ ¼ G0Q ðsP ; sQ Þ þ G0P ðsP ; sQ Þ ¼ þ

@g0 ðsP ; sQ Þr0 ðsQ Þ @rQ

@g0 @g0 ðsP ; sQ Þr0 ðsP Þ þ ðsP ; sQ Þ½z0 ðsP Þ  z0 ðsQ Þ ¼ g0 ðsP ; sQ Þ: @rP @zP

ð3:53Þ

Eq. (3.43) appears as ð1

½C0 ðsQ Þ  s0 ðsQ Þg0 ðsP ; sQ Þr0 ðsQ ÞdsQ ¼ 0;

ð3:54Þ

0

which, in turn, immediately implies that C0 ðsÞ ¼ s0 ðsÞ and dsðsÞ ¼ C0 ðsÞdq þ oðdqÞ ¼ s0 ðsÞdq þ oðdqÞ:

ð3:55Þ

This value is not the ‘‘genuine’’ surface electric charge perturbation yet because the variation of the disk radius does not preserve the surface element area. In accordance with Eq. (3.48)  0  r dr0 0  0 s dq ¼ s0 dq ð3:56Þ ds ¼ s dq  0 þ r ds Of note, this result is in full accordance with the fact that the nearsurface electric field is inversely proportional to the disk radius, provided that the boundary potential does not change, and remains true for the rescaling perturbation of any axisymmetric systems. Spatial distribution of the potential perturbation at a point R ¼ ðz; r; yÞ can be found from Eq. (3.46) and (3.47), which in our particular case take the form d’ðz; rÞ ¼ D0 ðz; rÞdq 2 3 ð1 0 @g D0 ðz; rÞ ¼ 4g0 ðz; r; sQ Þ þ ðz; r; sQ Þr0 ðsQ Þs0 ðsQ Þ5r0 ðsQ ÞdsQ : @rQ 0

ð3:57Þ

Obviously, the potential perturbation d’ðRÞ is fully determined by its zero harmonic and therefore does not depend on y. In the cross-sections z ¼ 0 and r ¼ 0, the exact calculation of the integral in Eq. (3.57) yields 8 2 > < pffiffiffiffiffiffiffiffiffiffiffiffi ; r>1 2 jzj 0 ð3:58Þ : D ð0; rÞ ¼ p r2  1 ; D0 ðz; 0Þ ¼ 2 > pz þ 1 : 0 ; r1

Geometry Perturbations

87

dq

z

x

FIGURE 49

Potential perturbation function for the boundary variation of disk rescaling.

The potential perturbation function (3.58) is shown in Figure 49 versus the two Cartesian coordinates x,z. The Lagrangian variations ds in Eq. (3.56) and d’ðz; rÞ in Eqs. (3.57) and (3.58) identically coincide with that obtained from the exact solutions (3.50), (3.49).

3.3.2. The Off-Axis Shift Let us now consider a small shift of the disk along the Ox axis by the distance dq. For this particular boundary variation, we have r1;1 ¼ 1 and all other coefficients in Eq. (3.32) equal zero. The general variational equation (3.43) at n ¼ 1; k ¼ 1 appears as ð1 0

ð1 C1;1 ðsQ Þg1 ðsP ; sQ Þr0 ðsQ ÞdsQ ¼  s0 ðsQ ÞFshift ðsP ; sQ Þr0 ðsQ ÞdsQ :

ð3:59Þ

0

The calculation of the function Fshift with regard to the properties of elliptic integrals gives Fshift ðsP ; sQ Þ ¼

@g1 @g0 g1 ðsP ; sQ Þ ; ðsP ; sQ Þ þ ðsP ; sQ Þ ¼  0 @rQ @rP r ðsQ Þ

ð3:60Þ

and the equation (3.59) appears as ð1 0

½C1;1 ðsQ Þr0 ðsQ Þ  s0 ðsQ Þg1 ðsP ; sQ ÞdsQ ¼ 0:

ð3:61Þ

88

Geometry Perturbations

This means that C1;1 ðsÞ ¼

s0 ðsÞ 1 pffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ 0 2 r ðsÞ p s 1  s2

dsðsÞ ¼ C1;1 ðsÞ cos y dq:

ð3:62Þ

The singularity 1=s in C1;1 ðsÞ appears whenever a boundary point belonging to the symmetry axis (r0 ¼ 0) is shifted off the axis. From the mathematical viewpoint, this singularity does not affect the convergence of the integrals in Eq. (3.43) and (3.47) thanks to the weight r0 ðsÞ in the integrand. As to the variation of the physical surface charge density, from Eq. (3.48) we obtain ds ¼

s0 r1;1 cos y dq  0 s0 cos y d q ¼ 0; 0 r r

ð3:63Þ

which is in full agreement with the absolutely predictable result that this type of variation does not affect the physical surface charge distribution on the disk. The potential variation takes the form d’ðz; rÞ ¼ D1;1 ðz; rÞcosy dq 2 3 ð1 1 @g D1;1 ðz; rÞ ¼ 4r0 ðsQ Þ ðz; r; sQ Þ þ g1 ðz; r; sQ Þ5s0 ðsQ ÞdsQ : @rQ

ð3:64Þ

0

The potential perturbation function (3.64) is shown in Figure 50 as function of two coordinates x; z.

dq

z x

FIGURE 50

Potential perturbation function for the case of the disk shift.

Geometry Perturbations

In the section z ¼ 0, we obtain 8 2 > < pffiffiffiffiffiffiffiffiffiffiffiffi ; r>1 1;1 D ð0; rÞ ¼ pr r2  1 : > : 0 ; r1

89

ð3:65Þ

Readers can easily discern that just the same relation follows from the exact solution given by Eq. (3.49).

3.3.3. The Disk Tilt Our goal now is to apply the axis misalignment perturbation to the unperturbed disk (see the left upper side in Figure 51). Putting r1;1 ðsÞ ¼ z0 ðsÞ ¼ 0;

z1;1 ðsÞ ¼ r0 ðsÞ ¼ s

ð3:66Þ

gives the particular case of Eq. (3.34). The corresponding integral variational equation for the surface charge density perturbation reads ð1

1;1

1

ð1

0

C ðsQ Þg ðsP ; sQ Þr ðsQ ÞdsQ ¼  s0 ðsQ ÞFtilt ðsP ; sQ Þr0 ðsQ ÞdsQ :

0

ð3:67Þ

0

A simple calculation shows that

 @g1 @g0 Ftilt ðsP ; sQ Þ ¼  r0 ðsQ Þ ðsP ; sQ Þ þ r0 ðsP Þ ðsP ; sQ Þ ¼ 0 @zQ @zP

ð3:68Þ

and, therefore, the coefficient C1;1 is zero. As soon as the tilt perturbation preserves the surface element area, the physical charge does not change x dq z z x

FIGURE 51 The field perturbation function in the case of disk tilt.

90

Geometry Perturbations

as well. The coefficient D1;1 of the field perturbation is given by Eq. (3.47), in which we have to put C1;1 ¼ 0. Thus, we obtain ð1 @g1 ðsP ; sQ Þðr0 ðsQ ÞÞ2 dsQ : D1;1 ðz; rÞ ¼  s0 ðsQ Þ @zQ

ð3:69Þ

0

The numerically calculated potential perturbation function (3.69) displayed in Figure 51 strictly coincides with that obtained from the exact solution (3.49).

3.3.4. Boundary Variation in the Cathode Lens ‘‘Spherical Capacitor’’ The cathode lens in question (Figure 52) comprises two concentric spheres with the radii Rc (the outer sphere) and Rg (the inner sphere). The outer sphere is grounded, whereas the inner one is supplied with the potential U. The left side of the outer sphere is considered as the cathode, the left side of the inner sphere as the anode transparent for electrons, and, finally, the right side of the outer sphere as the image receiver (screen). Physically, we may assume that the anode represents a spherically curved finestructure grid. We also assume that the potential distribution in the crosssection shown by the vertical dashed line exactly corresponds to that in the equivalent spherical capacitor. The space between the anode and screen is electrically isolated from the left part of the lens and has the grid potential U. This model electron-optical system creates a real image and in a definite sense represents an ideal electrostatic focusing cathode lens. It can be easily r

3 2 Rc z Rg

1

FIGURE 52 The cathode lens ‘‘spherical capacitor’’ (1, cathode; 2, anode (fine-structure grid); 3, image receiver).

91

Geometry Perturbations

shown that if Rc ¼ 3Rg , the electron-optical magnification of the lens is unit, and the surface of sharp focusing strictly coincides with the spherically curved surface of the screen. Due to spherical symmetry, the cathode lens in question is completely free of all off-axis geometrical aberrations. Another important peculiarity of the lens is that its crossover is located in the free-of-field region downstream from the fine-structure grid. The cathode lens ‘‘spherical capacitor’’ is well studied and actively used for software testing (for details see Ignat’ev and Kulikov, 1983; and Kolesnikov and Monastyrskiy, 1988). Here our sole interest is the potential perturbation caused by small variation of the cathode radius Rc. Our next goal is to compare the numerical results derived from the integral variational equation with those obtained from the exact axial potential distribution, which in the region between the cathode and grid (0  z  Rc  Rg ) appears as FðzÞ ¼

U Rg z : Rc  Rg Rc  z

ð3:70Þ

Thus, in the case under consideration, the structural parameter q responsible for geometric variation of the lens’ boundary is the cathode radius Rc. The Table IV shows the relative error d of the first-order variational derivative dFðzÞ=dq calculated by means of numerical solution of the variational integral equation with respect to the exact analytical solution (3.70). The data in Table IV correspond to the nominal values Rc ¼ 180 mm, Rg ¼ 60 mm, and U ¼ 1000 V. In the region of interest the relative error d does not exceed 0.1%.

3.3.5. Boundary Variation in Photoelectron Gun This example relates to the streak image tube and photoelectron gun simulation. Very briefly, the streak tube operation principle is as follows. The bunch of photoelectrons emitted from the photocathode under TABLE IV Comparison Between the Numerically Calculated and Exact Potential Perturbation Functions in the Cathode Lens ‘‘Spherical Capacitor’’

z,mm

35.2 59.4 83.4 108.0

dFðzÞ=dq, V mm1 (calculated numerically by means of the integral variation approach)

dFðzÞ=dq, V mm1 [calculated from the exact analytical solution (3.70)]

d

1.84871 4.09380 8.11533 16.6011

1.84884 4.09354 8.11973 16.6025

8  103 6  103 5  102 8  103

92

Geometry Perturbations

ultrashort (a few tens of femtoseconds) laser pulse excitation is accelerated up to 3-10 keV within the photocathode-grid gap of 1 mm width, and, having passed through the focusing electric field and additionally accelerated by the anode up to 15-30 keV, enters the free-of-field region, which contains a dynamic deflection system to sweep the electron bunch along the screen and thus to measure its temporal profile. Streak image tubes are commonly used in electron-optical photography as the most efficient tools to study the ultrashort physical events in infrared, visible, and X-ray wavelength ranges. Those tubes sometimes are also used as photoelectron guns or, in other words, ultrashort electron bunch generators, in the so-called time-resolved electron diffraction (TRED) experiments (see Chapters 7 and 8 for more details). The example below represents the calculation of the potential perturbation caused by the small tilt of a fine-structure grid positioned near the photocathode in the photoelectron gun shown in Figure 53. Numerical evaluation of the photocathode-grid misalignment tolerance is very important from the viewpoint of streak image tube manufacturing. Obviously, the simulation of potential perturbations represents the main and probably the most difficult part of this problem. The axis misalignment variation, with the tilt angle q playing the role of the small structural parameter to be varied (Figure 54a), has been used to numerically solve the variational integral equation (3.67). The numerically calculated axial distributions of the first harmonic of the perturbed potential and its sequential z-derivatives in the optically active area downstream from the grid are shown in Figure 54b. Linear potential distribution

Focusing electrodes Grid Uc = 0

Ug

Uf1

Uf2

Photocatode

Dynamic deflection system

Ua

Anode

0

100

Diaphragm

200

300

Screen

400

mm

FIGURE 53

Electron-optical structure of the femtosecond electron gun.

500

Geometry Perturbations

(a)

93

(b)

z

Relative units

δq

1.0 0.8 0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −0.8 −1.0

3

2

1 0

4

8

12

16

20

mm

FIGURE 54 (a) the grid tilt as boundary variation (the tilt angle dq is purposely enlarged for clearity); (b) axial distribution of the first harmonic of the perturbed potential (curve 1), and the sequential derivatives of the first harmonic (curves 2 and 3).

3.4. 3D PERTURBATIONS IN PLANAR SYSTEMS The planar electrostatic systems, with their geometries invariant with respect to translations along a given axis, are widely used in a variety of analytical electron/ion-optical devices. Examples include the quadrupole ion traps that serve as ion storage cells in mass analyzers (Makarov, 1996), the planar lenses used to form the band-shaped particle beams, the mirrors used in multireflection time-of-flight mass spectrometers (Yavor and Verenchikov, 2002; Wollnik et al., 2004), and the multipole radiofrequency ion guides intended for transporting the cooled ions (Tolmachev, Udseth, and Smith, 2000). The planar 2D fields generated by those systems are comparatively easy to calculate with high accuracy. However, such simplicity immediately vanishes as soon as the geometry is slightly perturbed. We consider here two types of such perturbations (Figure 55). The first type comprises the electrode deformations (e.g. bending, torsion, tilting) due to manufacturing imperfections or mechanical and thermal tensions. The second type includes perturbations resulting from the fringe effects that are principally unavoidable in any practical design of restricted dimensions. As soon as the fringe effects make a part of the system’s length unusable for practical applications, it is very important to evaluate the contribution of those effects correctly. Both types of perturbations lead to the field calculation problem beyond the 2D statement. As shown in this section, smooth 3D deformations of planar electrodes can be treated as a particular case of the general variation theory outlined in the section 3.1, quite similar to the case of 3D perturbations in axisymmetric systems previously considered. The case of fringe fields requires a somewhat different approach (see Section 3.6).

94

Geometry Perturbations

Fringe

dr

x

z

Small perturbation y w0 Γ0

FIGURE 55

Typical boundary variations in planar system.

Let us assume that the arc length parameter s on the unperturbed boundary G0 ¼ @o0 of the cross-section o0 runs in the interval ½0; smax , and the z-coordinate that characterizes the ‘‘extent’’ of the unperturbed geometry in the translation direction runs in the large enough interval L=2  z  L=2. In accordance with Section 3.1, we can represent the perturbed boundary geometry and potential in the parametrical form  0  x ðsÞ þ dxðs; zÞ ; rðs; zÞ ¼ r0 ðsÞ þ drðs; zÞ ¼ y0 ðsÞ þ dyðs; zÞ ð3:71Þ  ðs; zÞ ¼ ’  0 ðsÞ þ d ’ ’ ðs; zÞ:

Here, the z-independent functions with upper zero indexes describe the unperturbed (nominal) boundary geometry and potential distribution, and drðs; zÞ, d ’ ðs; zÞ are geometric and potential boundary variations. Let zðzÞ; VðrÞ; dq be, accordingly, a smooth function of z-coordinate, a smooth 2D vector-function of the two-dimensional vectorial argument r ¼ ðx; yÞ, and a small variation of the structural parameter q. Some important particular cases may be distinguished if we put drðs; zÞ ¼ zðzÞV½r0 ðsÞdq within a connected part of G0 (see Figure 56 and Table V). In general, we can define the geometric variations on the unperturbed boundary as the sum of harmonics " # N X n n n;1 n;2 0 r ðsÞ cosðx zÞ þ r ðsÞ sinðx zÞ dq drðs; zÞ ¼ r ðsÞdq þ n¼1 " # N X n;1 n n;2 n 0 d ’ ðs; zÞ ¼ c ðsÞdq þ c ðsÞ cosðx zÞ þ c ðsÞ sinðx zÞ dq n¼1

0  s  smax ; L  z  L;

ð3:72Þ

Geometry Perturbations

95

(a)

(b)

(c)

(d)

FIGURE 56 Some particular cases of the geometric variations. (a) shift; (b) tilt; (c) bending; (d) torsion.

TABLE V Analytical Description of Some Geometric Variations with Structural Parameters o0 , x , y . z¼1

O¼



cos o0 sin o0

O¼



O¼ O¼

z ¼ z=L

z ¼ 1z2 =L2

Shift Figure 56a

Tilt Figure 56b

Bending Figure 56c

y0 y x x0

Rotation

Torsion 1 Figure 56d

Torsion 2



x0 x y0 y

Rescaling

Conical deformation

Barrel deformation



x0 x y y0

Ellipticity

z-dependent ellipticity

z-dependent ellipticity

with fxn g denoting a set of different positive wave numbers. The twocomponent vector-functions rn;k and the scalar functions cn;k are assumed to be sufficiently smooth. In the first-order approximation, any individual harmonic generates a periodic potential distribution with the same wave number, and the contributions of those harmonics are additive. Thus, using the same set of wave numbers, the surface charge density sðs; zÞ on the perturbed boundary can be represented as

96

Geometry Perturbations

sðs; zÞ ¼ s0 ðsÞ þ dsðs; zÞ " # N X n n n;1 n;2 dsðs; zÞ ¼ C0 ðsÞdq þ C ðsÞ cosðx zÞ þ C ðsÞ sinðx zÞ dq: n¼1

ð3:73Þ

As seen in Chapter 1, the unperturbed charge density s0 obeys the first-kind Fredholm integral equation ð  0 ðsP Þ s0 ðsQ ÞG0 ðsP ; sQ ÞdsQ ¼ ’ ð3:74Þ G0

with the planar Green function G0 ðrP ; rQ Þ ¼ 2 ln

j rP  rQ j : D0

ð3:75Þ

Recall that G0 ðsP ; sQ Þ designates the contraction G0 ðrP ; rQ Þ jG0 of the Green function (3.75) onto the generatrix G0 of the nominal planar boundary @O0. As noted in Chapter 1, the value of the numerical parameter D0 should guarantee good conditionality of the integral equation (3.74). It is also noteworthy that the boundary variation (3.72) preserves the surface element area in the first-order approximation with respect to the small parameter dq; therefore, within the first-order perturbation theory, the surface charge densities s and s coincide. In the case under consideration, the full variation (3.16) of the Green function appears as 1 dGðP; QÞ ¼ d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrP  rQ Þ2 þ ðzP  zQ Þ2 ¼

< rP  rQ ; drP  drQ >

½ðrP  rQ Þ2 þ ðzP  zQ Þ2 3=2

:

ð3:76Þ

Substituting Eqs. (3.72), (3.73), and (3.75) into Eq. (3.15) and integrating over zQ produces a number of separate integral equations due to the orthogonality of the harmonics with different wave numbers. The equations for the zero harmonic (n ¼ 0) and the nonzero harmonics (n ¼ 1. . .N) are different: ð C0 ðsQ ÞG0 ðsP ; sQ ÞdsQ ¼ c0 ðsP Þ G0

ð < r0 ðsP Þ  r0 ðsQ Þ; r0 ðsP Þ  r0 ðsQ Þ > þ2 s0 ðsQ Þ dsQ ½r0 ðsP Þ  r0 ðsQ Þ2 G0

ð3:77Þ

97

Geometry Perturbations

ð   2 Cn;k ðsQ ÞK0 xn j r0 ðsP Þ  r0 ðsQ Þ j dsQ ¼ cn;k ðsP Þ þ G0

ð

2 s0 ðsQ Þ G

  < r0 ðsP Þ  r0 ðsQ Þ; rn;k ðsP Þ  S xn j r0 ðsP Þ  r0 ðsQ Þ j rn;k ðsQ Þ > ½r0 ðsP Þ  r0 ðsQ Þ2

0

dsQ :

ð3:78Þ In Eqs. (3.77), (3.78), the bounded continuous function S is  tK1 ðtÞ; t > 0 SðtÞ ¼ ; 1; t¼0

ð3:79Þ

and K0 and K1 are, respectively, the zero and first-order modified Bessel functions of the second kind, or the so-called MacDonald functions. When integrating over zQ to obtain Eqs. (3.77) and (3.78) from Eq. (3.15), we used the regularization procedure similar to that described in Section 1.7. The regularization ensures the correct transition from the 3D Green function with the asymptotics Oð1= j R jÞ; R ! 1 to the planar Green function with the asymptotics Oðln j r jÞ; r ! 1. It can be easily verified that, first, Eq. (3.77) is the limiting case of Eq. (3.78) when xn ! 0, and, second, the kernels K0 ðxn j r0 ðsP Þ  r0 ðsQ Þ jÞ in Eq. (3.78) possess logarithmic singularities in the coincidence limit. We leave it for the reader to show that the integrals in the right parts of Eqs. (3.77), (3.78) converge in the coincidence limit sQ ! sP under the condition that the boundary variations remain continuous within any electrode generated by the translation of the connected part of G0 . The numerical solution of Eqs. (3.77) and (3.78) gives the surface charge perturbation harmonics C0 and Cn;k ðn ¼ 1. . .N; k ¼ 1; 2Þ. The potential perturbation at any point R ¼ fr; zg 2 O0 in the first-order approximation with respect to the small parameter dq appears as d’ðRÞ ¼ D0 ðrÞdq þ

N  X Dn;1 ðrÞ cosðxn zÞ þ Dn;2 ðrÞ sinðxn zÞ dq: n¼1

ð3:80Þ

Here the coefficients D0 , Dn;k ðn ¼ 1; :::; N; k ¼ 1; 2Þ are determined by the relations # ð" 0 0 < r  r ðs Þ; r ðs Þ > Q Q C0 ðsQ ÞG0 ðr; sQ Þ þ 2s0 ðsQ Þ dsQ ð3:81Þ D0 ðrÞ ¼ ½r  r0 ðsQ Þ2 G0

98

Geometry Perturbations

n;k

D ðrÞ ¼ 2

ð(

G0

  Cn;k ðsQ ÞK0 xn j r  r0 ðsQ Þ j

  )

0 þ s ðsQ Þ dsQ : ½r  r0 ðsQ Þ2

ð3:82Þ

Generally speaking, there is a great latitude in choosing the set of the wave numbers fxn g, although for practical calculations it is more convenient to use the arithmetic progression xn ¼ ð2p=LÞn, which corresponds to the Fourier series within the interval ½L=2; L=2. In this case, the formal series in Eq. (3.72) may be considered as a finite-dimensional approximation to the corresponding Fourier integral. To avoid the artificial fringe effects resulting from the artificial ‘‘boundedness’’ of the system’s extent in z-direction, the length parameter L must several times exceed the characteristic width of the 2D cross-section.

3.4.1. Model Problem for Cylindrical Capacitor To illustrate the use of the variational integral equation approach in planar case, let us consider a cylindrical capacitor consisting of two coaxial, rather long, round electrodes. The electrodes radii are R1 ¼ 1 and R2 ¼ 2, and the boundary potentials are unit and zero, respectively. In the limit of infinitely long electrodes, the electric field between the cylinders appears as ’0 ðrÞ ¼ 1 

lnðr=R1 Þ ; R1  r  R2 : lnðR2 =R1 Þ

ð3:83Þ

Consider the boundary perturbation that slightly bends the straight axis of the capacitor into the arc with the radius R0 >> R2 (Figure 57). In the first-order approximation with respect to the small parameter dq ¼ 1=R0 , such perturbation can be determined by the boundary variation vector drðs; zÞ ¼ fz2 =2R0 ; 0g. Let us first construct a quasi-analytical solution to this problem and then compare that solution with the numerical one, obtained with the variational integral equations approach. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Denoting, as usual, r ¼ x2 þ y2 and cosy ¼ x=r, we can represent the perturbed potential distribution between the electrodes in the form ’ðr; y; zÞ ¼ ’0 ðrÞ þ

1 1 ’ ðr; y; zÞ þ OðR2 0 Þ: R0

ð3:84Þ

This sort of perturbation may be treated as a slight modification of the 2D Laplace operator by adding the small extra term ð1=R0 Þð@’=@xÞ þ OðR2 0 Þ. We leave it to the reader to show that this term

99

Geometry Perturbations

z x

R1

q

r R0

R2 y

FIGURE 57 capacitor.

The problem of potential perturbations in the slightly bent cylindrical

implies the perturbation (3.84) of the nominal solution (3.83), with the first-order perturbation coefficient ’1 ðr; y; zÞ given at the central plane z ¼ 0 by the formula

  r lnðr=R1 Þ 1 R22 R21 cosy: ð3:85Þ  r  ’1 ðr; y; 0Þ ¼ r 2 lnðR2 =R1 Þ 2 R22  R21 The function ’1 ðr; y; zÞ within the segment ½L=2; L=2 can be represented by the Fourier series 1 X

  2pm Cm Pm ðrÞcos z cosy; ’ ðr; y; zÞ ¼ L m¼0 1

ð3:86Þ

with Cm denoting the Fourier coefficients of the boundary perturbation function z2 =2R0 on the segment ½L=2; L=2, and Pm ðrÞ denoting the solutions of the Bessel equation   d2 Pm 1 dPm Pm 2pm 2  2  þ Pm ¼ 0 ð3:87Þ r dr L dr2 r with the boundary conditions Pm ðR1 Þ ¼ 

d’0 dr

j

r¼R1

; Pm ðR2 Þ ¼ 

d’0 dr

j

r¼R2

;

ð3:88Þ

100

Geometry Perturbations

which directly follow from the Bertein–Bruns theorem mentioned at the beginning of this chapter. The functions Pm ðrÞ can be expressed as linear combinations of the modified Bessel functions of the first and second kind, and the coefficients of those linear combinations, along with the coefficients Cm , can be numerically calculated if the parameters R1 ; R2 ; L are given. Thus, Eq. (3.86) represents a quasianalytical solution to the problem. We can compare this solution with that obtained by solving numerically the variational integral equations (3.77) and (3.78) at L ¼ 20 and the number of boundary elements on each of the two electrodes equal to 50. The first-order potential perturbation and its derivatives (Table VI) have been calculated at the point x ¼ 1.4, y ¼ 0, z ¼ 0 in three different ways: (1) analytically, according to Eq. (3.85); (2) as a finite sum of the first 40 harmonics, according to Eq. (3.86); and finally, (3) as a finite sum of the same first 40 harmonics, but this time derived from numerical solution of the variational integral equations (3.77) and (3.78). Figure 58 characterizes the convergence of the quasi-analytical solution to the analytical one.

3.5. LOCALLY STRONG 3D PERTURBATIONS IN AXISYMMETRIC SYSTEMS The deviations from axial symmetry considered in Sections 3.2 and 3.3 are assumed to be small in the C1 norm, which means that the corresponding geometric variations, along with their first derivatives with respect to Lagrangian coordinates, are assumed to be uniformly small over the entire boundary. At the same time, in a variety of practical problems we encounter the strong boundary perturbations, which are at the same time concentrated within a very restricted fraction of the boundary, so that the TABLE VI Comparison of the Exact Analytical, Quasi-Analytical and Numerical Solutions in the Problem of Potential Perturbations in the Slightly Bent Cylindrical Capacitor

Potential perturbation function and its derivatives

’1 @’1 =@x @ 2 ’1 =@x2 @ 2 ’1 =@z2

Exact analytical solution (3.85)

Quasi-analytical solution with 40 harmonics in Eq. (3.86)

Numerical solution of variational integral equations

0.117344 0.042741 1.001156 0.029339

0.117286 0.043402 1.010589 0.028838

0.117289 0.043404 1.010618 0.028808

101

Geometry Perturbations

0 j1(r,0,0) −0.02 Exact solution −0.04 −0.06 −0.08 −0.10

n = 30 n = 20

−0.12

n = 10 −0.14 −0.16 1

1.1

1.2

1.3

1.4

1.5 r

1.6

1.7

1.8

1.9

2

FIGURE 58 Convergence of the finite sums of n harmonics in Eq. (3.86) to the exact analytical solution ’1 ðr; 0; 0Þ given by Eq. (3.85).

contribution of such locally strong perturbation into the potential distribution in the main part of the system remains small enough. This happens, for instance, if an electrode with predominant axial symmetry has some apertures, slits, grooves, or other small-scale 3D details, which often are introduced for technological purposes. The question is to which extent such perturbations may affect operation of the device. In general, such perturbations formally make the geometry threedimensional, and the simplest but by far not the most accurate and reliable way to solve those problems is the ‘‘frontal’’ use of 3D numerical algorithms. The main problem here is that the effect produced by a locally strong boundary perturbation may be screened or twisted against the unavoidable computational noise. The perturbation theory suggests another, more sophisticated and efficient, solution that consists in separation of the small domain containing the perturbed fraction of the boundary from the rest of the computational volume and applying 3D algorithms only locally. After this is done, the perturbation effect may be expressed in terms of the boundary potential variations, thus reducing the problem to the already considered one. This idea can be illustrated by the example shown in Figure 59. Consider an immersion lens composed of two coaxial cylindrical electrodes with different voltages U ¼ 0 (grounded electrode) and U0 > 0. The gap between the cylinders contains a ring that is electrically connected with the

102

Geometry Perturbations

x A

A

R2 R1 z

∅ 0V

y

∅ U0

FIGURE 59 The example of locally strong boundary perturbation: coaxial cylindrical electrodes coupled with a ring with two slits (section A-A). The unperturbed equipotentials are shown.

grounded electrode. For some technological purposes, the ring is not solid but has two symmetrical slits. The presence of the slits is the source of the second-order axial harmonic of the electric potential, like d’ ¼ D2;1 ðz; rÞ cos2y, and our goal is to estimate the harmonic’s amplitude D2;1 in the vicinity of the main optical axis, where this harmonic, despite being relatively small, may nonetheless result in a noticeable astigmatism aberration. First, we separate the region of the slits (section A-A in Figure 59) from the rest of the computational volume by means of the virtual cylindrical surface r ¼ R1 (R1 is the inner radius of the cylindrical electrodes) on  ðzÞ ¼ ’0 ðz; R1 Þ with the which we impose the boundary condition ’ 0 unperturbed potential ’ taken from the solution of the corresponding axisymmetric problem as if there were no slits. Generally, there is no versatile ‘‘recipe’’ as to where the separating surface should be placed – the answer to this question is largely a matter of intuition. The electric field in the separated 3D domain is calculated by using the BEM as described in Chapter 1. To reduce the computational load, we use the reflection symmetry and restrict the 3D domain by a wide enough sector as shown in Figure 59. The potential distribution ’1 ðz; R2 ; yÞ in the A-A section on the cylindrical surface R ¼ R2 is presented in Figure 60. Now we are able to find the amplitudes 2 c ðzÞ ¼ p n;1

p=2 ð

p=2

’1 ðz; R2 ; yÞcosny dy

ð3:89Þ

Geometry Perturbations

103

A

U0

ϕ1

A

z 0V

θ

FIGURE 60 Three-dimensional treatment of the locally strong boundary perturbation and the potential distribution in the A-A cross-section at R ¼ R2 .

of the cosine-type harmonics with even n numbers (the other harmonics vanish due to symmetry). It should be noted that the linearity of the integral equation (3.43) with respect to cn;k allows consideration of finite (not obligatory small) boundary potential perturbations on the virtual cylindrical surface R ¼ R2 . In this particular case, we are interested in the second-order harmonic c2;1 , whose amplitude distribution at R ¼ R2 within the gap between the cylindrical electrodes is shown in Figure 61a. The continuation of this harmonic into the axisymmetric domain R < R2 is illustrated in Figure 61b by the set of logarithmic equipotentials. To obtain better smoothness and accuracy, we can now update the potential distribution on the virtual boundary R ¼ R1 with regard to the second and higher-order potential harmonics and then recalculate the potential distribution inside the 3D domain. Such procedure may be iteratively repeated until satisfactory convergence is obtained.

3.6. 3D FRINGE FIELDS IN PLANAR SYSTEMS Here we consider the boundary perturbations caused by finiteness of the electrodes comprising the geometry of any real quasi-planar system. The perturbations of this type are commonly present near the electrode fringes. In contrast to the smooth boundary perturbations considered in

104

Geometry Perturbations

(a)

(b)

r2

r1

Z

Z

FIGURE 61 The boundary distribution of the second-order harmonic amplitude c2;1 at R ¼ R2 (a) and its continuation into the work volume (b) (logarithmic equipotentials).

Section 3.4, the contribution of fringe perturbations is drastically different near the fringes themselves, where the perturbed field possesses a strongly expressed 3D character, and in the inner, ‘‘useful’’ part of the system where the difference (correction) between the perturbed and nominal field becomes essentially smaller compared with that in the fringe area but still remains very significant from the optical standpoint. In this section we propose a perturbation-based algorithm for numerical evaluation of such a correction. Our approach rests on the subdivision of the domain of interest into two overlapping subdomains, with subsequent smooth coupling of the potential distributions in both. As shown in Figure 62, the first subdomain - let us call it the 2D subdomain O0 - is restricted from the left side by the virtual plane S1 : z ¼ z1 and preserves the predominant planar symmetry. We assume that the right fringe of this domain is much farther away, so that O0 may be treated as being extended to infinity at z ! þ1. Let us denote its cross-section made by a plane orthogonal to the axis z as o0 and, accordingly, represent the 2D domain as O0 ¼ o0  ½z1 ; þ1Þ. Let ’0 ðx; yÞ denotes the potential distribution calculated in o0. We look for the perturbed potential distribution in O0 in the form of sum ’ðx; y; zÞ ¼ ’0 ðx; yÞ þ

1 X n¼1

Cn Cn ðx; yÞexp½ln ðz  z1 Þ

ð3:90Þ

of the nominal planar field and a 3D potential correction. Here Cn represents the eigenfunctions of the spectral problem  2  @ @2 Cn þ l2n Cn ¼ 0 ð3:91Þ þ @x2 @y2

Geometry Perturbations

105

w0

Ω1

Ω0

x

z

y S2: z = z2

S1: z = z1

Γ0

FIGURE 62 The numerical evaluation of the fringe fields in the systems with predominant planar symmetry.

with zero condition Cn ðx; yÞ j@O0 ¼ 0 on the cross-sectional boundary. It is important to note that the 2D cross-section o0 should be assumed finite; otherwise, the Laplace equation solution may not be unique. Under this assumption, the spectrum of the eigenvalues fln g is discrete and the sum in Eq. (3.90) converges at any z z1 . Hereafter we presume the eigenfunctions to be L2 -orthonormalized, so that ð Cn Cm dxdy ¼ dnm : ð3:92Þ o0

and numbered in the increasing order of the eigenvalues l1 < l2  l3 . . . The second subdomain, hereafter called the 3D subdomain O1, is restricted from the right side by the virtual plane S2 : z ¼ z2 > z1 and possesses an expressed 3D character. Let ’1 ðx; y; zÞ be the potential distribution calculated in the 3D subdomain O1 by means of solving the Laplace equation  2  @ @2 @2 ’1 ¼ 0 ð3:93Þ þ þ @x2 @y2 @z2 with the boundary condition ’1 ðx; y; z2 Þ ¼ ’ðx; y; z2 Þ ¼ ’0 ðx; yÞ þ

1 X n¼1

Cn Cn ðx; yÞ exp½ln ðz2  z1 Þ ð3:94Þ

on the virtual plane S2 . The coefficients Cn may help to satisfy the boundary condition ’ðx; y; z1 Þ ¼ ’1 ðx; y; z1 Þ

ð3:95Þ

106

Geometry Perturbations

on the other virtual plane S1 if we put ð Cn ¼ ½’1 ðx; y; z1 Þ  ’0 ðx; yÞCn ðx; yÞ dxdy

ð3:96Þ

O0

If this is the case, the uniqueness theorem for the solution of the Laplace equation in the overlapping domain O0 \O1 ¼ o0  ½z1 ; z2  guarantees smooth junction of the functions ’ and ’1 . To solve the self-consistent equations (3.93), (3.94), and (3.96), the iterative procedure can be used as follows. First, let us put all the coefficients Cn ¼ 0 and solve the 3D field problem (3.93) with the boundary condition on S2 induced by the unperturbed potential distribution ’0 . The next approximation to the coefficients Cn is determined by means of calculating the integrals in Eq. (3.96) and using the new Cn values to update the boundary condition (3.94) on the next iteration. The iterations are continued until the relative change of the coefficients Cn becomes less than a given discrepancy value. The number of those iterations is proportional to ½l1 ðz2  z1 Þ1 , where l1 is the minimal eigenvalue. In many cases, if the distance between the two virtual planes S1 and S2 is big enough, only one iterative step may be needed to estimate the fringe effect with sufficient accuracy. Because the higher eigenvalues terms in Eq. (3.90) vanish exponentially at shorter distances from the fringes, we may practically restrict the summation to a few terms with lowest ln . The boundary condition (3.95) for ’ is only approximately satisfied on S1 in this case, and the iteration process converges to an approximate solution. We use the finite element method to calculate the Laplace operator spectrum. The cross-section o0 is subdivided into a number of triangular elements. Within each of the elements the trial eigenfunction is represented in the form of a quadratic polynomial with unknown coefficients. The minimal eigenvalue l1 and the corresponding eigenfunction C1 are found numerically as a solution of the energy minimization problem ð ð3:97Þ W0 ¼ ðrCn Þ2 dxdy ! min o0

under the normalization condition ð Cn dxdy ¼ 1:

ð3:98Þ

o0

The subsequent eigenfunctions can be found using similar numerical procedure with the additional orthogonality condition

Geometry Perturbations

ð

Cn Cm dxdy ¼ 0;

m ¼ 0. . .n  1:

107

ð3:99Þ

o0

Example 3.6.1 The example of fringe field evaluation as applied to the conjunction region between two quadrupoles is shown in Figure 63. The equipotential lines of three of the fifteen eigenfunctions involved in the calculation process are shown. Due to predominant quadrupole symmetry, the eigenfunctions Cn with n ¼ 1; 5; 13 provide the most essential contributions to the sum (3.90). Example 3.6.2 Another example represents a quadrupole ion trap composed of four hyperbolic electrodes biased with the voltages U0 as shown in pffiffiffi Figure 64a. The distance between the opposite electrodes is 20 2 mm, and the virtual surface S2 is located at the distance z2 ¼ 30 mm from the fringe. The true quadrupole field ’0 ¼ 0:01U0 xy generated by the hyperbolic-shaped electrodes is perturbed in the vicinity of the fringe, where five additional electrodes are located. The round, 5-mm radius electrode is grounded, whereas the other four sector electrodes separated by 2-mm gaps are biased with the voltages bU0 . Our task here is to determine the value of the coefficient b 2 ½0; 1 to minimize the fringe perturbation.

(a)

(c)

(b)

Ψ1

Ψ5

Ψ13

FIGURE 63 The example of fringe field calculation. (a) the system of two coupled quadrupoles; (b) equipotentials of the most significant eigenfunctions; (c) equipotentials calculated within the conjunction region.

108

Geometry Perturbations

First, we solve the spectral problem (3.91) to derive a certain number of eigenfunctions and eigenvalues. We consider only the eigenfunctions with antireflection symmetry Cn ðx; yÞ ¼ Cn ðx; yÞ ¼ Cn ðx; yÞ ¼ Cn ðx; yÞ;

ð3:100Þ

and thus can be found by reducing the cross-section O0 to one of its quarters. Only some of the eigenfunctions found satisfy the additional requirement Cn ðx; yÞ ¼ Cn ðy; xÞ; three of them are shown in Figure 65. Strictly speaking, the cross-section o0 is not restricted in this case, and we should cut the hyperbolic electrodes at some distance D large enough to determine the eigenfunctions numerically. As our numerical experiments have shown, the eigenvalues have definite finite limits with the distance D increasing, which is reliable evidence that the spectrum of the Laplace operator is still discrete. The first three eigenfunctions that meet the symmetry requirements are C1 , C3 , and C6 , with the corresponding eigenvalues l1 ¼ 0:117, l3 ¼ 0:273, and l6 ¼ 0:458. The second of these eigenvalues appears more than two times bigger than the minimal one. Thus, if we manage to ‘‘kill’’ the coefficient C1 before C1 in Eq. (3.90), the fringe field vanishes more than two times faster. Having solved the 3D problem in the fringe region (Figure 64b), we discovered that the discrepancy ’1  ’0 becomes orthogonal to the eigenfunction C1 if the coefficient b  0:45. Figure 66 shows how the mixed

(a)

(b) y

−bU0

U0 j1

+U0 z

−U0

bU0 +bU0 −U0 +U0

x x

FIGURE 64 (a) the quadrupole system with hyperbolic electrodes and the auxiliary electrodes at the fringe; (b) potential distribution in the fringe plane calculated with the use of 3D algorithm (due to symmetry, only one quarter is shown).

Geometry Perturbations

Ψ1

FIGURE 65

Ψ2

109

Ψ6

The eigenfunctions with quadrupole symmetry.

0.014 1 0.012 2

⭸2f/⭸x⭸y, Uo /mm2

0.010 0.008 3 0.006 0.004 0.002 0 0

5

10

15

20

25

Z, mm

FIGURE 66 Fringe field penetration into the quadrupole at different voltages on the correcting electrodes: 1, b ¼ 1; 2, b ¼ 0:45; 3, b ¼ 0.

potential derivative (displaying the amplitude of the quadrupole field component) is approaching its nominal value @ 2 ’0 =@x@y ¼ 0:01 with the distance from the fringe increasing. It is clearly seen that the b value we have found is optimal.

CHAPTER

4 Some Aspects of Magnetic Field Simulation

Contents

4.1. Vector and Scalar Potential Approaches 4.2. Direct Integration Over the Current Contours 4.3. Current Contours in the Presence of Materials With Constant Permeability 4.4. Variational Principle in 3D, Planar, and Axisymmetric Cases 4.5. Finite Element Modeling of Magnetic Systems With Saturable Materials 4.6. Second-Order FEM and Curvilinear Elements 4.7. Magnetic Superelements 4.8. The Boundary Element Approach in Magnetostatics 4.9. Hybrid Computational Methods

112 114 116 117 121 129 130 133 138

This Chapter considers some methods for numerical simulation of magnetic fields. This problem is more complicated compared with electrostatic field simulation because the vector potential function is generally needed to describe magnetic field. Only in a few simplest cases is the scalar potential approach possible; this is the main topic of Section 4.1. In Section 4.2, the Biot-Savart law is used to calculate vector magnetic potential of the current-conducting circuits. General representation of magnetic field as a sum of the scalar and vector potential components makes a basis for analysis of the 3D magnetic problems considered in Section 4.3. The nonlinear problems for systems containing saturable magnetic materials can be effectively approached with the variational principle, which is introduced in Section 4.4. Sections 4.5 and 4.6 are devoted to magnetic fields simulation in planar and axisymmetric systems using the Advances in Imaging and Electron Physics, Volume 155 ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)00804-5

2009 Elsevier Inc. All rights reserved.

#

111

112

Some Aspects of Magnetic Field Simulation

finite element method (FEM). In Section 4.7 we consider the magnetic superelements method, which allows essential gain in the efficiency of finite element simulation in the nonlinear magnetic problems. Use of the BEM in vector magnetic potential simulation is detailed in Section 4.8. The most effective hybrid methods that combine the advantages of both the finite element and boundary element approaches are analyzed in Section 4.9.

4.1. VECTOR AND SCALAR POTENTIAL APPROACHES Static magnetic field modeling implies numerical solution of the pair of Maxwell equations div B ¼ 0 rot H ¼

ð4:1Þ

4p j: c

ð4:2Þ

Here B is the magnetic induction vector, H is the magnetic field strength vector, j is the current density vector, and c is the speed of light. Equations (4.1) and (4.2) should be supplemented with the material magnetization relationship B ¼ BðHÞ; which generally may be nonlinear. In this chapter, we consider only isotropic magnetic media, for which B ¼ mðjBjÞðH

MÞ:

ð4:3Þ

Thus, the magnetic permeability m > 0 may depend on the absolute value of magnetic induction B, and the coercive force M is introduced to describe permanent magnets. If G is the interface surface separating the materials with different magnetic properties, B1;2 , H1;2 are respectively B and H-vectors on both sides of the interface, and n is unit normal vector on G, the coupling conditions < B1 ; n >¼< B2 ; n >

ð4:4Þ

H1  n ¼ H2  n

ð4:5Þ

should be satisfied on G. These conditions reflect continuity of the normal component Bn and tangential component Ht on the interface G. The zero divergence of B-field implies the existence of the vector magnetic potential A; so that B ¼ rot A:

ð4:6Þ

We also will consider the scalar magnetic potential C satisfying the equation H ¼ rC: ð4:7Þ

Some Aspects of Magnetic Field Simulation

113

It should be noted that the scalar potential C is unsuitable to describe the magnetic field in entire space but only is used in the simply connected current-free domains. Equation (4.1) immediately leads to the secondorder elliptic equation divðmrCÞ ¼ divðmMÞ

ð4:8Þ

which resembles the electrostatic field equation, with magnetic permeability m playing the role of permittivity and the right part similar to electric charge density. The electrostatic analogy appears to be fruitful when calculating magnetic field in the vacuum domains between the magnetic yokes made of high-permeability material with no permanent magnetization – for instance, ferromagnetic alloys with typical m values of several thousands. In that case, the H-vector inside the yokes is negligibly small and the yoke surface may be considered as magnetically equipotential. In doing so, we obtain the Laplace equation DC ¼ 0 with the Dirichlet boundary condition C ¼ ð4p=cÞJ, where J is the winding current, as shown in Figure 67. In this situation, the electrostatic analogy makes the methods developed in Chapters 1 and 3 applicable to determine magnetic fields and their perturbations in a system with any kind of symmetry and with no symmetry at all (3D case). The BEM for scalar magnetic potential computation in axisymmetric systems was introduced by Kasper in 1987. However, the scalar potential method has essential limitations: (1) the yoke permeability should be high ðm >> 1Þ and the yokes cannot have permanent magnetization; (2) the computational domain cannot contain any currents, which, in turn, means that the windings should be ’’hidden’’ behind the yokes; and (3) the nonlinear materials with field-dependent permeability are not allowed. In the next section, we consider the vector potential and hybrid numerical methods that are free of some or even all of those limitations.

ψ2 J2 ψ1 J1

Computational domain

ψ0

FIGURE 67 The method of electrostatic analogy. The boundary conditions on the yokes are chosen according to the corresponding winding currents: C0 ¼ 0; C1 ¼ ð4p=cÞ J1 ; and C2 ¼ ð4p=cÞð J1 þ J2 Þ.

114

Some Aspects of Magnetic Field Simulation

4.2. DIRECT INTEGRATION OVER THE CURRENT CONTOURS This section considers the simplest magnetostatic problem – numerical modeling of the magnetic field induced by a system of infinitely thin conductive contours Ck ðk ¼ 1 . . . KÞ with electric currents Ik ; accordingly. We assume that there are no magnetic materials around the contours, which, in fact, represents rather specific case in charged particle optics applications. The coreless magnetic windings are used, for example, in dynamic deflectors operating at high frequencies or in some multipole aberration correctors. The contours are supposed to be either closed or have their ends located infinitely far from the calculation region, so that the current continuity condition is satisfied. The magnetic field generated by the currents is given by the Biot-Savart law ð 1X tðsÞds ; ð4:9Þ Ik AðRÞ ¼ c k jR Qk ðsÞj Ck

which is direct vector generalization of the Coulomb law (Landau and Lifshitz, 1984b). Here A is vector magnetic potential at the point specified by the radius-vector R. Let us assume that the continuous vectorfunctions Qk ðsÞ describing the contours’ geometry are parameterized by means of the arc-length parameter s, so that the unit tangential vector t ¼ dQk =ds indicates the currents direction. The magnetic induction B defined according to Eq. (4.6) automatically satisfies the Maxwell equation (4.1). Let us examine the validity of Eq. (4.2): rot B ¼ rot rot A ¼ rdivA DA 2 0 1 ð 1X 1 A Ik 4< t; r > @r ¼ c k jR Qk j Ck

tD

1 jR

3

5ds: Qk j

ð4:10Þ

The first term in square brackets vanishes as the integral of a gradient field taken over the closed contours, whereas the second term may be expressed through the Dirac delta-function. Thus, we obtain ð 4p X rot B ¼ Ik dD ðR Qk ðsÞÞt ds; ð4:11Þ c k Ck

which means that the B-field rotor turns into zero everywhere outside the contours R ¼ Qk 2 Ck and also satisfies Eq. (4.2) upon the contours with the singular current density X ð Ik dD ðR Qk ðsÞÞt: j¼ k

Ck

Some Aspects of Magnetic Field Simulation

115

In general, numerical integration in Eq. (4.9) can be performed by partitioning the contours into a set of small intervals and applying the Gauss quadrature formulas (see Appendix 1). The magnetic field components and their derivatives can be obtained by subsequent differentiation of the integrand. Consider the particular but important case when a contour, or its part located between the parameter values s1 and s2 , is straight. The contribution of the straight wire connecting the points P1 ¼ Qk ðs1 Þ and P2 ¼ Qk ðs2 Þ into the integrals in Eq. (4.9) is A  ðR Þ ¼

  Ik t < R P1 ; t > < P2 R; t > : asinh þ asinh c hðRÞ hð R Þ

ð4:12Þ

Here t ¼ ðP2 P1 Þ=jP2 P1 j is the direction unit-vector and h ¼ jðR P1 Þ  tj is the orthogonal distance from the point R to the conducting wire. In the exceptional case when one or both of the contour ends goes to infinity, the integral (4.9) diverges. Nevertheless, its derivatives are still finite and may be found from the ‘‘renormalized’’ formula for A , which may be derived in the following manner. Let us assume that one of the conductor endpoints is located in infinity - for example, let us put jP2 j ¼ 1. Identical transformation of the second term in Eq. (4.12) gives asinh

 < P2 R; t > ¼ ln h þ ln < P2 hð R Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R; t > þ < P2 R; t>2 þ h2

¼ ln h þ ln 2 < P2 ; t > þ oð1Þ

at P2 ! 1: As soon as the second term in the asymptotics (4.13) does not depend on R, we may consider   Ik t < R P1 ; t >  asinh ln h ð4:14Þ A ðRÞ ¼ c h as the ‘‘renormalized’’ magnetic vector potential for the case that one of the straight conductor ends goes to infinity. If both of the ends are infinitely far, the result is well-known logarithmic potential A ¼ ð2In t=cÞln h. This renormalization procedure is quite similar to the procedure used in Chapter 1 to transform the 3D Green function into a 2D one. Despite its simplicity, the method considered above is intrinsically 3D and therefore, as was shown by Rozenfeld, Vasichev, and Zotova (2000), may be used for numerical calculation of magnetic fields in coreless magnetic deflectors and multipole aberration correctors. A somewhat different computational technique for the cores or yokes made of nonsaturable magnetic materials is considered in the next section.

116

Some Aspects of Magnetic Field Simulation

4.3. CURRENT CONTOURS IN THE PRESENCE OF MATERIALS WITH CONSTANT PERMEABILITY The approach considered in the previous section for magnetic systems comprising a number of current contours in vacuum may be generalized to magnetic cores and yokes of any arbitrary shape in the presence of nonsaturable magnetic materials and in the absence of permanent magnetization. Following to Rouse and Munro (1989), we represent the Hfield as the sum of two vectors H ¼ H0 þ H1 ; H0 ¼ rot A; H1 ¼ rC:

ð4:15Þ

The vector potential A is given here by the Biot-Savart law (4.9), and the residual field H1 may be treated as a gradient of some scalar potential C because H1 is irrotational. Indeed, rot H1 ¼ rot H

rot H0 ¼

4p j c

4p j ¼ 0: c

ð4:16Þ

Considering that, in regions with constant permeability and no coercitivity, Eq. (4.1) appears as div B ¼ div ðm HÞ ¼ m div H0 þ m DC ¼ 0

ð4:17Þ

and div H0  0, we arrive at the Laplace equation DC ¼ 0 for the scalar magnetic potential. This equation holds true everywhere except for the interfaces separating the regions with different m values, where the coupling condition

ð4:18Þ mð2Þ H0n þ @nþ C ¼ mð1Þ H0n þ @n C ; H0n ¼< n; H0 >

is to be imposed. Here n is the interface’s normal vector pointing from the media with permeability mð1Þ to the media with permeability mð2Þ , @n C are normal derivatives of C on the separating interface, taken from the sides with the permeability values mð1Þ and mð2Þ , correspondingly. Instead of applying the FDM to magnetostatic problems as proposed by Rouse and Munro (1989), we further modify the BEM to solve the Laplace equation DC ¼ 0 with the coupling condition (4.18) on the separating interface and zero Dirichlet condition on some external boundary. The BEM advantages outlined in Chapter 1 in connection with electrostatic problems can be completely extended to the magnetostatic case in question. Similar to the electrostatic problems considered in Chapter 1, we represent the scalar magnetic potential in the ordinary-layer potential form ð sðQÞ ð4:19Þ dSQ ; CðRÞ ¼ jR Qj G

Some Aspects of Magnetic Field Simulation

117

in which integration is made over the join G of the interfaces described by the vector-function Qðu; vÞ parameterized in an appropriate (possibly local) curvilinear coordinates system ðu; vÞ. The unknown magnetic charge density sðQÞ should be found from the coupling condition (4.18), in which the magnetic field gradients on G take the form @n C ¼ Dn ½sŠ  2ps;

ð4:20Þ

with the direct normal derivative Dn representing the integral operator ð 1 Dn ½sŠðRÞ ¼ @n ð4:21Þ sðQÞ dSQ ; ðQ; RÞ 2 G: jR Qj G

This immediately bring us to the Fredholm second-kind integral equation with respect to sðQÞ on G 2psðQÞ ¼

mð2Þ mð1Þ fDn ½sŠðQÞ þ H0n ðQÞg; Q 2 G mð2Þ þ mð1Þ

ð4:22Þ

the numerical solution of which gives the magnetic ‘‘charges’’ distribution and thus unambiguously determines the magnetic field (4.15) as the sum of vector- and scalar-potential components. It is noteworthy that the integral equation kernel in Eq. (4.22) is integrable on the regular surfaces just as it was for dielectric interfaces in Chapter 1. The charge density singularities near the singular points of the interfaces should be explicitly introduced into the shape functions for s approximation. Thus, the permittivity e in the singularity indexes determined in Chapter 2 simply needs to be replaced by the magnetic permeability m. Figure 68 shows an example of 3D magnetic field calculation in the magnetic system with three circular current contours and an E-shaped core. The magnetic field is shown in two cross-sections: one is the symmetry plane and the other one is located at some distance above the magnetic poles. A similar method was proposed for axisymmetric yoke by Kasper and Stro¨er (1989). The algorithm assumes expansion of scalar magnetic potential into the Fourier series, which allows reducing the 3D field problem to a number of 2D problems.

4.4. VARIATIONAL PRINCIPLE IN 3D, PLANAR, AND AXISYMMETRIC CASES Having substituted Eqs. (4.6) and (4.3) into Eq. (4.2), we obtain a differential equation with respect to the vector magnetic potential A,   rot A 4p M ¼ j; ð4:23Þ rot m c

118

Some Aspects of Magnetic Field Simulation

2

1

FIGURE 68 Example of 3D magnetic field simulation for circular windings (1) with E-type core (2). Magnetic field vectors are shown.

which may be re-written in the form of the elliptic partial differential equation DA r div A 1 4p þ þ r  rot A ¼ j þ rot M m m m c

ð4:24Þ

to be solved in the domain O under the Dirichlet boundary condition  ðQÞ on @O. It is reasonable to eliminate the second term in the Aj@O ¼ A left-hand side of (4.24) using the calibration freedom in the vector potential definition. Using the Coulomb calibration condition div A  0, we obtain DA ¼

4pm

j þ jM B  rm c þ ; jM ¼ rot M; c m 4p

ð4:25Þ

where the operator D is here understood as the component-wise vectorial Laplace operator. Note that Eq. (4.25) is divided into three different equations in the case of nonsaturated piecewise homogeneous media. The integral flux of the magnetic vector potential through the closed boundary @O should vanish to be consistent with the Coulomb calibration, so we have ð  n ðQÞdSQ ¼ 0; ð4:26Þ A @O

Some Aspects of Magnetic Field Simulation

119

where n is outer normal to the boundary @O. The zero boundary condition  n ¼ 0 is obviously suitable here. A The Dirichlet problem for Maxwell equations (4.1) and (4.2) with regard to the material magnetization relationship (4.3) can be formulated in another, variational form. Denote ðB 0 1 B 0 UðBÞ ¼ 0
dB : 4p m B

ð4:27Þ

0

and consider the functional ð < M; B > UðjBjÞ þ W¼ 4p

 < A; j > dV; c

O

ð4:28Þ

which we, giving credit to tradition, will call the energy functional, although it is actually the integral of the Lagrangian density. Let us show that the solution of the magnetostatic problem in question brings a stationary point to Eq. (4.28) under the Dirichlet boundary condition imposed. Indeed, the first variation of the functional appears as ð ð 1 B 1 þ M; dB > dV < < j; dA > dV dW ¼ 4p mðjBjÞ c O

O

ð4:29Þ ð 1 < rot H ¼ 4p O

4p j; dA > dV c

ð 1 div½H  dAŠdV; 4p O

with H ¼ B=m þ M. The second term turns into the surface integral over @O and vanishes since dA ¼ 0 on the boundary. The first integral vanishes for any variation dA if and only if the Maxwell equations (4.1) and (4.2) are satisfied. Thus, we have shown that the differential equations (4.23)-(4.25) are necessarily satisfied on each of the solutions that deliver minimum value to the functional (4.28). This fact appears very useful in numerical modeling of magnetic systems, especially of those that contain the nonlinear magnetic materials and permanent magnets. It is noteworthy that the minimization of the functional (4.28) does not require the C2 -smoothness of field interpolation - only the field gradient square should be integrable in O: Let us consider the cases of planar and axisymmetric magnetic systems to illustrate more details of the variational approach discussed. In the planar case, assuming the presence of translation symmetry along the z-axis, we may impose the condition Ax ¼ Ay ¼ 0 on the vector potential A, with the third component Az to be found. For axisymmetric systems,

120

Some Aspects of Magnetic Field Simulation

the condition Az ¼ Ar ¼ 0 may be imposed, with only the angular component A’ ðz; rÞ being, generally speaking, nonzero and meeting the extra condition A’ ðz; rÞjr ¼ 0 ¼ 0 on the symmetry axis. It is easily verified that both of these conditions are fully consistent with Coulomb calibration. For writing simplicity, in both cases we will designate the corresponding nonzero field component as a scalar function of two coordinates, for example, we will put Az ¼ Aðx; yÞ or A’ ¼ Aðz; rÞ depending on whether the planar or axisymmetric system is considered. For planar systems, the B-field components and the field equation (4.25) take, accordingly, the form Bx ¼ DA ¼

@A ; @y

4pm

By ¼

@A @x

ð4:30Þ

j þ jM < rA; rm > þ ; c m

ð4:31Þ

where  c c @My ½rot MŠz ¼ jM ¼ 4p 4p @x

@Mx @y



ð4:32Þ

is the ‘‘current’’ induced by permanent magnetization. The axisymmetric case gives Bz ¼

div

1 @ ðrAÞ @A A ¼ þ ; r @r @r r

rðrAÞ ¼ r

4pm

Br ¼

@A @z

j þ jM < rðrAÞ; rm > þ ; c mr

ð4:33Þ

ð4:34Þ

where jM ¼

 c c @Mr ½rot MŠf ¼ 4p 4p @z

The energy functional (4.28) appears as   ð 1 @A @A Mx My UðjrAjÞ þ W¼ 4p @y @x O

W ¼ 2p

    ð  jrðrAÞj 1 @A A þ Mz þ U r 4p @r r O

 @Mz : @r

 Aj dxdy; c

Mr

 @A @z

ð4:35Þ

ð4:36Þ

 Aj r dz dr c ð4:37Þ

121

Some Aspects of Magnetic Field Simulation

in planar and axisymmetric cases, respectively. The magnetic field problem is thus reduced to determination of the functions Aðx; yÞ or Aðz; rÞ that deliver minimal value to the functional W under the Dirichlet boundary conditions. Hereafter in this chapter we concentrate on the axisymmetric case because the case of planar symmetry is simpler and the corresponding formulas for the planar case may be easily derived from the axisymmetric ones by omitting the terms that explicitly contain r, and redefining the coordinates in the way z ! x; r ! y. We denote the 2D meridian crosssection of the computational domain in the plane ðz; rÞ with the same letter O as in the 3D case. The one-dimensional interfaces separating different media in the mentioned cross-sections are denoted as G:

4.5. FINITE ELEMENT MODELING OF MAGNETIC SYSTEMS WITH SATURABLE MATERIALS Variational principle opens wide possibilities for applying the FEM in magnetic fields simulation for sophisticated magnetostatic systems. To the authors’ knowledge, it was Munro who pioneered in his PhD dissertation (1971) the use of the FEM in computational charged particle optics. In his monograph (1973) he developed this method toward the charge particle optics problems. The basic idea of the method consists in approximating the unknown function Aðz; rÞ by means of the linear combination Aðz; rÞ ¼

N X

k¼1

Ak Ck ðz; rÞ

ð4:38Þ

of N specially chosen shape functions Ck ðz; rÞ with the unknown coefficients Ak . The energy functional (4.37), being considered in the finite-dimensional subspace of the shape functions Ck , becomes a nonlinear function  W ðA1 . . . AN Þ. The variational problem in infinite-dimensional functional space is thus reduced to the finite-dimensional problem of finding out a finite set of unknown coefficients that bring minimal value to the function  W ðA1 . . . AN Þ. This leads to the set of N algebraic equations 

@W dW Wk ¼ ¼>¼ 0; @Ak dA

ð4:39Þ

in which dW=dA denotes the functional derivative, and the double angle Ð brackets denote the scalar product >¼ O Fðz; rÞGðz; rÞr dz dr. Solving Eq. (4.39) is equivalent to making the equation dW=dA ¼ 0 discrepancy orthogonal to the linear subspace ‘‘stretched’’ on the basis functions fCk g.

122

Some Aspects of Magnetic Field Simulation

Various FEM realizations, as applied to magnetostatic problems, differ in particular choice of the set of shape functions. Basically, the choice of those functions is limited by the following requirements: (1) the reduced finite-dimensional minimization problem should be well-conditioned, (2)  the computation of both the functional W and its derivatives Wk with respect to the coefficients Ak should be sufficiently fast, and (3) the set of the shape functions should ensure accurate enough numerical approximation of the exact solution. The first condition is met if the shape functions are linearly independent and, as a sequence, the matrix jjMkm jj ¼ jj > jj is not singular. This matrix also should be diagonally dominant to minimize numerical inaccuracies. The second condition is met if the shape functions have compact and mainly nonoverlapped supports. In this case, both the matrix jjMkm jj and the matrix of second derivatives



@2W

jjWkm jj ¼ ð4:40Þ

@A @A m k are sparse and thus easy to calculate and manipulate. The third condition claims that the linear subspace of approximating functions should be dense enough in the class of continuous and piecewise smooth possible solutions. The magnetic potential continuity immediately entails the continuity of the functions Ck . Nevertheless, the derivatives of the shape functions cannot be continuous everywhere since the tangential component of B-field experiences jumps on the interfaces separating the magnetic materials with different permeability. The set of shape functions meeting all these requirements may be constructed on a mesh of finite elements. Unlike the FDM, the FEM provides good approximation of the interfaces because the element shapes are not predefined. The comparison of FEM with finite difference and boundary elements methods can be found in the papers by Cubric et al. (1999) and Lencova´ (2004). At the same time, finite element mesh generation is an additional problem to be solved before proceeding to the field computation itself. The mesh generation complicity is the main factor restricting the use of FEM in 3D field problems, although computer algorithms oriented to some particular cases do exist. As example, an automatic subdivision procedure for a 3D region capable of being represented in the form of a union of parallelepipeds is proposed by Tanizume, Yamashita, and Nakamae (1999); a more general case is described by George, Hecht, and Saltel (1990). Having in view the axisymmetric and planar field problems, later in this section we make emphasis on the construction of the 2D meshes. The FEM meshes differ in element shapes, which can be quadrilateral, triangular, and so on. The elements should cover the entire computational

Some Aspects of Magnetic Field Simulation

123

domain in such a way that the interfaces, if any present, should coincide with the mesh lines. Under this condition, each element contains a homogeneous material with continuous magnetic properties, whereas the boundary conditions (including all possible field discontinuities) may be set only on the element sides. In 1971 Munro used the triangular elements resulting from diagonal bisection of the quadrilateral cells comprising a quasi-regular mesh. The computational domain was first divided into a set of quadrilaterals, with their sides located on the interfaces. Then the opposite sides of the quadrilaterals were connected by a number of straight lines to form a quasiregular quadrilateral mesh, as shown in Figure 69. Similar quadrilateral meshes were used by Lencova´ and Lenc (1996, 1999) with no extra subdivision into triangles. The major disadvantage of quasi-regular meshes of this sort is that the mesh density is strictly dictated by the boundary shape and the interfaces which the mesh should be matched with. In practical problems, other significant requirements can be placed on the mesh nodes distribution. For instance, from the viewpoint of approximation accuracy, the mesh in region (A) in Figure 69 should be made denser near the axis, whereas the number of nodes in region (B) could be made smaller to reduce the computational expenses. Generally, caution should be observed in arranging the mesh density gradient because, as confirmed by Lencova´ and Lenc (2004), the abrupt change in mesh density may represent an additional source of approximation errors. We consider further the construction of irregular triangular meshes whose density may vary smoothly from one point to another. The first step represents selection of a set of simply-connected subdomains whose boundaries contain all the interfaces to be considered. If necessary, some multiply-connected regions with only one magnetic-type material are

B

A

FIGURE 69 Constructing a quasi-regular quadrilateral mesh (example given by Munro, 1970). The marks (A) and (B) denote the regions with different mesh densities.

124

Some Aspects of Magnetic Field Simulation

further subdivided. Then, as shown in Figure 70, the subdomains are split by means of straight lines into the smaller subdomains whose boundaries are divided into a number of intervals. The lengths of those intervals are chosen in accordance with the local mesh density required. Then the triangular elements are constructed inside each of the subdomains. To do so, the algorithm ‘‘walks’’ around the perimeters of the subdomains (Figure 70b) until the entire computational domain is triangulated, as shown in. The mesh constructed in this manner usually is imperfect because some triangles may differ too much from the equilateral ones, and the areas of two adjacent elements may also be too different. To eliminate such disadvantages, we apply a special regularization procedure that shifts the mesh node coordinates ðzi ; ri Þ in a certain way to minimize the
2
2 P quadratic form zi zj þ ri rj , in which the summation is made i; j

over the pairs of nodes connected by one mesh line. This quadratic form is minimized numerically with the use of the gradient search method. This mesh modification procedure is applied along with the diagonal swap regularization procedure proposed by Tarnhuvud, Reichert, and Skoczylas (1990). The essence of the regularization procedure is replacement of the longer diagonal of a quadrilateral by the shorter (shown in Figure 71). The regularized triangular finite element mesh is shown in Figure 72. We now proceed to constructing the shape functions. Let us introduce the local coordinates ðu; vÞ for each element individually, as shown in Figure 73, and define three linear functions c1 ¼ 1

(a)

u

v; c2 ¼ u; c3 ¼ v;

ð4:41Þ

(b)

FIGURE 70 (a) The first step of the triangular mesh construction procedure is subdividing the computational domain into a set of the single-connected subdomains; (b) the triangulation procedure.

Some Aspects of Magnetic Field Simulation

(a)

125

(b)

FIGURE 71 Mesh fragment before (a) and after (b) the diagonal swap regularization procedure.

FIGURE 72

Regularized triangular finite element mesh with density gradient.

P3

V

v=1

u=1 P1

u P2

FIGURE 73 Local coordinates for the first-order finite element approximation.

126

Some Aspects of Magnetic Field Simulation

each of which takes unit value at one of the corners of the triangular element and vanishes at the two others. The coordinate representation Pðu; vÞ ¼ c1 ðu; vÞP1 þ c2 ðu; vÞP2 þ c3 ðu; vÞP3 :

ð4:42Þ

parameterizes the element surface with the local coordinates running inside the domain restricted by the inequalities u  0; v  0, and u þ v  1. The variants of the first-order finite element method (FOFEM) differ in the use of the linear functions ck for vector magnetic potential interpolation. For example, in the studies by Lencova´ and Lenc (1996, 1999), the potential A is represented as a linear combination of the functions like (4.41), with the additional condition A ¼ 0 imposed on the axis. Edgcombe (1999) proposed considering the value 2prA as unknown function; the physical meaning of this function is the magnetic flux through
the ring of the radius r. The use of the transformed coordinates z; r2 was proposed by Melissen and J. Simkin (1990) to make the magnetic flux an even function of r and thus improve the field approximation accuracy in the vicinity of the symmetry axis. Let us assume that the function A=r is linear within each of the finite elements constructed and can be represented as a linear combination of the functions (4.41). Under this assumption, the extra condition A ¼ 0 is met automatically, so there is no difference between the near-axis elements and the off-axis ones. As seen from Eq. (4.33), the doubled value of A=r immediately gives the z-component of magnetic field on the axis: Bz ðz; 0Þ ¼ 2 lim A=r. From the methodological viewpoint, it is better to r!0

include the factor r into the shape functions (so that Ck ðu; vÞ ¼ r ck ðu; vÞ), and approximate the magnetic potential with the linear combination Aðu; vÞ ¼ A1 C1 þ A2 C2 þ A2 C2 ;

ð4:43Þ

in which the coefficients A1;2;3 correspond to the element corners. The functions Ci defined on the elements adjacent to a fixed mesh node, and being nonzero at that node, determine a single shape function associated with that node. With such definition, the shape functions are continuous everywhere within the computational domain. We can now evaluate the contribution of a single element to the energy integral. We need the components of the covariant metric tensor



guu guv U2 < U; V >

¼

ð U ¼ P2 P1 ; V ¼ P 3 P1 Þ g¼

guv gvv < U; V > V2

ð4:44Þ

to be expressed through the local coordinates ðu; vÞ. The Jacobian of the coordinate transformation

127

Some Aspects of Magnetic Field Simulation

J¼

@ ðz; rÞ pffiffiffiffiffiffiffiffiffiffiffi ¼ det g ¼ jU  Vj @ ðu; vÞ

ð4:45Þ

is the double area of the element. The absolute value B of magnetic induction may be expressed through the values Ai ; i ¼ 1; 2; 3 as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u

 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X u @ ðrAÞ @ ðrAÞ @ ðrAÞ=@u  rðrAÞ 1 t 1

¼

 j¼ bkm Ak Am ; g B¼ @v @ ðrAÞ=@v r r @u k;m¼1

ð4:46Þ

where the coefficients bkm 8 1 < @ ðrCk Þ @ ðrCm Þ @ ðrCk Þ @ ðrCm Þ bkm ¼ 2 2 gvv þ guu rJ : @u @u @v @v 2 39 = @ ð rC Þ @ ð rC Þ @ ð rC Þ @ ð rC Þ m m k k 5 guv 4 þ ; @u @v @v @u

ð4:47Þ

depends on the basis functions and metric tensor components. The energy integral over a single element appears as W ðelÞ ¼ 2p

ð1 1ð u  0

0

U ð BÞ

ð1 1ð u  Aj 1 @ ðrAÞ r J dvdu þ Mu c 2r @v 0

Mu ¼< U; M >

0

Mv ¼< V; M > :

Mv

 @ ðrAÞ dvdu @u ð4:48Þ

By substituting Eqs. (4.43) and (4.46) into Eq. (4.48), we express the value of W ðelÞ through the coefficients Ai ; i ¼ 1; 2; 3; which correspond to the element corner, and derive the computations formulas for the first and second-order variations (4.48) with respect to these coefficients as follows: 3 dW ðelÞ X ðelÞ okm Am ¼ dAk m¼1

ðelÞ

gk

ð4:49Þ

d2 W ðelÞ ðelÞ ðelÞ ¼ okm þ wkm : dAk dAm

ð4:50Þ

and

The coefficients involved in Eq. (4.49) and (4.50) are calculated as

128

Some Aspects of Magnetic Field Simulation

ðelÞ okm

1 ¼ 2

ð1 1ð u 0

ðelÞ gk

¼

ð1 1ð u  0

2pr J Ck j c

0

ðelÞ wkm

¼

3 X

0

bkm r J dvdu mðBÞ

 1 @ ðrCk Þ Mu 2r @v

As At

s;t¼1

ð1 1ð u 0

@ ðrCk Þ Mv @u

ð4:51Þ



dvdu

bks bmt dm rJdvdu 2m2 B dB

ð4:52Þ

ð4:53Þ

0

by using the Gauss quadrature formulas for a triangular domain as described in Appendix 1. The coefficients marked with the superscript index ðelÞ, with their indexes running the values 1, 2, and 3, indicate the contribution of a single finite element. To find the variations of the energy integral over the entire ðelÞ ðelÞ computational domain, let us sum the matrixes okm ; wkm and the vector ðelÞ gk corresponding to all the mesh elements with respect to continuous numbering of the coefficients A1 . . . AN . By doing so, we obtain the N  N sparse matrixes okm ; wkm and the N-dimensional vector gk , through which the magnetic energy integral may be represented in the form of Taylor expansion in the vicinity of a trial solution A1 . . . AN : ! N N X X W ½A þ dAŠ ¼ W ½AŠ þ okm Am gk dAk k¼1

m¼1

N 1 X þ ðokm þ wkm ÞdAk dAm 2 k;m¼1

ð4:54Þ

It is worth noting here that the coefficients before dAk give the discrete analog of the field equation N X

okm Am

gk ¼ 0

k ¼ 1 . . . N:

m¼1

We use the gradient search method with preconditioning to minimize the energy functional. The zero boundary conditions are satisfied by forcing the coefficients that correspond to the mesh nodes lying on the boundary @O to be zero. The other coefficients, including those corresponding to the on-axis point, are subjected to the iteration procedure described by the recursive formula

129

Some Aspects of Magnetic Field Simulation

ðqþ1Þ Ak

¼

N CðqÞ X okm AðmqÞ okk m¼1

ðqÞ Ak

ðqÞ

!

gk ;

k ¼ 1...N

ð4:55Þ

where Ak is the try solution at the q-th iteration and the relaxation constant 0 < CðqÞ < 2 is determined as the ‘‘golden mean’’ between slow convergence at low values of C and computational instability at high values. We have found empirically that CðqÞ  0:8 0:9 is suitable in most practical cases.

4.6. SECOND-ORDER FEM AND CURVILINEAR ELEMENTS The use of the second-order interpolants instead of the first-order ones allows further advance in numerical accuracy and smoothness. Instead of the three linear functions (4.41), let us introduce the six second-order polynomials c1 ¼ ð1 u vÞð1 c3 ¼ vð2v 1Þ c5 ¼ 4uv

2u

2vÞ

c2 ¼ uð2u c4 ¼ 4uð1 c6 ¼ 4vð1

1Þ u vÞ u vÞ

ð4:56Þ

rt ct ðu; vÞ

ð4:57Þ

and define the local parameterization in the form zðu; vÞ ¼

6 X t¼1

zt ct ðu; vÞ;

rðu; vÞ ¼

6 X t¼1

with ðzt ; rt Þ denoting the coordinates of the six nodal points Pt ðt ¼ 1; . . . 6Þ. The first three points are, as before, the triangular element corners (shown in Figure 74). The three other points, namely P4 ; P5 , and P6 , are located on

u = 0, v = 1

P3

P5 u = 1/2, v = 1/2

P6 u = 0, v = 1/2

P4

P1

u = 1/2, v = 0

u = 1, v = 0 P2

u = 0, v = 0

FIGURE 74

Local coordinates for the second-order finite element approximation.

130

Some Aspects of Magnetic Field Simulation

the element sides, which may be either straight or curved. For the straight sides, let us place the on-side nodal point at the side middle. If a side is curved (as is the side P1 P3 in Figure 74), let the on-side node split the side into two intervals of equal length. For sufficiently smooth boundaries, such parameterization ensures the third-order approximation with respect to the element size. In the case of the second-order approximation, the basis vectors U ¼ f@z=@u; @r=@ug; V ¼ f@z=@u; @r=@ug, the metric tensor (4.44), and the Jacobian (4.45) explicitly depend on the local coordinates. Equations (4.49) through (4.54) remain valid with the node indexes running in the range 1 . . . 6. The second-order approximation of magnetic potential requires the six functions (4.56) to be used for constructing the shape functions. Accordingly, in this case the number of nodal points within one finite element is 6, which is twice as large as with the first-order FEM. It is possible to simultaneously apply different approximation schemes for mesh geometry and magnetic potential representation. For example, the second-order interpolation functions (4.56) can be used to construct curvilinear elements and, at the same time, the first-order magnetic potential approximation can be used to reduce the number of the mesh nodes involved. Xieqing Zhu and Munro (1989) have shown that the second-order interpolation allows reduction of the approximation errors. However, the higher-order shape functions do not guarantee smoothness of the solution’s derivatives at the element edges, which is vital if the aberration coefficients are to be calculated. As shown by Khursheed (1996), even the use of the C1 -smooth interpolants, whose implementation is much more complicated, appears insufficient. We will return to this problem when considering the BEM and the hybrid methods. Automatic mesh generation algorithms have greatly encouraged and stimulated the development of FEM software oriented to the needs of charged particle optics. The magnetic field simulation program was presented by Lencova´ and co-workers (1990, 2006). The programs by Rapotsevich (1996) and by Melnikov and Vasichev (2003) also use the first-order FEM and are capable of solving the nonlinear problems. The magnetic field simulation program package INTMAG was presented by Becker in 1990. Hodkinson and Tahir (1995) developed a magnetic simulation program based on the use of the second-order finite element approximation.

4.7. MAGNETIC SUPERELEMENTS In applications of the FEM to the problems of charged particle optics, only a small fraction of the total number of finite elements is commonly located in the saturable region of yokes. In this regard, it is noteworthy that at

Some Aspects of Magnetic Field Simulation

131

least two types of regions within the computational domain, namely the vacuum region and the solenoids, do not suffer from saturation. The status is quite different in electromechanical problems from which the ‘‘magnetic’’ FEM was originated. At the same time, the general iteration procedure described by Eq. (4.55) operates with the magnetic potentials defined on the set of all nodes, including those located in the ‘‘linear’’ subdomains that do not contain any saturable materials. Following Zhong-Qing You et al. (1988), we show further how such linear elements may be gathered to exclude a great number of nodes from the iteration process. ðelÞ First, we note that the matrix okm in Eq. (4.51) does not depend on ðelÞ magnetic field and the matrix wkm given by Eq. (4.53) vanishes provided that magnetic permeability inside any element is constant. This fact allows precalculation of all the coefficients for such elements before the iterative procedure starts. We may go even further and consider a linear subdomain OðLÞ  O containing M nodes on its boundary and N M nodes inside (as shown in Figure 75). This subdomain can be treated as a single finite element with complex internal structure. Following Strakhovskaya and Fedorenko (1979) and Fedorenko (1994), we use the term ‘‘superelement’’ to describe that situation. Let us assume that the indexes of the M boundary nodes lie in the range 1; . . .; M. The matrix jjokm jjð1  k; m  NÞ is thus separated into four submatrixes – two square ones, oA and oB with the dimensions M  M

9

8

7

6 5

10

4 3

11

2

12 N

1

13

M

14

25

15 24 16 17

FIGURE 75

18

19

20

21

22

23

Node numbering in the linear subdomain. See text for details.

132

Some Aspects of Magnetic Field Simulation

and ðN MÞ  ðN submatrixes oC :

MÞ accordingly, and two rectangular M  ðN

oA

o ¼ ðoC ÞT

oC





oB

MÞ

ð4:58Þ


T (here oC is the matrix being transposed with respect to oC ). We may start minimization of the energy integral W ðLÞ ðA1 . . . AN Þ ¼

N X

N 1 X okm Ak Am 2 k;m ¼ 1

gk Ak

ð4:59Þ

k¼1

taken over the linear subdomain from minimizing it with respect to the internal node coefficients AMþ1 . . . AN . With the boundary values A1 . . . AM fixed, the conditional minimum is reached at ! M N X X
1 An ¼ oCkm Ak ; n ¼ M þ 1 . . . N: oB nm gm ð4:60Þ m ¼ Mþ1

k¼1


1 Here the matrix oB is reciprocal of the submatrix oB . By using Eq. (4.60), the energy integral (4.59) may be expressed as a function of the boundary magnetic potential only. Introducing the modified coefficients N X

okm ¼ oA km

oCki oB

i; j ¼ Mþ1

gm ¼ gm

N X

i; j ¼ Mþ1

oCki oB







1 ij

1 ij

oCmj

gj ;

ð4:61Þ

we obtain (with the accuracy up to an unessential additive constant) W ðLÞ ½A1 . . . AM Š ¼

M 1 X o Ak Am 2 k; m ¼ 1 km

M X

gk Ak ;

ð4:62Þ

k¼1

The coefficients in Eq. (4.61) may be easily found by means of the Gauss elimination procedure. It is useful to note that the matrix jjokm jj is no longer sparse but still diagonally dominant. The internal nodes belonging to the linear subdomains are excluded in Eq. (4.62), which allows substantial reduction of the number of equations to be solved in the iterative procedure. After the solution of the nonlinear magnetic problem is found, Eq. (4.61) ensures reconstruction of the magnetic potential distribution in the inner nodes of the linear subdomain using the potential distribution on its boundary.

Some Aspects of Magnetic Field Simulation

133

4.8. THE BOUNDARY ELEMENT APPROACH IN MAGNETOSTATICS From the mathematical standpoint, the finite element method allows replacement the boundary-value problem for the second-order elliptical partial differential equation with the problem of numerical minimization of the energy functional expressed in terms of the field gradient. In doing so, the smoothness requirements on the magnetic potential prove to be essentially reduced – the function Aðz; rÞ should be only continuous, with its first derivatives square-integrable. This facilitates the field calculation but renders the FEM rather disadvantageous when not only field strength but higher field derivatives are also needed. The BEM described in Chapter 1 as applied to electrostatic problems is essentially ‘‘smoother’’. Nevertheless, this method confronts two serious difficulties when applied to magnetostatics: (1) even for homogeneous materials, the saturation effect makes the permeability dependent on coordinates, and (2) the field sources such as currents or coercitivity usually are distributed over the solenoids or permanent magnets volume. These two factors necessitate calculation of not only surface but also volume integrals, which essentially complicates the computational model and makes it too cumbersome for practical implementation. As mentioned previously, the BEM for scalar magnetic potential calculation was introduced by Kasper in 1987 and then developed in 1989 in collaboration with Stro¨er. Because (1) the opportunities the scalar potential method offers in magnetostatic are limited, and (2) this approach is quite analogous to that described in Chapter 1 for electrostatic problems, we dedicate this and the following sections to the vector potential numerical approaches. Some methods for solving magnetostatic problems in the vector potential statement may be found in Kasper’s (2000) more recent work. By analogy with the electrostatic BEM, let us introduce the Green function Gðz; r; z0 ; r0 Þ satisfying the equation div

r½rGðz; r; z0 ; r0 Þފ ¼ r

4p dD ðz

z0 Þ dðr

r0 Þ

ð4:63Þ

and vanishing in infinity. The Green function allows the representation 0

0

Gðz; r; z ; r Þ ¼ r

0

2p ð 0

cos’ d’ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðz z0 Þ þ ðr r0 cos’Þ2 þ r0 2sin2 ’

4r0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K1 ðkÞ ðz z0 Þ2 þ ðr þ r0 Þ2 through the elliptic integral

ð4:64Þ

134

Some Aspects of Magnetic Field Simulation

K 1 ðkÞ ¼

p=2 ð 0

where k¼

cosð2cÞdc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k cos2 c

ð4:65Þ

4rr0 ðz

ð4:66Þ

z0 Þ2 þ ðr þ r0 Þ2

is its module. The physical meaning of the Green function Gðz; r; z0 ; r0 Þ is the magnetic field of the infinitely thin circular winding with the radius r0 (Figure 76). The parameter k takes its values in the interval ½0; 1Š and turns to unit in the coincidence limit z ¼ z0 ; r ¼ r0 : The function K1: reveals ð RÞ ðRÞ logarithmic singularity at k ! 1 : K1 ¼ lnð1 kÞ þ K1 ; with K1 being a regular residual.The Green function (4.64) inherits this type of singularity: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gðz; r; z0 ; r0 Þ ¼ 2 ln ðz z0 Þ2 þ ðr r0 Þ2 þ GðRÞ ; jGðRÞ j < 1; ð4:67Þ

To speed up numerical calculation, the regular residual GðRÞ may be tabulated. We will use the field equation (4.34) as the starting point for our considerations. Consider the magnetic field as the sum A ¼ Að jÞ þ AðmÞ þ AðIÞ of three terms originating from the sources of different nature. The first source is the winding current and nonhomogeneous coercitivity: ð 1 ðjÞ mðz0 ; r0 Þ½ jðz0 ; r0 Þ jM ðz0 ; r0 ފGðz; r; z0 ; r0 Þ dz dr: ð4:68Þ A ðz; rÞ ¼ c O

ds

Or

r⬘

z⬘ z

FIGURE 76 Definition of the Green function as the magnetic field induced by a thin circular winding.

Some Aspects of Magnetic Field Simulation

135

The second source is the permeability gradient, which may be nonzero due to either material nonhomogeneity or saturation effect: AðmÞ ðz; rÞ ¼

ð 1 < rðr0 A0 Þ; rmðz0 ; r0 Þ > Gðz; r; z0 ; r0 Þ dz dr: 4p r0 m

ð4:69Þ

O

Calculating the first two terms Að jÞ and AðmÞ is rather laborious because Eqs. (4.68) and (4.69) involve double integrals. However, the term Að; jÞ may be beforehand calculated and tabulated, whereas the term AðmÞ vanishes for the nonsaturable homogeneous materials. If we introduce the unknown virtual current density I ðsÞ on the interface G separating the magnetic media with different permeability, we can represent the third term as ð ð4:70Þ AðIÞ ðz; rÞ ¼ I ðsÞGðz; r; zðsÞ; rðsÞÞds: G

Here, as before, z ¼ zðsÞ; r ¼ rðsÞ is the piecewise smooth parameterization on G in terms of the arc-length s. The distribution of the virtual current density I ðsÞ should be found from the coupling condition 1 @n ðrAÞ 1 @ þ ðrAÞ þ < t; M1 >¼ ð2Þ n þ < t; M2 > ð 1 Þ r r m m

ð4:71Þ

being valid for tangential component of the H-vector on the interface G separating two neighboring media with different permeability values mð1Þ ; mð2Þ and coercitivity vectors M1 ; M2 (Figure 77). Using properties of the Green function (4.64), the two one-sided limits of the normal derivatives upon the interface may be expressed as @n ðrAÞ ¼ BðnjÞ þ BðnmÞ þ Dn ½I Š  2pI; r

r Γ

n τ

m (1)

m (2) z

FIGURE 77

Setting the coupling condition on the interface.

ð4:72Þ

136

Some Aspects of Magnetic Field Simulation

where

i 1 h BðnjÞ ¼ @n rAð jÞ ; r

i 1 h BðnmÞ ¼ @n rAðmÞ r

ð4:73Þ

and the integral operator Dn is the normal B-field component, which is the direct derivative of the magnetic potential (4.70) on G: ð 1 I ðs0 Þ@n ½rGðzðsÞ; rðsÞ; zðs0 Þ; rðs0 Þފds0 : ð4:74Þ Dn ½I ŠðsÞ ¼ rðsÞ G

It should be noted that the kernel in Eq. (4.74) may have an integrable singularity in the coincidence limit s ! s0 , which, however is not stronger than logarithmic at the regular points of the interfaces. Indeed, let fzðsÞ; rðsÞg and fzðs0 Þ; rðs0 Þg represent two close enough points on G; d ¼ fzðs0 Þ zðsÞ; rðs0 Þ rðsÞg is the vector joining these points, and n ¼ fnz ; nr g is normal vector at the point fzðsÞ; rðsÞg on G (Figure 78). At s0 ! s, we have the asymptotic equality @n ½rGŠ ¼ nr G þ r@n G 

2nr lnjdj

2r

< n; d > jdj2

;

ð4:75Þ

the second term of which tends to a constant in the coincidence limit because the vectors n and d become orthogonal, and their scalar product < n; d >¼ jdjcosn  jdj2 =2Rc , where Rc is the curvature radius at the point fzðsÞ; rðsÞg. As a result, @n ½rGŠ  2nr lnjdj þ r=Rc at s0 ! s: After substituting Eq. (4.72) into Eq. (4.71), we arrive at the secondtype Fredholm integral equation o mð1Þ mð2Þ n mð1Þ mð2Þ Dn ½I ŠðsÞ þ BðnjÞ ðsÞ þ BðnmÞ ðsÞ þ ð2Þ DMt ðsÞ; 2pI ðsÞ ¼ ð2Þ ð 1 Þ m þm m þ mð1Þ ð4:76Þ from which the unknown current distribution I ðsÞ can be found. It should be noted that the coercitivity enters both two terms in the right-hand side of Eq. (4.76) - implicitly through the ‘‘current’’ jM in the integral operator BðnjÞ and directly through the term DMt ¼< t; M2 M1 >. Γ

s′

n n d s Rc

FIGURE 78

The coincidence limit singularity in @n ½rGŠ. See text for details.

Some Aspects of Magnetic Field Simulation

137

Numerical solution of Eq. (4.76) requires partitioning of the boundary G into a set of intervals with the nodal points fst g; t ¼ 0 . . . T between them. For the nodes located at the regular points of the boundary, we define the shape functions in the form ð0Þ

ð4:77Þ

Ct ðsÞ ¼ rðsÞCt ðsÞ 8s s t 1 > ; > > > s s t t 1 > > > > < ð0Þ stþ1 s C t ðsÞ ¼ ; > st s > tþ1 > > > > > > : 0;

st

1

< s  st

st < s < stþ1 otherwise

ð4:78Þ

to approximate the unknown current density distribution (see Figure 79). The multiplier rðsÞ in Eq. (4.77) improves the near-axis approximation with regard to the fact that the current density vanishes at r ¼ 0: The sharp corners require special treatment because the current density may possess a powerlike singularity near the singular boundary points. We ð0Þ use the shape functions Ct ðsÞ ¼ rðsÞCt js st j g for such nodes. The singularity index g > 0 can be determined from the corner geometry and permeability constants according to Chapter 2. The example of the shape functions behavior is given in Figure 79. Having associated the set of unknown coefficients It with the nodal point st , we interpolate the current density distribution between the nodes as T X I t C t ðsÞ ð4:79Þ I ð sÞ ¼ t¼0

and find the It values by equating the weighted residuals 8 sð tþ1 <  mð1Þ mð2Þ  ð jÞ ð mÞ 2pI ðsÞ D ½ I Š ð s Þ þ B ð s Þ þ B ð s Þ Rt ¼ n n n : mð2Þ þ mð1Þ st 1 9 = ð1Þ ð2Þ m m ð0Þ C ðsÞds DM ð s Þ t ; t mð2Þ þ mð1Þ

ð4:80Þ

to zero according to the Galerkin technique. Numerical integration in Eq. (4.80) is performed by means of a special version of the Gauss quadrature scheme that require the integrand
to be calculated at certain inner
points inside each of the intervals st 1; st and st; stþ1 ; thus avoiding the laborious calculation of the kernel G and its gradient at the corner points.

138

Some Aspects of Magnetic Field Simulation

S5

S4

Ψ3

Ψ5

S3 S2 S1

Ψ2 Ψ1

Ψ4

Ψ0

0

S1

S2

S3

S4

S5

FIGURE 79 Example of the shape functions for virtual surface current approximation. The dashed line is the function rðsÞ on the boundary. See text for details.

4.9. HYBRID COMPUTATIONAL METHODS The preceding sections have shown that both the FEM and BEM can be used to calculate magnetic fields in axisymmetric and planar systems. The finite element approach is advantageous if magnetic saturation and volume currents must be taken into account. The BEM is less efficient in this case because it requires integration not only over 1D interfaces but also over some 2D subdomains. At the same time, only the BEM can provide the proper solution smoothness, which is critical for imaging charged particle optics applications. In addition, the BEM provides better accuracy with a smaller number of nodes and is well-suited for open magnetic lenses modeling, wherein correct restriction of the computational domain of interest is difficult. Although the advantages of BEM have made this method widely used in electrostatic problems of charged particle optics, further efforts are still required to extend the method to magnetostatics, especially nonlinear. The best approach is to combine the advantages of the FEM and the BEM in socalled hybrid methods. Kasper and Stro¨er (1990) proposed to calculate the vector magnetic potential distribution with FEM and use this distribution to form a Dirichlet condition for BEM calculation in vacuum domains. The hybrid BEM-FDM method was proposed by Weth and Becker (1999). This paragraph shows how the FEM and BEM can be combined in the framework of the variational approach. The main idea lies in decomposition of theScomputational domain, possibly infinite, into three subdomains S O ¼ OðLÞ OðSÞ OðBÞ (shown in Figure 80). Let us suppose that the

Some Aspects of Magnetic Field Simulation

Ω(B)

139

Γ⬘

m (1) m (2) Ω(L)

n

Γ

Ω(s)

FIGURE 80 Decomposition of the calculation domain into three subdomains: solenoid, saturable part of the yoke, and the linear subdomain, which includes the nonsaturable part of the yoke and the infinite vacuum volume.

materials contained in the subdomain OðLÞ are nonsaturable. At the same time, the materials may bear the distributed currents, coercitivity, or permeability gradient. Physically, the subdomain OðLÞ may represent a solenoid or a weak permanent magnet. The subdomain OðSÞ contains the saturable magnetic materials (perhaps, also with nonzero coercitivity), such as magnetic polepieces or strong magnets. Finally, the subdomain OðBÞ comprises the regions with piecewise constant permeability, including the vacuum ones. The first two subdomains are to be covered with a finite element mesh; the magnetic field within the third subdomain OðBÞ is to be found with the BEM. With this goal in mind, the boundary G ¼ @OðBÞ should be partitioned into a set of 1D intervals coinciding with the sides of the finite elements wherever G touches the subdomains OðLÞ or OðSÞ . The finite element and the boundary element solutions should be properly coupled on the interface G. This may be done quite naturally if we represent the energy functional W as the sum of integrals taken over the three subdomains: W ¼ W ðLÞ þ W ðSÞ þ W ðBÞ ;

ð4:81Þ

and minimize it with respect to the magnetic potential distribution in the entire computational domain. The finite element approximation of the energy functional (obtained in Section 4.5) is suitable for the subdomains OðLÞ and OðSÞ . Moreover, the superelement method (described in Section 4.7) allows representation of the term W ðLÞ in the form of a functional dependent on the magnetic potential distribution on the boundary @OðLÞ : The magnetic energy concentrated within the subdomain OðBÞ also can be calculated if we only know the magnetic potential on the boundary, as demonstrated by Lencova´ and Lenc (1984) in their work devoted to

140

Some Aspects of Magnetic Field Simulation

open magnetic lenses. Our aim here is to generalize this result to the subdomains containing magnetic materials with piecewise constant permeability, including the nonsaturated parts of the yokes. K S Ok Let the boundary element subdomain OðBÞ be a union OðBÞ ¼ k¼1

of the smaller subdomains O1 ; . . .; OK filled with materials having the permeability constants m1 ; . . .; mK , respectively. The magnetic energy concentrated within the subdomains Ok ; 1  k  K is ð ð 2 B 1 rðrAÞ2 dz dr ð4:82Þ r dz dr ¼ Wk ¼ 2p r 8pmk 4mk Ok

Ok

(the relation B2 ¼ ðrðrAÞ=rÞ2 has been used here). The 2D integral in Eq. (4.82) can be reduced to a contour integral over the boundary @Ok with the use of the Gauss theorem. Indeed, this yields   rðrAÞ rðrAÞ2 rðrAÞ þ rA div ¼ ; ð4:83Þ div½ArðrAފ ¼ div ðrAÞ r r r where the second term vanishes according to Eq. (4.34) if one puts the right-hand side of this equation zero. Thus, if n is the inward normal to @Ok ; I is the distribution of virtual currents, and the integral operator Dn ½I Š is given by Eq. (4.74), we have r 1 @n ðrAÞ ¼ Dn ½I Š 2pI, and, accordingly, ð ð 1 1 A@n ðrAÞds ¼ f2pI ðsÞ Dn ½I ŠðsÞgAðsÞrðsÞds: ð4:84Þ Wk ¼ 4mk 4mk @Ok

@Ok

0

Let us denote G the union of the interfaces separating the subdomains Ok ; 1  k  K from each other, in contrast to the interface G ¼ @OðBÞ separating the OðBÞ subdomain from the subdomains OðLÞ and OðSÞ . By summing the energy integrals Wk ; we obtain: ð 2pI ðsÞ Dn ½I ŠðsÞ W ðBÞ ¼ AðsÞrðsÞds 4mð2Þ G

ð 2p mð2Þ þ mð1Þ I ðsÞ þ mð2Þ mð1Þ Dn ½I ŠðsÞ  þ A ðsÞrðsÞds: ð4:85Þ 4mð2Þ mð1Þ G0

Here AðsÞ is the magnetic potential distribution on the boundary G; A ðsÞ is the magnetic distribution on the boundary G0 ; I ðsÞ is the current S potential 0 ð1Þ density on G G , and m and mð2Þ are the piecewise constant functions of the parameter s. The normal vector n in the designation of the integral operator Dn ½I Š on G0 is directed from the subdomain with permeability mð1Þ to the neighboring subdomain with the permeability mð2Þ (shown in Figure 80).

Some Aspects of Magnetic Field Simulation

141

Minimization of the functional (4.85) with respect to A (at the current densities fixed) leads to the homogenous second-kind Fredholm integral equation on G0 , being analogous to Eq. (4.76): 8 ð ð2Þ ð1Þ m m 1 < 2pI ðsÞ þ ð2Þ I ðs0 Þ@n ½rðsÞGðzðsÞ; rðsÞ; zðs0 Þ; rðs0 Þފds0 m þ mð1Þ rðsÞ : G 9 = Ð þ G0 I ðs0 Þ@n ½rðsÞGðzðsÞ; rðsÞ; zðs0 Þ; rðs0 Þފds0 ¼ 0 fzðsÞ; rðsÞg 2 G 0 : ;

ð4:86Þ

Equation (4.86) should be supplemented by the first-kind Fredholm integral equation on G ð I ðs0 ÞGðzðsÞ; rðsÞ; zðs0 Þ; rðs0 ÞÞds G

þ

ð

G

I ðs0 ÞGðzðsÞ; rðsÞ; zðs0 Þ; rðs0 ÞÞds0 ¼ AðsÞ fzðsÞ; rðsÞg 2 G:

ð4:87Þ

0

It should be remembered that the conditionality of Eq. (4.87) is guaranteed by the presence of the integrable logarithmic kernel singularity if we look for possible solutions in the class of rather smooth functions with prescribed singularities at irregular points of the boundary. Equations (4.86) and (4.87) determine the current density I as a linear operator ℜ whose argument is the magnetic potential A on the boundary G: I ðsÞ ¼ ℜ½AŠðsÞ;

s 2 G:

ð4:88Þ

The second integral in Eq. (4.85) vanishes in accordance with Eq. (4.86). Having substituted Eq. (4.88) into the first integral, we obtain the magnetic energy as an integral functional of the boundary potential distribution AðsÞ: ð 2pℜ½AŠðsÞ Dn ½AŠðsÞ AðsÞrðsÞds: ð4:89Þ W ðBÞ ¼ 4mð2Þ G

After boundary partitioning, the energy integral W ðBÞ may be represented in quadratic form as 1 X ðBÞ o Ak Am ; ð4:90Þ W ðBÞ ¼ 2 k;m km

142

Some Aspects of Magnetic Field Simulation

where Ak are the coefficients of magnetic potential approximation with the shape functions on the finite element mesh boundary. With no loss of ðBÞ generality, the matrix kokm k may be considered symmetrical. The next step consists in numerical minimization of the sum (4.81) with respect to the coefficients Ak that correspond to (1) the mesh nodes of the saturable subdomain OðSÞ , and (2) the boundary nodes of the nonsaturable subdomains OðLÞ and OðBÞ . The solution having been originally found on that set of nodes then should be reconstructed inside Oð LÞ according to Eq. (4.61), and the current density I should be determined from Eq. (4.88). The potential distribution in the subdomain OðBÞ , including the vacuum region in which the trajectories are to be traced, is then calculated as the integral

(a)

60 mm

P

40 mm

70 mm

100 AT

Z

10 −3

∆B/B

(b)

10 −4

10 −5

10 −6 1/8000 1/2000

1/1000

1/500

1/N

FIGURE 81 Test problem for a solenoid with homogeneous current distribution. (a) axial cross-section; (b) dependence of the field calculation error at the point P on the number of finite element N inside the solenoid subdomain.

Some Aspects of Magnetic Field Simulation

Aðz; rÞ ¼ G

ð

S

I ðsÞGðz; r; zðsÞ; rðsÞÞds

fzðsÞ; rðsÞg 2 OðBÞ

143

ð4:91Þ

G0

which may be consequently differentiated to obtain the B-field components and their higher derivatives with no substantial loss in accuracy. The other advantage of the hybrid approach over the conventional FEM is the substantial reduction of the number of mesh nodes involved in calculations. This reduction occurs because the linear subdomains should not be partitioned at all or, at the most, only their boundaries participate in the iterative minimization procedure. In practice, the hybrid method using the meshes with a few thousand elements ensures the same accuracy as the conventional FEM with as many as hundreds thousands elements. A test problem for a solenoid with rectangular cross-section is shown in Figure 81. Figure 82 illustrates the use of the hybrid method for simulation of the image tube with a wide photocathode work area. In this case, we need to construct a set of aberration expansions in the vicinity of different principal trajectories that originate from different points of the photocathode. This immediately implies rather high requirements for magnetic field accuracy and smoothness, which are to be met not only in the paraxial region near the main optical axis but also within the entire work volume of the tube. Of note, in the open-type magnetic system simulation, the hybrid FEM-BEM allows one to restrict the triangulation to the solenoid itself and the fraction of the yoke in which the saturation effect is essential.

FIGURE 82 Simulation of the image tube with combined electrostatic and magnetostatic focusing. The solenoid and saturated yoke region is covered with mesh as shown on the right side.

CHAPTER

5 Aberration Approach and the Tau-Variation Technique

Contents

5.1. Some Instructive Facts of The History of Aberration Theory 5.2. The Essence of the Tau-Variation Technique 5.3. The Tau-Variation Equations in Tensor Form 5.4. Arrival Time Variations and Contact Transformation 5.5. Jump Condition for Aberration Coefficients 5.6. Multiple Principal Trajectories Approach 5.7. Tolerance Analysis Using the Aberration Theory 5.8. Tracking Technique 5.9. Charged Particle Scattering

146 150 155 158 162 164 166 169 172

This chapter is devoted to the tau-variation technique – the versatile approach, which allows construction of a unified aberration theory of narrow and wide bunches in static and dynamic problems of charged particle optics. In Section 5.1, we recall some instructive facts of the history of aberration theory and show the preconditions that brought us to the developing of the tau-variation technique. Section 5.2 briefly describes the most essential advantages of this approach, whereas Sections 5.3 and 5.4 provide a detailed representation of the tau-variation technique in tensor form. Section 5.5 considers a special but practically important case when the electromagnetic field in the domain of the charged particles’ motion is non-smooth. Section 5.6 introduces the multiple principal trajectory approach, which combines the ideas of direct ray-tracing and aberration analysis. The last three sections represent the examples of application of the tau-variation technique to different problems of charged particle optics, including the tolerance analysis (Section 5.7), tracking technique (Section 5.8), and charged particle scattering (Section 5.9). Advances in Imaging and Electron Physics, Volume 155 ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)00805-7

2009 Elsevier Inc. All rights reserved.

#

145

146

Aberration Approach and the Tau-Variation Technique

5.1. SOME INSTRUCTIVE FACTS OF THE HISTORY OF ABERRATION THEORY The beginning of theoretical electron optics as a self-dependent branch of science may be conditionally ascribed to the year 1926, when Bush derived his famous expansion of axially symmetrical potential into the power series with respect to the distance from the symmetry axis (Bush and Bru¨che, 1937). Indeed, this remarkable result, which now seems so clear and simple from the state of our present knowledge, laid a basis for the aberration theory in charged particle optics. The ideas and methods of aberration theory had been originally borrowed from classical mechanics and light optics, and then, during the 1930s and 1940s, were substantially advanced by Bush, Bru¨che, Scherzer, Glaser, Recknagel, Picht, and other founders of classical electron optics. The transmission electron microscope originally was the primary object of studies on regularities of electron image formation. The possibility of expanding any particle trajectory into the power series with respect to a set of small parameters that describe the initial state of charged particles comprising the bunch rested on the strong mathematical basis of regular perturbations theory developed by Poincare. It was revealed that the coefficients of those expansions (the aberration coefficients) could be represented in the form of nonlinear integral functionals dependent on the axial potential distribution and its derivatives. The design of any electron-optical system with prescribed characteristics (in modern terms, the problem of electron-optical synthesis) thus could be divided into three separate problems to be solved in turn: (1) calculation of optimal axial distributions of electric and magnetic fields to obey the electron image requirements, (2) continuation of the optimal fields from the symmetry axis into space, and (3) shaping the appropriate geometry of electrodes and magnets using some properly chosen ‘‘pieces’’ of the solutions found. Formally, such an approach freed the researcher of the necessity of solving the boundary-value problems for the corresponding field equations, which, in itself, was very important at those times because of the lack of computation tools capable of calculation the electrostatic and magnetic fields numerically. The problem statement assumed that the desired (or optimal) electron-optical system geometry should be found directly from the axial field distributions fulfilling the prescribed requirements. This explains why the primary problem of imaging computational electron optics – calculating the geometry of electrodes and magnetic system along with the boundary voltages and currents to ensure the prescribed values of image quality characteristics – was named the direct problem of electron optics. When today’s powerful computers allow very accurate

Aberration Approach and the Tau-Variation Technique

147

direct calculation of electric and magnetic fields in any system with most sophisticated geometry, we commonly call the ‘‘old’’ direct problem the inverse one. This historical fact is a good example of how scientific terminology may change in relation to the current possibilities of science. From the modern viewpoint, the first, ‘‘paraxial’’ part of the direct (in the ‘‘old’’ sense) problem of electron optics constituted what we call now the nonclassical variational problem, or the optimal control problem. The second problem, connected with the continuation of axial distributions of electric and magnetic fields into space, in modern terminology represents a typical ill-conditioned problem of mathematical physics. Nothing can deny the obvious fact that the advances in those comparatively new areas of science have been largely based on numerical methods and computers. However, if we look back to the early 1930s, we can clearly see that the possibility of defining image characteristics in terms of only two functions of one variable (those are axial distributions of electric potential and magnetic induction) on the one hand, and the lack of powerful enough computational tools, on the other, provided almost unrestricted room for the sophisticated analytical skills of many authors. It would not be an exaggeration to say that the dawn of electron optics was purely ‘‘analytical’’, but what a beautiful dawn it was! The bounds of this monograph are too narrow to adequately review all the outstanding analytical results obtained in charged particle optics between the 1930s and 1960s. We touch on only two results that laid a basis for many important investigations in the field of theoretical electron optics and related applications. The first of the analytical results whose importance cannot be overestimated constitutes the famous Scherzer theorem (1936) on the unavoidability of spherical aberration in axisymmetric smooth fields. It is well known that spherical aberration represents one of the most significant aberrations in electron microscopy, and it is quite understandable that Scherzer’s theorem became a starting point for many investigations aimed at minimizing the spherical aberration in practice. The second example of the striking analytical abilities of old masters is the study by Tretner (1959) on the boundaries of chromatic aberration in electron lenses. By considering the extreme properties of chromatic aberration under the condition that the axial field strength is uniformly restricted, Tretner was able to derive not only rather accurate analytical estimations but also to forestall (at least, from the methodological standpoint) some general statements of modern nonclassic variation theory. As previously mentioned, the overwhelming majority of studies were based on the well-grounded regular aberration theory of narrow beams. By the end of the 1930s, it seemed that all principal problems of the aberration theory had been already solved, and researchers should only

148

Aberration Approach and the Tau-Variation Technique

classify the aberration coefficients and study their properties as applied to the focusing and deflection systems of various types. The comparatively smooth evolution of theoretical electron optics was disturbed by two papers published in the early 1940s. The first, by Recknagel (1941), was devoted to imaging problems of electron mirrors. The second paper, by the Russian physicist Artsimovich (1944), first raised the question whether it is principally possible to describe the regularities of image formation in the wide-beam emission-imaging electron-optical systems (image tubes) in terms of classic aberration theory. Since the work by Artsimovich played an important role in subsequent studies of emissionimaging electron optics, this work merits further detailed discussion. The main peculiarities of emission-imaging electron-optical systems are as follows: 1. The source of emitted electrons (cathode) is immersed into the focusing field. 2. The initial energy of emitted electrons is much less compared with the energy the electrons gain, having passed from the cathode to image receiver (commonly, the ratio is  10-3 or less). 3. The full angular aperture of elementary electron beams emitted from a fixed point of emitter amounts to 180 degrees. Artsimovich was the first to reveal that in this case the routine aberration procedure based on regular perturbation theory necessarily leads to some disturbing oddities in the aberration coefficients behavior near the emitter surface. From the mathematical standpoint, those singularities are directly connected with the presence of the ‘‘moving’’ singular point in the wellknown stationary trajectory equation 0 1 0 0 0 1 þ r r r @’ @’ @ A r00 þ ’ ’0 þ e 2 @z @r 3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 u u e 1 þ r0 r0 4 0 @’m @’ @’ þ r02 m ð2 þ r0 r0 Þ m 5 ¼ 0; ð5:1Þ r þ it @z @r @r 2m ’ ’0 þ e which can be easily derived from the Lorenz equations by using the energy integral. Here r ¼ x þ iy is a two-dimensional vector describing the charged particle position on the plane perpendicular to the main optical axis Oz; ’; ’m are, accordingly, the electrostatic and magnetostatic potentials; ’0 is the electrostatic potential at the starting point x0 ; y0 ; z0 ; e is the initial energy of a charged particle, expressed in potential units (the asterisk signifies complex conjugation and the upper prime signifies the differentiation with respect to the coordinate z).

Aberration Approach and the Tau-Variation Technique

149

It is easily seen that Eq. (5.1) degenerates in the vicinity of the starting point x0 ; y0 ; z0 at small initial energies e. Physically, the presence of such mathematical singularity points to the existence of some image-formation peculiarities that are intrinsic to the emission-imaging system and cannot be satisfactory described within the framework of regular aberration procedures of the narrow beam theory. The fact that this singularity was revealed by Artsimovich using not the general equation (5.1) but the paraxial equation being a linear approximation to Eq. (5.1) does not change the essential – Artsimovich’s paper had substantially shaken the well-established foundation of aberration theory and thus opened a new era in theoretical electron optics. During the period of 1950s and 1960, and especially in the 1970s, a large number of papers were aimed at overcoming the difficulties revealed by Artsimowich and constructing a correct aberration theory of wide beams. As mentioned by Hawkes and Kasper (1989), the main contributions to this area were made by Russian and Chinese scientists. Concurrently and practically independently, the narrow-beam theory confined by the assumption of small initial beam apertures and high initial energies of the charged particles comprising the beam had been in development and various stages of perfection. As the result, two very different and strongly isolated versions of the aberration theory – one for narrow and another one for wide charged particle beams – were clearly outlined by the mid-1970s. The difference between the narrow- and widebeam theories became so essential that the experts working in different areas of charged particle optics barely understood each other when speaking the aberration theory. Such a paradoxical state resembled that described in the famous novel by Jonathan Swift: The dwellers of the Lilliput Country were divided into two uncompromising belligerent groups – the ‘‘Little-endians’’ and the ‘‘Big-endians’’ – in regard to which end of the boiled egg should be eaten first! As often happens in science, the possibility of constructing a unified aberration theory valid for both the narrow and wide electron beams was found when an entirely new problem, with no outward connection to the general aberration problems, was formulated. The end of the 1970s was marked by a distinctive breakthrough in electron-optical recording of ultrafast events: Temporal resolution of streak image tubes approached and then, during the 1980s, steadfastly penetrated into the sub-picosecond range (1 ps ¼ 10-12 s). This created the opportunity to conduct unique experiments never feasible before, including, as example, studies on the thin structure of laser radiation and time-resolved electron diffraction (TRED) experiments on direct observations of atomic/molecular dynamics in solid and gaseous matter. The advances in temporal resolution brought to the fore the problem of accurate theoretical and numerical description of spatiotemporal properties of ultrashort electron bunches

150

Aberration Approach and the Tau-Variation Technique

emitted from the photocathode under laser pulse excitation. The nonstationary nature of this problem, which is related to the use of timedependent electric fields for the bunch focusing and deflection; Coulomb interaction between the charged particles comprising the bunch; transient electromagnetic effects localized in the immediate vicinity of the photocathode; and so forth, render it impossible to use the stationary trajectory equation (5.1) as a basis for the aberration theory in dynamic case.

5.2. THE ESSENCE OF THE TAU-VARIATION TECHNIQUE The problem of sub-picosecond photoelectronic imaging provided a powerful incentive to develop a new general aberration approach valid both for stationary and nonstationary charged particle optics. The preprint version of such approach termed the tau-variation technique, as applied to the theory of temporal aberrations in cathode lenses, was published by Monastyrskiy and Schelev in 1980. More extended versions of the tauvariation technique, along with applications of the method to different problems of emission-imaging charged particle optics, can be found in Kulikov et al. (1985), Kolesnikov and Monastyrskiy (1988), Il’in et al. (1990), Degtyareva, Monastyrskiy, et al. (1995, 1998). The most versatile tensor representation of the tau-variation technique was given in Greenfield, Monastyrskiy and Tarasov (2006). Before proceeding to details, we would like to emphasize the more essential features of the method in question. According to the tau-variation technique, the computation of spatial and temporal aberrations consists of two numerical procedures that may be carried out ‘‘synchronously’’: (1) numerical integration of a system of differential equations for isochronous variations (tau-variations) of the Lorenz equation, and (2) transformation of the tau-variations into the aberration coefficients according to certain algebraic correlations (the so-called contact transformation). The tau-variation technique is most versatile. It allows unified construction of the aberration expansions for narrow and wide beams in any stationary or non-stationary electromagnetic field. The tau-variation technique has high computational stability; this follows from the fact that the tau-variations represent regular components of the aberration coefficients. According to the tau-variation procedures, the regular aberration components are calculated separately from the singular ones which, as was first noticed by Recknagel and Artsimovich, may have some singularities in the low potential region near the starting points on the emitter surface or turning points in electron mirror. Another important advantage of the tau-variation technique is that it can be implemented in the most versatile tensor form that allows effective

Aberration Approach and the Tau-Variation Technique

151

combining of the ‘‘direct ray-tracing’’ and ‘‘aberration’’ ideas in charged particle optics. Now let us consider the charged particles motion described by the Lorenz equation in tensor representation   q 1 ijk
k j i i i
ð5:2Þ x¨ ¼ f ðx; x; tÞ ¼ G E þ e x B ; m c where xi ði ¼ 1; 2; 3Þ are homogeneous Cartesian coordinates of the 3D vector x; t is time; Ei ; Bi are, respectively, electric and magnetic field components; q=m is charge-to-mass ratio; and eijk is the alternator tensor (e.g. the tensor being antisymmetrical with regard to all its components and fulfilling the condition e123 ¼ 1). Hereafter, summation is made over all repeated ‘‘dumb’’ indexes i; j; k; . . .. The relativistic factor G is " # 1=2 3   1X i 2

x ; ð5:3Þ Gðx Þ ¼ 1 c2 i¼1 where c is the speed of light. The initial conditions for Eq. (5.2) xi ðt ðxÞ; xÞ ¼ Xi ðxÞ; x
i ðt ðxÞ; xÞ ¼ V i ðxÞ; 0

0

ð5:4Þ

are dependent on the parameters vector x ¼ fxa g; a ¼ 1; 2; . . .; P, belonging to a set  of the P-dimensional vectorial space RP . The functions  i  i XðxÞ ¼ X ðxÞ ; V ðxÞ ¼ V ðxÞ determine the initial phase state of a particle on the emitter surface, and the function t0 ðxÞ determines the start time moment. The electric and magnetic field vectors Ei ðx; t; xÞ, Bi ðx; t; xÞ in the right-hand side of Eq. (5.2) are assumed to be dependent not only on the coordinates and time but also on the parameters vector x ¼ fxa g, which makes both the optimization problem and the problem of mechanical tolerances computation automatically included in the tau-variations procedure set forth below. Consider a fixed vector xð0Þ together with its vicinity Uxð0Þ  . We will say that the set of parameters x ¼ fxa g unambiguously determine the bunch of perturbed charged trajectories in the vicinity of the particles

principal trajectory xi ¼ xi t; xð0Þ if each of the trajectories xi ¼ xi ðt; xÞ, x 2 Uxð0Þ represents a unique (for given x) solution of the Lorenz equation (5.2) and is defined on a common (for all such trajectories) time interval. Without any loss of generality, we may put xð0Þ ¼ 0 and consider fxa g as small parameters  0 that determine the bunch in the vicinity of the principal trajectory xi ¼ xi ðt; 0Þ. Depending on the particular electron-optical problem in question, the parameters fxa g may specify the initial positions of the charged particles comprising the bunch; the initial velocities, energies, and angles; the particles’ start time; the structural parameters responsible for geometric and potential perturbations of the system’ boundary; the excitation currents in magnetic coils, and so on.

152

Aberration Approach and the Tau-Variation Technique

As an example, let us assume that the emitter surface equation may be 1 2 explicitly resolved in the vicinity  1of 2the  point x ¼ x ¼ 0 with respect to 3 3 the third coordinate x : x ¼ S0 x ; x . In this case, we may parameterize the initial position of the bunch’s particles on the emitter surface by means of the parameters x1 ; x2 by putting X 1 ð x Þ ¼ x1 ;

X 2 ð x Þ ¼ x2 ;

X3 ðxÞ ¼ S0 ðx1 ; x2 Þ:

ð5:5Þ

i ¼ 1; 2; 3

ð5:6Þ

The definition V i ð xÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2jqj=m xiþ2

for

ascribes the square roots of the corresponding initial energy components (expressed in potential units) to the parameters x3 ; x4 ; x5 (Figure 83a). Such definition is most convenient in the wide-beam problems of emission-imaging charged particle optics, when the systems with either photocathode or thermionic cathode are considered. Another choice of the parameters x3 ; x4 ; x5 is more reasonable if we consider a narrow beam of particles with the nonzero average velocity relativistic. Let us introduce the relativistic factor u0 that  may be  1=2 and the corresponding kinetic energy e0 ¼ G0 ¼ 1 u20 =c2 ðG0 1Þmc2 on the principal trajectory. Using the small parameter x5 , the perturbed particle energy can be expressed as eðx5 Þ ¼ e0 ð1 þ x5 Þ. Correspondingly, the absolute value of the particle velocity in this case is pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi  1 þ x5 2 þ ð G 0 1Þ ð 1 þ x 5 Þ e 0 x5 þ oðx5 Þ uðx5 Þ ¼  u0 1 þ 1 þ ðG0 1Þð1 þ x5 Þ G0 ð1 þ G0 Þ m ð5:7Þ

(a)

(b)

x1

x5 x3

x 3,4,5

u0

x1

S0 x2

x2

S0

FIGURE 83 The small parameters definition in wide-beam (a) and narrow-beam (b) cases. See text for details.

Aberration Approach and the Tau-Variation Technique

153

the nonrelativistic limit, we get the simpler correlation uðx5 Þ ¼ pIn ffiffiffiffiffiffiffiffiffiffiffiffiffi u0 1 þ x5  u0 ð1 þ x5 =2Þ. By putting V 1 ðxÞ ¼ uðx5 Þsinx3 V 2 ðxÞ ¼ uðx5 Þsinx4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 3 ðxÞ ¼ uðx5 Þ 1 sin2 x3 sin2 x4 ;

ð5:8Þ

we ascribe the meaning of initial angular deviations from the principal trajectory to the small parameters x3 and x4 (Figure 83b). One of the small parameters, say x6 , may be reserved to designate the start time perturbation. With this goal in mind, assuming that the principal trajectory starts at the time moment t0 ð0Þ ¼ 0, let us put t 0 ð x Þ ¼ x6 :

ð5:9Þ

As mentioned previously, some other small parameters may represent small deviations of the structural parameters of electrodes and magnetic system from the nominal values and thus be responsible for variations of the electromagnetic field that governs the bunch motion. According to the well-known Poincare theorem (for an example, see Nayfeh, 1981), under the condition that the electromagnetic field in the domain of the bunch motion is smooth enough, the solutions xi ðt; xÞ of the Lorenz equation (5.2) may be expanded into the Taylor power series xi ðt; xÞ ¼ xi0 ðtÞ þ

P X a¼1

xia ðtÞxa þ

P P 1X 1 X xiab ðtÞxa xb þ xi ðtÞxa xb xg þ . . .; 2 a;b¼1 6a;b;g¼1 abg

ð5:10Þ

which uniformly converges within a finite time interval t 2 ½0; tmax Š. The coefficients of this expansion are the partial derivatives

@ n xi ðt; xÞ

ð5:11Þ xiab... ðtÞ  @xa @xb . . . x¼0

calculated on the principal trajectory at a fixed time moment t. We call the coefficients xiab... ðtÞ in Eq. (5.11) the tau-variations (or isochronous variations) because the expansion Eq. (5.10) determines the phase state of the particles comprising the bunch at a fixed time moment t, as shown in Figure 84. Going farther, we obtain the differential equations for the tau-variations (5.11), and slightly forestalling the events to come, we may say that the numerical solution of those regular equations represents the first of the two tau-variation procedures. The series like those shown in Eq. (5.10) are widely used in celestial mechanics where they serve (as example) to describe the deviation of a heavenly body motion from some nominal trajectory at any fixed time moment, or in other words, in any temporal plane t ¼ const. In contrast to

154

Aberration Approach and the Tau-Variation Technique

Initial parameters set Ξ

t = const x(t, x )

ξ

x0(t)

Principal trajectory

FIGURE 84

On the tau-variations definition.

that application, in charged particle optics we are primarily interested in the particle distribution on a smooth surface S, which we commonly call the particle collector or image receiver. Let tS ðxÞ be the time for an individual particle ‘‘marked’’ by the parameter x to arrive at the surface S, and xS ðxÞ ¼ xðtS ðxÞ; xÞ the particle’s position on the surface S. Under the same general assumptions   of smoothness, we may try to represent the function xS ðxÞ ¼ xiS ðxÞ as a series xiS ðxÞ ¼ xiS ð0Þ þ

P   P   P   X 1X 1 X xiS a xa þ xiS ab xa xb þ xiS abg xa xb xg þ . . .; 2 6 a¼1 a;b¼1 a;b;g¼1

ð5:12Þ

in which  i @ n xiS ðxÞ j ; xS ab...  @xa @xb . . . x¼0

n ¼ 1; 2; . . .

ð5:13Þ

are consecutive derivatives of the function xiS ðxÞ calculated on the principal trajectory at x ¼ 0. Obviously, Eq. (5.12) is nothing more than the aberrational representation of the beam on the surface S with respect to the set of small parameters fxa g. By introducing special designations for the aberrational coefficients  i xS ab...  i  ð5:14Þ xS jxa xb . . . ¼ m1 !m2 ! . . . mP !

where the integer numbers mp ð1  p  PÞ reflect how many times the  parameter xp is encountered in the parameters combination xa xb . . . , we attach a more habitual form to the expansion in Eq. (5.12): X X    xiS jxa xb xa xb xiS jxa xa þ xiS ðxÞ ¼ xiS ð0Þ þ X1aP  i  1abP ð5:15Þ xS jxa xb xb xa xb xg þ . . . þ 1abgP

155

Aberration Approach and the Tau-Variation Technique

i Now, the question is how the tau-variations according to  i xab... defined  Eq. (5.13) and the aberrations coefficients xS jxa xb . . . in Eq. (5.14) are interconnected? The next section shows that the aberrations coefficients  i  xS jxa xb . . . may be easily obtained from the tau-variations xiab... by an algebraic transformation that we call the contact transformation. That transformation comprises the second of the tau-variation procedures. In the simplest case, such transformation is performed only on the particle collector or image receiver surface to evaluate the ‘‘final’’ aberrations of the bunch. Nevertheless, often it is of interest to know the aberration coefficients not only at the very end of the bunch ‘‘history’’ but also on some intermediate virtual surfaces (virtual screens) being intentionally placed in the path of the bunch motion. Moreover, the possibility of displaying the aberration coefficients in the form of some continuous curves originating from a physical or virtual emitter may offer the researcher vital information regarding the optical properties of the system in question. In this case, a virtual surface (or simply a plane) to ‘‘record’’ the aberration coefficients is to be conceived moving together with the principal trajectory of the bunch, and the contact transformation of the tau-variations into the aberration coefficients are to be made on every integration step of the differential equations for tau-variations.

5.3. THE TAU-VARIATION EQUATIONS IN TENSOR FORM Consecutive varying of the Lorenz equation (5.2) on the principal trajectory with respect to xa immediately yields a set of ordinary differential equations for the tau-variations xiab... ðtÞ. The first-order tau-variations xia meet the system of linear non-homogeneous differential equations h i

i q q h x¨ ia ¼ G Da Ei þ eijk x
k Da B j þ x
ka B j þ ðDa GÞ Ei þ eijk x
k B j ; ð5:16Þ m m

in which the symbol Da denotes full variation on the trajectories of Eq. (5.2): Da Ei ¼ Eia þ Eij xja ; " @G
j Da G ¼
j x a ¼ 1 @x

Da Bi ¼ Bia þ Bij xja ; 3   1X 2 x
i c2 i¼1

#

3=2

ð5:17Þ

x
jx
ja : c2

ð5:18Þ

Hereafter, the symbols Eij and Bij denote spatial partial derivatives of the field components, so that Eij ¼ @Ei =@xj and Bij ¼ @Bi =@xj , respectively. The tensors with additional Latin and Greek indexes signify, accordingly, the higher-order derivatives with respect to the coordinates and parameters. It is easy to see that, due to the field equations, those tensors are symmetrical with respect to the transposition of any Greek n indexes. Hereafo n two Latin or o ijk

ijk

ter, we refer to the sets of the tensors Ei ; . . . Eab ; . . . and Bi ; . . . Bab ; . . . , with the total number of indexes in each set up to K þ 1 inclusively, as the

156

Aberration Approach and the Tau-Variation Technique

(Kþ1)-order field tensors. The components of those tensors enter the differential equations needed to obtain the tau-variations of the K-th order. The variation of the initial conditions (5.4) at x ¼ 0 gives the initial conditions for the first-order tau-variation equations (5.16):  i   i  @X @t0

@V @t0

i i i i
; x ð 0Þ ¼ V ð xÞ f ðXðxÞ; V ðxÞ; t0 ðxÞÞ xa ð 0Þ ¼ @xa x¼0 a @xa x¼0 @xa @xa ð5:19Þ

As an example, in the particular case of the wide-beam parameters fxa g defined according to Eq. (5.5), (5.6), and (5.9), the nonzero components of initial conditions (5.19) are @S0 3 @S0 ; x2 ð 0Þ ¼ 2 1 @x @x

x11 ð0Þ ¼ 1; x22 ð0Þ ¼ 1; x31 ð0Þ ¼ x
13 ð0Þ ¼ x
24 ð0Þ ¼ x
35 ð0Þ ¼ x
i6 ð0Þ ¼

q i Ej m x¼0;t¼0

sffiffiffiffiffiffiffiffi 2jqj m

ði ¼ 1; 2; 3Þ

ð5:20Þ




All other components xia ð0Þ; xa ið0Þ are zero. In the narrow-beam case defined by Eq. (5.7) and (5.8), we get another set of non-zero components x11 ð0Þ ¼ 1;

x22 ð0Þ ¼ 1

x
13 ð0Þ ¼ x
24 ð0Þ ¼ u0 ; x
35 ð0Þ ¼ x
36 ð0Þ ¼

u0 ; x
i6 ð0Þ ¼

u0 G0 ð1 þ G0 Þ

f i jx¼0;t¼0

ði ¼ 1; 2; 3Þ

ð5:21Þ

Now, only some slight change in initial conditions for the differential equations describing the tau-variations is needed to switch the computation process from the narrow- to the wide-beam case. By varying the first-order tau-variation Eq. (5.16) with respect to xb , we obtain the linear nonhomogeneous differential equation for the secondorder tau-variations x¨ iab ¼



i q h G Dab Ei þ eijk x
kab B j þ 2 < x
kb Da B j > þ x
k Dab B j m h

i q þ 2 < ðDa GÞ Db Ei þ eijk x
k Db B j þ x
kb B j > m i h q Dab G Ei þ eijk x
k B j þ m

ð5:22Þ

Aberration Approach and the Tau-Variation Technique

157

where ij

j

j

j

ij

j

j

j

Dab Ei ¼ Eiab þ 2 < Ea x b > þ Eijk x a x bk þ Eij xab Dab Bi ¼ Biab þ 2 < Ba x b > þ Bijk xa xkb þ Bij x ab Dab G ¼

@G
i @2G
i
j x x ax b þ ab @ x
i @ x
i @ x
j

ð5:23Þ ð5:24Þ

  and the tensor symmetrization by two indexes < fab >¼ fab þ fba =2 is used. For the third-order tau-variations, we have

i q h x¨ iabg ¼ G Dabg Ei þ eijk x
kabg Bj þ 3 < x
kab Dg B j > þ 3 < x
ka Dbg B j > þ x
k Dabg B j m h

i q þ 3 < ðDa GÞ Dbg Ei þ eijk x
k Dg Bj þ 2x
kb Dg Bj þ x
kbg Bj > m

i h q  þ 3 < Dab G Dg Ei þ eijk x
k Dg Bj þ x
kg Bj > m i h q Dabg G Ei þ eijkx
k Bj ð5:25Þ þ m where ij

j

ij j

j

Dabg Ei ¼ Eiabg þ 3 < Eab xg > þ 3 < Ea xbg > þ Eij xabg

j

j

ijk j

þ 3 < Ea xb xkg > þ 3 < Eijk xab xkg > þ Eijkm xa xkb xm g ij

j

ij j

j

Dabg Bi ¼ Biabg þ 3 < Bab xg > þ 3 < Ba xbg > þ Bij xabg

þ 3 < Ba xb xkg > þ 3 < Bijk xab xkg > þ Bijkm xa xkb xm g

ð5:26Þ

@G
i @2G
i
j @3G j x
ia x
b x
kg x x x þ 3 < > þ abg ab g @ x
i @ x
i @ x
j @ x
i @ x
j @ x
k

ð5:27Þ

ijk j

Dabg G ¼

j

j

and the tensor symmetrization by three indexes < fabg >¼

fabg þ fagb þ fbag þ fbga þ fgab þ fgba 6

ð5:28Þ

is used. The initial conditions for the differential equations (5.22) and (5.25) can be easily derived from Eq. (5.4) by double and triple varying with respect to xa (we leave this calculation for the reader). It is noteworthy that the equations are not as complicated as they may seem at the first glance – the derivation of these equations is no more difficult than the differentiation of a composite function. Let us evaluate

158

Aberration Approach and the Tau-Variation Technique

the number of the tau-variation equations to be numerically solved and the corresponding number of the field tensor components to be calculated. The total number of the tau-variation equations needed to construct the aberrational series of the K-th order with respect to P parameters fxa g is NK;P ¼

3ðP þ 1Þ! K!ðP þ 1 KÞ!

3

ð5:29Þ

(three equations describing the principal trajectory itself are not included). Each of the ðK þ 1Þ-order fields tensor contains MK;P ¼

K ðK

1 Þ þ 3 ð P þ 3 Þ ð K þ P þ 2Þ ðK þ P þ 1Þ! K!ðP þ 3Þ!

ð5:30Þ

independent components, where P is the total number of parameters xa . Thus, the numerical integration of one principal trajectory itself, with no tau-variations involved ðK ¼ 0Þ, requires M0;P ¼ 3 components of each of the first-order field tensors Ei and Bi to be calculated (it is readily understood that MK;P do not depend on P in this case). With P ¼ 6 parameters defining the initial conditions, we need M1;6 ¼ 27 components to find the first-order aberrations ( M0;P plus six second-order derivatives like Eij and 3  6 ¼ 18 mixed derivatives like Eia ). Accordingly, M2;6 ¼ 136 and M3;6 ¼ 505. The number of independent nonzero tau-variations and the corresponding field tensor components are essentially smaller if the principal trajectory and/or the electromagnetic field possess some kind of symmetry. For example, in the axisymmetric electrostatic cathode lenses, there are only 9 nonzero independent aberration coefficients in the third-order aberrational expansion (5.15) constructed in the vicinity of the main optical axis (see Chapter 7 for more details).

5.4. ARRIVAL TIME VARIATIONS AND CONTACT TRANSFORMATION The key point of the contact transformation is the arrival time variations. Let us represent the time tS ðxÞ for any particle with a given set of small parameters x ¼ fxa g to arrive at the particles collector S in the form of the Taylor expansion tS ð xÞ ¼ t S ð 0Þ þ

P X a¼1

t a xa þ

P P 1X 1 X tab xa xb þ tabg xa xb xg þ . . . ; ð5:31Þ 2 a;b¼1 6 a;b;g¼1

in which tS ð0Þ is the arrival time on the principal trajectory, and the coefficients ta ; tab ; tabg ; . . . are the arrival time variations

159

Aberration Approach and the Tau-Variation Technique

tab... 

@ n tðxÞ

; n ¼ 1; 2; . . . @xa @xb . . . x¼0

ð5:32Þ

calculated on the principal trajectory at x ¼ 0. If Sðx; xÞ ¼ 0 is the equation of the particle collector or image receiver surface S (in the case in question, the use of the same letter S both for the surface itself and its equation would not result in any misunderstanding), the equality SðxðtS ðxÞ; xÞ ¼ 0

ð5:33Þ

holds identically true for any x being small enough. The consecutive varying of Eq. (5.33) with respect to x on the principal trajectory at x ¼ 0 gives the chain of equalities ð5:34Þ Si x
i t þ F ¼ 0; Si x
i t þ F ¼ 0; Si x
i t þ F ¼ 0; . . . : a

a

ab

ab

abg

abg

in which Fa ¼ Si xia þ Sa

ð5:35Þ



Fab ¼ Si f i ta tb þ 2 < x
ia tb > þ x iab

j j þ Si j x
ix
j ta tb þ 2 x
i < x a tb > þ x ia x b þ 2 < S i t > x
i þ 2 < S i x i > þ S a b

a b

ð5:36Þ

ab



Fabg ¼ Si Dt f i ta tb tg þ 3 < Da f i tb tg > þ3 < x
ia tbg > þ3 < x
iab tg > þxiabg



j j þ Sij f i x
j ta tb tg þ 3 < x a tb tg > þ 3Sijx
i < x
ja tb tg > þ < x ab tg >

j j þ 3Sij < x
ia xb tg > þ < xia x bg >

j j þ Sijk x
ix
jx
k ta tb tg þ 3 x
ix
j < x ka tb tg > þ3 x
i < xa x kb tg > þxia x b x kg

ij þ 3 < Sia tb tg > xi þ 6 < Sia x
ib tg > þ 3 < Sia xibg > þ 3 < Sa tb tg > x
ix
j j ij ij þ 6 < S xi t > x
j þ 3 < S xi x > þ 3< Si t > x
i þ 3 < Si xi >þS a b g

a b g

ab g

ab g

abg:

ð5:37Þ

Here f i are the right parts of the motion equations (5.2) and Dt f i ¼

@f i @f i
j @f i j þ x þ
j f @t @xj @x

Da f i ¼

@f i @f i j @f i
j x ; þ x þ @xa @xj a @ x
j a

ð5:38Þ

160

Aberration Approach and the Tau-Variation Technique

are, correspondingly, the full time derivatives and variations with respect to x of the right-hand side of the Lorenz equation (5.2) on the principal trajectory. It can be directly seen from Eqs. (5.35)–(5.38) that each of the variations Fa ; Fab ; Fabg of, respectively, the first, second, third, and higher orders explicitly depends on all the tau-variations xa ; xab ; xabg ; . . . up to the given order inclusive and all the arrival time variations ta ; tab ; tabg ; . . . up to the order being by unit less. Let us assume that the principal trajectory is not tangential to the surface S at the arrival point xS ðtð0ÞÞ, which implies that < ▽S; x
>¼ Six
i 6¼ 0 at x ¼ 0 on S. In this case, Eq. (5.34) can be consecutively resolved with respect to ta ; tab ; tabg ; . . . ta ¼

Fa ; Si x
i

tab ¼

Fab ; Si x
i

tabg ¼

Fabg ;...: Si x
i

ð5:39Þ

which gives the explicit polynomial representation of the arrival time variations ta ; tab ; tabg ; . . . through the tau-variations xa ; xab ; xabg ; . . . of the Lorenz equation (5.2). Now we need only express the aberration coefficients xiS jxa xb . . . or (what is more convenient if we wish to avoid the factorial multipliers), the on-surface variations xiS ab... in Eq. (5.13) through the arrival time variations ta ; tab ; tabg ; . . . and the tau-variations xa ; xab ; xabg ; . . . Varying of the equality xS ðxÞ ¼ xðtS ðxÞ; xÞ yields the chain of correlations ðxi Þ ¼ xi þ x
i t S a

a

a

xiSab ¼ xiab þ x
i tab þ 2 < x
ia tb > þf i ta tb xiSabg ¼ xiabg þ x
i tabg þ 3 < x
ia tbg > þ3 < x
iab tg >

þ 3f i < tab tg > þ 3 < Da f i tb tg > þ Dt f i ta tb tg ;

ð5:40Þ

determine the polynomialtranswhich, when combined with Eq. (5.32),   formation ℵ: xa ; xab ; xabg ; . . . ! ðxiS jxa Þ; ðxiS jxa xb Þ; ðxiS jxa xb xg Þ; . . . converting a set of tau-variations into the set of aberration coefficients of the same maximum order. The transformation ℵ is called the contact transformation on the surface S in the vicinity of the principal trajectory x ¼ 0. The geometrical essence of the contact transformation is shown in Figure 85. Thus, as mentioned generally in Section 5.2, the calculation of aberration coefficients on the arbitrary surface S; under the condition that Si x
i 6¼ 0 on the principal trajectory upon S; comprises the numerical integration of the tau-variation equations (5.16), (5.22), and (5.25) and the contact transformation of the corresponding tau-variations into the aberration coefficients on the surface S according to Eqs. (5.39) and (5.40). special consideration. The The case Si x
i ¼ 0 is singular and merits  scalar product of the gradient rS ¼ Si being normal to the surface S and the particle’s velocity vector x
¼ x
i may be zero in two cases: (1) when the principal trajectory comes tangentially onto the surface S,

Aberration Approach and the Tau-Variation Technique

S=0

161

t = t S0 = const ∇S = {S i} . {xi}

FIGURE 85

The definition of contact transformation.

and (2) when the principal trajectory velocity x
is zero on S. The first case is specific to the turning points in electron mirrors whereas the second case is common to cathode lenses. (We direct readers’ attention to the fact that exactly these special cases were the main concern of the previously cited works by Recknagel and Artsimovich). In both cases, the differential equations for tau-variations remain regular, whereas the contact transformation is not defined because Eq. (5.34) can not be resolved with respect to the arrival time variations. This clearly indicates that the aberration expansions (5.12) and (5.31) in the form of Taylor series does not exist in the vicinity of the singular points, where the scalar product Si x
i turns to zero. This peculiarity of cathode lenses and electron mirrors in due time created a barrier between the aberration theories of narrow and wide beams; now this barrier is completely removed. The presence of singular points on the principal trajectory does not affect the process of aberration coefficients calculation in the frame of the tau-variation technique. Indeed, when integrating the differential equations for tau-variations being regular everywhere, we can simply ‘‘pass through’’ the singularities, not transforming the tauvariations to aberration coefficients in the singular point’s vicinity. Thus, the tau-variation technique offers a regular means of calculating the aberration coefficients in any electron/ion-optical system, including cathode lenses and electron mirrors. It is naturally to ask the following question: if the Lorenz equation does not allow the regular, Taylor-type, aberration expansions in the vicinity of the points where the scalar product Si x
i on the principal trajectory turns to zero, what is the true asymptotics of the charged particle trajectories xðt; xÞ with respect to x in the vicinity of those points? The answer to this question requires some concepts of the boundary layer theory to be invoked, which is provided in Section 7.4.

162

Aberration Approach and the Tau-Variation Technique

5.5. JUMP CONDITION FOR ABERRATION COEFFICIENTS All our previous considerations in this Chapter have been based on the assumption that the electromagnetic field in the domain of charged particle motion is smooth. This condition is certainly broken, for example, if the bunch of charged particles passes through a fine-structure grid considered in the macroscopic approximation (e.g. as a solid electrode being transparent for charged particles; see the Section 1.11). In this case, the electric field tensor components are discontinuous on the grid surface. This fact, as we will show in this section, entails the corresponding discontinuity of the tau-variations. (Before proceeding, we recommend that readers acquaint themselves with Appendix 5, in which the problem of isochronous variations discontinuity is considered in more general form). The starting point for deriving the jump condition for tau-variations is the assumption that the individual charged particle trajectories are continuously differentiable as functions of time on the surface of electromagnetic field discontinuity (we omit the exceptional case of double surface charge layer). Let Sðx; xÞ be theequation of the discontinuity surface S; on  which the right-hand side f ¼ f i of the motion equation (5.2) undergoes   i ðf i ÞS , where ðf i Þþ the discontinuity f i S ¼ ðf i Þþ S S ; ðf ÞS are limiting values of the components of the function f on both sides of S (in this ðAÞS of any section we use the symbol ½AŠS to designate the jump ðAÞþ S function A on the surface S). If x ðt; xÞ is the perturbed trajectory of the Lorenz equation (5.2) before the arrival of the trajectory at the surface S; tS ðxÞ is the arrival time, and xþ ðt; xÞ is the trajectory continuation through the surface S; we have by definition xþ ðtðxÞ; xÞ ¼ x ðtðxÞ; xÞ; x
þ ðtðxÞ; xÞ ¼ x
ðtðxÞ; xÞ:

ð5:41Þ

The tau-variation equations for xa ; xab ; xabg ; . . . are valid and can be numerically integrated up to the time moment tS ð0Þ of the principal trajectory’s arrival at the discontinuity surface Sðx; 0Þ. Our immediate þ þ task is to derive the initial condition for the continuation xþ a ; xab ; xabg ; . . . of the tau-variations through S. The first variation of Eq. (5.41) with respect to xa gives x
þ t þ xþ ¼ x
t þ x ; x
þ t þ x
þ ¼ x
t þ x
: ð5:42Þ a

a

a

a

a

a

a

a

With regard to the fact that x
þ ¼ x
on the discontinuity surface S, we obtain from Eq. (5.42) the jump (coupling) conditions for xia ; x
ia on S (see Figure 86)  i    þ  i   i  þ 
i  xa ¼ 0; x
ia S ¼ x
ia f S ta : ð5:43Þ xa ¼ xa S ¼ xia

163

Aberration Approach and the Tau-Variation Technique

(a)

(b) S X S

Jump Jump

t = t S0 = const Jump Jump

X

FIGURE 86 The jump condition for tau-variations on the discontinuity surface. (a) in the coordinate space; (b) in the phase space.

In the same manner, the jump conditions for xiab ; xiabg can be obtained from Eq. (5.41) by, accordingly, double and triple varying with respect to xa : ½xiab ŠS ¼ 2 < ½x
ia ŠS tb > ½ f i ŠS ta tb ½x
i Š ¼ 2 < ½D f i Š t > ½D f i Š t t ð5:44Þ ab S

a

S b

t

S a b

½xiabg ŠS ¼

3 < ½x
iab ŠS tg > 3 < ½x
ia ŠS tbg > 3½ f i ŠS < ta tbg > 3 < ½Da f i ŠS tb tg > ½Dt f i ŠS ta tb tg

½x
iabg ŠS ¼

3 < ½D2ab f i ŠS tg > 3 < ½Da f i ŠS tbg > 3½Dt f i ŠS < ta tbg > 3 < ½D2ta f i ŠS tb tg > ½D2t f i ŠS ta tb tg

ð5:45Þ

Here, D2t f i ; D2ta f i ; D2ab f i designate the full second-order derivatives D2t f i ¼

D2ta f i ¼

@2f i @2f i
j @2f i j @2f i x þ2 þ2 f þ j k x
jx
k j j 2
@t @t@x @x @x @t@ x

2 i 2 i i i @ f
j f k þ x
k f j þ @ f f j f k þ @f f j þ @f D f j x þ t @xj @ x
j @xj @ x
k @ x
j @ x
k

ð5:46Þ

@2f i @2f i
j @2f i j @2f i j @2f i
j @2f i x þ x a þ j k xja x
k þ xa þ f þ j j j j

@t@xa @xa @x @t@x @x @x @xa @ x @t@ x þ

@2f i  j k
j
j  @2f i @f i @f i xa f þ x x a þ
j
k x
ja f k þ j x
ja þ
j Da f j ð5:47Þ k
j @x @x @x @ x @x @x

164

Aberration Approach and the Tau-Variation Technique

D2ab f i ¼

@2f i @2f i j @2f i
j @2f i j k x x x > þ 2 < þ2< x > þ b b @xa @xb @xa @xj @xj @xk a b @xa @ x
j

þ2

@2f i @ 2 f i
j
k @f i j @f i
j j
k x ab x x x þ x < x > þ þ b a b a ab j @x @ x
j @xj @ x
k @ x
j @ x
k

ð5:48Þ

5.6. MULTIPLE PRINCIPAL TRAJECTORIES APPROACH The convergence properties of the aberration series in Eq. (5.10), (5.12) often do not allow accurate enough description of charged particle trajectories over the entire parameters domain  with the use of a single principal trajectory. In this situation, several aberration expansions can be constructed in the vicinity of different principal trajectories respondent to different points of the domain . Such calculation technology combines the ideas of direct ray-tracing and aberration analysis, and we call it the multiple principal trajectory approach. In the most general form, this approach assumes that the entire parameters domain  is covered by the set of subdomains XðkÞ  , with the nodal points xðkÞ chosen within each of the subdomains to determine the corresponding principal trajectories TðkÞ : It is also assumed that the aberration series constructed on the principal trajectories TðkÞ ensures acceptable accuracy within the corresponding subdomain XðkÞ : Now we must consider not the vector x but the deviations dxðkÞ ¼ x xðkÞ as small parameters. Consider the chain of overlapping domains XðkÞ o  as shown in n ð kÞ

Figure 87, with the nodal points xðkÞ ¼ x1 ; 0; 0 . . .

parameterized

by the parameter x1 . This allows us to expel dx1 from the set of small parameters. Indeed, having calculated the tau-variations xia ; xiab ; . . . ða; b . . .  2Þ as functions of time on the set of principal trajectories T ðkÞ , we can interpolate between the nodal points xðkÞ to make the tau-variation expansion xðt; xÞ ¼ xð0Þ ðt; x1 Þ þ

P X a¼2

xa ðt; x1 Þxa þ

P 1X x ðt; x1 Þxa xb þ . . . 2 a;b¼2 ab

ð5:49Þ

cover the ‘‘big’’ domain [  ðkÞ of the parameters’ space. k Let us consider one practical example of the multiple principal trajectories approach. The example represents the calculation of spatial resolution in the axisymmetric electrostatic X-ray image intensifier intended for medical purposes. The image intensifier structure is shown on the left side of Figure 88. The main point is that in this case the photocathode

Aberration Approach and the Tau-Variation Technique

165

x1 x2 , x3 ,... Ξ Ξ

(4) T(4)

(3) T(3)

Ξ

(2) T(2)

Ξ (1) T(1)

FIGURE 87 The multiple principal trajectory approach. TðkÞ are the principal trajectories originating from the nodal points xðkÞ chosen in the subdomains ðkÞ : 3

4

5

6

r0

4

1

2

7

8

FIGURE 88 Image curvature calculation in X-ray image intensifier. 1, photocathode; 2, focusing electrode; 3, anode; 4, image receiver; 5 and 6, the meridian and sagittal image curvature surfaces calculated with the third-order paraxial aberration expansion; 7 and 8, the same surfaces calculated with the multiple principle trajectory approach.

work area is very large, and it is impossible to correctly describe the distribution of spatial resolution along the entire work area using only a single third-order aberration expansion constructed nearby the main optical axis of the device (the so-called paraxial aberration expansion). The multiple principal trajectories approach, with ten principal trajectories originating from different equidistant points of the photocathode work area (four of them are shown in Figure 88) and the corresponding aberration expansions being ‘‘attached’’ to the principal trajectories chosen,

166

Aberration Approach and the Tau-Variation Technique

1.5 2

N(r0)/N(0)

1.0

0.5

0.0

1

0

40

80

120

160

200

r0 (mm)

FIGURE 89 The relative value of meridian spatial resolution along the photocathode work area. 1, calculated with the third-order paraxial aberration expansion; 2, calculated with the multiple principal trajectory approach.

ensures the correct solution to this problem. In the case in question, the parameters’ vector x has three components x1 ; x2 ; x3 , so that x1 plays the role of the particle’s initial radial coordinate r0 ; and the other two compopffiffiffiffiffi pffiffiffiffi nents are defined as x2 ¼ em ; x3 ¼ et , with em ; et designating the initial energy components in meridian and saggital directions, respectively. As seen in Figure 88, the meridian and saggital image curvature surfaces constructed with the multiple principal trajectory approach and those calculated with the third-order paraxial aberration expansion are satisfactorily close in the paraxial region but become much different toward the work area edge. The corresponding distributions of meridian spatial resolution along the photocathode work area, calculated on the spherically shaped image receiver surface, prove to be essentially different as well (Figure 89). More details on image surface curvature and spatial resolution in image tubes are provided in Chapter 7.

5.7. TOLERANCE ANALYSIS USING THE ABERRATION THEORY If the field perturbations caused by the boundary geometry variations are known (see Chapter 3), the inclusion of the corresponding structural parameters into the number of the vector x components makes it possible to use the aberration theory and tau-variation technique to

Aberration Approach and the Tau-Variation Technique

(a) 1

2

3

4

dq

5

167

(b)

3 mm 3 mm 6

R 5 mm R 2 mm

FIGURE 90 Tolerance calculation for the objective lens of low-voltage scanning electron microscope. (a) The lens construction (1 – solenoid, 2 – magnetic yoke, 3 – electric field equipotentials, 4 – booster electrode, 5 – magnetic flux lines, 6 – sample). (b) Electron trajectories calculated in the ideal axisymmetric field and with the booster electrode shifted off the axis (the trajectories and geometry are given in different scales).

evaluate the impact of geometry imperfections on device operation. The mechanical tolerance is commonly estimated as the maximum value of the structural parameter variation, for which the contribution of the additional ‘‘tolerance’’ aberration is still acceptable. (Readers may also refer to the works by E. Munro, 1988; Liu, Zhu and Munro, 1990; and Rouse, Zhu and Munro, 1991; which illustrate the concept of tolerance analysis by estimating some field perturbation-induced electron-optical aberrations). In this paragraph, we consider one example of the use of aberration theory for tolerance calculation. Our topic of concern is the objective lens of the low-voltage scanning electron microscope (LVSEM) shown in Figure 90. The focusing magnetic field is generated by the solenoid with yoke. The sample and the yoke are grounded, whereas the booster electrode is biased with Ub ¼ 10 kV voltage. Thus, the electron beam is decelerated from its initial energy eðUb þ Us Þ to the landing energy eUs ¼ 1 kV. (Readers can find other designs of the decelerating objective lenses in the works by Tsuno et al., 1995, 1996; and Hordon, Boyer, and Pease, 1995). The focal distance f and the most significant aberration coefficients of the lens in question are presented in Table VII. With the energy spread de ¼ 0:1 eV typical of a field emission gun with monochromator, and the angular aperture g ¼ 15 mrad at the sample, which corresponds to the entering beam radius dr0 ¼ gf  0:068 mm, the contribution of spherical and chromatic aberrations to the electron probe radius

168

Aberration Approach and the Tau-Variation Technique

TABLE VII Unperturbed Objective Lens Characteristics

Focal distance Spherical aberration coefficient Chromatic aberration coefficient

f ¼ ðajr0 Þ 1 Cs ¼ ðrjr0 r0 r0 Þf 3 Cc ¼ ðrjder0 ÞeUs f

4:51 mm 4:75 mm 2:66 mm

proves to be Ds ¼ Cs g3 =4  4:0 nm and Dc ¼ 0:5Cc deg=ðeUs Þ  2:0 nm, respectively. The numerical value of the electron diffraction spread at the half-maximum level constitutes Dd ¼ 0:51ð2phÞð2m e Us Þ 1=2 g 1  1:3 nm. Our immediate task is to evaluate the shift and coma aberrations resulted from the booster electrode shift off the symmetry axis. The calculations have shown that the electron probe shift in the firstorder approximation with respect to the geometric variation dq (Figure 90) appears as Dx ¼ ðxjqÞdq ¼ 0:15 dq;

Dy ¼ ðyjqÞdq ¼

0:29 dq:

ð5:50Þ

This type aberration does not lead to any deterioration of spatial resolution and may be easily compensated by appropriate adjustment of the deflector. The situation with coma aberration is quite different. The corresponding third-order coma aberration can be characterized by the matrix of coefficients

   

xjx2 q yjx2 q



0



0 D ¼ f 2

ðxjx0 y0qÞ ðyjx0 y0 qÞ

¼

xjy20 q yjy20 q



0:34 0:15 0:19



0:39

0:46

0:23

ð5:51Þ

Having calculated this matrix, we can estimate the contribution of the coma aberration as Dcoma ¼ Dmax g2 dq, where Dmax ¼ 0:46 is the maximum absolute value of the coma matrix components, and compare it with technical requirements. For example, if we stipulate that the contribution of the additional ‘‘parasitic’’ aberrations into spatial resolution should not exceed that of the spherical aberration: Dmax g2 dq < Ds  4 nm, the tolerance for booster electrode displacement is dqmax  0:04 nm. Using the maximal element of the matrix D in Eq. (5.51) may lead to an overvalued contribution of the coma aberration. More accurate estimation can be derived by ‘‘honest’’ calculation of the current density distribution in the probe using the multiple principal trajectories approach. The halfmaximum value of such distribution versus the electrode displacement is shown in Figure 91 (diffraction is not taken into account). The unperturbed spatial resolution of  4 nm decreases twice down to 8 nm with the electrode displacement dqmax  0:07 mm.

Aberration Approach and the Tau-Variation Technique

169

15

12.5

δx, nm (FWHM)

C

y, nm

30

B

20

10

10

7.5

5

2.5 A 0 10

20 x, nm

30

0

0.05

0.1 d q, mm

0.15

FIGURE 91 Intensity profiles of the electron probe (darker regions correspond to higher current densities; the probe shifts are not in scale). (a) ideal axial symmetry, (b) the booster electrode is shifted by dq ¼ 0:1 mm; C dq ¼ 0:2 mm. The probe FWHM is shown versus the booster electrode displacement.

5.8. TRACKING TECHNIQUE The aberration series can be used for trajectory calculations only if they ensure uniform accuracy within the time interval required. As a rule, this basic assumption is broken if the time interval is too large or expanded to infinity. This can be clearly seen from the trivial example of the particle motion in the field-free space – the slightest difference in the initial velocities results in arbitrarily large separation between the particles with time increasing. This explains why the aberration theory is applicable only if the electric and magnetic fields prevent too large spatial expansion of the charged particle bunch, or in other words, ensure the bunch motion stability. Fortunately, namely those focusing fields are commonly used in charged particle optics instruments. In some cases the bunch of charged particles is not disintegrated at a long distance, but the standard aberration procedures are nevertheless unusable because of growing temporal spread within the bunch. To illustrate this point, let us consider a simple example of the oscillating motion of charged particles in a non-quadratic 1D potential well, assuming that the trajectories remain confined within a certain phase-space domain. However, as soon as the oscillation period T generally depends on the particle’s initial energy e, some particles find themselves on different sides of the well after a certain (perhaps, very large) number of

170

Aberration Approach and the Tau-Variation Technique

S2

T0

Ξ[S1]

S1

S0

Ξ[S0] T1

FIGURE 92

The tracking technique scheme. See text for details.

oscillations. This means that the separation between those particles can be barely described in terms of a uniformly convergent Taylor expansion with respect to the spread in initial velocities. In this case, we face the appearance of the so-called secular terms in the aberration coefficients, which is well-known fact in nonlinear oscillation theory and its applications. In charged particle optics, we encounter such situations when a charged particle bunch undergoes multiple oscillations in spatially or temporally periodic fields. A special tracking technique may be used in this case. Consider the particle emitter S0 (Figure 92) and the K0 -order aberration series constructed with respect to the parameters xS0 2 ½S0 Š in the vicinity of the principal trajectory Tð0Þ . The designations xS0 ; ½S0 Š are used here to emphasize that the initial parameters set X is associated with the emitter surface S0 . Let the surface S1 be a virtual particle receiver surface. In this case, the bunch of aberrational charged particle trajectories xS1 ðxÞ ¼ xðtS1 ðxÞ; xÞ having arrived at the surface S1 determines the oneset ½S0 Š onto the set to-one map ½S0 Š ! ½S1 Š of the initial parameters  ½S1 Š of the new ‘‘initial’’ parameters xS1 . To be distinct, let us assume that each of the sets ½S0 Š; ½S1 Š includes six parameters that unambiguously characterize the phase state of the bunch upon any surface - for example, two coordinates, three initial velocity components, and start (arrival) time. Now, by considering S1as a virtual emitter surface and using the new set of initial parameters xS1 , we can construct a new K1 order aberration expansion in the vicinity of the trajectory Tð1Þ that is a continuation of the trajectory Tð0Þ : Formally, taking into account that the parameters xS1 themselves represent K0 -order aberration polynomial with respect to xS0 , we arrive at a K0 K1 -order polynomial with respect to xS0 . It should be emphasized that, despite the fact that this ‘‘combined’’ polynomial is incomplete and therefore cannot be considered as a true K0 K1 order aberration expansion of the bunch in the vicinity   of the principal trajectory Tð1Þ with respect to the set of parameters xS0 , it is much more

Aberration Approach and the Tau-Variation Technique

171

accurate and stable than the original aberration expansion, directly continued onto the principal trajectory Tð1Þ : This procedure can be continued for other virtual surfaces S2 ; . . .; Sn ; . . .. In doing so, we obtain a chain of transformations ½S0 Š ! ½S1 Š ! ½S2 Š ! . . . with the corresponding aberration expansions constructed in the vicinity of the principal trajectories Tð0Þ ! Tð1Þ ! Tð2Þ !. . .. The properties of such transformations are important to make conclusions as to the stability of charged particle motion. The tracking technique is especially effective in the case of periodic motion with the principal trajectory turning into itself: Tð0Þ ! Tð0Þ ! Tð0Þ ! . . .. In this case, there is no need to recalculate the coefficients of aberration expansions on each stage. A simple example as follows illustrates the use of the tracking technique. Consider the sequence of einzel lenses with decelerating central electrode (shown in Figure 93). The stability condition for charged particle beam propagation along the symmetry axis is well known: The electronoptical magnification of each individual lens should be less than unit (see the paper by Verenchikov and Yavor, 2004, as an example). Although the stability condition in our example is fulfilled, direct application of the third-order tau-variation technique to ten periods of the structure results in beam disintegration (Figure 94a). The reason is quite obvious. The traveling time along the symmetry axis between the cross-sections S0 and S1 separated by 40 mm is t0 ¼ 2:338 ns for 1-keV electron, whereas the traveling time between the same cross-sections for the electron started at the distance r0 ¼ 2 mm from the symmetry axis is tðr0 Þ ¼ t0 þ Dtðr0 Þ where Dt ¼ ðtjr0 r0 Þr20 ¼ 0:014 ns. This difference, although relatively small on one period, accumulates on multiple periods, which makes the spatial separation Dz between the particles with slightly different initial conditions continuously grow. When the spatial spread of the bunch becomes comparable with the axial length of electric field generated by an individual lens, the bunch can no longer be described

r T0

S0

z

S1 40 mm

FIGURE 93 Periodic structure of axisymmetric electrostatic einzel lenses (the field period is located between the cross-sections S0 and S1 ).

172

Aberration Approach and the Tau-Variation Technique

(a) 5 0 −5 0

100

200

300

0

100

200

300

(b) 5 0 −5

FIGURE 94 Trajectory simulation with the use of the third-order tau-variation technique (the proportions are distorted). (a) direct calculation; (b) tracking technique.

by the Taylor expansion constructed on the principal trajectory, and the aberrational expansion inevitably fails after a number of periods. At the same time, the tracking technique, which in the case under consideration represents repetitive application of the aberration series having been found for a single period in between the virtual planes S0 and S1 , provides the results in excellent agreement with direct tracing of the individual trajectories (Figure 94b). Finally, it is noteworthy that the tracking technique allows significant enhancement of the computational stability and reduction of the computational load in the case of a lengthy system consisting of a number of field-independent subsystems. Indeed, in this case each subsystem possesses its own set of aberration coefficients, and if the field of one of the subsystems alters (say, as a result of optimization), we must re-calculate the aberration coefficients in that subsystem only, keeping the aberration coefficients in other subsystems unchanged.

5.9. CHARGED PARTICLE SCATTERING The aberration analysis assumes that the electric and magnetic fields are piecewise smooth, with the characteristic extent of the field smoothness much larger than the bunch dimension. The situation changes if the charged particles comprising the bunch experience scattering on the field peculiarities, the characteristic dimension of which is much smaller

Aberration Approach and the Tau-Variation Technique

173

than that of the bunch itself. For example, such scattering may occur on the microscopic electron lenses created by the fine-structure grid cells (Williams, Read, and Bowring, 1995; Read et al.,1999), residual gas molecules, or as a result of interparticle collisions - the so-called Bo¨rsch and Lo¨ffler effects (see Bo¨rsch, 1954; Stickel, 1995). In all those cases, the scattering process is stochastic and may be described in terms of a probability distribution function sðt; R; V ; Vþ Þ, where R is the particle’s radius vector and V ; Vþ are, correspondingly, the particle velocities before and after the scattering event. Obviously, the scattering events themselves cannot be described in terms of smooth solutions of the tau-variation equations. At the first consideration, the aberration theory completely fails in this case, and one of the two "good old" approaches should be used instead. The first approach consists in tracing the individual particles, with the scattering events sampled randomly using the Monte-Carlo method (see, for example, the papers by Mkrtchyan et al., 1995; and Harriot et al., 1995). The second approach assumes the Boltzmann kinetic equation to be solved numerically with respect to the six-dimensional distribution function. The main disadvantage of those approaches (not to mention the obvious fact that both of them are rather bulky in practical implementation and lead to most time-consuming algorithms) is that the obligatory requirements of emission-imaging system simulation – high accuracy and stability – can barely be achieved. The method suggested in this section allows inclusion of the lowangled scattering case DV ¼ jVþ V j min ’ðk1Þ > min ’ðk2Þ ; min ’ðkÞ < min ’ðk1Þ < min ’ðk2Þ O

O

O

O

O

O

ð6:14Þ is satisfied. On the contrary, if none of the conditions (6.14) is satisfied, lðkÞ is decreased by means of multiplying by the factor b. It has been confirmed

Space Charge in Charged Particle Bunches

183

in a series of numerical experiments that this strategy allows effective diminishing  of potential oscillations   during the iterative process. The absolute DðkÞ ’ and relative dðkÞ ’ errors of potential calculation on the k-th iteration are evaluated as   DðkÞ ’ ¼ max j’ðkÞ ðri ; zj Þ ’ðk 1Þ ðri ; zj Þj ð6:15Þ ðri ; zj Þ 2 O

dð k Þ ’ ¼

DðkÞ ’ lðkÞ max ’ðkÞ min ’ðkÞ 1

O

ð6:16Þ

O

and serve to control the convergence. A number of model problems have been solved to examine the numerical properties of the algorithm and to choose the optimal values of the numerical parameters involved. Model problem 1: Axisymmetric diode with a finite-size flat emitter. Considered is a simplest axially symmetric diode with a flat, finite-size, cathode as shown in Fig. 97. The system in question is a practical analog of the well-known ideal 1D model of the flat diode, for which the exact solution was obtained by Child (1911) and Langmuir (1913), and then generalized by Bursian (1921) and Langmuir and Blodgett (1923) to take the initial energy distribution into account. The exact solution of this problem in the case of Maxwellian initial energy distribution can be found in the monograph by Beck (1953). The solution represents the correlation between two dimensionless variables x and , (a) r

(b) j,V Emitter region

Anode 2.0

60 mm

1 Z

1.0 12

18

0.0

Ξ Ω\Ξ

8

−1.0 5 mm

0.0

1.0

2.0

3.0

4.0

5.0

z, mm

FIGURE 97 The test problem for flat diode with a finite-size cathode. (a) The system configuration. (b) Convergence to the exact axial potential distribution (the 1st, 8th, 12th, and 18th iterations are shown).

184

Space Charge in Charged Particle Bunches

ð

in which

and

d x ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ;  ffi   erf pffiffi e 1  e   2 =p 0  mp 1=4 qffiffiffiffiffiffiffiffiffiffiffiffi x¼2 4pjqjj ðz 2kT

zmin Þ;  ¼ jqj

pffiffiffi 2 erf  ¼ pffiffiffi p

pffiffi ð

e

u2

du:

U

Umin ; kT

ð6:17Þ

ð6:18Þ

ð6:19Þ

0

Here j is emission current density, zmin is position of the potential minimum, kT is cathode temperature expressed in energy units, and q and m are electron charge and mass, respectively. A general 1D charge limited emission model was numerically considered by Preston and Stringer (2005). The test system configuration we have analyzed is shown in Figure 97a. The anode voltage ’a ¼ 2:5V was intentionally chosen comparable with the thermal energy of electrons distributed according to the Maxwellian law, with the effective temperature T ¼ 2400 0 K being typical for thermionic cathodes. The 60  40 mesh was used in the near-cathode domain X of 80-mm diameter, containing the emitter region of 60-mm diameter. The number of random trajectories calculated per iteration was about 50,000. The exact self-consistent solution of the ideal one-dimensional flat diode problem with the same cathode-anode gap of 5 mm gives the following performances: the diode current density j ¼ 1:6055 mA/mm2; the potential minimum ’min ¼ 1:4283 V is distanced for zmin ¼ 0:7403 mm from the cathode. Figure 97b shows rather fast convergence to the exact potential distribution in the problem in question. The main characteristics of the iteration process are given in Table VIII. In comparison with the exact solution, the relative accuracy of axial potential calculation gained on the last 18-th iteration is 0.14%, current density – 1%, and potential minimum location – 5.4%. It is of interest to emphasize strong deviations from the ideal 1D solution, which can be observed in the periphery of the emitter area. The topological structure of the equipotential lines in this region reveals the potential minimum extending to the emitter area edge r ¼ 30 mm (Figure 98a). The radial distribution of the emission current density (defined as the difference between the number of electrons emitted from the cathode per time unit and that of the electrons reflected back to the

Space Charge in Charged Particle Bunches

TABLE VIII

185

The Iteration Process in The Flat Diode Problem

Iteration index k

Potential minimum value

Current density at the cathode center, j (mA/mm2)

’min (V)

zmin (mm)

Relaxation parameter l

1600 0.00 -0.0670 0.246 1.49 1.44 1.59

0.00 -75.26 -3.5826 -1.4783 -1.4156 -1.4152 -1.4149

0.00 2.25 1.84 0.80 0.70 0.70 0.70

0.618 0.618 0.618 0.236 0.236 0.145 0.0557

1 3 6 9 12 15 18

(a)

Relative error df

1.0 1.6 1.9 0.23 0.017 0.014 0.0014

(b)

r, mm 40.0 5

j, µA/mm2 6

4

10.0

30.0

5.0

20.0 3 2

1 0.0

10.0

−5.0

0.0 0.0

0.5

1.0 z, mm

1.5

2.0

25.0

r, mm

30.0

FIGURE 98 (a) The topological structure of equipotential lines in the emitter vicinity ð1; ’ ¼ 1:36 V; 2; ’ ¼ 1:28V; 3; ’ ¼ 1:19V; 4; ’ ¼ 0; 5; ’ ¼ 0:11V; 6; ’ ¼ 0:34VÞ; (b) The emission current density versus the radial coordinate at the edge of the emitter area.

cathode by the potential barrier) is practically constant up to r  28 mm (Figure 97b). This current is due to the small portion of electrons belonging to the ‘‘tail’’ of the Maxwellian distribution, whose energies are large enough to penetrate through the potential barrier. The region, in which the numerical and exact solutions are in very good agreement, comprises about 90% of the total emitter area. The important advantage of the numerical approach applied is that it provides the opportunity to investigate the subtle effects that occur in the vicinity of the emitter area edge, where the potential barrier disappears and allows the electrons emitted at sufficiently large angles to escape the

186

Space Charge in Charged Particle Bunches

potential barrier. Those electrons constitute a sharp current density peak one order of magnitude higher than the current density at the center. In addition, the diode current density becomes negative in the immediate vicinity of the emitter edge outside the emitter region. This effect is accounted by two coincidental factors: first, no electrons are emitted outside the emitter region, and, second, some of the electrons reflected by the potential barrier are returned to the cathode surface. Model problem 2: The Pierce electron gun. There exists one exceptional case when the singular behavior of the emission current density observed in the preceding example does not take place even if the emission abruptly terminates at some point of the cathode. This happens if the emitter plane and the adjacent electrode of the same voltage constitute the angle of 67.5 degrees. The exact solution for this exceptional case was given by Pierce (1954). The Pierce electron gun generates a parallel beam, in which the internal Coulomb field acting on electrons is compensated by the external field created by the electrodes of special shape shown in Figure 99a. In the numerical example of the Pierce gun calculation considered below, the cathode temperature is zero and the anode potential is 10 V. The number of mesh points along the radial and axial directions between the electrodes is 40 and 50, correspondingly. As soon as the partitioning of the initial velocity space not required in this particular case, the amount of 2,000 electron trajectories calculated on each iteration step proves to be sufficient to ensure the reliable convergence. In the course of the calculations made, only 7 iterations were needed to diminish the relative error of axial potential distribution to 0.02% compared with the exact solution given by Pierce. The axial space charge density distribution obtained on

(a)

(b)

r 0.0 −4.0

Z Electron beam

r, pc/mm3

∅ 2 mm

67,5o

−8.0 −12.0 −16.0

8 mm

FIGURE 99

−20.0 0.0

2.0

4.0 z, mm

6.0

(a) The Pierce electron gun; (b) The axial space charge distribution.

8.0

187

Space Charge in Charged Particle Bunches

r, mm

(a)

(b)

3.2

3.2

2.8

2.8

2.4

2.4

2.0

2.0

1.6

1.6

1.2

1.2

0.8

0.8

0.4

0.4

0.0 0.0

2.0

4.0 z, mm

6.0

8.0

0.0

0.0

2.0

4.0 z, mm

6.0

8.0

FIGURE 100 Electron trajectories in the Pierce electron gun: (a) The first iteration; (b) The seventh iteration.

the seventh iteration is shown in Figure 99b. The numerically determined current density value j ¼ 1:1512 mA/mm2 is in good agreement with the exact solution. One reasonable criterion of numerical solution reliability in the problem under consideration is the parallelism of electron trajectories in the self-consistent mode. As follows from Figure 100, the electron beam is essentially convergent if the space charge effect is not taken into account (first iteration) whereas the beam turns to be practically parallel to the z-axis in the self-consistent mode (seventh iteration).

6.2. COLD-CATHODE APPROXIMATION: SEMI-ANALYTICAL APPROACH This Section considers simulation of electron guns in the limiting case of ‘‘cold’’ cathode, when the initial energies of emitted electrons can be neglected in first approximation. This approach is suitable for thermoemission devices at anode voltages much exceeding kT=jqj and potential minimum located very close to the emitter surface, so that the approximate assumption as to the strict coincidence of potential minimum with emitter surface is well grounded. In this case, the charge density tends to infinity in the emitter vicinity. The cold-cathode approximation offers the possibility of analytical treatment of that singularity, thus improving the stability of numerical calculations. Child (1911) showed that, at the anode voltage given, the emission current could not exceed a certain value; otherwise the space charge of emitted electrons would be so great that the field strength on the cathode

188

Space Charge in Charged Particle Bunches

would become negative and suppress the emission. If the initial thermal energies of electrons are nonzero, some of them can penetrate through the potential barrier formed by the space charge concentrated near the emitter. The distance between the cathode surface and potential minimum location may be roughly estimated as d  La ðkT=jqjUa Þ3=4 , where Ua and La are, respectively, the anode voltage and anode separation from the cathode. For instance, with the anode voltage Ua ¼ 10 kV and the thermal energies kT less than 1 V, the distance d appears less than 103 La , which justifies the limiting cold-cathode approximation, in the frame of which the electric field strength is zero at the potential minimum located exactly on the emitter surface. The last circumstance violates the conditions of the uniqueness theorem for the Cauchy problem as applied to the solutions of the Lorenz equation with zero initial velocity, which makes the current density of emitted electrons dependent on the field distribution in the near-emitter region. A possible approach to the problem of thermionic gun simulation, capable of considering the dependence of the emission current density upon the field, consists in separation of the near-emitter layer from the rest of the computational volume. For example, Beluga (1997) constructed a special set of virtual flat diodes located in the emitter vicinity to directly determine the emission current from the Child-Langmuir law. A general asymptotic theory of the near-emitter layer, the so-called anti-paraxial expansions embracing both C- and T-modes, was developed by Syrovoy (1991, 2004) and then used by other authors in software design and numerical simulations (e.g. see Sveshnikov, 2004a,b and Greenfield, 2006). Before proceeding to application of the near-cathode asymptotic solutions found by Syrovoy to axisymmetric thermionic electron gun simulation, let us briefly outline some points of Syrovoy’s theory. Let us construct the curvilinear coordinate system fs; h; cg in the vicinity of the emitter (Figure 101), so that s is the arc length along the generatrix z ¼ z0 ðsÞ, r ¼ r0 ðsÞ of the axisymmetric cathode surface S, h is the distance to the cathode surface along the normal, and c is the azimuthal angle. We will use the hydrodynamic approximation, in the frame of which each point R occupied by the electron beam is characterized by a certain vector of electron velocity VðRÞ. Formally, this assumption implies that the particle distribution function in the phase space may be reduced to the form f ðR; VÞ ¼ nðRÞdD ðV  VðRÞÞ, where nðRÞ ¼ rðRÞ=q is space particle density. Although such simplification is too inaccurate if applied to the entire electron gun volume, the close vicinity of the cold cathode is an exceptional region, in which the trajectories started from different points of the smooth emitter surface with zero initial velocities do not commonly intersect within, at least, a thin enough near-cathode layer. This fact makes the hydrodynamic model useful to construct the

Space Charge in Charged Particle Bunches

189

r s

h

z f

FIGURE 101

The near-cathode curvilinear coordinates system fs; h; cg.

asymptotic theory in the limit h ! 0 and study the near-cathode space charge density singularity. The electron flood in the hydrodynamic model meets the stationary Euler equation q r’ ð6:20Þ < V; r > V ¼ m which immediately entails two important conservation laws. First, integration of Eq. (6.20) under the condition of zero starting velocity and zero cathode potential gives the energy conservation law mV2 ¼ 2

q’:

ð6:21Þ

Second, the circulation of the velocity field is constant along the crosssection of any fluid tube (e.g. see the monograph by Landau and Lifshitz, 1959), which, with regard to the motion equation (6.20) and the boundary condition V ¼ 0 on the emitter surface, immediately implies that the velocity field is irrotational ðrotV ¼ 0Þ almost everywhere in the near cathode region and, therefore, can be represented by the gradient of a scalar velocity potential function UðRÞ ¼ 0: ð6:22Þ V ¼ rU; U h¼0

To proceed further, we should note that the curvilinear coordinate system ðs; h; cÞ is orthogonal, with the diagonal components of the metric tensor gss ¼ ð1 þ k0 hÞ2 ; ghh ¼ 1; gcc ¼ r2 ð1 þ k1 hÞ2 :

ð6:23Þ

190

Space Charge in Charged Particle Bunches

The coefficients k0 ðsÞ ¼ z00 r000 z000 0 r 0 and k1 ðsÞ ¼ z00 =r0 represent the meridian and sagittal curvatures of emitter surface. Let us write a full set of equations to describe the electron flux in the emitter vicinity. With regard to the metric tensor components (6.23), the Laplace operator expressed in the curvilinear coordinates fs; h; cg appears as

 @2 k0 k1 @ 1 þ þ D¼ 2þ 1 þ k0 h 1 þ k1 h @h ð1 þ k0 hÞ2 @h  2 0 @ k1 h k00 h @ þ k þ 2 ð6:24Þ @s2 1 þ k1 h 1 þ k0 h @s and the Poisson equation (6.1) takes form D’ ¼

4pr;

ð6:25Þ

where r ¼ qn is the negative space charge density of electrons. The term with the derivative with respect to the angle c is omitted in Eq. (6.24) due 0 to axial symmetry, and the third curvature coefficient k2 ¼ r0 =r0 is introduced. The energy conservation law expressed by Eq. (6.21) now appears as

2 2 1 @U @U 2q þ ¼ ’: ð6:26Þ 2 @s @h m ð 1 þ k 0 hÞ We also need the flux continuity equation divðrrUÞ ¼ rDU þ rrrU ¼ 0

ð6:27Þ

which in the curvilinear coordinates fs; h; cg takes the form rDU þ

@r @U 1 @r @U þ ¼ 0: 2 @h @h ð1 þ k0 hÞ @s @s

ð6:28Þ

The solution ð’; U; rÞ to Eqs. (6.25), (6.26), and (6.28) should meet the following boundary conditions on the cathode surface h ¼ 0: ’ ¼ 0;

@’ ¼ EðsÞ; @h

ð6:29Þ

@U ¼ 0; @h

ð6:30Þ

U ¼ 0;

lim

h!0

r

@U ¼ jðsÞ: @h

ð6:31Þ

191

Space Charge in Charged Particle Bunches

Here EðsÞ is on-cathode electric field strength, and jðsÞ is emission current density taken with opposite sign to be a nonnegative value. We immediately conclude that Eqs. (6.30) and (6.31) can be simultaneously met for a nonzero emission current density only if the space charge density r is infinite on the emitter surface. Our next goal is to construct a solution to Eqs. (6.25), (6.26), and (6.28) with the boundary conditions (6.29) – (6.31) in the form of the power expansions with respect to the distance h from the cathode surface. Depending on the sign of the on-cathode electric field EðsÞ, we obtain three quite different types of local solutions. Mode 1 ðE < 0Þ. In this case, emission is impossible and jðsÞ  0. The electric potential expansion reads 2

k0 þ k1 k0 þ k0 k1 þ k21 E00 ðsÞ þ k2 E0 ðsÞ 3 2 EðsÞh þ Eð s Þ h þ ... ’ðs;hÞ ¼ EðsÞh 2 3 6

ð6:32Þ

This formula was earlier obtained in Section 1.8. Mode 2 ðE > 0Þ. The positive field strength E corresponds to the temperature limited emission mode (T-mode), with the saturated current density jsat determined by the cathode temperature according to the Richardson thermoemission law (e.g. see Dobretsov and Gomoyumova, 1966) jsat ¼ A0 ð1 #ÞT2 expð W=kT Þ, where W is the cathode material work function, A0 ¼ 120:4 A cm 2 K 2 is the Sommerfeld constant, and # < 1 is the empiric constant characterizing the reflection of electrons from the surface potential barrier. The potential expansion appears as ’ðs; hÞ ¼ EðsÞh þ E1 ðsÞh3=2 þ E2 ðsÞh2 þ E3 ðsÞh5=2 þ E4 ðsÞh3 þ . . . ; where

pffiffiffi rffiffiffiffi 8 2p m jsat pffiffiffiffiffiffiffiffiffi ; E1 ðsÞ ¼ 3 q EðsÞ

ð6:33Þ

ð6:34Þ

and the other coefficients being expressed through E; E1 and its derivatives:

E4 ¼

3 ðE1 Þ2 ; 16 E

k0 þ k1 E 2

E2 ¼ k20

þ k0 k1 þ 3

k21

E3 ¼

3 ðE1 Þ3 32 E2

2 1 ðE1 Þ 49 Eþ ðk 0 þ k 1 Þ 16 E 5 2

11 ðk0 þ k1 ÞE1 20 3 2  ðE1 Þ 5 1  00 E1 þ k2 E01 2 E 6

ð6:35Þ

Mode 3 ðE ¼ 0Þ. In this case, the space-charge limited emission mode (C-mode) is realized. The emission current in this mode may be less than

192

Space Charge in Charged Particle Bunches

jsat , and its self-consistent value j is directly interrelated with the first coefficient in the potential expansion ’ðs; hÞ ¼ E1 ðsÞh4=3 þ E2 ðsÞh7=3 þ E3 ðsÞh10=3 þ . . .

ð6:36Þ

by the formula E1 ¼



1=3 81p2 m j2=3 : 2 e

ð6:37Þ

The two other coefficients in Eq. (6.36) are 8 ðk0 þ k1 ÞE1 15

E2 ¼ 2

83 E3 ¼ 4 ðk0 þ k 1 Þ2 225

3 7 k0 k1 5E1 18

 2 1 E01 600 E1

 2  00 E1 þ k2 E01 15

ð6:38Þ

It should be noted that the coefficient E3 becomes infinitely large at the points separating the emitter regions operating in the C-mode from those in which the emission is absent. The current density j is zero at such points; therefore, the coefficient E1 in the denominator of the second term of Eq. (6.38) vanishes in accordance with Eq. (6.37). It may be shown that the hydrodynamic approximation fails in the vicinities of such singular emitter points; nonetheless, the contribution of those points to the beam current and charge density appears negligible in most practical cases. Let us now proceed to some aspects of numerical simulation based on the theory by Syrovoy. The expansions (6.32), (6.33), and (6.36) define the electric potential distribution near the cathode in the three emission modes with accuracy up to h3 . The coefficients of these expansions are expressed through the functions EðsÞ and jðsÞ and their derivatives. Our aim here is to find these functions numerically in the form of smooth interpolants under the condition of coupling of the analytical near-cathode expansions with the electric potential distribution in the remaining volume of the electron gun. We use the finite element technique to solve the Poisson equation in the region O\X obtained by the exclusion of the near-cathode finite elements from the computational domain O, as illustrated in Figure 102. The potential distribution is found in the subdomain O\X by numerical minimization of the energy functional # ð " ðr’Þ2 ’r ð2prÞdzdr ð6:39Þ W ¼ 2p 8p O=X

under the Dirichlet condition on the subdomain boundaries, including the B-type nodal points (see Figure 102), at which the potential is assumed

Space Charge in Charged Particle Bunches

C

B

A

C B

A

Ξ

Ω\Ξ

C

Emitter

A

FIGURE 102

193

A

B B

C

Finite-element mesh in the near-cathode region.

E

s

j

A1

A2

A4

A3

A5

jsat

s Mode 2

Mode 3

Mode 1

FIGURE 103 Example of the interpolated field strength and emission current density distributions over the emitter area.

to be zero (equal to emitter potential) on the first iteration. The space charge density r is also zero in the beginning of the iteration process. Let us denote the solution as ’0 ðz; rÞ, ðz; rÞ 2 OnX. The values Eðsi Þ and jðsi Þ are ascribed to the A-type mesh nodes located on the emitter surface. Then the unknown functions EðsÞ and jðsÞ are defined by smooth interpolation as shown in Figure 103. Each of the intervals located between two adjacent nodes is assumed to operate in one of the three emission modes previously described, which imposes some limitations on the nodal values of the two unknown functions to be found. If both of the values Eðsi Þ; Eðsiþ1 Þ are nonpositive but not equal to zero simultaneously, the interval ðsi ; siþ1 Þ is assumed to operate in Mode 1 and, therefore, jðsi Þ ¼ jðsiþ1 Þ ¼ 0. If both of the values Eðsi Þ; Eðsiþ1 Þ are nonnegative but also not equal to zero simultaneously,

194

Space Charge in Charged Particle Bunches

then Mode 2 is assumed. In this case, jðsi Þ ¼ jðsiþ1 Þ ¼ jsat . If Eðsi Þ ¼ Eðsiþ1 Þ ¼ 0, Mode 3 takes place, and the values jðsi Þ, jðsiþ1 Þ are allowed to take any value between 0 and jsat . The case when Eðsi Þ have opposite signs at the adjacent mesh nodes is prohibited – the intervals with the positive and negative values of the normal electric field component must be separated by at least one node with Eðsi Þ ¼ 0. With the pair of functions EðsÞ and jðsÞ given, the Eqs. (6.32), (6.33), and (6.36) define the electric potential ’1 ðz; rÞ in the domain X containing the strip of elements adjacent to the emitter. We extend these expansions even farther P  to the next layer of elements and define the functional 2 R ¼ i ’1 ðzi ; ri Þ ’0 ðzi ; ri Þ , where summation is made over the C-type nodes. Then we are seeking such set of Eðsi Þ and jðsi Þ at the A-type nodes that meets the above-stated limitations and gives the best agreement between the solutions ’0 and ’1 by minimizing the discrepancy R. This is a classical problem of nonlinear programming with the inequality-type constraints. Then a representative set of electron trajectories is traced, starting from those regions of the emitter that operate in Modes 2 and 3. The space charge density distribution for the next iteration is calculated only within the subdomain OnX, thus avoiding the difficulties connected with the infinite values of r on the emitter itself. Each consequent iteration consists of three steps: (1) solving the Poisson equation in OnX by minimizing the functional Eq. (6.39) with the boundary condition given by the Eqs. (6.32), (6.33), and (6.36) at the B-type nodes; (2) finding the EðsÞ and jðsÞ distributions that minimize the discrepancy R in the C-type nodes; and (3) calculating the electron trajectories and space charge density distribution in OnX. After 10 to 20 iterations, we normally arrive at a stationary selfconsistent solution for the electric potential and emission current density. Examples of electron gun modeling are given in Figures 104 – 106. Figure 104 shows the geometry of a triode gun, which includes the cathode (0 V), negatively biased forming electrode (-200 V), and anode (10 kV). The electric field equipotentials and electron bunch trajectories are shown. As seen in Figure 105, the region adjoining the cathode center operates in C-mode, whereas the negative electric field suppresses emission from the cathode periphery. The electron current density versus the forming electrode voltage is given in Figure 105c. Two other examples are given in Figure 106.

6.3. COULOMB FIELD IN SHORT BUNCHES: THE TECHNIQUE OF TREE-TYPE PREORDERING This and the following sections are devoted to the nonstationary dynamics of charged particle bunches subjected to simultaneous action of the external electromagnetic field and Coulomb field generated by the bunch itself.

Space Charge in Charged Particle Bunches

195

Anode (+10 kV)

(−0.2 kV)

Electron beam

Cathode

Forming electrode

r, mm

z, mm

8 mm

FIGURE 104

The triode electron gun with negatively biased forming electrode.

(a) j, mA/mm2

4 3

(c)

2 120

1 0

E0, V/mm

(b)

0.5

1.0 r, mm

1.5

2.0

j, mA

0

80 40

0 −100

0 0

−200

100

200 300 -Uforming, V

400

−300 0

0.5

1.0 r, mm

1.5

2.0

FIGURE 105 Emission current density (a) and electric field (b) on the cathode for the triode electron gun shown in Figure 104. (c) The current-voltage characteristic of the electron gun.

Calculation of the Coulomb potential induced by N particles with the charges qp and coordinates Rp ; p ¼ 1; . . .; N at a given point S consists in the summing up of N terms

196

Space Charge in Charged Particle Bunches

(a)

(b)

r1 r0

2.0

50

j, mA/mm2

j, mA/mm2

100

r1 r0

0.0 0.0

0 5.0

6.0 7.0 r, mm

1.0

8.0

1.0

2.0

3.0

r, mm

FIGURE 106 Examples of electron gun simulation. (a) Diode gun with ring-shaped cathode (anode voltage 50 kV); (b) triode gun with spherical cathode (anode voltage 1 kV, forming electrode voltage 0.5 kV). The insets show the emission current distribution on the cathode operating in Mode 3.

’ ð SÞ ¼

N X

qp ; jS Rp j p¼1

ð6:40Þ

the calculation of each involving the most time-consuming operations of square root extraction and floating point division. Thus, if we need to know the Coulomb field at the location of each of the N particles of interest to integrate the set of N motion equations, we need to calculate as many as NðN 1Þ=2 of such terms for all possible pairs of the particles comprising the bunch. If the number N is large enough, the calculation becomes exceedingly time-consuming even for today’s fast computers. As an example, the Coulomb field calculation for a cloud containing N  105 charged particles with a 3-GHz computer takes approximately 7 minutes for a single Runge-Kutta integration time step, which means that tracing the entire set of trajectories over 100 integration steps would take more than 10 hours. The Coulomb field calculation may be substantially accelerated if we combine the particles, depending on their positions with respect to the point S of field calculation, into groups and treat each group as a single source of field. Obviously, the particles located far from the point S may be combined into larger groups, whereas the particles located close to the point of field calculation should be combined in smaller groups or even considered individually. In any case, unless special measures are

Space Charge in Charged Particle Bunches

197

undertaken, we must work at the level of individual particles when regrouping them depending on the location of point S; and such regrouping is to be fulfilled N times per one integration step, if the field at N different points is needed. Barnes and Hut (1986) proposed more effective realization of the general idea of particle grouping to calculate the gravitational field of star clusters in celestial mechanics. Instead of grouping the individual particles with respect to a given point of field calculation, the Barnes-Hut algorithm assumes ranking of the cells of a tree-type structure, which should be prepared from the whole particles cloud on each integration step before the field calculation. Since mathematical descriptions of the gravitational and Coulomb fields are similar, this algorithm proved fruitful in computational charged particle optics, as shown simultaneously by Munro et al. (2006) and the authors of this monograph (2006). The Barnes-Hut algorithm is set forth below with some modifications to improve its accuracy. Let us first describe the details of the tree-type construction procedure. Consider a bunch of N indexed particles and imagine that some of them are surrounded by a cube with the center R and the rib length L. Denote by J ¼ J ðR; LÞ the subset of the indexes fpg ¼ ½1 . . . N  that correspond to the particles located inside the cube. We denote by RðJÞ the center of any cube respondent   to a set of the particle indexes J. Now substitute the single term that correspond to ’ðJÞ S; RðJÞ ¼ QðJÞ =jS  RðJÞ j for all the terms in Eq. (6.40)P the indexes belonging to J, as if the total charge QðJÞ ¼ p2J qp were concen-

trated at the cube center RðJÞ . In doing so, we introduce the discrepancy   X q p ð6:41Þ DðJÞ ðSÞ ¼ ’ðJÞ S; RðJÞ  jS  Rp j p2J into Eq. (6.40), the relative value of which can be estimated as w ð J Þ ð SÞ ¼

L : jS  RðJÞ j

ð6:42Þ

The tree-type structure we seek to construct is aimed at grouping the particles depending on their proximity to each other. The structure will comprise a number of nested cells to be generated by the recursive procedure as follows. At the start, the entire domain occupied by the particles is inscribed into a minimal cube with the rib length L0 . This cube is hereafter referred to as the cell of zero (upper) level of the tree-type structure to be constructed. Then, by means of three planes – each parallel to one face of the cube – the cube is divided into eight equal first-level cells that also are cubic (see Figure 107). If any of the cells thus produced contains more than one particle, the total charge of the particles inside each of those cells, as well as the dipole and quadrupole moments of the particles

198

Space Charge in Charged Particle Bunches

with respect to the cell center, are calculated and stored. Then each of those cells is again divided into eight smaller cells, and this procedure is repeated until each cell of the deepest level contains only a single particle. To avoid unlimited subdivision, we restrict the recursive procedure to some maximum depth M (say, M ¼ 10), so that the rib length of the smallest cell is L0 =2M . The deepest level has NM  N cells containing at least one particle, whereas the number Nm of the cells constructed at the depth level m  M does not exceed minðN; 8m Þ. If the particle cloud is smooth enough, this estimation proves to be overstated and can be replaced with a more realistic one: Nm  N=8Mm . Generally speaking, the total number of the cells constructed with the algorithm does not exceed N M, but normally, if dealing with a smooth particle distribution, it may be estimated as Nþ

N N N 8 þ þ . . . M < N: 8 82 7 8

ð6:43Þ

More accurate estimations can be found from numerical experiments. Let us denote Si and Ri ; i ¼ 1; 2; 3, the Cartesian coordinates of the vectors S and R, respectively, and let Dxip ; i ¼ 1; 2; 3 be Cartesian coordinates of the deviation vector Rp  RðJÞ , p 2 J. If we consider the dipole and quadrupole moments of the cell " # 3  2 X X dij X ðJ Þ ðJ Þ i j k i Dxp qp Dxp Dxp  qp Dxp ; Qij ¼ ; ð6:44Þ Qi ¼ 3 k¼1 p2J p2J we can diminish the discrepancy DðJÞ ðSÞ by using a more accurate representation  i ðJÞ 3 i X   QðJÞ ðJ Þ S  R ðJ Þ ðJ Þ Qi ¼ þ ’ S; R jS  RðJÞ j i ¼ 1 jS  RðJÞ j3 h i h  ðJÞ  ðJ Þ i ð6:45Þ 3 S i  Ri Sj  Rj X ðJ Þ þ3 Qij jS  RðJÞ j5 i;j ¼ 1   for the potential ’ðJÞ S; RðJÞ induced at the point S by the cell respondent to the subset J. In this case, the discrepancy DðJÞ ðSÞ has the third order of smallness with respect to wðJÞ . The total potential ’ðSÞ at the point S induced by all the cells thus generated takes the form   X ’ðJÞ S; RðJÞ : ð6:46Þ ’ ð SÞ  J

In the limiting case, when a cell contains only one particle, the potential ’ðJÞ should be put equal to the corresponding term in Eq. (6.40). Let us assume that the tree-type structure is constructed and the field calculation point S is given. Now, guided by some criterion, we must

Space Charge in Charged Particle Bunches

199

e3

R(J)

e2

e1 S

FIGURE 107

Coulomb field calculation with the tree-type structure.

arrange a procedure of extraction of different cells from different levels to put the contribution of the cells’ potential into Eq. (6.46), so that the entire set of particles is exhausted. The reasonable criterion for such extraction may be obtained by imposing the upper limitation wðJÞ  wmax R0

where Q ¼ qN is the total charge of the cloud of particles. The number of populated cells of the tree-type structure proved to be 1:45 105 , which only slightly exceeds the estimation given by Eq. (6.43). The Coulomb field has been calculated at each of the 20 points uniformly distributed in the domain 0 < r < 2R0 , half of them located inside the sphere and the rest outside. The mean-square deviation of the numerical results obtained with the tree-type algorithm from the exact solution (6.47) and the results of direct summation according to Eq. (6.40) are presented in Table IX for different wmax values. Table IX also presents the average number of the cells involved in the field calculation at a single point. The last column of the table represents the average computation time for an Athlon 1.2 GHz computer. The numerical results are closer to the direct sum rather than to exact solution because the tree-type algorithm takes into account the stochastic noise of random particle distribution. The accuracy is monotone increasing to some limit with the parameter wmax decreasing, although the calculations become longer. This is understandable – a larger number of the TABLE IX Accuracy and productivity of the tree-type structure algorithm as applied to the Coulomb field calculation inside and outside the uniformly charged sphere Relative error (mean-square deviation, %)

wmax

1/2 1/3 1/4 1/5 1/6

Average With respect to the With respect to the calculation exact solution given direct summation Average number of cells time (ms) by Eq. (6.47) in Eq. (6.40)

5.3 4.8 3.3 3.5 3.7

3.9 2.7 1.3 1.5 1.8

175 295 810 1300 2200

80 140 400 710 1540

Space Charge in Charged Particle Bunches

201

cells involved allow more detailed description of the space charge distribution. Nevertheless, as the last two lines of Table IX show, setting the parameter wmax too small reduces the accuracy slightly, which apparently is caused by the contribution of the round-off errors resulting from summing too many terms with opposite signs. The time needed for Coulomb field calculation at a fixed point is proportional to the total number of the cells involved, which, according to our numerical experiments, can be fitted by the empiric formula NG ¼ 

lnN ; w2m

ð6:48Þ

with the coefficient   3:0  3:5. In our example, only 600  800 terms like (6.45) ensured the relative error of 3:5% with respect to exact solution at wmax ¼ 1=4. Accordingly, the calculation speed proved to be about two orders of magnitude higher than that of direct summation. Now consider another, more complicated, model problem of Coulomb dynamics of a spherically symmetric charged particle bunch with uniform and nonuniform initial space charge distributions. Let us assume that the charged particle cloud at the initial time moment t ¼ 0 is at rest and has initial spherically symmetrical distribution n0 ðr0 Þ. The Lorenz equation describing the particles’ trajectories rðr0 ; tÞ under the Coulomb repulsion may be written in the form of the integro-differential equation @ 2 rðr0 ; tÞ 4pq2 1 ¼ m r2 @t2

1 ð 0

    n0 r0 r20 Y r0  rðr0 ; tÞ dr0

with the initial conditions at t ¼ 0 rðr0 ; tÞjt ¼ 0 ¼ r0 ;

@rðr0 ; tÞ jt ¼ 0 ¼ 0: @t

ð6:49Þ

ð6:50Þ

Here q and m are, correspondingly, the particle charge and mass, and YðÞ is the step function. If we assume the function rðr0 ; tÞ to be monotone with respect to r0 , the solution of Eq. (6.49) can be easily derived in the implicit form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi rðr  r0 Þ r r  r0 2q2 þ ¼t þ ln N ðr0 Þ; ð6:51Þ r0 r0 r0 mr30 Ðr0   where N ðr0 Þ ¼ 4p n0 r00 r00 dr00 is the number of particles within the 0

sphere of the radius r0 at the initial time moment t ¼ 0. Consider the results of testing the tree-type structure algorithm using the cloud comprising N0 ¼ 5 104 protons uniformly distributed at the initial time moment inside a sphere of r0 < R0 ¼ 1 mm radius. The space

202

Space Charge in Charged Particle Bunches

15 t = 0 ms r, mm

t = 1 ms

10

Ion density (rel. units)

5 0

t = 2 ms

0

t = 3 ms

1

2

3 4 t, ms

5

t = 4 ms t = 5 ms

0

5

10 r, mm

15

20

FIGURE 108 Coulomb dynamics of the spherically symmetric charged particle cloud in the case of uniform initial distribution (the dashed lines denote exact solution of Eq. (6.51)). Crosses in the inset show the full width at half maximum of the particle distribution versus time (the line denotes the exact solution).

charge density evolution calculated numerically with the 3D tree-type structure algorithm is presented in Figure 108. The inset shows the external radius of the expanding cloud compared with the exact solution for the peripheral trajectory rðr0 ; tÞ. The numerical solution error proved to be less than 3%. In the case of essentially nonuniform initial distribution, the Coulomb dynamics may be essentially different. Let us consider the results of numerical experiments for the same number N0 ¼ 5  104 of protons with spherically symmetrical Gaussian initial distribution with the mean-square radius R1 ¼ 0:2 mm. The monotony condition @r=@r0 > 0 is violated in this case, and the geometrical points at which the derivative @r=@r0 vanishes constitute a peculiar caustic surface expanding in time. This effect, called the overtaking catastrophe, is well known in the general bifurcation theory (e.g. see Arnold, 1998). The charged particle density evolution in time calculated with the tree-type structure algorithm is shown in Figure 109. Apparently, no exact solution exists in this case, so we can compare the tree-type numerical solution only with the direct numerical solution of the integro-differential equation (6.49). Starting from the time moment t  0:5ms, the particles originally located in the vicinity of r0  2R0 overtake their neighbors and thus

203

Space Charge in Charged Particle Bunches

30

r, mm

t = 0 ms

20

Ion density (rel. units)

10 0 0

t = 1 ms

0

t = 3 ms

t = 2 ms

10

t = 4 ms

20

1

2

3 t, ms

4

5

t = 5 ms

30

r, mm

FIGURE 109 Coulomb dynamics of the spherically symmetric charged particle cloud in the case of essentially nonuniform initial particle distribution (the dashed line shows the solution of Eq. (6.49)). Crosses in the inset show the peak location at different time moments (the line indicates the caustic surface position calculated from Eq. (6.49)).

form a moving caustic surface on which the space charge density attains its maximum. The inset in Figure 108 displays the peak location as a function of time, calculated with the tree-type structure algorithm and taken from the direct numerical solution of Eq. (6.49). It is noteworthy that the charge density peak on the caustic surface obtained with the tree-type algorithm is blunt, in contrast to that obtained with Eq. (6.49). Seemingly, the reason is that the tree-type algorithm takes into account the particleto-particle interaction, while Eq. (6.49) does not. On the one hand, this restricts the comparison accuracy ( 7 % in the case in question) but, conversely, shows the capabilities of the tree-type algorithm. Finally, we note that the problem on Coulomb dynamics of nonuniformly charged sphere was first formulated by Bykov and Turin (1995). Based on another, semi-analytical, approach, these authors revealed and studied the charged particle cloud disintegration caused by the overtaking catastrophe. A few years earlier, a similar effect was numerically observed by Monastyrskiy in his doctoral thesis (1992) and later confirmed by Monastyrskiy et al. (1999) in their numerical studies on Coulomb dynamics of electron bunches in picosecond streak tubes.

204

Space Charge in Charged Particle Bunches

6.4. EXCLUSION OF THE EXTERNAL FIELD IN SPACE CHARGE PROBLEMS This section shows how the perturbation theory can essentially simplify evaluation of space charge and scattering effects in charged particle optics and make it more accurate. We can represent the motion equations that determine the particles distribution at the time moment t > 0 in the form  R_ ¼ V ð6:52Þ _ ¼ Fext ðR; V; tÞ þ Fint ðR; tÞ; V where Fext is the force experienced by a charged particle from the external electromagnetic field, and Fint is the force of ‘‘internal’’ Coulomb interaction between the particles (both forces are divided by the particle mass). As usual, Eqs. (6.52) should be supplemented by the initial conditions Rjt¼0 ¼ fx1 ; x2 ; x3 g and Vjt¼0 ¼ fx4 ; x5 ; x6 g on the emitter surface. The difficulties of solving this problem numerically are obvious. First, we can no longer speak about calculating the individual trajectories; due to the particles interaction inside the bunch, we have to consider the bunch as a whole. Second, it is not possible to apply any aberrational approach directly to Eq. (6.52) because the differential properties of the two terms in the right-hand side of the second equation are very different, especially if the bunch is short. Indeed, the external electromagnetic force Fext ðR; V; tÞ is almost constant on the characteristic length of the bunch, whereas the Coulomb force Fint ðR; tÞ varies strongly. At the first sight, nothing but a huge computational burden can prevent a frontal attack of this problem – using the general method of macroparticles – although it would revert to direct integration of the motion equations for all the macroparticles comprising the bunch. Unfortunately, another difficulty awaits us in this way, which is especially unpleasant if very high calculation accuracy is needed. Typical example is imaging charged particle optics, when the external electromagnetic field exceeds the internal Coulomb field by many orders of magnitude, and it is barely possible to distinguish the Coulomb field with acceptable accuracy on the external field background. At the same time, it is well known that even such comparatively weak space charge effects may significantly contribute to the image properties. The resolution of this dilemma is provided by one of the oldest and frequently used approaches of the perturbation theory: the method of initial parameters variation. This method goes back to Cauchy, and during its long history has served as a basis for many important results in nonlinear mechanics and differential equations theory. In our case, we will use the generalized method of initial parameters variation to decompose the motion equations into two differential equations, one of which

Space Charge in Charged Particle Bunches

205

(unperturbed) contains the external force Fext ðR; V; tÞ, while the other one contains the internal force Fint ðR; tÞ. As a sequence, this offers the possibility of active use of the aberration theory in the problem under consideration. Let us denote by R0 ðt; xÞ; V0 ðt; xÞ the solution of the unperturbed motion equation  R_ 0 ¼ V0 ð6:53Þ _ 0 ¼ Fext ðR0 ; V0 ; tÞ V which describes the particle motion with no regard to the space charge effects. The 6D vector of parameters x ¼ fx1 . . . x6 g plays the role of initial conditions R0 jt ¼ 0 ¼ fx1 ; x2 ; x3 g and V0 jt ¼ 0 ¼ fx4 ; x5 ; x6 g for the unperturbed solutions of Eq. (6.53). Let us introduce the unknown vectorfunction xðtÞ ¼ fxa ðtÞg and determine it in such a way that the combination RðtÞ ¼ R0 ðt; xðtÞÞVðtÞ ¼ V0 ðt; xðtÞÞ

ð6:54Þ

represents a solution of the full motion equation (6.52). Substitution of Eq. (6.54) into Eq. (6.52) gives 6 X @R0 @R0 dx ðt; xÞ þ ðt; xÞ a ¼ V0 ðt; xÞ @t @xa dt a¼1

6 X @V0 @V0 dx ðt; xÞ þ ðt; xÞ a ¼ Fext ðR0 ðt; xÞ; V0 ðt; xÞ; tÞ þ Fint ðR0 ðt; xÞ; tÞ; dt @t @xa a¼1

ð6:55Þ

whence, taking into account that the pair fR0 ; V0 g obeys Eq. (6.53) for any value of the parameters fxa g and introducing the 66 Jacobian matrix ℘, ℘ðt; xÞ ¼

@ ðR0 ; V0 Þ ; @ ð x1 . . . x6 Þ

ð6:56Þ

we immediately arrive at a system of two first-order differential equations for the unknown functions xa ðtÞ

 dx 0 1 ; ð6:57Þ ¼ ℘ ðt; xÞ Fint ðR0 ðt; xÞ; tÞ dt the right-hand side of which is fully determined, provided that the unperturbed solution R0 ðt; xÞ; V0 ðt; xÞ is known for any x 2 X. The reversibility of the matrix ℘ is guaranteed by the phase volume conservation law, according to which jdet℘j  1. Thus, we have decomposed the original motion equation (6.52) into Eqs. (6.53) and (6.57) determined by the external and internal fields, accordingly. Geometrically, the correlation RðtÞ ¼ R0 ðt; xðtÞÞ shows that, the perturbed trajectory RðtÞ coincides with just one of the unperturbed

206

Space Charge in Charged Particle Bunches

R0(x,t)

Ξ R(t)

x(t) t

FIGURE 110 The procedure of external field exclusion. At any time moment, the perturbed trajectory R ðtÞ ¼ R0 ðt; xðtÞÞ (solid line) intersects one of the unperturbed trajectories (dashed lines) originating from the phase-space domain X of initial parameters.

trajectory set fR0 ðt; xÞ; x 2 Xg at each time moment t, namely, with the trajectory corresponding to x ¼ xðtÞ. Thus, with time increasing, the perturbed trajectory RðtÞ is continuously ‘‘sliding’’ from one unperturbed trajectory R0 ðt; xÞ to another one, as shown in Figure 110. This situation is similar to that discussed in Section 5.9 devoted to the charged particle scattering. The only difference is that in the scattering case the transition of the perturbed trajectory from one unperturbed trajectory to another was discrete (which reflected the discrete nature of the scattering events), whereas, in the present case, such transition is continuous due to the continuous nature of Coulomb interaction. Such similarity becomes even more transparent if we, by way of the internal force Fint , consider the scattering ‘‘force’’ Fint ¼ DVdD ðt  ts Þ

ð6:58Þ

where DV is the random abrupt change of particle velocity at the time moment ts , and dD ðÞ is the Dirac delta function. Let us highlight some important numerical aspects of the external field exclusion procedure outlined above. First, the tau-variation technique (see Chapter 5) allows us to easily calculate the unperturbed solution R0 ðt; xÞ and the matrix ℘ðt; xÞ for any initial x 2 X. Second, the subsequent numerical integration of Eq. (6.57) describing the trajectory xðtÞ in initial parameters space is much easier and faster because the evolution of initial parameters xa in time is much slower than that of the real phase coordinates.

6.5. SOME EXAMPLES OF ION BEAM SIMULATION The practical examples presented in this section illustrate the numerical approaches described in this chapter. Figure 111 shows the spatial resolution deterioration caused by Coulomb repulsion effects in the bunch of

Space Charge in Charged Particle Bunches

(b)

207

(c)

1 mm

1 mm (a)

X, mm

1

0

−1 0

10

20

30

Z, mm

FIGURE 111 Spatial resolution deterioration caused by space charge effect in the test object representing 1 mm  1 mm square with five slits. (a) The stroboscope picture (not in scale); (b) ion distribution on the test object; (c) ion distribution after 24-mm travel length).

105 argon ions (m/q ¼ 40) with 500 eV energy. The bunch duration is 15 ns. The ion image of the square mask comprising 0.1-mm wide slits gradually loses its sharpness during the bunch propagation. Such simulation may be used to look inside different processes of ion beam technology. The next example refers to the space charge problem in time-of-flight (TOF) mass-spectrometer design. It represents a reflectron-type mass analyzer similar to those described by Verenchikov and Yavor (2004) and Pedersen et al. (2002). The ‘‘heart’’ of the analyzer represents two coaxial ion mirrors (see Figure 112) optimized so that the ion bunch is able to make hundreds of oscillations between them. The TOF (or, equivalently, the oscillation frequency) of the particles comprising the bunch is measured and the ion mass spectrum is thus obtained. The principal requirement to the mirrors design, which determines the limiting mass resolution, is that the dependence of the halfoscillation period T ðr0 ; eÞ (the time for the ion to return to the middle plane after one reflection) on the initial energy e of the ion and its initial

208

Space Charge in Charged Particle Bunches

5

6

0.6 eV 3.1 eV r, mm e 0 = 2000 eV

3

2

1

4

e

2

Φ (z) z, mm

0 −2

1mm

2000 mm

FIGURE 112 The reflectron TOF mass-spectrometer design. 1, Electrodes; 2, momentary ‘‘snapshot’’ of the ion bunch; 3, ion trajectories; 4, axial distribution of electric potential FðzÞ; 5, 6, Coulomb-induced energy distributions in the ion bunch after a half-oscillation with the numbers of ions 104 and 5  104 , respectively.

distance r0 from the middle plane should be as weak as possible. One way to fulfill this requirement is to make zero the corresponding temporal aberration coefficients  2  k ð6:59Þ Tjr0 jr0 ¼ 0; e ¼ e0 ¼ 0: Tje jr0 ¼ 0; e ¼ e0 ¼ 0 ðk ¼ 1; 2; 3Þ;

Here e0 is initial average energy of the bunch. If we assume the full temporal spread of the bunch equal to zero, the mass resolution ℜ0 of such an ideal mass-spectrometer would be proportional to the number M of the bunch oscillations: ℜ0 ¼ MT ð0; e0 Þ=t0 , with t0 denoting the initial duration of the bunch. In reality, even if  Eq.(6.59) is obeyed, the nonzero higher-order aberrations ðTjr0 r0 eÞ and Tje4 resulting from the energy spread may noticeably restrict the mass resolution. Coulomb repulsion plays a very negative role here – it leads to a continuous increase of the energy spread in the course of ion bunch oscillations. The mechanism of this effect is rather simple: the ions traveling at the leading edge of the bunch acquire the energy because they experience the Coulomb force being codirectional with their velocity. At the same time, the ions located at the bunch’s tail are continuously decelerated.

Space Charge in Charged Particle Bunches

209

Figure 112 illustrates the energy spread resulting from the Coulomb repulsion effect after the half-oscillation time of the bunch traveling between the mirrors. For the bunch comprising N ¼ 5  104 ions (initial average energy of the bunch e0 ¼ 2; 000 eV, initial duration t0 ¼ 60 ns), the energy spread appears to be  3 eV. The Coulomb-induced energy spread acquired after multiple oscillations may become crucial.

CHAPTER

7 General Properties of Emission-Imaging Systems

Contents

7.1. Charged Particle Density Transformations and Electron Image 7.2. Spatiotemporal Spread Function: Isoplanatism Condition 7.3. Modulation and Phase Transfer Functions: Spatial and Temporal Resolution

212 222 228

This chapter is devoted to most general characteristics of the images formed by the bunches of charged particles moving in electromagnetic fields. In Section 7.1, we derive the integral correlations between the initial (perhaps, nonstationary) probability distributions of charged particles on the emitter surface (the input image) and the probability distribution of charged particles at a given time moment or on a given smooth surface. Thus we come to the concept of the output electron or ion image and its principal characteristic – the spatiotemporal spread function. The general isoplanatism condition is introduced in Section 7.2 as a condition for the spread function to be invariant with respect to the shifts in a sufficiently small spatiotemporal region of the emitter. Section 7.3 considers the Fourier representation of the spread function and introduces the concept of modulation and phase transfer functions, which leads to the definition of the spatiotemporal resolution for static and dynamic emission-imaging systems.

Advances in Imaging and Electron Physics, Volume 155 ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)00807-0

2009 Elsevier Inc. All rights reserved.

#

211

212

General Properties of Emission-Imaging Systems

7.1. CHARGED PARTICLE DENSITY TRANSFORMATIONS AND ELECTRON IMAGE We will study the charged particle motion in a 3D domain O with the boundary @O. Let us assume that the charged particles enter the domain O through the surface E  @O which we term the emitter, and choose a global Cartesian coordinate system {x,y,z} with the center O, so that the z-axis coincides with the normal vector nO to the emitter surface at the point O, as shown in Figure 113. We also assume that the emitter surface allows unambiguous projecting onto the xy-plane, which is equivalent to the condition that any straight line parallel to the z-axis has no more than one point of intersection with E. We denote projection of the emitter E onto the xy-plane by Exy. Under this condition, the emitter surface equation can be explicitly resolved with respect to z: z0 ¼ z0 ðx0 ; y0 Þ ¼ z0 ðr0 Þ:

ð7:1Þ

Here {x0, y0, z0} 2 E are coordinates of the points on the emitter surface, and r0 ¼ {x0, y0} 2 Exy is a two-dimensional vector in the xy-plane. Consider an individual particle that enters the domain O at the time moment t0 (0  t0  T) at the initial point A 2 E with the radius-vector

∂Ω

z Ω

x

R0 nO z0 = z0(r0)

O r0

E

Exy

y

FIGURE 113 The charged particles entering the domain O through the emitter surface E. See text for details.

General Properties of Emission-Imaging Systems

213

R0 ¼ {r0, z0(r0)}, and let v0njr0 be the projection of initial velocity of the particle onto the normal vector nA at the point A. The entire bunch of the particles entering the domain O through the emitter E during the time interval 0  t0  T (further, we call the time T pulse duration) can be described by the set of independent initial parameters
 ð7:2Þ X ¼ r0 ; v0 ; t0 : r0 2 Exy ; 0  t0  T; v0n jr0  0 :

It is very convenient to interpret initial conditions of a particle as realization of the 6D random variable x ¼ {r0, v0, t0} taking values in X. With this aim in view, consider the differential distribution ðTÞ

ðTÞ ðTÞ

d6 Q0 ðr0 ; v0 ; t0 Þ ¼ L0 px ðr0 ; v0 ; t0 Þd2 r0 d3 v0 dt0

ð7:3Þ

of the electric charge brought by the bunch of charged particles through the emitter E during the time interval 0  t0  T into the domain O. ðTÞ The probability density function px ðr0 ; v0 ; t0 Þ is determined on X and normalized so that ð ðTÞ ð7:4Þ px ðr0 ; v0 ; t0 Þd2 r0 d3 v0 dt0 ¼ 1: X

ðTÞ

According to this normalization, L0 represents the total charge of the bunch. Along with the distribution (7.3), consider the distribution ðTÞ

ðTÞ ðTÞ

d5 J0 ðr0 ; v0 ; t0 Þ ¼ L0 px ðr0 ; v0 ; t0 Þd2 r0 d3 v0

ð7:5Þ

of the current passing through the emitter surface. If the charged particles’ emission into the domain O is stationary (e.g. does not depend on time) we consider the stationary differential distribution of emission current d5 J0 ðr0 ; v0 Þ ¼ I0 px ðr0 ; v0 Þ d2 r0 d3 v0 ;

ð7:6Þ

with I0 denoting the total stationary current through the emitter surface and the stationary probability density distribution px(r0, v0) obeying the normalization condition ð px ðr0 ; v0 Þd2 r0 d3 v0 ¼ 1: ð7:7Þ X

0

Here X ¼ fr0 ; v0 : r0 2 Exy; v0n jr0  0g is the ‘‘coordinate-velocity’’ 0 component of the initial parameters set X, so that X ¼ X  [0, T]. Now we must define a law that governs the motion of charged particles in the domain O. Assuming that an electromagnetic (generally, nonstationary) field {E(R,t), B(R,t)} exists in the domain O, consider the 0

214

General Properties of Emission-Imaging Systems

trajectories of the particles with charge q and mass m as solutions of the Lorenz equations : ¨ ¼ q ½EðR; tÞ þ R  BðR; tފ ð7:8Þ R m with initial conditions on the emitter surface R jt ¼ 0 ¼ fr0 ; z0 ðR0 Þg;

R jt ¼ 0 ¼ v 0 :

ð7:9Þ

Denote R(t, t0, r0, v0) the solution of the Cauchy problem (7.8), (7.9) as function of time and initial parameters. It should be noted that the ‘‘internal’’ electromagnetic fields resulting from the interaction between the charged particles can be also ascribed to the field {E, B}. In the particular case that the electromagnetic field {E, B} in the domain O does not depend on time, the charged particle trajectories are invariant with respect to any shift of initial time moment t0 in the sense that Rðt; t0 ; r0 ; v0 Þ  Rðt

t0 ; 0; r0 ; v0 Þ;

0  t0  T;

t  t0

ð7:10Þ

which means that any trajectory R(t, t0, r0, v0) with the initial parameters t0, r0, v0 is, in fact, the shift in time by t0 of the corresponding trajectory R (t, 0, r0, v0) starting at the time moment t0 ¼ 0. It is clear that any distribution function characterizing the coordinates and velocities of the charged particles in the domain O at any time moment can be expressed in terms of the initial distribution ðTÞ px ðr0 ; v0 ; t0 Þ on the emitter surface and the set of trajectories {r(t, t0, r0, v0)}. We start our consideration of distribution functions describing the charged particle dynamics from the space charge density distribution. As is well known from elementary physics, the space charge density r(R, t) is defined as a ratio of the charge contained in a small enough region that involves the point R at the time moment t to the volume DV of that region in the limit DV ! 0. Let us fix the time moment t > 0 and consider a transformation of the 6D random variable x $ {r0, v0, t0} into the 3-dimensional random variable  $ R(t, t0, r0, v0). In accordance with the general equation (see Appendix 6), we derive ðTÞ ðTÞ

rðTÞ ðR; tÞ ¼ L0 p ðR; tÞ ð ðTÞ ðTÞ p ðr0 ; v0 ; t0 ÞdD ½R x ¼L 0

ð7:11Þ Rðt; t0 ; r0 ; v0 ފd2 r0 d3 v0 dt0 :

It is worthwhile to note that the integration over t0 should be made here within the time interval 0  t0  min(t, T). This is manifestation of the general causality principle: Only the particles emitted not later the time moment t0 ¼ t may contribute to r(T)(R, t). If the electromagnetic field does not depend on time, we get

215

General Properties of Emission-Imaging Systems

rðTÞ ðR; tÞ ¼

ðTÞ L0

ð ð minðt;TÞ ðTÞ px ðr0 ; v0 ; t0 ÞdD ½R

X

¼

ðTÞ L0

0

ð

X

Rðt

t0 ; r0 ; v0 ފd2 r0 d3 v0 dt0

0

2

3

d r0 d v 0

0

ðt

ðTÞ

px ðr0 ; v0 ; t

tÞdD ½R

Rðt; r0 ; v0 ފdt: ð7:12Þ

t minðt;TÞ ðTÞ

If, additionally, the emission probability density px ðr0 ; v0 ; t0 Þ does not depend on time within the pulse interval 0 < t0 < T, for any fixed ðTÞ T > 0 it can be represented in the form px ðr0 ; v0 ; t0 Þ ¼ T 1 px ðr0 ; v0 ÞYT ðt0 Þ, where the stationary probability density px(r0, v0) is normalized according to Eq. (7.7), and YT is unit function of the time interval [0, T]. Denoting ðTÞ I0 ¼ L0 T 1 the constant current incoming from the emitter surface into the domain O during the time interval [0, T], from Eq. (7.12) we derive ðTÞ

r

ðR; tÞ ¼ I0

ð

X

0

2

3

px ðr0 ; v0 Þd r0 d v0

ðt

dD ½R

Rðt; r0 ; v0 ފdt:

ð7:13Þ

t minðt;TÞ

It should be emphasized that it is the invariance condition (7.10) that makes it possible to proceed in Eq. (7.12) and (7.13) from the integration over the initial time moment t0 to the integration over time t on the stationary trajectories R(t, r0, v0). Let us briefly analyze the space charge density in Eq. (7.12), (7.13) as function of time. Having fixed a point R 2 O, consider a set of initial parameters {r0, v0} ¼ X(R)  X, for which the equation R ¼ R(t, r0, v0) is resolvable with respect to t, and denote t*(R, r0, v0) the solution of this equation for any fixed r0, v0 2 X0(R). For the sake of simplicity, here we assume that the solution t*(R, r0, v0) is unique for any fixed r0, v0, which means that the corresponding trajectory arrives at the point R only once (we leave it for the reader to consider more general case). Obviously, X(R) represents the set of initial parameters {r0, v0} that correspond to the stationary trajectories R(t, r0, v0) passing at different time moments t*(R, r0, v0) through the point R 2 O. The set X(R) is called the accessibility set of the point R. It may happen that the accessibility set X(R) is empty for a fixed R 2 O, which means that there are no trajectories passing through the point R, and therefore the space charge density is zero at such point for any t. Taking this into account, we can define a maximal subdomain ~  O, so that X(R) is not empty for all R 2 O. ~ Obviously, the subdomain O ~ O is completely filled with the charged particle trajectories R(t, r0, v0). If we denote

216

General Properties of Emission-Imaging Systems

t min ðRÞ ¼

min 0

t∗ ðR; r0 ; v0 Þ;

fr0 ;v0 g2 X ðRÞ

t max ðRÞ ¼

max0

t∗ ðR; r0 ; v0 Þ

fr0 ;v0 g2 X ðRÞ

ð7:14Þ the minimal and maximal time for the trajectories {R(t, r0, v0)} to arrive at the fixed point R, the evolution in time of the space charge density r(T)(R, t) will be determined by the relative position of the time interval of integration ∗ [t min(t, T), t] and the time interval ½t∗ min ðRÞ; tmax ðRފ characterizing the delta-function support in Eqs. (7.12) and (7.13). Dependence r(T)(R, t) on time t is qualitatively shown in Figure 114 for the emission pulse being rather long, so that T > DT ∗ ðRÞ ¼ ∗ t∗ t∗ min ðRÞ. For t  tmin ðRÞ, the space charge density at the point max ðRÞ R is zero because none of the particles belonging to the charged particles ‘‘cloud’’ {R(t t0, r0, v0)}, r0, v0 2 X0(R), t0 2 [0, T] is able to reach the point R for the time lesser than t∗ min ðRÞ. During the time interval ∗ t∗ min ðRÞ  t  tmax ðRÞ, the number of particles able to contribute to the space charge density at the point R increases, and, accordingly, r(T)(R, t) increases from zero to its maximum value. Within the time interval ∗ t∗ max ðRÞ  t  tmin ðRÞ þ T all the particles belonging to the accessibility set X0(R) contribute to the space charge density, and therefore it remains ∗ constant. Then, for T þ t∗ min ðRÞ  t  T þ tmax ðRÞ, the charged particles cloud is gradually leaving the point R, and r(T)(R, t) decreases from its maximum to zero. The space charge density is again zero for t  T þ t∗ max ðRÞ because all particles of the cloud have already left the vicinity of the point R. It can be easily seen that the presence of the ‘‘plateau’’ in the dependence r(T)(R, t) in the case under consideration is directly connected with the additional assumption that the emission pulse is long enough. We leave it for the reader to consider the case when the electromagnetic field

r (R, t)

t*min

FIGURE 114

t*max

t*min + T

t*max + T (T)

t

Qualitative view of the space charge density r (R, t) as a function of time.

General Properties of Emission-Imaging Systems

217

ðTÞ

is still stationary and the emission probability density px does not depend on time within the pulse interval 0 < t0 < T, but T  DT*(R). Directing the pulse duration T in Eq. (7.13) to infinity, we come to the stationary space charge density in the domain O rðRÞ ¼ I0

ð

X

2

3

px ðr0 ; v0 Þd r0 d v0

0

1 ð

dD ½R

Rðt; r0 ; v0 ފdt:

ð7:15Þ

0

The infinite upper limit in the inner integral can be replaced by any finite value larger than max0 T∗ ðRÞ. Simple physical sense can be attached to fr0; v0 g2X ðRÞ

Eq. (7.15) by means of constructing its discrete analog. Consider three simplest rectangular computational meshes: {r0i}, {v0j} in the initial parameters region {r0, v0}, and {Rk} in the domain O of charged particle motion. Replacing approximately the integrals in Eq. (7.15) by finite sums, we obtain I0 X px ðr0i ; v0j ÞmðDR0i ÞmðDv0j ÞDt∗ ðRk ; r0i ; v0j Þ: ð7:16Þ rðRk Þ  mðDRk Þ i;j




Here m( ) denotes the volume of the corresponding elementary parallelepiped, and Dt*(Rk, r0i, v0j) represents the time that the trajectory R(t, r0i, v0j) spends inside the elementary cell DRk with the center at the point Rk (Figure 115). For each k fixed, summation over i, j is spread over only those trajectories that pass through the cell DRk. This simple formula is frequently used for space charge density calculation in static charged particle optics. ∆Rk

B A R (t, r0i, V0j)

FIGURE 115 Numerical interpretation of the Eq. (7.15): DRk is the elementary cell with the center at the point Rk; A and B are, relatively, the points at which the trajectory R(t, r0i, v0j) enters and exits the elementary cell DRk.

218

General Properties of Emission-Imaging Systems

Now we proceed to constructing the spatiotemporal current density distribution on a smooth surface S. It is well known that the current density can be defined as a ratio of the charge DQS carried by the particles through an elementary area DS during the time Dt to the product DS Dt at DS and Dt tending to zero independently. Let us consider a smooth surface S located in the domain O and placed on the pathway of the bunch of charged particles. Assuming, as before, the surface S to allow unambiguous projecting onto the xy-plane, we may represent the equation of the surface S as zS ¼ zS(rS). Denote tS ¼ tS(t0, r0, v0) the absolute time (counted from the time moment t ¼ 0) at which the trajectory R(t, t0, r0, v0) arrives at the point RS on the surface S. Obviously, tS ¼ t0 þ tS(t0, r0, v0), where tS(t0, r0, v0) is the transit time needed for an individual particle to reach the surface S. It should be emphasized that the transit time tS in the nonstationary field depends itself upon the start time moment t0. The solution of the Lorenz equation (7.8) appears in the form of two components R(t, t0, r0, v0) ¼ {r(t, t0, r0, v0), z(t, t0, r0, v0)}, therefore, the position of the particle arrival point RS ¼ {rS, zS} 2 S appears as rS ¼ rS ðt0 ; r0 ; v0 Þ ¼ r ðtS ðt0 ; r0 ; v0 Þ; t0 ; r0 ; v0 Þ zS ¼ zS ðt0 ; r0 ; v0 Þ ¼ z ðtS ðt0 ; r0 ; v0 Þ; t0 ; r0 ; v0 Þ:

ð7:17Þ

We again encounter a transformation of random values: the transformation of the 6D random value x $ {r0, v0, t0} into the 3D random value  $ {rS, tS}, as shown in Figure 116. Again, using the general Eq. (A6.11),

x

x

v0 rs

R0 r0

RS O1

O

E y

z

S

R(t, t 0, r0, v0) y

(T)

FIGURE 116 Transformation of the input charged particle distribution px (r0, v0, t0) from the emitter E to the surface S by the charged particle trajectories R(t, t0, r0, v0).

General Properties of Emission-Imaging Systems

219

we immediately get the time-dependent spatiotemporal current density distribution on the surface S: JS ðrS ; tS Þ

ðTÞ ðTÞ

¼ L0 p ðrS ; tS Þ ðTÞ

¼ L0 tS

Ð

ðTÞ

px ðr0 ; v0 ; t0 ÞdD ½rS 2

rS ðt0 ; r0 ; v0 Þ;

3

tS ðt0 ; r0 ; v0 ފd r0 d v0 dt0 :

ð7:18Þ

Denote tS; min ¼ fr0 ; vmin tS ; tS; max ¼ fr0 ;vmax ts the minimal and 0 ; t0 g 2 X 0 ;t0 g 2 X maximal transit time needed for the emitted particles to arrive at the surface S. It is clear that the current density JS(rS, tS) is a compactly supported function being nonzero within the interval tS,min  tS  T þ tS,max because none of the particles emitted within the time interval [0, T] could reach the surface S at the time tS lesser than t∗ S;min , and all the particles will have already passed through the surface S by the time moment T þ tS,max. Let assume that we must accumulate all the particles emitted during the pulse duration T on the image receiver surface S. Integrating Eq. (7.18) over any time interval {tS} containing the support of the function JS(rS, tS) - in other words, summing up all the particles having reached the elementary area DS of the surface S during the pulse time interval 0  t0  T - provides the output surface charge density on the image receiver S: ð ðTÞ ðTÞ sS ðrS Þ ¼ L0 px ðr0 ; v0 ; t0 ÞdD ½rS rS ðt0 ; r0 ; v0 ފd2 r0 d3 v0 dt0 : ð7:19Þ Further sS(rS) represents an important object that contains basic information on spatiotemporal characteristics of electron image. To clarify the physical sense of the relation obtained, let us consider, as above, its discreet analog. The simplest piecewise approximation of Eq. (7.19) upon the meshes {r0i}, {v0j}, {t0k}, {rSl} gives ðTÞ

sS ðrSl Þ 

L0 X px ðr0i ; v0j ; t0k ÞmðDr0i ÞmðDv0j ÞDt0k : mðDrSl Þ i;j;k

ð7:20Þ

With the index l fixed, the summation is spread over those indexes i,j,k, which correspond to the trajectories hitting the elementary area DrSl centered at the point rSl, and m( ) again denotes the volume of the corresponding elementary cells. Let us now consider the limiting stationary case when both particle emission and electromagnetic field in the domain O do not depend on time. It follows from Eq. (7.10) that the time tS needed for the trajectory R(t, t0, r0, v0) to reach the surface S in stationary field is




tS ¼ tS ðt0 ; r0 ; v0 Þ ¼ t0 þ tS ð0; r0 ; v0 Þ ¼ t0 þ tS ðr0 ; v0 Þ;

ð7:21Þ

220

General Properties of Emission-Imaging Systems

with the transit time tS(r0, v0) independent of t0. Obviously, the point RS of the particle arrival on the surface S also does not depend on t0. Indeed, according to Eq. (7.10) and (7.21),     RS ¼ R tS ðt0 ; r0 ; v0 Þ; t0 ; r0 ; v0 ¼ R tS ðt0 ; r0 ; v0 Þ t0 ; 0; r0 ; v0   ¼ R tS ðr0 ; v0 Þ; r0 ; v0 ¼ frS ðr0 ; v0 Þ; zS ðr0 ; v0 Þg: ð7:22Þ Let us consider a pulse of big enough duration T and transform Eq. (7.18) with regard to Eq. (7.21), (7.22) as follows: ðTÞ

ðTÞ

JS ðrS ;tS Þ ¼ L0

Ð

Ð

ðTÞ

t0 tS ðr0 ;v0 ފd2 r0 d3 v0 dt0 ðT rS ðr0 ;v0 ފd2 r0 d3 v0 dD ½tS t0 tS ðr0 ;v0 ފdt0 :

px ðr0 ;v0 ÞdD ½rS

¼ I0 px ðr0 ;v0 ÞdD ½rS

rS ðr0 ;v0 Þ;tS

0

ð7:23Þ

The inner integral in Eq. (7.23) is equal to either unit or zero, depending on the interrelation among the values T, tS, and tS(r0, v0). It can be easily shown (we recommend it to the reader as an exercise) that evoluðTÞ tion of the current density JS in time is quite similar to evolution of the (T) space charge density r (R, t) in Eq. (7.13). Directing T to infinity in Eq. (2.29) or, in other words, considering a sequence of increasingly long pulses, we come to the stationary current density through the surface S ð ð7:24Þ JS ðrS Þ ¼ I0 px ðr0 ; v0 ÞdD ½rS rS ðr0 ; v0 ފd2 r0 d3 v0 : The discrete analog of this expression is quite similar to Eq. (7.20), with the only exception that the mesh {t0k} and the corresponding summation should be omitted. Until now we have made no special assumptions respecting the propðTÞ erties of the input probability density px ðr0 ; v0 ; t0 Þ (in the stationary case px(r0, v0)). Now, in order to proceed to the question of how the charged particles may form an image, we need to accept some additional assumption regarding the properties of charged particles emission. For definiteness and brevity sake, we will speak here about electron image, although all our considerations may also be in full measure referred to ion image. Consider a local Cartesian coordinate system {U,V,W} in the velocity space, attached to an arbitrary chosen point R0 ¼ {r0, z0(r0)} on the emitter surface E (Figure 117). The W-axis is directed along the internal (with respect to the domain O) normal nE to the emitter surface E at the point R0. Accordingly, the axes {U,V} determine the tangential plane to the emitter surface E at the point R0. Provided that the axes {U,V} are defined, we can

General Properties of Emission-Imaging Systems

221

U V

V0t

x

V0

w Ω R0

r0

nE

V0n

z

O W

y

E

FIGURE 117 Local coordinate system {U, V, W} in the velocity space, attached to a point R0 on the emitter surface (nE is normal vector, the plane (U, V) is tangential to the emitter surface E at the point R0).

establish a linear relationship v0 ¼ P(R0)u0 between the components v0 ¼ (v0x, v0y, v0z) of the initial velocity vector of a particle in the global coordinate system {x, y, z} and the components u0 ¼ (U0, V0, W0) of the same vector in the local coordinate system {U,V,W}. The transformation matrix P(R0) is unambiguously determined by the differential properties of the emitter surface E at the point R0 and smoothly dependent on R0 if we consider the local coordinate system {U,V,W} as a shift (in the sense of Riemann geometry) of the global coordinate system {x, y, z} along the geodetic joining the origin O and the point R0 on the emitter surface. Let us assume that the initial distribution of particles in velocities is independent of that in initial coordinates and time, so that the distribution ðTÞ density px ðr0 ; u0 ; t0 Þ (we further use the same designation for the distribution density transformed to the local coordinate system in the velocity space) can be factorized as ðTÞ

px ðr0 ; u0 ; t0 Þ ¼ f ðTÞ ðr0 ; t0 Þhðu0 Þ;

ð7:25Þ

with the functions f (T)(r0, t0), h(u0) being integrally normalized by unit as ð ð f ðTÞ ðr0 ; t0 Þd2 r0 dt0 ¼ hðu0 Þdu0 ¼ 1: ð7:26Þ Such an assumption, usually made when emission-imaging systems are considered, implies that the local emission properties characterized by

222

General Properties of Emission-Imaging Systems

the distribution density h(u0) are the same for all points of emitter and do not change in time. The role of the local coordinate system {U,V,W} attached to a fixed point of the emitter surface is quite clear: This coordinate system is well suited to describe the particle distribution in initial velocities in the case of a non-flat emitter surface. ðTÞ ðTÞ The function J0 ðr0 ; t0 Þ ¼ L0 f ðr0 ; t0 Þ describing the distribution of emitted particles in coordinates and time represents the input spatiotemporal electron image. Thus, the condition (7.25) makes it possible to separate the properties of input electron image from the initial distribution of emitted particles in initial velocities, which, in fact, is connected with intrinsic physical properties of emitter. With regard to the assumptions made, the basic Eqs. (7.18) and (7.19) appear as ð ðTÞ ðTÞ JS ðrS ; tÞ ¼ J0 ðr0 ; t0 Þhðu0 ÞdD ½rS rS ðr0 ; u0 ; t0 Þ; t

tS ðr0 ; u0 ; t0 ފd2 r0 d3 u0 dt0

ð ðTÞ ðTÞ sS ðrS Þ ¼ J0 ðr0 ; t0 Þhðu0 ÞdD ½rS

ð7:27Þ

rS ðr0 ; u0 ; t0 ފd2 r0 d3 u0 dt0 :

ð7:28Þ

ðTÞ

We call the output electron image the distribution sS ðrS Þ being the integral of the distribution JS(rS, t) over the pulse recording time tS. If the emission characteristic h(u0) is given, we may consider Eq. (7.28) as a ðTÞ transformation of the input nonstationary electron image J0 ðr0 ; t0 Þ into ðTÞ the output electron image sS ðrS Þ, performed by the emission-imaging system S. The properties of this transformation are entirely determined by the properties of electromagnetic fields and charged particle trajectories inside the domain O. In the stationary case, assuming that px(r0, u0) ¼ f(r0)h(u0), we come to the input stationary electron image J0 (r0) ¼ I 0 f (r0). The transformation of the input electron image J0(r0) into the output electron image JS(rS) reads ð JS ðrS Þ ¼ J0 ðr0 Þ h ðu0 Þ dD ½rS rS ðr0 ; u0 ފ d2 r0 d3 u0 ð7:29Þ

7.2. SPATIOTEMPORAL SPREAD FUNCTION: ISOPLANATISM CONDITION Let us consider the question of whether the emission-imaging system S performing the transformation of input electron image according to Eq. (7.29) is linear. With this goal, we first must specify the structure of the electromagnetic field in the domain O of charge particle motion. For the sake of simplicity, we restrict ourselves in this section to the consideration of two types of electric fields acting on the charged particles in O.

General Properties of Emission-Imaging Systems

223

To the first type, we assign the external static or time-dependent electric fields generated by a system of electrodes that form the boundary @O of the domain O. For the second type of fields, we consider the internal Coulomb electric fields resulting from interaction between the charged particles comprising the charged particle cloud. Within the quasi-stationary approximation inherent to the majority of charged particle optics problems, the electric field E(R, t) describing both the first and second types of fields possesses the time-dependent potential ’(R, t), so that EðR; tÞ ¼

r’ðR; tÞ;

ð7:30Þ

and the closed system of equations describing the charged particles motion in the domain O appears as D’ ¼

4prðR; tÞ;

ð

Rðt; t0 ; r0 ; u0 ފ d2 r0 d3 u0 dt0

rðR; tÞ ¼ J0 ðr0 ; t0 Þhðu0 ÞdD ½R ¨ ¼ R

 ðR; tÞ ’j@O ¼ ’

q r’ðR; tÞ m

R jt¼t0 ¼ fr0 ; z0 ðr0 Þg;

_ jt¼t ¼ PðR0 Þu0 ; R 0

ð7:31Þ ð7:32Þ ð7:33Þ ð7:34Þ

 ðR; tÞ is a given potential distribution on the boundary @O. This Here ’ system of equations is quite similar to that given in Chapter 6, with the only exception that both the emission current density and the external field in the domain O are here assumed time-dependent. Now it is clear that the transformation J0 (r0, t0) ! sS (rS) in Eq. (7.28) of the input electron image into the output one cannot generally be considered as linear because, as the Eqs. (7.31) – (7.34) show, the charged particle trajectories and, consequently, the positions rS(r0, u0, t0) of charged particles on the image receiver surface S depend themselves on the input electron image J0(r0, t0). From a physical standpoint, such nonlinearity springs from the contribution of the Coulomb interaction between the charged particles, which makes the right-hand side of the Poisson equation (7.31) nonzero. The transformation J0(r0, t0) ! sS(rS) may be treated ðTÞ as asymptotically linear if the total emitted electric charge L0 (in the static case, the total emission current density I0) is small in the sense that the contribution of the Coulomb interaction to the charged particles’ motion is small enough compared with the external electric field action. The expression ‘‘contribution to the charged particles’ motion’’ seems somewhat indistinct, and it would have been more correct to allude not to the charged particle trajectories but to the characteristics of output

224

General Properties of Emission-Imaging Systems

electron image in speaking about the emission-imaging system linearity. Therefore, remaining within the assumption of negligibly small spacecharge effects, we first consider the general criteria of image quality in emission-imaging systems on the basis of the linear system theory (see Appendix 5) and then discuss possible generalizations. Thus, let us put r(R, t) ¼ 0 in Eq. (7.31). In this case, the self-consistent field/trajectory problem (7.31) – (7.34) disintegrates into two independent problems, the first of which - the field problem - is calculating the external electric field as a solution of the Dirichlet problem for the Laplace equation D’ ¼ 0;

 ðR; tÞ ’ j@O ¼ ’

ð7:35Þ

with boundary conditions on the electrodes surface, whereas the second one - the trajectories problem - is solving the Cauchy problem for the Lorenz equation (7.33) with initial conditions (7.34) on the emitter surface for a representative enough set of initial parameters {r0, u0, t0}. Following the methodology of Appendix 5, let us arbitrarily fix any ∗ initial point r∗ 0 2 Exy and any initial time moment t0 2 ½0; TŠ, and define the spatiotemporal point spread function of the emission-imaging system S as ∗ the system’s response to the input dD-pulse located at the point ðr∗ 0 ; t0 Þ: ð ∗ IS ðrS ; r∗ r∗ t∗ rS ðr0 ; u0 ; t0 ފd2 r0 d3 u0 dt0 0 ; t0 Þ ¼ dD ðr0 0 ; t0 0 Þhðu0 ÞdD ½rS ð ¼ hðu0 ÞdD ½rS

∗ 3 rS ðr∗ 0 ; u0 ; t0 ފd u0 :

ð7:36Þ

According to the ‘‘image-oriented’’ terminology accepted in this chap∗ ter, the spatiotemporal point spread function IS ðrS ; r∗ 0 ; t0 Þ represents the output electron image generated by the emission-imaging system S on the image receiver S as a transformation of the pulse electron image t∗ dD ðr0 r∗ 0 Þ with the total charge equal to unit. Inasmuch as 0 ; t0 the system S is supposed to be linear, for an arbitrary input electron image J0(r0, t0) we get the output electron image ð sS ðrS Þ ¼ J0 ðr0 ; t0 ÞIS ðrS ; r0 ; t0 Þd2 r0 dt0 : ð7:37Þ Similarly, the correlation between the input and output electron images in the stationary case is given by the relation ð ð7:38Þ JS ðrS Þ ¼ J0 ðr0 Þ IS ðrS ; r0 Þ d2 r0 ; with ð h IS ðrS ; r0 Þ ¼ hðu0 ÞdD rS

i rS ðr0 ; u0 Þ d3 u0

ð7:39Þ

General Properties of Emission-Imaging Systems

225

being the spatial point spread function of the stationary emission-imaging system S and representing the response of the system S to the input pulse image dD ðr0 r∗ 0 Þ with the total current I0 equal to unit. Obviously, due to the normalization (7.26) of the particle distribution function h(u0) in initial velocities, the point spread functions (7.36) and (7.39) are also normalized ð ð IS ðrS ; r0 Þd2 rS ¼ IS ðrS ; r0 ; t0 Þd2 rS dt0 ¼ 1: ð7:40Þ The point spread functions IS ðrS ; r0 ; t0 Þ, IS ðrS ; r0 Þ contain in themselves complete information needed to characterize any emissionimaging system from the viewpoint of image quality transformation. The most widely used criteria that allow correct judgment of emissionimaging system quality are the modulation and phase transfer functions (MTF and PTF). These characteristics are introduced and analyzed in Appendix 5 for general linear systems under the important assumption of isoplanatism in a broad sense. Generally speaking, no reason exists to consider any linear emission-imaging system as isoplanatic in a broad sense because the spatiotemporal point spread function (7.36) may ∗ essentially depend on the particular choice of the initial parameter r∗ 0 ; t0 on the emitter. Nonetheless, as mentioned in Appendix 5, we can ‘‘localize’’ the isoplanatism conception by considering the transformations of the input electron images located in a sufficiently small spatiotemporal region of emitter. ∗ Let us choose a small enough vicinity Wðr∗ 0 ; t0 Þ of the arbitrary fixed ∗ ∗ initial point r0 2 Exy , t0 2 ½0; TŠ (Figure 118) and consider the spatiotemporal spread function ð i h ð7:41Þ IS ðrS ; r0 ; t0 Þ ¼ hðu0 Þ dD rS rS ðr0 ; u0 ; t0 Þ d3 u0 ∗ ∗ in Wðr∗ 0 ; t0 Þ. Having also fixed any initial velocity vector u0 ¼ u0 , let us expand the function rS(r0, u0, t0) with respect to the differences t∗ u∗ r0 r∗ 0 , picking out explicitly the linear part of the 0 ; t0 0 ; u0 expansion with respect to r0 r∗ t∗ 0 ; t0 0 ∗ ∗ rS ðr0 ; u0 ; t0 Þ ¼ rS ðr∗ 0 ; u0 ; t 0 Þ þ

þ

@rS jr∗ ;u∗ ;t∗ ðt0 @t0 0 0 0

@rS jr∗ ;u∗ ;t∗ ðr0 @r0 0 0 0 ∗ t∗ 0 Þ þ AS ½r0

r∗ 0Þ r∗ 0 ; u0

u∗ 0 ; t0

t∗ 0 Š: ð7:42Þ

All velocity-dependent terms in Eq. (7.42) starting from the linear one (we will see farther that this linear term, being proportional to u0 u∗ 0,

226

General Properties of Emission-Imaging Systems

t0 Isoplanatism region W(r*0,t *0)

Image receiver region

{rS} T

rS

(∆r0, ∆t 0) R(t , t 0, r0, u0) (r*0, t* 0)

(r0, t 0)

r*S Diffusion figure

∆rs

Exy {r0} Emitter region

R(τ, t* 0, r* 0, u0)

FIGURE 118 Schematic view of isoplanatic transformation of the charged particle ∗ bunches emitted from the spatiotemporal region Wðr∗ 0 ; t0 Þ.

plays a special role in image formation) are included into the aberration term A∗ S ½
Š. u∗ Let us introduce local variables Dr0 ¼ r0 r∗ 0 ; Du ¼ u0 0 ; Dt0 ¼ ∗ t0 t0 in the initial parameters region and local variables ∗ ∗ ∗ ∗ DrS ðr0 ; u0 ; t0 Þ ¼ rS ðr0 ; u0 ; t0 Þ rS ðr∗ rS ðr∗ 0 ; u0 ; t0 Þ, DrS ¼ rS 0 ; u0 ; t0 Þ in ∗ the image receiver region. The square matrix MS and the vector V∗ S defined as M∗ S ¼

@rS jr∗ ;u ;t∗ ; @r0 0 0 0

V∗ S ¼

@rS jr∗ ;u ;t∗ @t0 0 0 0

ð7:43Þ

are called the matrix of local magnifications and image streak speed on the ∗ ∗ principal trajectory Rðt; t∗ 0 ; r0 ; u0 Þ. With no loss in generality, we may further put u∗ 0 ¼ 0 and use u0 instead of Du0 in all the ‘‘local’’ equations below. With regard to the notations introduced, we can bring Eq. (7.42) to the form ∗ ∗ DrS ðr0 ; u0 ; t0 Þ ¼ M∗ S Dr0 þ VS Dt0 þ AS ½Dr0 ; u0 ; Dt0 Š;

ð7:44Þ

whereas the argument of the dD-function in Eq. (7.41) appears as rS

rS ðr0 ; u0 ; t0 Þ ¼ DrS

M∗ S Dr0

V∗ S Dt0

A∗ S ½Dr0 ; u0 ; Dt0 Š:

ð7:45Þ

Equation (7.44) describes transformation of the local spatiotemporal ∗ ∗ ∗ ∗ region Wðr∗ 0 ; t0 Þ on the emitter into a vicinity of the point rS ðr0 ; u0 ; t0 Þ on the image receiver surface S. The first-order terms in Eq. (7.44) with

General Properties of Emission-Imaging Systems

227

respect to Dr0, Dt0 describe the linear (and, therefore, the univalent) part of the transformation, whereas the aberration term A[ ] is responsible for charged particle scattering on the surface S, resulted from the spread in initial velocities. The mechanism of such scattering is that every fixed ∗ point ðr0 ; t0 Þ 2 Wðr∗ 0 ; t0 Þ is mapped by means of the charged particle trajectories into the figure of diffusion {rS(r0, u0, t0)}, all points of which are parameterized by the initial velocity vector u0 that runs over all its possible values on the emitter (Figure 118). Generally, the figure of diffusion {rS(r0, u0, t0)} depends on the particular choice of the point (r0, t0), but in the frame of the local isoplanatism ∗ approximation, the region Wðr∗ 0 ; t0 Þ is assumed to be physically small enough for all its points (r0, t0) to have the same figures of diffusion, so that those figures can be obtained, one from another, by means of shift by the vector DrS. This immediately implies that the aberration term A∗ S ½Dr0 ; u0 ; Dt0 Š in Eqs. (7.44) and (7.45) may be replaced by the isoplanatic ∗ aberration term Aloc S ½u0 Š ¼ AS ½0; u0 ; 0Š related to the principal trajectory ∗ ∗ ∗ Rðt; t0 ; r0 ; u0 Þ. Within this approximation, we have




rS

rS ðr0 ; u0 ; t0 Þ ¼ DrS

M∗ S Dr0

V∗ S Dt0

Aloc S ½u0 Š

ð7:46Þ

∗ for all ðr0 ; t0 Þ 2 Wðr∗ 0 ; t0 Þ and, denoting the localized spatiotemporal ∗ ∗ point spread function as Iloc S ðDrS ; Dr0 ; Dt0 Þ ¼ IS ðrS þ DrS ; r0 þ Dr0 ; t0 þ Dt0 Þ, we obtain from Eq. (7.41) ð 3 loc V∗ Aloc IS ðDrS ; Dr0 ; Dt0 Þ ¼ hðu0 ÞdD ½DrS M∗ S Dr0 S Dt0 S ½u0 ŠŠd u0

¼ Iloc S ðDrS

M∗ S Dr0

V∗ S Dt0 ; 0; 0Þ:

ð7:47Þ

Thus, we have shown that the localized spatiotemporal spread function Iloc S ðDrS ; Dr0 ; Dt0 Þ obeys the isoplanatism condition with the rectan~ ∗ ¼ kM∗jV∗ k. According to the definition of local gular (2  3)-matrix M S S S isoplanatism (see Appendix 5), the emission-imaging system in question ∗ is locally isoplanatic in the spatiotemporal region Wðr∗ 0 ; t0 Þ with the spatiotemporal spread function ð i h 3 ðDr Þ ¼ hðu0 ÞdD DrS Aloc ð7:48Þ Iloc S S S ½u0 Š d u0 : We say that a transformation of the electron image is ideal in the ∗ isoplanatism region Wðr∗ 0 ; t0 Þ if Iloc S ðDrS Þ ¼ dD ½DrS Š;

ð7:49Þ

∗ and the figure of diffusion {rS(r0, u0, t0)} of any point fr0 ; t0 g 2 Wðr∗ 0 ; t0 Þ is ∗ ∗ reduced to the point rS(r0, 0, t0), so that the transformation Wðr0 ; t0 Þ ! S is univalent. It follows from Eq. (7.48) that the ideal image may exist in the two limiting cases:

228

General Properties of Emission-Imaging Systems

1. Aloc S ½u0 Š ¼ 0 for those u0 that obey h(u0) 6¼ 0 2. h(u0) ¼ dD(u0). The first condition is directly connected with the properties of electromagnetic field and charged particle trajectories in the domain O and requires all the image aberrations resulting from the charged particle spread in initial velocities to be zero on the image receiver surface. The second condition refers to the properties of emitter and signifies the complete monochromaticity of the latter. Both conditions cannot be strictly realized in practice and rather represent general criteria for emissionimaging system optimization. According to Eq. (7.37) and (7.48), the relationship between the ‘‘localized’’ electron images ∗ J0loc ðDr0 ; Dt0 Þ ¼ J0 ðr∗ 0 þ Dr0 ; t0 þ Dt0 Þ;

takes the convolution form ð loc sS ðDrS Þ ¼ J0loc ðDr0 ; Dt0 ÞIloc S ðDrS

∗ sloc S ðDrS Þ ¼ sS ðrS þ DrS Þ ð7:50Þ

M∗ S Dr0

2 V∗ S Dt0 Þd Dr0 dDt0

~ MS ð7:51Þ Iloc S ðDrS Þ  The stationary case can be considered analogously, with the difference that the isoplanatism region W does not contain time component t0, the aberration term in Eq. (7.44) does not depend on Dt0, and V∗ S ¼ 0. Formally, finite expression for the localized point spread function in the stationary case coincides with Eq. (7.48), and the corresponding convolution equation can be derived from Eq. (7.51) by putting V∗ S ¼ 0 and omitting integration over Dt0 as follows ð 2 loc M∗ JS ðDrS Þ ¼ J0loc ðDr0 ÞIloc S ðDrS S Dr0 Þd Dr0 ¼ J0loc ðDr0 ; Dt0 Þ

∗ ¼ J0loc ðDr0 Þ MS Iloc S ðDrS Þ: 

ð7:52Þ

7.3. MODULATION AND PHASE TRANSFER FUNCTIONS: SPATIAL AND TEMPORAL RESOLUTION We now proceed to consideration of the MTF and PTF in emissionimaging systems. We start from the stationary case as the simplest one. loc Assuming the matrix M∗ S nondegenerate and denoting A0 ½u0 Š ¼ loc ∗ 1 loc ðMS Þ AS ½u0 Š, we can refer the point spread function IS ðDrS Þ to the emitter by putting ð ðrÞ;loc ∗ loc ∗ 3 IS ðDr0 Þ ¼jdet MS jIS ðMS Dr0 Þ ¼ hðu0 ÞdD ½Dr0 Aloc 0 ½u0 ŠŠd u0 ð7:53Þ

General Properties of Emission-Imaging Systems

229

and define the point transfer function as the Fourier transform  ð rÞ; locðvÞ ¼ w∗ ðvÞexp½if∗ ðvފ of the referred point spread function I S ðrÞ;loc

IS ðDr0 Þ. Correspondingly, w*(v) and f*(v) are here the modulation and the phase transfer functions (MTF and PTF) associated with the isoplanatism region Wðr∗ 0 Þ. Consider the input electron image JSloc ðDr0 Þ ¼ 1 þ cos < o0 k; Dr0 >

ð7:54Þ

representing a local 1D test pattern with harmonic current density distributed along the unit vector k in the isoplanatism region Wðr∗ 0 Þ. As shown in Appendix 5, the MTF and PTF for the test pattern (7.54) represent, relatively, the k-cross-sections of w(v) and f(v) in 2D v-space, and coincide with the MTF and PTF of the spread function constructed for the plane L(k) being perpendicular to the vector k, so that ∗ ∗ ∗ w∗ k ðo0 Þ ¼ w ðo0 kÞ; fk ðo0 Þ ¼ f ðo0 kÞ. Let us put o0 ¼ 2pN, where N is the number of full oscillations (periods) of the test pattern density on the unit of length. In other words, N is spatial frequency of the test pattern. Following the definition accepted in optics, we will say that a stationary, locally isoplanatic,  at the contrast emission-imaging system possesses the spatial resolution N Þ along the direction k, if level  w in the isoplanatism region Wðr∗ 0  ¼ w∗ ð2pNkÞ  ¼ w: ð7:55Þ w∗ ð2pNÞ k

 ¼ Nðk;  wÞ Thus, spatial resolution represents the spatial frequency N of the k-oriented harmonic test pattern, which the emission-imaging system in question can transform with the prescribed contrast level w. Now we must consider image quality characteristics in dynamic emission-imaging systems. Let us first note that, in contrast to static emission-imaging systems intended mainly for transformation and recording of stationary electron images, dynamic emission-imaging systems are used for transformation and recording of electron images dependent on time. As we will show, thanks to the combination of the static focusing principle and the principle of image sweep along a given direction, the dynamic emission-imaging systems allow, under definite conditions, high-resolution recording of both spatial and temporal input current density modulation. The latter property makes the dynamic emissionimaging systems very effective in ultrafast processes investigations. Unlike the stationary case considered previously, the dimensions of arguments of input and output electron images in dynamic case are different: The 3D vector of initial parameters {DR0(Dx0, Dy0), Dt0} is transformed (in linear approximation) by the rectangular (23)-matrix ~ ∗ MS ¼ kM∗ S jVS k into a 2D image space vector ∗ DRS ¼ M∗ S DR0 þ VS Dt0 :

ð7:56Þ

230

General Properties of Emission-Imaging Systems

First, this means that, if we wish to study the properties of input electron images distributed both in space and time, we must (by analogy with the stationary case) introduce into consideration the input spatiotemporal test pattern, the current density of which is harmonically modulated both along a given spatial direction on the emitter Exy and in initial time t0. Second, it is clear that the direction of the streak speed vector V S should be correctly specified so that it would be possible to discern those two input current density modulations in the output electron image. In our theoretical considerations aimed at constructing the spatiotemporal MTF and PTF, we will follow the principal scheme commonly used in ultrafast recording experiments. In the vicinity of the point r∗ 0 (in practice the point r∗ usually coincides with the geometrical center of 0 ¼ 0), let us consider a vertical slit of the width l along the emitter r∗ 0 local coordinate Dx0 and the length L >> l along the local coordinate Dy0 (Figure 119). Consider the input electron image J0loc ðDr0 ; Dt0 Þ ¼ pl ðDx0 Þj0 ðDt0 ; Dy0 Þ

ð7:57Þ

being spatially localized within the slit, so that pl(Dx0) ¼ 1/2l for jDx0j  l and pl(Dx0) ¼ 0 for jDx0j > l. The representation (7.57) assumes that the input electron image may have current density modulation along the coordinate Dy0 (the longest side of the slit) and in initial time Dt0. We must now refer the spatiotemporal spread function (7.48), along with the general representation (7.51) of the output electron image, to the emitter. As already mentioned, in principle it is impossible to perform such a transformation for all three variables {Dx0, Dy0, Dt} ∆y0

∆yS

h M*S e2

V*S

I

∆xS

e2 x0, ∆z

e1

M*S e1

2 1

FIGURE 119 Linear transformation of input spatiotemporal current density modulation within the rectangular slit on the emitter to the image receiver (1, spatial modulation along the slit; 2, temporal modulation; the streak speed V∗ S is directed along the equiphase lines of spatial modulation).

General Properties of Emission-Imaging Systems

231

independently because only the 2D recording space {DxS, DyS} is available. In our case, having purposely specified the slit geometry, we can record the spatiotemporal current density modulation of the input electron image in terms of the variables Dy0, Dt0. Within the frame of the formalism used, it means that we must choose the square matrix   M12 V1S ∗ ; ð7:58Þ CS ¼ M22 V2S which is the 2  2 minor of the augmented rectangular matrix   M11 M12 V1S ~ M∗ ¼ S M21 M22 V2S

ð7:59Þ

describing the linear transformation {Dx0, Dy0, Dt0} ! {DxS, DyS} in Eq. (7.56). Of course, in this case the matrix c∗ S is assumed nondegenerate. Of note, we have come, at least formally) to a difference in the manner of referring the spatiotemporal point spread function to emitter in stationary and dynamic cases. In the stationary case, we used for that purpose the ∗ matrix of local magnifications M∗ S , instead of the matrix CS representing ~∗ the second and third columns of the matrix MS . Nonetheless, as we will show, the already mentioned requirement of independent recording of spatial and temporal current density modulation of the input electron image makes these two approaches quite equivalent. Indeed, if at any time moment the streak speed V∗ S is directed strictly along the equiphase lines of spatial current density modulation of the swept image on the image receiver surface, the image sweep itself will not (at least, in linear approximation) affect the dynamic spatial resolution along the slit. (The reservation ‘‘in linear approximation’’ is essential because dynamic spatial resolution along the slit may, in principle, be affected by dynamic deflection aberrations, as we will see later). Let us attach a simple mathematical form to this statement. Consider the basis vectors e1,e2 of the coordinate system {Dx0, Dy0} on the emitter. According to Eq. (7.57), the direction of the equiphase lines of the input current density modulation along the slit coincides with the direction of the vector e1 on the emitter. Obviously, the vector M*Se1 gives the direction of the same equiphase lines transformed to the image receiver surface, so that the requirement of independent recording may be written in the form ∗ V∗ S ¼ lMS e1

or

1 ∗ < ðM∗ S Þ VS ; e2 >¼ 0;

ð7:60Þ

with l being a nonzero real number (similar to the stationary case, the matrix M∗ S is assumed non-degenerated). This immediately implies that the first column (M11, M21) of the matrix of local magnifications M∗ S and the∗ streak speed vector (V1S, V2S) are proportional, so that the matrices ~ MS and C∗ S appear as

232

General Properties of Emission-Imaging Systems

~ M∗ S ¼

M11

M12 lM11

M21

M22 lM21

!

;

C∗ S

¼

M12 lM11 M22 lM21

!

:

ð7:61Þ

Now it is clear that the operations of transformation to emitter, being ∗ made either with the matrix M∗ S or with the matrix CS , are equivalent up to the proportionality factor l. Therefore, to preserve the analogy with the stationary case, we will further use for the matrix M∗ S for this purpose. From the Eq. (7.60), it is easy to realize physical sense of the proportion∗ 1 ∗ ality factor l. Let us call the vector V∗ 0 ¼ ðMS Þ VS the streak speed vector referred to the emitter. It follows from Eq. (7.60) that the vector V∗ 0 is perpendicular to the basis vector e2 (or, in other words, to the long side of the slit), and its module V0∗ ¼jV∗ 0 j is equal to l. Thus, in the frame of the assumptions made, the real sweeping of the output electron image of the slit along the image receiver surface with the streak speed V∗ S may be regarded as a virtual sweeping of the input electron image described by Eq. (7.57) along the emitter surface with the referred streak speed V∗ 0 directed perpendicularly to the slit. Let us transform the Eq. (7.48), (7.51) to the emitter with the use of the local magnifications matrix M∗ S . Denoting, as in the static case, ðrÞ;loc

IS

ðrÞ;loc

loc ∗ ðDr0 Þ ¼jdet M∗ S jIS ðMS Dr0 Þ; sS

loc ∗ ðDr0 Þ ¼j detM∗ S j sS ðMS Dr0 Þ;

ð7:62Þ we derive ðrÞ; loc IS ðDr0 Þ

ðrÞ;loc sS ðDr0 Þ

ð

ð

¼ hðu0 ÞdD bDr0

ðrÞ; loc

¼ J0loc ðDr00 ; Dt0 ÞIS

ðDr0

Dr00

3 Aloc 0 ½u0 Šcd u0

ð7:63Þ

2 0 V∗ 0 Dt0 Þd Dr0 dDt0 : ð7:64Þ

Although the expressions for spatiotemporal point spread functions in stationary and dynamic cases are formally similar, the external fields that determine the matrix of local magnifications M∗ S and the referred aberra½u Š are essentially different, which is closely connected with tion term Aloc 0 0 the principal difference in electron image formation in these cases. The transformation and recording of the 2D, stationary, spatially modulated electron images can be performed with stationary focusing electromagnetic fields, whereas the transformation and recording of the electron images being modulated both in space and time requires, along with static focusing, dynamic deflection (sweeping) to transform temporal modulation of the input electron image on the emitter to spatial modulation of the output electron image on the image receiver. This requirement is considered in the next chapter in more detail.

General Properties of Emission-Imaging Systems

233

The response of any dynamic emission-imaging system to the input electron image given by Eq. (7.57) appears as ð 0 0 ðrÞ;loc sl ðDx0 ; Dy0 Þ ¼ pl ðDx0 Þ j0 ðDt0 ; Dy 0 Þ IS ðDx0 0

0

0

0

Dx0 V0∗ Dt0 ; Dy0 Dy0 Þ dðDx0 Þ dðDy0 ÞdðDt0 Þ ð 0 0 0 ðrÞ ¼ j0 ðDt0 ; Dy 0 ÞIl ðDx0 V0∗ Dt0 ; Dy0 Dy0 ÞdðDy0 ÞdðDt0 Þ

ð7:65Þ

Here the function ðrÞ Il ðDx0 ; Dy0 Þ

1 ¼ 2l

ðl

ðrÞ;loc

IS

ðDx0

0

0

Dx0 ; Dy0 ÞdðDx0 Þ

ð7:66Þ

l

may be treated as the spatiotemporal spread function of the infinitely thin segment (stroke) of the length l parallel to the axis Dx0 and centered at the ðrÞ local coordinate system’s origin. Obviously, the spread function Il turns ðrÞ;loc when the slit width l tends to zero. over to the point spread function IS Similar to the stationary case, the normalization condition for the initial ðrÞ;loc ðrÞ velocity distribution h(u0) implies that the spread functions IS ; Il are also normalized ð ð ðrÞ;loc ðrÞ 2 ð7:67Þ IS ðDr0 Þd Dr0 ¼ Il ðDr0 Þd2 Dr0 ¼ 1:

If we introduce the time-like spatial coordinate DB0 ¼ V0∗ Dt0 and the vectors Dr0B ¼ {DB0, Dy0} being isomorphic to the vectors Dr0 ¼ {Dx0, Dy0} on the emitter, and also denote J0loc ðDB0 ; Dy0 Þ ¼ ð1=V0∗ Þj0 ðDB0 =V0∗ ; Dy0 Þ, Eq. (7.65) may be rewritten as ð ðrÞ sl ðDr0 Þ ¼ J0loc ðDr0B ÞIl ðDr0 Dr0B Þd2 Dr0B ; ð7:68Þ which makes Eq. (7.68) quite similar to the corresponding relation in the stationary case. Obviously, the spatiotemporal harmonic test pattern j0 ðDt0 ; Dy0 Þ ¼ 1 þ cosðot Dt þ oy Dy0 Þ

ð7:69Þ

is equivalent to the ‘‘spatial-like’’ harmonic test pattern J0loc ðDB0 ; Dy0 Þ ¼ 1 þ cosðoB DB0 þ oy DyÞ ¼ 1 þ cos < o0 k; Dr0B > ð7:70Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the space Dr0z. Here o0 ¼ o2B þ o2y is module of the frequency

vector ðoB ; oy Þ and k is unit vector with the components qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðoB = o2B þ o2y ; oy = o2B þ o2y Þ. The angular frequencies related to the

234

General Properties of Emission-Imaging Systems

pure temporal (Dt0) and time-like (DB0) coordinates are interconnected by the relation ot ¼ oB V0∗ . Similar to the stationary case, we can define the  ðrÞ ðvB Þ related to the infinitely thin spatiotemporal transfer function I l stroke of length l within the slit on the emitter as the Fourier transform ðrÞ of the spread function Il ðDr0B Þ:  ðrÞ ðvB Þ ¼ w∗ ðvB Þexp½if∗ ðvB ފ I l l l

ð7:71Þ

∗ with w∗ l ðvB Þ and fl ðvB Þ representing the spatiotemporal MTF and PTF, respectively. The response to the input electron image (7.70) takes the form ∗ sl ðDr0B Þ ¼ 1 þ w∗ l ðo0 kÞ cos ½ < o0 k; Dr0B > þfl ðo0 kÞ Š :

ð7:72Þ

This relation completely characterizes the streaked image quality in terms of spatial and temporal frequencies of the test patterns (7.69) and (7.70). Indeed, as it follows from Eq. (7.72), the streaked image of the slit represents a 1D harmonic current density modulation along the vector k with the frequency o0. The equiphase lines of the streaked image referred to the emitter are perpendicular to the vector k and turned by the angle arctg(oy / oB) with respect to the positive direction of the axis Dy0. The relative change of the frequencies oB, oy of the test pattern (7.69) is equivalent to the turn of the vector k in the frequency space, or, equally, to the turn of the test pattern (7.70) in the ‘‘spatial-like’’ plane (DB0, Dy0). Quite similar to the stationary case, such a turn of the test pattern results in the corresponding turn of the equiphase lines of the streaked image on the image receiver. Particularly, in the case that there is no spatial modulation of the current density along the slit and the image sweeping is performed in the presence of temporal modulation only (oz 6¼ 0, oy ¼ 0), the referred equiphase lines are parallel to the slit and perpendicular to the streak speed vector V∗ S . This case corresponds to recording of pure temporal resolution along the sweep direction. On the contrary, the referred equiphase lines of the streaked image formed by sweeping a spatially modulated input electron image having no modulation in time (oB ¼ 0, oy 6¼ 0) are perpendicular to the slit and parallel to the sweep direction. This case corresponds to recording of dynamic spatial resolution along the slit. There is a definite methodological difference between experiments on electron image recording in static and dynamic cases, which is conditioned by the possibility of practical realization of the spatial test pattern (7.54) and the spatiotemporal test pattern (7.69). The spatial test pattern (7.54) is comparatively easy to realize in practice. Commonly, experimental studies on image quality in stationary emission-imaging system consist in the analysis of the output electron image formed by a test chart being used as the input image. Such charts contain some sets of the test patterns like (7.54) or, more often, the

General Properties of Emission-Imaging Systems

FIGURE 120

235

Electron image of the test chart in static image tube.

piecewise-constant approximations of the test pattern (7.54), representing the sets of black and white strokes of equal widths with different frequencies and orientations, as shown in Figure 120. On the contrary, practical realization of the spatiotemporal test pattern (7.69) represents a rather difficult physical and engineering problem. This is the reason why experimentalists, when studying dynamic emission-imaging system properties, usually prefer to work not with spatiotemporal test patterns but with spatiotemporal spread functions directly. Let us consider in more detail the two aforementioned limiting cases, which are of frequent use in practice. 1. Temporal Resolution Recording (oB 6¼ 0, oy ¼ 0; k ¼ e1 ¼ (1,0)). In this case (the most typical for time-analyzing image tubes), electron image of the ‘‘uniform’’ slit having no current density modulation along its long side is swept on the screen. The corresponding spread function in the sweep direction e1 IðrÞ e1 ðDx0 Þ

¼

þ1 ð

ðrÞ

Il ðDx0 ; Dy0 ÞdDy0

ð7:73Þ

1

can be obtained experimentally if the input electron image is rather short in time, or, in other words, the correlation j0(Dt0, Dy0) ¼ dD(Dt0) holds true with sufficient accuracy.  ðrÞ ðo0 Þ ¼ w∗ ðo0 Þexp½if∗ ðo0 ފ is the corresponding transfer funcIf I e1 e1 e1 tion, then

236

General Properties of Emission-Imaging Systems

 w∗ e1 ðo0 Þ ¼ I

ðrÞ e1 ðo0 Þ



∗  ðrÞ f∗ e1 ðo0 Þ ¼ argIe1 ðo0 Þ ¼ fl ðo0 e1 Þ:

¼ w∗ l ðo0 e1 Þ;

ð7:74Þ

Putting o0 ¼ 2pN0B, where N0B is referred spatial frequency along the sweep direction, from the relation ∗ ∗ w∗ e1 ð2pN0B Þ ¼ w

ð7:75Þ

∗ ∗ we can define the referred dynamic spatial resolution N0B ¼ N0B ðV0∗ ; w∗ Þ along ∗ the sweep direction at the referred streak speed V0 and the prescribed contrast level w*. ∗ is invariant with respect to the It is obvious that the product V0∗ N0B transformation from the image receiver to emitter in the sense that, if VS∗ ; NS∗ are, respectively, the streak speed module and spatial frequency ∗ are along the sweep direction on the image receiver surface, and V0∗ ; N0B the corresponding values referred to emitter by means of the matrix M∗ S, ∗ . Thus, if DT ¼ 2p/ot is the oscillation period then VS∗ NS∗ ¼ V0∗ N0B of temporal component of the test pattern (7.69), with regard to the correlation between the frequencies oB and ot we obtain

DT∗ ¼

1 ∗ V0∗ N0B

¼

1 VS∗ NS∗

:

ð7:76Þ

This relation defines temporal resolution DT* of the dynamic emissionimaging system and is of special importance because it allows determination of temporal resolution directly from experimental data. At first glance, it may be concluded from Eq. (7.76) that the higher is the streak speed, the higher is the temporal resolution. However, in reality this is not true. More detailed analysis shows that dynamic defocusing of the streaked image, which inevitably occurs when the streak speed becomes high enough, may result in the rather fast fall of the spatial resolution NS∗ , and further streak speed increase leads not to improvement but to deterioration of temporal resolution (Figure 121). 2. Recording of Dynamic Spatial Resolution Along the Slit (oB ¼ 0, oy 6¼ 0; k ¼ e2 ¼ (0,1)). In this case, the output spread function represents the streaked electron image of the infinitely thin stroke of the finite length l. The output electron image quality can be expressed in terms of the transfer function  ðrÞ ðo0 Þ ¼ w∗ ðo0 Þexpif∗ ðo0 Þ which is the Fourier transform of the I e2

e2

e2

spread function IðrÞ e2 ðDy0 Þ: The latter, in turn, can be obtained from the ðrÞ

spread function Il ðDr0B Þ by means of integration over the time-like coordinate Dx0B. We derive for the direction k ¼ e2 ¼ (0,1)

237

General Properties of Emission-Imaging Systems

∆T*

2 ∆T* min 1 ∆T* phys

(VS*)opt

VS*

FIGURE 121 Temporal resolution versus streak speed (1, ideal dynamic deflection system; 2, contribution of dynamic deflection aberrations).

∗  ðrÞ w∗ e2 ðo0 Þ ¼j Ie2 ðo0 Þ j¼ wl ðo0 e2 Þ;

∗  ðrÞ f∗ e2 ðo0 Þ ¼ argIe2 ðo0 Þ ¼ fl ðo0 e2 Þ:

ð7:76Þ

Putting o0 ¼ 2pN0y, where N0y is spatial frequency of the input current density modulation along the slit direction, we can define dynamic spatial ∗ ðw∗ Þ along the slit versus the prescribed contrast level w* from resolution N0y the relation ∗ ∗ w∗ e2 ð2pN0y Þ ¼ w :

ð7:77Þ

Due to the above-mentioned dynamic defocusing effects, dynamic spatial resolution along the slit commonly proves to be somewhat worse compared with that measured in static mode. A special method is used to reduce the contribution of dynamic deflection aberrations to temporal and spatial resolution in streak mode: some preliminary defocusing of the static image is intentionally introduced to compensate, at least partially, for dynamic defocusing Rather often but far from always reasonably, experimentalists estimate temporal resolution by means of the FWHM value of the corresponding spread functions, defining, for instance, temporal resolution as ∗ ¼ DTFWHM

½IS ŠFWHM ; j V∗ S j

ð7:78Þ

where ½IS ŠFWHM is experimentally measured FWHM of the spread function IS on the image receiver. Such estimation does not take into

238

General Properties of Emission-Imaging Systems

consideration the contrast level w* and may lead to rather perceptible errors in some cases. Indeed, assuming that IS has Gaussian shape, it is easy to obtain that   DT∗ ðw∗ Þ C p ¼ pffiffiffiffiffiffiffiffiffiffiffi ð7:79Þ C ¼ pffiffiffiffiffiffiffi ¼ 1:887::: : ∗ DTFWHM jlnw∗j 2 ln2

It follows from Eq. (7.79) that the temporal resolution DT*(w*) calculated according to Eq. (7.75) strictly coincides with the estimation (7.78) at the contrast  w∗ ¼ 0:028 which is approximately two times lower than the limiting contrast capable of being recorded by the human eye. Within the w∗ Þ, the estimation (7.78) gives the overregion of large contrast ðw∗ >  valuated temporal resolution compared with DT*(w*). In particular, at ∗ amounts w* ¼ 0.05 the relative difference between DT*(w*) and DTFWHM to 10%, at w* ¼ 0.1 to 25%, and at w* ¼ 0.2 to 50%. The correlation between ∗ could have been considered as acceptable in the DT*(w*) and DTFWHM region w* ¼ 0.03 0.1 but for the fact that the real temporal spread functions are, as a rule, asymmetric, and may substantially differ from Gaussian distributions. Therefore, certain caution is required when using Eq. (7.78) for temporal resolution estimation. As to the FWHM estimation of spatial resolution in the stationary case, the error may be even greater because the shape of the stationary spread functions in the optimal focusing region is, as a rule, substantially different from Gaussian distribution, especially in the central, spike-like part of the spread functions. This difference makes the FWHM-based resolution estimation too rough in the low-contrast region (w* < 0.1), or, equivalently, in the region of comparatively high spatial frequency of the test pattern. We recall that all our constructions as to the frequency properties of the spread functions are based on the assumption of emission-imaging system linearity. The validity of this assumption depends on whether the space charge effects can be neglected or, at least, considered as a perturbation. The emission-imaging systems in question commonly obey this condition: In those systems, the internal Coulomb field resulting from charged particle interaction proves to be by many orders of magnitude less compared with the external focusing and accelerating fields generated by electrodes. Even in this case, however, the space charge effects may substantially contribute to spatial and temporal characteristics of the output electron image. The perturbation methods considered in Chapter 6 as applied to the short bunches of charged particles make it possible to include a theoretical description of the bunch motion in the presence of space charge effects into the image quality evaluation scheme presented above. In fact, this represents a particular case of the general quasi-linearization principle that is widely used in many problems of numerical analysis.

General Properties of Emission-Imaging Systems

239

The light power density range of input radiation, within which a photoemission-imaging system remains quasi-linear, is called the dynamic range of the system in question. With regard to the emission properties of the photocathode, this range can be expressed in terms of the total number of charged particles comprising the bunch. The quasi-linearity criterion, commonly accepted in practice, says that the relative increase of the light power density, compared with its minimal value which still allows recording of the swept image above the noise level (or, correspondingly, the relative increase of the total charge within the bunch), should not result in more than 20% spread of the spatiotemporal spread function. The dynamic range, as one of the most important characteristics of dynamic emission-imaging system, determines the upper limit of the input light power density that may be actually used in the experiments on ultrafast processes recording. The problems connected with ultrashort bunch dynamics are considered in more detail in Chapter 9.

CHAPTER

8 Static and Time-Analyzing Image Tubes With Axial Symmetry

Contents

8.1. Spatial Aberrations of the Electron Image Formed by Electrostatic Systems 8.2. Temporal Aberrations in Streak Image Tubes 8.3. High-Frequency Asymptotics of MTF and PTF in Image Tubes 8.4. Examples of the Spread Functions, MTF and PTF in Image Tubes 8.5. The Boundary-Layer Effect in Cathode Lenses and Electron Mirrors

245 255 258 265 270

In the preceding chapter we have shown that the quality of an electron image formed by a linear emission-imaging system may be expressed, at least locally, in terms of the frequency properties of corresponding spread functions. According to Eq. (7.63), being the distribution h(u0) in initial velocities given, the point spread function of any emission-imaging system is completely determined by the aberration term Aloc 0 ½u0 Š, which characterizes scattering of the particles emitted from a common point on the cathode with different velocities. Our main aim in this chapter is to consider in detail how the aberration term Aloc 0 ½u0 Š is formed in static and time-analyzing emission-imaging systems with axial symmetry and provide a classification of the aberration coefficients in this most important practical case. Before proceeding to this problem, we need to establish a general and physically transparent interconnection between the optical transfer  ðrÞ;loc function (OTF) I S ðvÞ, the initial distribution in charged particle

Advances in Imaging and Electron Physics, Volume 155 ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)00808-2

2009 Elsevier Inc. All rights reserved.

#

241

242

Static and Time-Analyzing Image Tubes With Axial Symmetry

velocities h(u0), and the aberration term Aloc 0 ½u0 Š. Applying the Fourier transform to Eq. (7.63), we have ð ð ðrÞ;loc 2 3 I S ðvÞ ¼ expði < v; Dr0 >Þd r0 hðu0 ÞdD ½Dr0  Aloc 0 ½u0 d u0 ð ð 2 ¼ hðu0 Þd3 u0 expði < v; Dr0 >ÞdD ½Dr0  Aloc 0 ½u0 d r0 ð 3 ¼ hðu0 Þexpði < v; Aloc ð8:1Þ 0 ½u0  >Þd u0 : Equation (8.1) offers the opportunity to calculate the optical transfer  ðrÞ;loc function I S ðvÞ directly from the initial energy distribution h(u0) and the aberration term Aloc 0 ½u0 , omitting the intermediate calculation of the ðrÞ;loc spread function IS ðDr0 Þ. The reason wherefore this relation is seldom used in practical calculations is simply that the spread function itself, as a rule, is of significant interest in practice. Nonetheless, (8.1) may be regarded, in a definite sense, as a fundamental equation in the emissionimaging system theory. Indeed, this equation directly correlates the electron image quality expressed in terms of frequency properties of the optical transfer function with emission properties of the cathode as a source of electrons, on the one hand, and aberration properties of the local isoplanatic transformation realized by electron trajectories, on the other. It immediately follows from Eq. (8.1) that the ‘‘best’’ optical transfer  ðrÞ;loc function I S , being identically equal to unit at all frequencies, corresponds to the ideal limiting cases already mentioned in Chapter 7: Either the emitter should be absolutely monochromatic (h(u0) ¼ dD(u0)) or all the image aberrations in a given local isoplanatism region should be strictly zero (Aloc S ½u0  ¼ 0). If at least one of these conditions is met, the linear isoplanatic emission-imaging system transforms the local input electron image to the image receiver with no phase shift and loss in contrast. In fact, those limiting conditions can never be achieved in reality, and the problem of optical transfer function improvement usually represents a multiparametric nonlinear optimization problem in regard to the geometry and voltages of the electrodes comprising the system in question, cathode emission properties, and so on. Let u0 ¼ (U0, V0, W0) be the components of the initial velocity vector in the local coordinate system connected with the normal to the photocathode surface at an arbitrary fixed point {r0, z0(r0)} (Figure 117), and e the particle’s kinetic energy expressed in potential units, so that m m ð8:2Þ ju0 j2 ¼ ðU02 þ V02 þ W02 Þ ¼ ee: 2 2

Static and Time-Analyzing Image Tubes With Axial Symmetry

243

Putting U0 ¼

pffiffiffiffiffiffiffiffiffiffiffi 2e=m U;

V0 ¼

pffiffiffiffiffiffiffiffiffiffiffi 2e=m V;

pffiffiffiffiffiffiffiffiffiffiffi 2e=m W

ð8:3Þ

W ¼ e1=2 cosO:

ð8:4Þ

W0 ¼

and denoting, as shown in Figure 117, O the angle between the vector u0 and the positive normal to the emitter surface, and o the angle between the projection of the vector u0 onto the tangent plane (U, V) and the positive direction of the U axis, we obtain in the spherical coordinate system {e1/2, O, o} U ¼ e1=2 sinO coso;

V ¼ e1=2 sinO sino;

It is sometimes convenient to consider the initial energy as a sum of two components in tangential and normal directions with respect to the emitter surface. Denoting et ¼ ðm=2eÞðU02 þ V02 Þ, en ¼ ðm=2eÞW02 , so that e ¼ et þ en, we obviously have 1=2

et

¼ ðU2 þ V 2 Þ1=2 ¼ e1=2 sinO;

en1=2 ¼ W ¼ e1=2 cosO:

ð8:5Þ

We further assume that the distributions of emitted electrons in initial energies e and angles O, o are independent and the distribution in o uniform. This implies that the differential distribution of electrons h(U, V, W) on the emitter E can be factorized as hðU; V; WÞ dUdVdW ¼

1 FðeÞGðOÞ sin O de d O do: 2p

ð8:6Þ

The emitter E is assumed homogeneous, so that the distribution of electrons in initial velocities does not depend upon the emission point location on the cathode’ surface. This assumption is usually well-grounded in imaging charged particle optics. The distributions of photo- and thermoelectrons in initial energies are rather well studied experimentally, and we refer the reader to the extensive literature on this subject (see, for example, Beck, 1953; Berkovsky, Gavanin, and Zaidel, 1976; and Butslov, Stepanov, and Fanchenko, 1978). In the case of thermoemission, the function F(e) is, generally speaking, nonzero for all 0  e < 1 and under rather general assumptions represents the Maxwellian distribution  ee  ð8:7Þ FðeÞ ¼ Ae exp kT

with a maximum at the point of the most probable energy e0 ¼ kT/e, which for different types of thermocathodes commonly lies in the range of some tenths of volt. As to photocathodes, the function F(e) differs from zero within a given interval of initial energies 0  e  emax, with the maximal energy emax determined, according to the Einstein photoemission law, by the so-called red-boundary wavelength of initiating optical radiation.

244

Static and Time-Analyzing Image Tubes With Axial Symmetry

For the majority of the known photocathodes operating in the infrared, visible and ultraviolet wavelength ranges, the value emax does not exceed a few volts. Along with experimental data from special measurements, some model approximations for energy distributions are also used in calculation practice. This is, for instance, so-called ‘‘binomial’’ distribution FðeÞ ¼ Apq e p ðe

emax Þq ;

ð8:8Þ

the fitting parameters p, q of which serve to vary both the distribution’s shape and position of its maximum within the interval 0  e  emax. The Lambert law G(O) ¼ B cos O is commonly used to describe the thermoemission angular distribution. The angular distribution of photoelectrons is usually described by a slightly more complicated analytical approximation GðOÞ ¼ G1 ðcosOÞ;

G1 ð0Þ ¼ 0:

ð8:9Þ

Putting, in particular, G1 ðtÞ ¼ Bm tm ;

m > 0;

ð8:10Þ

we arrive at the set of ‘‘cosine-power’’ angular distributions, with the parameter m governing the effective width of angular spread. The case m ¼ 1 corresponds to the Lambert law, but, due to some physical reasons, the approximations with m > 1 appear more adequate to describe photoemission in rather strong electric fields. It follows from the normalization condition for h(u0) that the functions F(e), G(O) may be normalized as 1 ð 0

FðeÞde ¼

p=2 ð

GðOÞsinO dO ¼ 1:

ð8:11Þ

0

In the case of photoemission, the infinite upper limit in the first integral in Eq. (8.11) can be replaced by emax. With the initial energy and angular distribution functions given, the corresponding distribution of electrons in initial velocities appears as ! 1 FðU2 þ V 2 þ W 2 Þ W G1 : hðU; V; WÞ ¼ p ðU2 þ V 2 þ W 2 Þ1=2 ðU2 þ V 2 þ W 2 Þ1=2 This representation will be helpful in the upcoming sections.

ð8:12Þ

Static and Time-Analyzing Image Tubes With Axial Symmetry

245

8.1. SPATIAL ABERRATIONS OF THE ELECTRON IMAGE FORMED BY ELECTROSTATIC SYSTEMS We should notice that the restriction of our consideration by the electrostatic case is not a major concern and is accepted only to simplify and clarify the material. Many facts related to the theory of cathode lenses with combined electric and magnetic fields are given in the monograph by Il’in, et al. (1990) and the papers cited therein. Let us consider an electrostatic emission-imaging system possessing axial symmetry pffiffi with respect to the z-axis.locFor the sake of simplicity, we ½ e; O; oŠpthe denote Aloc 0 ffiffi aberration term A0 ½u0 Š expressed in terms of the initial parameters f e; O; og. It is convenient to use complex representation for the input and output 2D vectors r0 ¼ x0 þ iy0 and rS ¼ xS þ iyS in the axisymmetric case. The position of a charged particle on the emitter surface can be represented as r0 ¼ r0 eib ;

z0 ¼ z0 ðr0 Þ;

ð8:13Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r0 ¼ jr0 j ¼ x20 þ y20 ; b ¼ argðr0 Þ are, respectively, the module and argument of the vector r0 being interpreted as a complex number, and z0 ¼ z0(r0) is a smooth function that determines the cathode’ shape. The local coordinate system {U, V, W} described in Section 7.1 (see Figure 117) can be specified here in the manner shown in Fig. 122. The normal vector nE to the emitter surface representing a spherical segment centered at the origin O of the global coordinate system {x, y, z} gives the positive direction of the axis W. Denote U0 the auxiliary axis that corresponds to the positive direction of the tangent at the point r0 to the meridian cross-section of the emitter surface, passing both through the point r0 and the z-axis, and let V0 be the auxiliary axis that corresponds to the binormal direction. The axes (U0 , V0 ) define a tangent plane to the emitter surface E at the point r0. Now let us rotate the tangent plane around the W-axis by the angle ( b) and denote (U, V) the axes obtained from (U0 , V0 ) by means of such turn. We have constructed a local coordinate system {U, V, W} that smoothly turns into the global one {x, y, z} at r0 ! 0. The latter is of importance if we wish to include r0 in the list of small aberration parameters. We leave it to the reader to obtain the explicit view of the asymptotic expansion of the matrix P(R0) at R0 ! 0 for the case under consideration. Under the condition that the parameters e1/2, r0 are small enough, the tauvariation technique described in Chapter 5 allows representation of any charged particle trajectory in the form of the aberration expansion (see also Kolesnikov and Monastyrskiy, 1988)

246

Static and Time-Analyzing Image Tubes With Axial Symmetry

U x b w b

Ω

R0

r0

v0 z

O

nE V

W

y

FIGURE 122

The local coordinate system {U, V, W} in the axisymmetric case.

pffiffi rðz; e; r0 ; b; O; oÞ ¼ eib fr0 ½wðzÞ þ EðzÞr20 þ . . .Š   þ e1=2 sin O½vðzÞeio þ DðzÞeio þ CðzÞe io r 20 þ . . .Š   þ e½HðzÞeio sin O cos O þ GðzÞ þ FðzÞe2io r0 sin2 O þ QðzÞr0 cos2 O þ . . .Š þ e3=2 eio sin O ½BðzÞsin2 O þ PðzÞcos2 O þ . . .Šg þ . . .

ð8:14Þ

The aggregate maximum power of the terms in Eq. (8.14) with respect to the parameters {e1/2, r0} is three; therefore Eq. (8.14) is commonly called the third-order aberration expansion with respect to the main optical axis. At any z fixed, the coefficients v, w, E, D, C, H, G, F, Q, B, P represent some nonlinear functionals dependent on the axial potential distribution F(x) and its derivatives up to the fourth order inclusively in the region 0  x  z. Without loss of generality, hereafter we may suppose that F(0) ¼ 0. The first-order coefficients v(z), w(z) are the linear-independent solutions of the limiting paraxial equation 1 1 PðrÞ  Fr00 þ F0 r0 þ F00 r ¼ 0 2 4 and have the asymptotic behavior at z ! 0 2 pffiffiffi vðzÞ ¼ pffiffiffiffiffiffiffi z þ Oðz3=2 Þ; 0 F0

wðzÞ ¼ 1

ð8:15Þ

00

F0 2 0 z þ Oðz Þ 2F 0

ð8:16Þ

Static and Time-Analyzing Image Tubes With Axial Symmetry

0

247

00

Here F 0 ; F 0 are the first and second derivatives of the axial potential distribution F(z) at the photocathode center (z ¼ 0). The paraxial trajectories v(z), w(z) are interconnected by the Lagrange-Helmholtz invariant i pffiffiffiffiffiffiffiffiffiffih ð8:17Þ FðzÞ vðzÞw0 ðzÞ v0 ðzÞwðzÞ ¼ 1:

pffiffi In the parameter region f e; r0 g within which the representation (8.14) allows sufficiently accurate description of charged particle trajectories, the aberration coefficients completely determine the output image characteristics. Obviously, the error of the third-order aberration expansion (8.14) increases with the increase in initial energy e and initial distance r0 of the point of particle’s emission from the photocathode center. We now turn our attention to the electron image properties, which directly follow from the expansion (8.14) and the localization principle established in the previous chapter. Let us consider two directions {sm, st} in the local isoplanatism region Wðr∗ 0 Þ associated with an arbitrary point , as shown in Figure 123. We call the first of these directions, collinear r∗ 0 , the meridian direction and the second one, orthogonal to with the vector r∗ 0 , the sagittal direction. r∗ 0 Any arbitrary point r0 belonging to the isoplanatism region Wðr∗ 0Þ allows representation ∗

∗ ib r0 ¼ r∗ ; 0 þ Dr0 ¼ ðr0 þ s0 Þe

ð8:18Þ

ig

with s0 ¼ s0e being a local vector in Wðr∗ 0 Þ and the angle g counted anticlockwise from the axis sm. After substituting Eq. (8.18) into the aberration expansion (8.14), we can construct a local isoplanatic y0

S0 St g

r*0

Sm

W(r*0)

b* O

x0

FIGURE 123 Meridian (sm) and sagittal (st) directions in the local isoplanatism region W ðr∗ 0 Þ.

248

Static and Time-Analyzing Image Tubes With Axial Symmetry

transformation of the photocathode region Wðr∗ 0 Þ to the image receiver surface in the third-order approximation with respect to the small parameter e1/2. The complex form of such transformation appears in the local coordinate system {sm, st} as pffiffi ð8:19Þ s ¼ M∗ ðgÞs0 þ Aloc S ½ e; O; oŠ; where

M∗ ðgÞ ¼ w þ Er∗2 0 ð2 þ e

2ig

Þ

ð8:20Þ

is the local complex electron-optical magnification at the photocathode point r∗ 0 in the direction g0, and h i pffiffi 1=2 sin O vðzÞeio þ ðDðzÞeio þ CðzÞe io Þr∗2 Aloc S ½ e; O; oŠ ¼ e 0 h 2 þ e HðzÞeio sin O cos O þ ðGðzÞ þ FðzÞe 2io Þr∗ 0 sin O i h i 2 3=2 io e sin O BðzÞsin2 O þ PðzÞcos2 O þ QðzÞr∗ 0 cos O þ e

ð8:21Þ

is a pcomplex representation of the third-order local ffiffi pffiffi aberration term ½ e ; O; oŠ with respect to the small parameter e at the distance r∗ Aloc S 0 fixed. It follows from Eq. (8.20) that the matrix of local magnifications M∗ S related to the point r∗ 0 proves to be diagonal in the local coordinate system {sm, st}:   0 wðzÞ þ 3EðzÞr∗2 ∗ 0 ; ð8:22Þ MS ¼ 0 w þ EðzÞr∗2 0 which means that the pure geometrical part of the transformation (8.19) represents a ‘‘stress-strain’’ transformation along the local coordinate axes sm, st with the coefficients   ∗ ∗ ¼ wðzÞ þ 3EðzÞr∗2 M∗ ¼ wðzÞ þ EðzÞr∗2 M∗ m ¼ M  s ¼ M  0 ; 0 g¼0

g¼p=2

ð8:23Þ

representing the local magnifications along the meridian and sagittal directions, correspondingly. The value E(z) is called the scale distortion coefficient with respect to the main optical axis. It can be easily seen that the relative contribution of the scale distortion to the local magnification M∗ m is three ∗ . At r ¼ 0, the local complex magnificatimes bigger compared with M∗ s 0 tion and, correspondingly, the matrix of local magnifications take especially simple form   wðzÞ 0 ∗ ∗ ; ð8:24Þ M ðgÞ ¼ wðzÞ; MS ¼ 0 wðzÞ

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249

which shows that the geometrical part of the transformation (8.19) is reduced to a similarity transformation with the coefficient w(z). This explains why w(z) is usually called the electron-optical magnification at the work area center. Due to axial symmetry, the initial vectors r0 ¼ r0eib, after having been transformed to the image region, maintain their orientation while the transformation of their lengths   EðzÞ 2 ib  ð8:25Þ r þ ... rðr0 Þ e¼0 ¼ e wðzÞr0 1 þ wðzÞ 0 becomes nonlinear with r0 increasing. The real value   rðr0 Þe¼0 EðzÞ 2 ¼ wðzÞ 1 þ r þ ... Mðr0 Þ ¼ r0 wðzÞ 0

ð8:26Þ

is called the average electron-optical magnification at the point r0, and the ratio Aðr0 Þ ¼

Mðr0 Þ wðzÞ ; wðzÞ

ð8:27Þ

which characterizes the dependence of average magnification M(r0) on the distance of the point r0 from the cathode center, is called the scale distortion. Obviously, Aðr0 Þ  ½EðzÞ=wðzފr20 at small enough r0. From Eqs. (8.23), (8.26) and (8.27), we can see that the local magnifications Mm, Ms, the average magnification M(r0), and the distortion A(r0) depend quadratically on r0 within the third-order aberrational approximation. The limiting relations @rðz; e1=2 ; r0 ; b; O; oÞ jO¼p=2;r0 ¼0 ; e!0 @e1=2

vðzÞeiðbþoÞ ¼ lim

@rðz; e1=2 ; r0 ; b; O; oÞ je¼0 r0 !0 @r0

wðzÞeib ¼ lim

ð8:28Þ ð8:29Þ

that directly stem from Eq. (8.14) induce the following definitions of the most important first-order parameters of the emission-imaging system. The plane z ¼ zG where the electrons emitted from the cathode center with infinitely small tangential velocities are focused is called the limiting image plane, or the Gauss plane. It follows from Eq. (8.28) that zG is a nonzero solution of the equation vðzG Þ ¼ 0;

ð8:30Þ

or, in other words, the plane z ¼ zG is conjugated to the plane z ¼ 0. An emission-imaging system is called the focusing system if Eq. (8.30) has at

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least one nonzero root. In this case, the value MG ¼ w(zG) determines the electron-optical magnification at the center of work area. Some authors call Eq. (8.30) the first-order focusing  the center of work area in the pffiffi condition at sense that the equality rðzG ; e; r0 ; b; O; oÞr0 ¼0; O¼p=2 ¼ 0 is obeyed with the accuracy to the first-order terms with respect to e1/2. The plane z ¼ zcr where the electrons emitted with zero velocities from the infinitely small vicinty of the cathode center are focused is called the limiting crossover plane. It follows from Eq. (8.29) that zcr is a root of the equation wðzG Þ ¼ 0:

ð8:31Þ

The first-order parameters zG, MG, zcr characterize the emission-imaging system properties in paraxial approximation. It is known from ordinary differential equations theory that the linear-independent solutions of the linear differential equations such as Eq. (8.15) have their zeros in an alternating sequence. Therefore, the limiting image plane is necessarily located between two limiting crossover planes, and vice versa. ð1Þ In particular, since v(0) ¼ 0, a single crossover plane zcr is necessarily located between the cathode and the first nonzero limiting image ð1Þ plane zG . Imagine (pure theoretically!) that all the second and third-order aberration coefficients D, C, H, F, G, Q, B, P, E in Eq. (8.21) are equal to zero in some limiting image plane z ¼ zG where we place the center of the image 1=2 ; O; oŠ is idenreceiver surface. In this case, the aberration term Aloc S ½e tically zero for all emitted particles, independently of the particular values of the initial parameters r0, e1/2, O, o. This implies that each point of the cathode is imaged onto the corresponding point on the image receiver surface in the third-order approximation, and the local magnifications ∗ M∗ m , Ms , independently of r0, are equal to the magnification MG at the center. Thus, we have an ideal output electron image in the plane z ¼ zG. Unfortunately, as already mentioned, such an ideal case cannot be implemented in practice. To describe the influence of the aberration term in the axisymmetric case, we proceed to studying the partial figures of diffusion. According to the definition given in Section 7.2, a figure of diffusion represents a locus 1=2 ; O; oŠ on the image receiver surface when covered by the vector Aloc S ½e the initial parameters {e, O, o} are running through the corresponding intervals 0  e  emax, 0  O  p/2, 0  o  2p. Analysis of these figures related to different aberration coefficients allows qualitative estimation of partial contribution of those coefficients to electron image formation. We start our analysis from the figures of diffusion related to the cathode center. Putting r∗ 0 ¼ 0 in Eq. (8.31), we obtain

Static and Time-Analyzing Image Tubes With Axial Symmetry

 pffiffi 1=2  Aloc vðzÞeio sin O þ eHðzÞeio sin O cos O S ½ e; O; oŠ r∗ ¼0 ¼ e 0

þ e3=2 PðzÞeio sin O cos2 O þ e3=2 BðzÞeio sin3 O:

251

ð8:32Þ

First-Order Image Defocusing at the Center. This is the first-order aberration responsible for the spread in tangential components of charged particle velocities and represented by the first term in the expansion (8.21). It can be easily seen that the corresponding figure of diffusion is a circle with the 1=2 radius emax jvðzÞj, which, obviously, turns into a point in the Gauss plane z ¼ to the Lagrange-Helmholtz invariant (8.17), v0 ðzG Þ ¼ zG. According pffiffiffiffiffiffiffi 1=MG FG where FG ¼ F(zG) is the axial potential value in the Gauss plane zG. Thus, taking into account that v(z)  v0 (zG)(z zG) in the Gauss plane’ vicinity, we obtain in linear approximation with respect to e1/2  pffiffi e1=2 ðz zG Þ io  pffiffiffiffiffiffiffi e sin O þ . . .  A loc ð8:33Þ S ½ e; O; oŠ ∗ r0 ¼0 MG F G This simple asymptotic relation will be useful later. Second-Order Spherochromatic Aberration. This aberration, given by the second term of the expansion (8.21), is of special importance for static emission–imaging systems because it represents the main factor that restricts spatial resolution of electron optics at the center of work area in the focusing image tubes. The second-order spherochromatic aberration 1=2 1=2 results from initial spread both in tangential (et ) and normal (en ) components of charged particle velocities. As far back as in the beginning of 1940s, a very simple formula for the aberration coefficient H(z) was independently obtained by Recknagel (1941) and Artsimovich (1944): HðzÞ ¼

2wðzÞ ; 0 F0

ð8:34Þ

0

where F 0 is the field intensity at the cathode center, so that the secondorder term with respect to e1/2 appears in Eq. (8.21) as  pffiffi wðzÞ io  Aloc e e sin 2O þ . . . ð8:35Þ 0 S ½ e; O; oŠ r∗ ¼0 ¼ . . . 0 F0 The figure of diffusion associated aberration in question repre with  the 0 sents a circle of the radius emax wðzÞ=F 0 . Thus, the contribution of this aberration to image quality is inversely proportional to the field intensity at the cathode center, which indicates the simplest way to improve spatial resolution at the center of work area in static emission-imaging systems. Third-Order Spherochromatic Aberration. The electron-optical nature of this aberration, represented by the aberration coefficient P(z), is quite

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similar to that of the second-order one. Indeed, as can be seen from 1=2 Eq. (8.21), this aberration linearly depends on the tangential (et ) compo1=2 nent and quadratically on the normal (en ) component of the particle’s initial velocity. The figure of diffusion peculiar to p the spherffiffiffi third-order 3=2 ochromatic aberration is a circle of the radius ð2=3 3Þemax jPðzÞj. Spherical Aberration. This aberration is caused by the third-order 1=2 spread in tangential (et ) component of initial velocities of charged particles. The figure of diffusion is represented by a circle of the radius 3=2 emax jBðzÞj. It is noteworthy that in static photoemission-imaging systems the contribution of the third-order spherochromatic and spherical aberrations to spatial resolution is generally insignificant. This fact essentially distinguishes the wide-beam electron optics of photoemission-imaging systems from the narrow-beam optics of electron microscopes. As is well known, spherical aberration represents in the latter case one of the main factors restricting spatial resolution. As to the sub-picosecond time-analyzing streak image tubes, with the 0 near-cathode field intensity F 0 raised up to some tens of kilovolts per millimeter, the contribution of spherical and third-order spherochromatic aberrations to spatial resolution may exceed that of the second-order spherochromatic aberration, which, as we have seen above, is inversely 0 proportional to F 0 . If the contribution of the third-order spherochromatic and spherical aberrations can be neglected, both the spread function and spatial resolution at the center of work area are determined mainly by the first two terms of the expansion (8.21). It follows from the location of zeros of the linear-independent solutions v, w of the paraxial equation (8.15) that in the right-hand vicinity of the Gauss plane z ¼ zG those solutions ð1Þ are of the same sign. In particular, if z ¼ zG 6¼ 0 is the first Gauss plane, ð1Þ v(z) and w(z) are negative in the right vicinity of zG . This means that the individual terms of the binomial expansion  pffiffi wðzÞ io 1=2  vðzÞeio sin O e e sin2 O; ð8:36Þ Aloc 0 S ½ e; O; oŠ r∗ ¼0 ¼ e 0 F0 considered as functions of z, partially compensate each other in the righthand vicinity of the Gauss plane. In turn, this implies that a plane exists, in which the best focusing conditions for imaging the work area center are realized. With regard to statistical nature of the initial parameters e, O, o, it is more accurate to express this statement in the MTF terms: The best MTF related to the center of work area, which ensures the highest output contrast in the high-frequency region, is attained in the best focusing plane for the work area center z ¼ zF > zG located in the right-hand vicinity of the Gauss plane z ¼ zG. Location of this plane can be calculated either numerically or with the stationary phase method presented in paragraph 7.4.

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Now we proceed to analysis of the figures of diffusion associated with the off-axis aberrations given by the coefficients D, C, G, F, Q. Astigmatism and Image Curvature. Let us consider the relationship between the aberration coefficients D, C and the focusing conditions for the off-axis beams. The corresponding aberration term reads   pffiffi 1=2 io io io ½ e ; O; oŠ ¼ e sin O ½vðzÞe þ DðzÞe þ CðzÞe r20 Š þ . . . ; ð8:37Þ Aloc S

which means that the figure of diffusion represents an ellipse with the semi-axes    1=2  amer ðzÞ ¼ emax vðzÞ þ DðzÞ þ CðzÞ r20     1=2  bsag ðzÞ ¼ emax vðzÞ þ DðzÞ CðzÞ r20  ð8:38Þ

being parallel to meridian and sagittal directions, correspondingly. Consider a focusing emission-imaging system with the limiting image plane z ¼ zG. The considerations of continuity bring us to the conclusion that, for any small enough r0 , the planes z ¼ zmer(r0), z ¼ zsag(r0) exist, in which the semi-axes amer(z), bsag(z) correspondingly vanish and the ellipse of diffusion turns into a segment. Obviously, those planes may be considered as the off-axis Gauss planes for the beams emitted from the cathode in the meridian (o ¼ 0) and sagittal (o ¼ p/2) directions, correspondingly. By denoting DG ¼ D(zG), CG ¼ C(zG), replacing pffiffiffiffiffiffiffir0 by rS/MG, and using again the approximation vðzÞ  ðz zG Þ=MG FG , we easily obtain from (8.38) the explicit view of the limiting surfaces upon which the meridian and sagittal beams emitted in the vicinity of the work area center are focused. Those surfaces, which are called the surfaces of meridian and sagittal curvature, appear as the paraboloids of revolution zmer ðrS Þ ¼ zG þ

r2S ; 2rmer

zsag ðrS Þ ¼ zG þ

r2S 2rsag

ð8:39Þ

with the curvature radii at the center rmer ¼

MG pffiffiffiffiffiffiffi ; 2 FG ðDG þ CG Þ

rsag ¼

MG pffiffiffiffiffiffiffi 2 FG ðDG

CG Þ

;

ð8:40Þ

correspondingly. Let us introduce into our consideration the surface of mean curvature 1 r2S ð8:41Þ zmean ðrS Þ ¼ ½zmer ðrS Þ þ zsag ðrS ފ ¼ zG þ 2rmean 2 located strictly in between the surfaces of meridian and sagittal curvature. The curvature radius of this surface is rmean ¼




rmer rsag rmer þ rsag

¼

MG pffiffiffiffiffiffiffi : 2 FG DG

ð8:42Þ

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Obviously, in the frame of the approximation considered, on the surface of mean curvature the ellipse of diffusion turns into a circle with  1=2  the radius of emax CG r20 . The aberration coefficient DG characterizing the curvature radius of the surface of mean curvature is usually called the image curvature coefficient. Coma Aberration. This aberration is determined by the coefficients G(z), F(z) which are called the coefficients of coma aberration. Omitting, as before, all other terms in (8.21) except those connected with G(z), F(z), we have pffiffi 2io Šer0 sin2 O þ . . . : ð8:43Þ Aloc S ½ e; O; oŠ ¼ . . . þ ½GðzÞ þ FðzÞ e In order to analyze the figure of diffusion resulting from coma aberration, let us consider the following cases.

1. jG(z)j > jF(z)j. In this case the figure of diffusion represents a part of a circular cone (Figure 124a), the center of which is located at the origin of the local coordinate system {sm, st}, the axis of symmetry coincides with the meridian direction, and the apex angle is equal to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ 2 arctanjFðzÞj= G2 ðzÞ2 F2 ðzÞ. On the side of large jsmj, the figure of diffusion is restricted by the arc of the circle being inscribed into the cone and having its center at the point {sm ¼ emaxr0 G(z), st ¼ 0}. The figure of diffusion resembles a comet, the ‘‘tail’’ of which for G > 0 is directed toward increasing sm, and for G < 0 in opposite direction. 2. jGjjFj. In this case the figure of diffusion represents a circle with its center at the point {sm ¼ emaxr0 G(z), st ¼ 0} and the radius r ¼ emaxr0jF(z)j (Figure 124b). It should be emphasized that, regardless of the correlation between the coefficients G(z), F(z), the coma aberration results in asymmetry of the figure of diffusion in the meridian direction. Chromatic-Position Aberration. This aberration is characterized by the aberration coefficient Q(z). The partial contribution of the aberration in question into the third-order aberration term (8.21) is pffiffi 2 ð8:44Þ A loc S ½ e; O; oŠ ¼ . . . þ eQðzÞr0 cos O þ :::; (a) y0

sm

(b) y0

sm st

st

x0

x0

FIGURE 124 Figure of diffusion for coma aberration. (a), jG(z)j > jF(z)j; (b), jG(z)j < jF(z)j.

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255

and, consequently, the corresponding figure of diffusion represents a segment of the length emaxjQ(z)jr0, stretched along the meridian direction sm. Thus, similar to the coma aberration, the chromatic-position aberration results in asymmetry of the figure of diffusion in the meridian direction.

8.2. TEMPORAL ABERRATIONS IN STREAK IMAGE TUBES Now we proceed to the analysis of aberration properties of emissionimaging systems with electron-image sweeping. Such systems (shortened in name to streak image tubes) are commonly used in photoelectronic imaging of ultrafast events. Principal scheme of streak image tube operation is shown in Figure 125. By illuminating the photocathode with an optical (say laser) pulse, we gain an electronic ‘‘response’’ in the form of an electron bunch containing complete information as to the properties of the incident optical radiation. Temporal modulation of the light power density within the incident optical pulse is transformed at this stage into spatial modulation of the space charge density within the electron bunch emitted. The photoelectron bunch produced by the photocathode is considered a carrier of information on the incident optical pulse structure, and all subsequent transformations of the bunch within the streak tube (accelerating, focusing, and dynamic deflection) are aimed at preserving and then revealing that information on the image receiver. It is important to notice that the latter requirement inevitably involves the use of stationary fields only for accelerating and focusing, because any time-dependent field would essentially destroy the temporal structure of the bunch. The streak image tube usually contains two functionally different subsystems, one of which is purposed at charged particles accelerating and focusing, while the other sweeps the electron bunch along the image

1

Accelerating and focusing fields area

2

4

7

6

8 3

5

FIGURE 125 Principal scheme of streak image tube operation: 1 – input optical radiation, 2 – photocathode, 3 – fine-structure grid, 4 – electron bunch, 5 – diaphragm, 6 – dynamic deflection system, 7 – image receiver, 8 - electron image of the swept bunch.

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receiver to reveal temporal resolution. These subsystems are commonly separated by a diaphragm to avoid the unwanted superposition of the focusing and deflecting fields. The aberrational scattering of charged particles on the image receiver surface is conditioned not only by spatial but also (especially, at high streak speed) by temporal aberrations of the electron bunch. As far back as in the mid-1950s, a simple expression for the first-order temporal aberration caused by the charged particles spread in initial axial velocities (the first-order temporal chromatic aberration) was obtained in the pioneering works by Zavoisky and Fanchenko (1956, 1976), which laid the grounds for the physical principles of ultrafast electron-optical photography. The systematic theoretical and numerical studies on the temporal aberrations in cathode lenses were initiated in the end of 1970s by Monastyrskiy and Schelev (1980, 1989), and we refer the reader to relevant papers for details. Here we touch only on some final results of those investigations. Using the tau-variation technique from Chapter 5, it can be shown that the aberration expansion of the transit time t of charged particles in any static emission-imaging system with respect to the small parameters {e1/2, r0} appears as 2 2  t½z; e1=2 ; r0 ; O; oŠ ¼ t0 ðzÞ þ n ðt r0 ÞðzÞr0 o  1=2 1=2 þ e1=2 ðten ÞðzÞcos O þ ðtet r0 ÞðzÞr0 sin O cos o þ . . . 

  þ e ðtet ÞðzÞsin2 O þ ðten ÞðzÞcos2 O þ . . . þ . . . ð8:45Þ

Here t0S ðzÞ is the transit time along the main optical axis of the particle 1=2 emitted with zero initial velocity, ðtjen Þ is the coefficient of the first-order 1=2 temporal chromatic aberration, ðtjet r0 Þ is the coefficient of the secondorder temporal spherical-position aberration, (tjet) is the coefficient of the second-order spherical aberration, (tjen) is the coefficient of the secondorder chromatic aberration, and finally, ðtjr20 Þ is the coefficient of the second-order temporal distortion. It is assumed that the image receiver surface S intersects the symmetry axis at the coordinate z. According to the Zavoisky-Fanchenko formula originally established by Zavoisky and Fanchenko (1956) for homogeneous electric field and then generalized by Monastyrskiy and Schelev (1980) for arbitrary stationary electromagnetic field, rffiffiffiffiffiffiffi  2m 1 Þ ¼ ð8:46Þ ðte1=2 0 : n e F0

This simple but very important relation, which is similar to the ArtsimovichRecknagel formula (8.34), shows that, with the initial energy and angular distribution given, the first-order temporal chromatic aberration in a static

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0

electromagnetic field is fully determined by the field intensity F 0 at the cathode center and does not depend on the image receiver position. The situation is quite different if the electric field depends on time; this case is considered in Chapter 9. 1=2 The second-order aberration coefficients (tjen), (tjet), ðtjet r0 Þ, ðtjr20 Þ represent rather complicated nonlinear functionals, which, similar to the spatial aberration coefficients, depend on the axial potential distribution F and its derivatives. To reveal some important points of electron-image sweeping, let us consider the simplest dynamic deflection system represented by a flat capacitor with a linear electric voltage ramp Udef (t) applied to its plates. If we neglect the fringe effects at the deflection system edges and assume the electric field between the plates at each timepmoment to be strictly ffiffi homogeneous, the dynamic aberration term Aloc S ½ e; O; oŠ describing the total scattering of charged particles on the image receiver surface appears as the sum of two terms as follows: 1=2 1=2 ; O; oŠ ¼ ðAloc ; O; oŠ þ VS Dtstat ½e1=2 ; O; oŠ: Aloc S ½e S Þstat ½e

ð8:47Þ

The first term is the static spatial aberration conditioned by the charged particles traveling through the static field region from the cathode to the image receiver. The second, dynamic, term is directly proportional to the streak speed VS and is conditioned by electron-image sweeping by the deflector. The time spread Dtstat[e1/2, O, o] is fully determined by the temporal aberrations at the center z ¼ zdef of deflection area Dtstat ½e1=2 ; O; oŠ ¼ t½z; e1=2 ; O; oŠ

t0 ðzÞjz¼zdef :

ð8:48Þ

Within the assumptions made, the streak speed value on the image receiver surface is given by the simple relation VS ¼

1 Ll dUdef ; 2 Uh dt

ð8:49Þ

where l is the capacitor plates’ length, h is the distance between the plates, L is the distance along the main optical axis between the deflection center zdef and the image receiver, U is the accelerating voltage that characterize the energy of the particles entering the deflection region, and dUdef/dt is the deflection voltage change rate. The Eq. (8.47) clarifies the role of streak speed in temporal resolution recording. Projecting Eq. (8.47) onto the sweep direction and dividing by VS, we obtain DtS ½e1=2 ; O; oŠ ¼

1=2 ðAloc ; O; oŠ S Þstat ½e þ Dtstat ½e1=2 ; O; oŠ: VS

ð8:50Þ

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If we hypothetically assume the system to be ideal from the viewpoint 1=2 ; O; oŠ ¼ 0), then for any of spatial focusing in the static field (ðAloc S Þstat ½e nonzero streak speed VS DtS ½e1=2 ; O; oŠ ¼ Dtstat ½e1=2 ; O; oŠ;

ð8:51Þ

which means that, in the ideal case, any - even arbitrarily small - streak speed is acceptable to record physical temporal resolution determined by the static temporal aberration term (8.48). Inasmuch as ideal focusing in reality is impossible, the spatial aberrations comprising the aberration term 1=2 ; O; oŠ would inevitably spoil temporal resolution. Under the ðAloc S Þstat ½e assumption made, the streak speed high enough to obey the condition   loc ðA Þ ½e1=2 ; O; oŠ   S stat ð8:52Þ þ < Aa x; x > þ’a ðxÞ 2

ð8:60Þ

of the aberration term (8.21). As soon as the distribution in initial energies F(e) in the photoemission case differs from zero only within the finite interval 0  e  emax, the corresponding dimensionless distribution in initial velocities (8.12) differs from zero within the hemisphere Oþ ðmax Þ ¼ fx1 ; x2 ; x3 jx21 þ x22 þ x23  max ; x3  0g, where max ¼ emax =~e. For simplicity, we use the same letters h, F,. . . for the corresponding distributions brought to the dimensionless form. We also need to introduce the dimensionless spatial frequency n, which is connected with the real spatial frequency N on the 0 photocathode by the relation n ¼ ð~e=F 0 ÞN. The optical transfer function (8.1), as applied to the meridian and sagittal directions a ¼ m, s, appears now in the form of the phase integral I

ðrÞ;loc ðnÞ a

¼

þ

ð

hðxÞexp½2pin aa ðxފd3 x

ð8:61Þ

O ðmax Þ

taken over the domain Oþ(max). The algebraic structure of the phase function aa(x) in the meridian direction is essentially different from that in the sagittal one: the matrix Am is, generally speaking, nondegenerate, whereas the matrix As is definitely degenerate. This fact reflects a fundamental physical difference in image formation properties in the meridian and sagittal directions. Thus, to apply the stationary phase method (see the Appendix 7) to study the asymptotic behavior of IaðrÞ;loc ðnÞ in the high-frequency range, we need to consider the cases a ¼ m and a ¼ s separately. Meridian direction (a ¼ m). Let us first assume that the third-order spherical and spherochromatic aberration coefficients pm, qm are so small that we can set them equal to zero and make the phase function am(x) quadratic 1 am ðxÞ ¼ < q; x > þ < Am x; x > : 2

ð8:62Þ

The determinant of the matrix Am is 2ym ðlm mm h2m Þ, and, as soon as lm ðr0 Þmm ðr0 Þ > 0 for small enough r0, the matrix Am is nondegenerate in the deleted vicinity of the cathode center (0 < r0  rmax) under the only condition that the coma coefficients G, F are not exactly equal (ym 6¼ 0). The exceptional case ym ¼ 0 merits special consideration not provided here.

h2m

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261

According to Eq. (8.62), the phase function am(x) possesses a single nondegenerate stationary point xð0Þ ¼ Am1 q being the solution to the equation ram(x) ¼ 0. In the coordinate form, the stationary point is x01 ¼

xm mm ; 2ðh2m lm mm Þ

x02 ¼ 0;

x03 ¼

xm hm : 2ðh2m lm mm Þ

ð8:63Þ

Our main concern is the case that the stationary point x(0) is located strictly inside the domain Oþ(max): x(0) 2 int Oþ(max) as illustrated in Fig.126. In this case, according to the multidimensional stationary phase method outlined in Appendix 7, the main term of the OTF asymptotics in the high-frequency range takes the form h i 1 p I aðrÞ;loc ðnÞ  3=2 jdet Am j 1=2 hðx0 Þexp 2pin am ðx0 Þ þ i sgn Am ; ð8:64Þ 4 n wherefrom, denoting Cm ¼ jdetAm j

1=2

hðx0 Þ;

sm ¼ sgn Am ;

a0m ¼ am ðx0 Þ;

ð8:65Þ

we obtain the high-frequency asymptotics of MTF and PTF for the meridian direction w∗ m ðnÞ 

Cm ; n3=2

p 0 f∗ m ðnÞ  2pnam þ sm : 4

ð8:66Þ

The coefficients Cm ; a0m ; sm are explicitly determined by the aberration coefficients and the initial distributions of electrons in energies and angles. Of note, the condition x(0) 2 int Oþ(max) may be considered as a sort of optimization criterion or requirement that should be imposed on the x1

1 2 O

x3

x2

FIGURE 126 Location of stationary points of the dimensionless aberration expansion aa(x) in the domain Oþ(Zmax) for meridian (1) and sagittal (2) directions.

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aberration coefficients to improve the image quality, because, as it follows from the stationary phase method, if this condition is not met, the corresponding MTF decreases in high-frequency range faster compared with (8.66). Generally, when bm, pm differ from zero, the phase function am(x) may have, depending on particular values of the aberration coefficients, several stationary phase points belonging to int Oþ(max). According to the localization principle of the stationary phase method, the contributions of those stationary points to the asymptotics in question may be calculated separately and then summed. When the stationary points are nondegenerate, this could change only the coefficient Cm in Eq. (8.66) but not the asymptotics’ order O(n 3/2). The calculation formulae for the case of several nondegenerate stationary phase points are easy to derive, but they are rather cumbersome (we leave it to the reader to reconstruct all necessary details). Sagittal direction (a ¼ s). This case is slightly more complicated because the matrix As is degenerate, and the solutions of the equations ras(x) ¼ 0 are not isolated but continuously fill in a certain curve l in the space {x1, x2, x3} (Figure 126). Such a curve may be called the stationary curve of the phase function as(x). Let us consider the nonlinear transformation z ¼ L(x) z 1 ¼ x1 ;

z3 ¼ xs þ 2fs x1 þ 2hs x3 þ bs ðx21 þ x22 Þ þ ps x23

z2 ¼ x2 ;

ð8:67Þ

so that the transformed phase function ~as ðzÞ ¼ z2 z3 does not depend on the coordinate z1. As can be easily seen, the Jacobian J0 ¼ detkDz/Dxk ¼ 2(hs þ ps x3) of this transformation differs from zero in Oþ(max) for ps being small enough. The transition to the variables z1, z2, z3 in the phase integral (8.61) for a ¼ s, and the change of the order of integration gives ð ðrÞ; loc ðnÞ ¼ hðxÞexp½2pinas ðxފ d3 x Ia Oþ ðmax Þ

¼

ð

ℜðz2 ; z3 Þexp½2pinz2 z3 Š dz2 dz3 :

ð8:68Þ

~ þ ð ފ Pr½O max þ

~ ð ފ onto the Here the integration is made over the projection Pr½O max space {z2, z3} of the image of the hemisphere Oþ(max) by the transformation z ¼ L(x), and the pre-exponent ℜðz2 ; z3 Þ represents the single integral zþ ðz2 ;z3 Þ 1

ℜðz2 ; z3 Þ ¼

ð

z1 ðz2 ;z3 Þ

  h L 1 ðz1 ; z2 ; z3 Þ J0 1 ðz1 ; z2 ; z3 Þ dz1

ð8:69Þ

taken along the segment z1 ðz2 ; z3 Þ  z1  zþ of which 1 ðz2 ; z3 Þ, the endpoints ~ þ ð ފ fixed. ~ þ ð Þ for any ðz2 ; z3 Þ 2 Pr½O belong to the boundary @ O max max

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Obviously, the transformed phase function ~as ðzÞ ¼ z2 z3 in the phase integral (8.68) has the single nondegenerate stationary point z2 ¼ z3 ¼ 0 in ~ þ ð Þ. Applying the stationary phase method to the phase the domain O max integral (8.68), we obtain the high-frequency asymptotics in sagittal direction Cs ; n

w∗ s ðnÞ 

ð8:70Þ

with the coefficient Cs being the integral ðzþ Þ 1 0

Cs ¼

ð

ðz1 Þ0

 h L 1 ðz1 ; 0; 0Þ J0 1 ðz1 ; 0; 0Þ dz1 :

ð8:71Þ

 taken between the integration limits ðz 1 Þ0 ¼ z1 ð0; 0Þ which correspond to the endpoints of the stationary curve l ¼ {x1 ¼ z1, x2 ¼ 0, x3 ¼ x3(z1, 0, 0)}, located on the boundary of the domain Oþ(max). Those points can be easily calculated with a simple computational procedure. The sagittal PTF is identically equal to zero because the corresponding figure of diffusion is symmetrical with respect to meridian direction. Thus, in high-frequency range the sagittal MTF is ‘‘better’’ (e.g. decreases slower) than the meridian one. This represents one of the fundamental properties of image formation in cathode lenses. Let us now consider more details of the high-frequency asymptotics of MTF at the cathode center. For simplicity, we again restrict our consideration to the case when the aberration coefficients p, b can be neglected. As easily seen from the algebraic structure of the aberration expansion aa(x), the sagittal asymptotics (8.70) at r0 ! 0 smoothly turns into the asymptotics at the center. The index ‘‘s’’ is unessential at the cathode center, and we omit it further. The defocusing parameter x defined according to Eq. (8.56) appears as 0

x ¼ xðzÞ ¼

F 0 vðzÞ wðzÞ~e1=2

ð8:72Þ

;

where v(z), w(z) are the linear-independent solutions of the limiting paraxial equation (8.15). The coefficient C in the asymptotics (8.70) takes the form 1 C ¼ CðxÞ ¼ 2p

max ð

  FðtÞ x p ffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G   1 2 t dt; x2 t t 4 x2 =4

pffiffiffiffiffiffiffiffiffiffi ð0  x  2 max Þ:

ð8:73Þ

Inasmuch as G1(0) ¼ 0 by definition, the coefficient C(x) vanishes at the pffiffiffiffiffiffiffiffiffiffi endpoints of the interval 0  x  2 max , which is connected with degeneration of the asymptotics (8.70) near the boundary of the domain

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Oþ(max). The function C(x) is non-negative, therefore it attains its maxipffiffiffiffiffiffiffiffiffiffi mum at a point x ¼ xopt located strictly inside the interval 0  x  2 max . Location of this maximum determines the position of the best focusing plane and, correspondingly, the asymptotic behavior of the optimal MTF in the high-frequency region. Consider the case that the energy and angular distributions are taken in the analytical form (8.8), (8.9). In this case FðtÞ ¼ Apq tp ðt

tmax Þq ;

G1 ðtÞ ¼ Bm tm

ðm > 0Þ;

where, according to the normalization condition (8.11), 2 3 1 max ð tp ðmax tÞq dt5 ; Bm ¼ m þ 1: Apq ¼ 4

ð8:74Þ

ð8:75Þ

0

The coefficient C(x) (which we denote as Cpqm(x) in the case in question), appears as   Apq ðm þ 1Þ x m Cpqm ðxÞ ¼ 2p 2

max ð

ðmax tÞq qffiffiffiffiffiffiffiffiffiffiffi dt: 2 mþ1 p 2 t x4 t 2 x =4

ð8:76Þ

This relation makes it possible to study the MTF asymptotics as a function of the parameters p, q, m, max. As an example, consider two most important particular cases: 1) p ¼ q ¼ m ¼ 1, and 2) p ¼ q ¼ 1, m ¼ 3. Let us put the normalization energy ~e equal to the most probable energy e0 of the parabolic energy distribution, so that max ¼ 2 in both cases. Integration in Eq. (8.76) gives 0 13=2 2 1x@ xA 2 C111 ðxÞ ¼ 4 p2 2 0 12 0 vffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 13 vffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u u 2 3 4 @xA 2u x2 A @x A u t2 x 5 : C113 ðxÞ ¼ arctg@ t2 2 4 4 p 2 x 2 ð8:77Þ

The coefficients C111(x), C113(x) are shown as functions of the defocusing parameter x in Figure 127. Thepffiffifunction C111(x) attains its maximum exactly at the point ffi xopt ¼ 2  1:41 . . . which can be found ‘‘by hand’’ from Eq. (8.77), whereas the maximum point xopt  1.64. . . for the function C113(x) can be easily calculated numerically. Thus, the optimal MTF possessing maximal output contrast in the high-frequency region takes the form   0:413 0:616 ; w113 ðnÞz¼zF  : ð8:78Þ w111 ðnÞz¼zF  n n

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0.6

C(x)

C113

0.4

C111

0.2

0.0 0.0

1.0

2.0

3.0

x

FIGURE 127 Dependence of the coefficients C111(x) and C113(x) on the defocusing parameters x.

Inasmuch as the best focusing plane zF ¼ zF(xopt), in which the MTF becomes optimal, is commonly located nearby the Gauss plane, we can express zF(xopt) in the explicit form pffiffiffiffiffiffiffipffiffi M2G FG ~e zF  zG þ xopt ð8:79Þ 0 F0 pffiffiffiffiffiffiffi if we put vðzÞ  ðz zG Þ=MG FG in Eq. (8.72). As expected, the asymptotic given by the second of Eqs. (8.78) is somewhat higher compared with the first one because of a narrower angular distribution of photoelectrons. Figure 128 shows the MTF w113(n) calculated according to the asymptotics (8.78) and the same MTF calculated by Kulikov (1975) with the use of a rather complicated numerical procedure of integration in velocity space. The discrepancy does not exceed 5-7% in the frequency range n  2, where the output contrast falls down from 0.3 to zero.

8.4. EXAMPLES OF THE SPREAD FUNCTIONS, MTF AND PTF IN IMAGE TUBES The high-frequency asymptotics obtained in the previous section becomes invalid when the stationary phase points are located on the boundary of the domain Oþ(max) or outside it. The first case can also be investigated with the stationary phase method, but such investigation is more complicated, and we not provide it here. In the second case, direct numerical calculation of the phase integral (8.1) should be applied.

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1.0

0.8 2 R(n)

0.6 1 0.4

0.2

0.0 0.0

1.0

2.0 n

3.0

4.0

FIGURE 128 The comparison between the MTF w113(n) calculated by means of a numerical procedure (Kulikov, 1975) (1) and the asymptotics (8.78) (2).

In this section, we consider some examples of MTF and PTF that allow complete analytical representation. 1. The Gauss plane case. Let us put p ¼ q ¼ 1, m ¼ 3 and restrict ourselves by the binomial expansion (8.36), which takes the form pffiffi MG io sin2 O ð8:80Þ Aloc 0 e e ¼0; z¼zG ¼ S ½ e; O; oŠjr∗ 0 F0

in the Gauss plane z ¼ zG. In addition to the dimensionless parameters 0 0 x0 ¼ ðF 0 =~eÞx0 . For writing simused above, let us denote r0 ¼ ðF 0 =~eÞr0 ,  plicity, we omit the indexes ‘‘loc’’ and ‘‘S’’ in the notation of the point ðrÞ;loc referred to the cathode, and denote the spread function IS x0 Þ. We leave it to the reader corresponding line spread function as J ðrÞ ð as an exercise to show that the exact representations for IðrÞ ðr0 Þ, J ðrÞ ðx0 Þ, and w113(n) in the approximation considered appear as 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2max r20 7  max 6 ðrÞ 2 2   ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q I ðr0 Þ z¼z ¼ C1 4 max r0 þ ln 5 ð0  r0  max Þ G max þ 2max r20  x0 Þz¼zG ¼ C2 ðmax j x0 jÞ2 ; ð0  j x0 j  max Þ J ðrÞ ð  w113 ðnÞ

z¼zG

¼

6

ð2pnmax Þ

 1 2

sinð2pnmax Þ 2pnmax



ð8:81Þ

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1.0 0.8

1

0.6

2 0.4 0.2 0.0 −2.0

−1.0

0.0

1.0

2.0

x 1.0

0.8

0.6

3

0.4

0.2

0.0 0.0

0.5

1.0 n

1.5

2.0

FIGURE 129 The point spread function (curve 1), line spread function (curve 2), and MTF (curve 3) in the Gauss plane (3) for p ¼ q ¼ 1, m ¼ 3.

Here and in the examples below, the positive constants Ci ensure normalization of the corresponding functions. The functions (8.81) are shown in Figure 129. We can see that the MTF w1,1,3(n)jz ¼ zG in the high-frequency range has the asymptotics O(1/n2), which means that this MTF decreases in the Gauss plane z ¼ zG faster than the MTF calculated in the best focusing plane z ¼ zF. It is interesting to note that the point spread function IðrÞ ðr0 Þjz¼zG possesses, within the approximation used, the logarithmic singularity Oðjlnr0 jÞ at r0 ! 0, and, as a consequence, the line spread function

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J ðrÞ ð x0 Þjz¼zG possesses the ‘‘beak-shaped’’ singularity Oðjx0 jÞ at x0 ! 0. Vendt (1955) was apparently the first to notice this phenomenon as peculiar to the image formation in cathode lenses. In fact, the point/line spread functions and MTF derived in Vendt’s paper devoted to the cathode lens with combined homogeneous electric and magnetic fields in the approximation (8.80) have the singularities and asymptotic behavior similar to (8.81). It is useful to note in this connection that the approximation (8.80) corresponds to the case that the defocusing parameter x turns to zero and the sagittal stationary curve l lies on the boundary of the domain Oþ(max). More detailed analysis shows that the asymptotics O(1/n2) is connected in this case with the asymptotics F(e)  e of initial energy distribution at small energies, which is peculiar both to our case (p ¼ q ¼ 1) and the case of Maxwellian distribution (8.7) considered by Vendt. As soon as the point and line spread functions at the cathode center are symmetrical in the case in question, and the MTF w113(n)jz ¼ zG never turns to zero, the PTF is identically equal to zero. 2. The case of strong defocusing. Let us put now p ¼ q ¼ m ¼ 1 and consider the case of strong defocusing (x >> 1) when all the terms of the aberration expansion (8.21), except the first one, may be neglected. In this approximation pffiffi 1=2 vðzÞeio sinO þ . . . ð8:82Þ A loc S ½ e; O; oŠ ¼ e and the corresponding point/line spread functions and MTF appear as IðrÞ ðr0 Þjx>>1 ¼ C3 ðr2max

r20 Þ2 ;

x0 Þjx>>1 ¼ C4 ðr2max J ðrÞ ð

 x20 Þ5=2 ;

w111 ðnÞjx>>1 ¼

48 ð2pnrmax Þ3

ð0  r0  rmax Þ ð0  jx0 j  rmax Þ

jJ3 ð2pnrmax Þj;

ð8:83Þ

pffiffiffiffiffiffiffiffiffiffi where rmax ¼ xðzÞ max and J3(Z) is the third-order Bessel function. The functions (8.83) are shown in Figure 130 (curves 1,2). With regard to the well-known asymptotic properties of the Bessel function J3(Z) at large Z, it follows from Eq. (8.83) that at high frequencies w111(n)jx>>1 ¼ O(1/n7/2). Thus, the high-frequency MTF in the strong defocusing region decreases essentially faster compared with that in the Gauss plane, and all the more faster compared to that in the plane of best focusing. The characteristic peculiarity of the case in question is that the contrast w111(n)jx>>1 vanishes at the frequencies nk ¼ Zk/2prmax, where Zk 6¼ 0, k ¼ 1,2,. . . are the zeros of the Bessel function J3(Z), numerated in increas ðrÞ ð2pnÞj ing order. Inasmuch as the imaginary part ImI x>>1 of the OTF is

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1.2 2 1.0 1

0.8 0.6 0.4 0.2 0.0 −1.0

0.0

1.0 x/r max

1.0 2p 0.8 3 0.6

4 p

0.4

0.2

0.0 0.0

0.5

1.0

1.5

2.0 n r max

FIGURE 130 The point spread function (curve 1), line spread function (curve 2), MTF (curve 3), and PTF (curve 4) for the case of strong defocusing at p ¼ q ¼ m ¼ 1.

identically equal to zero due to symmetry of the line spread function  ðrÞ ð2pnÞj IðrÞ ðr0 Þjx>>1 , and the real part ReI x>>1 changes its sign at the ðrÞ  ð2pnÞj ¼ argI represents a steppoints n ¼ nk, the PTF f ðnÞj 111

x>>1

x>>1

wise function equal to zero within the interval 0  n < n1 and undergoing the abrupt increment by p at the points n ¼ nk (Figure 130, curve 3). This means that a phase shift by p occurs in the output image at each characteristic frequency n ¼ nk. This is the well-known ‘‘false contrast’’ effect in optics (see, for example, the monograph by O’Neil, 1963).

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If in the example under consideration, the parabolic distribution in initial energies is replaced by the Maxwellian one, both the point/line spread functions and MTF prove to be of Gaussian type:  2  2 r0 x0 ðrÞ ðrÞ ; I ðr0 Þjx>>1 ¼ C6 exp I ðr0 Þjx>>1 ¼ C5 exp 2 x x2  1 ð2pnxÞ2 : wðnÞjx>>1 ¼ exp ð8:84Þ 4 In this case, the contrast falls exceedingly fast with the frequency increase. The Eq. (8.84) makes it clear wherefore the spread function approximations in the form of Gaussian distributions are so popular among the experimentalists (see also paragraph 7.2 on this matter).

8.5. THE BOUNDARY-LAYER EFFECT IN CATHODE LENSES AND ELECTRON MIRRORS As seen in Chapter 5, the formal procedure of the tau-variation technique allows correct and computationally effective construction of aberration expansions of charged particle trajectories in the vicinity of any arbitrarily chosen principal trajectory. The essence of the tau-variation approach lies in the possibility of performing two numerical procedures synchronously: solving the differential equations for the isochronous variations (tauvariations) and transforming the tau-variations into the aberration coefficients according to the contact transformation formulas. The first of these procedures is regular everywhere on the principal trajectory, including the low-potential region in cathode lenses and the turning point vicinity in electron mirrors, where the particle velocity on the principal trajectory tends to zero. The second procedure is commonly of interest in the region where the particle velocity is high enough. This section shows that the singular behavior of aberration coefficients in the low-potential region, first outlined by Artsimovich and then for an extended period been the ‘‘watershed’’ between the narrow and wide beam theories, is that the spatiotemporal properties of electron trajectories in the low- and high-potential regions are so different that it is basically impossible to describe them using the same regular aberration expansions so common to charged particle optics. First, we will show by a few simple examples that the regular power aberration expansions do not exist in the low-potential region and should be replaced by the so-called boundarylayer aberration expansions, which rapidly turn into the regular ones outside the boundary-layer region. Then we will touch on the question of how the uniformly accurate boundary-layer aberration expansions can be constructed in general case.

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We reiterate that the aberration expansions similar to those given by Eqs. (8.21) and (8.45) can be used only if they possess asymptotic character with respect to the small parameters e1/2 and r0. Let us consider the asymptotic properties of those expansions in the simplest case of 0 the homogeneous static electric field FðzÞ ¼ F 0 z. To avoid dealing with the initial angles O, o (which are unnecessary here), we will use the 1=2 1=2 normalized tangential et ¼ e1=2 sin O and normal en ¼ e1=2 cos O components of the charged particle velocity. In the case in question, exact representations of the trajectory and transit time for the particle emitted at the cathode center (r0 ¼ 0) are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1=2 io 0 rex ¼ 0 et e ð en þ F 0 z en1=2 Þ; ð8:85Þ F0 rffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m 1 0 en þ F 0 z tex ¼ 0 ð e F0

en1=2 Þ:

The paraxial trajectories v, w are also easy to derive: 2 pffiffiffi z; w ¼ 1: v ¼ pffiffiffiffiffiffiffi 0 F0

ð8:86Þ

ð8:87Þ

Let us check if the second-order binomial expansion (8.36) in this case is asymptotical everywhere in the region z  0. Let us estimate the difference between the exact solution (8.85) and the expansion (8.36) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi   2 1=2 0 0 1=2 1=2 F 0 zÞ Heio et en1=2  ¼ 0 et ð en þ F 0 z jDrj ¼ rex veio et F0 ð8:88Þ

Introducing the boundary layer parameter B ¼ F/en, we can transform Eq. (8.88) to 1=2 1=2

et en ffi pffiffiffi ; jDrj ¼ pffiffiffiffiffiffiffiffiffiffi 1þBþ B

ð8:89Þ

which shows that the difference jDrj has the third order of smallness 1=2 Oðet en Þ in the high-potential region F >> en, whereas this difference in the low-potential region F  en has the second order of smallness 1=2 1=2 Oðet en Þ. Thus, we can deduce that the binomial expansion (8.36), with the second-order chromatic aberration coefficient H calculated according to the Artsimovich-Recknagel formula (8.34), is asymptotical in the region F >> en and is not within the boundary-layer region F  en. Similar considerations allow us to extend this deduction to the aberration expansion of transit time given by Eq. (8.45), with the first-order temporal 1=2 chromatic aberration coefficient ðtjen Þ calculated according to the

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Zavoisky-Fanchenko formula (8.46). As a result, we can see that both the Artsimovich–Recknagel and Zavoisky-Fanchenko formulas are valid in the high-potential region F >> en and are not valid in the boundary-layer region F  en. The latter is obvious because, in contravention of the physical sense, the aberration coefficients (8.34) and (8.46) do not turn into zero at the starting point z ¼ 0. Let us consider the boundary-layer function 1 BðBÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ; 1þzþ B

ð8:90Þ

which tends to zero as B 1/2 at B ! 1 and turns into unit at B ! 0. A simple identical transformation allows representation of the exact solutions (8.85) and (8.86) in the form of ‘‘improved’’ aberration expansions 1=2 ~ 1=2 ; ~ io e1=2 e1=2 ; tex ¼ t0 þ Te ð8:91Þ rex ¼ veio et þ He t n n p ffiffiffiffiffiffiffiffiffiffi ffi p ffiffi ffi 0 in which t0 ¼ 2m=e z=F 0 is the transit time of a particle emitted with zero initial energy from the cathode center and traveling along the main optical axis, and

 H ¼  T ¼

2 2 BðBފ ¼ H þ 0 BðBÞ 0 ½1 F0 F0 sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi 2m 1 2m 1 1=2 BðBފ ¼ ½tjen Š þ 0 ½1 0 BðBÞ e F0 e F0

ð8:92Þ

are the ‘‘corrected’’ aberration coefficients which differ from the aberra1=2 tion coefficients H; ðtjen Þ by the presence of the boundary-layer function (8.90), which is essential in the low-potential region F  en and negligibly small in the high-potential region F >> en. Inasmuch the aberration expansions (8.91) represent the exact solution, they ensure uniform accuracy for all z  0, and we can see that the ‘‘corrected’’ aberration coeffi  cients H, T vanish at the starting point z ¼ 0. It should be emphasized that the aberration expansions (8.91) are not of the classic Poincare sense because the coefficients of these expansions are not regular functions of the variable z but represent a sum of the regular functions of z (being constant in the case in question) and the regular boundary-layer functions of the variable B ¼ F/en. In fact, this reflects the general situation. Below we analyze the reason why the boundary-layer components appear in the aberration expansions similar to those represented by Eq. (8.91) and discuss a general way to construct the uniformly accurate aberration expansions valid in both low- and high-potential regions. Consider the small parameters vector x ¼ fxi g3i¼1 with the components 1=2 1=2 x1 ¼ et , x2 ¼ en , x3 ¼ r0. As before, we denote gi the derivative of any

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273

function g(x) with respect to xi. For any vector x fixed, we can consider the charged particle trajectory as one-to-one function of time r ¼ r ðt; xÞ;

z ¼ z ðt; xÞ;

ð8:93Þ

where, extending our deductions to the case of electron mirrors, we use the sign ‘‘ ‘‘ and ‘‘þ’’ for the direct and inverse branches of the trajectory, respectively. If t ¼ t(z, x) is the inverse function being the particle transit time at any x fixed, we have the obvious identity z ½t ðz; xÞ; xŠ

z ½t ðz; 0Þ; 0Š ¼ 0

ð8:94Þ

valid for all z, x. Denote Dt ¼ t(z, x) t(z, 0) the temporal dispersion in the vicinity of the principal trajectory and expand the left part of the identity (8.94) in the Taylor series 1 1=2  2   1=2 z_  Dt þ z zðDt Þ2 þ z þ z 11 et þ z22 en þ z33 r0 2 en þ z3 r0 þ ½€ 1 et 2  1=2 1=2   1=2  1=2 z þ 2_z þ 2_z 3 Dt r0 þ 2z12 et en 2 Dt en þ 2_ 1 Dt et 1=2

 1=2 þ 2z 13 et r0 þ 2z23 en r0 Š þ . . . ¼ 0

ð8:95Þ

(either upper or lower signs should be taken simultaneously). The first-order and second-order tau-variations zi, zij are studied in Kulikov et al. (1985), but the reader can easily see that 2 pffiffiffiffi F; z3 ¼ 0; z12 ¼ 0; z23 ¼ 0 ð8:96Þ z1 ¼ 0; z 2 ¼  0 F0

in the axially symmetric static cathode lenses. Considering the case of the electron mirror, we assume that F(zP) ¼ 0, F0 (zP) 6¼ 0 at the turning point trajectory. zP of the principalp ffiffiffiffiffiffiffiffiffiffiffipffiffiffiffi Substituting Eq. (8.96) along with the obvious relation z_ ¼  2e=m F in Eq. (8.95), we obtain rffiffiffiffiffi 2epffiffiffiffi  2 pffiffiffiffi 1he 0  2  F ðDt Þ2 þ z FDt  0 Fen1=2 þ  11 et þ z22 en þ z33 r0 m 2 m F0 # rffiffiffiffiffi 0 2e F  1=2  1=2 þ 2z13 et r0 þ . . . ¼ 0: ð8:97Þ þ2 0 Dt en mF0 This equation makes possible constructing a uniformly accurate asymptotic expansion of the temporal dispersion Dt with respect to the 1=2 1=2 set of small parameters et , en , r0. First, we note the important fact that in the low-potential region F  en the second term in Eq. (8.97) has not the first (as in the high-potential region F >> en) but the second order of smallness. This implies that, wishing to obtain a uniformly accurate second-order asymptotics of Dt, we must make the sum of the terms in Eq. (8.97) vanish as a whole but not equate to zero the linear and quadratic

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parts of Eq. (8.97) separately. The latter would have immediately brought us to the nonuniform aberration coefficients considered above. Second, we can see that Eq. (8.97) degenerates at the point F0 ¼ 0 at which the particle’s acceleration on the principal trajectory turns to zero. The presence of such a point (which necessarily exists in the electron mirror) by itself does not cause any difficulties, but mandates caution about the choice of the roots of Eq.(8.97) in the regions where F0 preserves a constant sign. Let us denote the second-order term connected with the parameters 1=2 1=2 et , en , r0 on both branches as 1=2

  2   2L 2 ¼ z11 et þ z22 en þ z33 r0 þ 2z13 et r0

ð8:98Þ

and consider the direct branch first. We can bring Eq. (8.97) to the standard quadratic equation ! rffiffiffiffiffiffiffi pffiffiffiffi F0 1=2 2 2m 4m 1 pffiffiffiffi 1=2 2m L2 2 Fen þ F þ 0 en ðDt Þ þ ðDt Þ þ 0 0 0 ¼ 0; e e F e F F0 F

ð8:99Þ

the solution of which, vanishing at the starting point z ¼ 0, is rffiffiffiffiffiffiffi 1=2 rffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi! F 2m en 2m en F L2 : þ 02  Dt ¼ 0 0 0 02 e F0 e F F F0 F

ð8:100Þ

Here, the sign ‘‘þ’’ should be chosen in the region F0 > 0 and the sign ‘‘ ’’ in the region F0 < 0 to ensure a continuous transition of Eq. (8.100) through the degeneration point F0 ¼ 0. Conversely, the sign ‘‘þ’’ should be chosen on the inverse branch Dtþ in the region F0 < 0 and the sign ‘‘ ’’ in the region F0 > 0. These cases can be easily combined (see below). The calculation formulas for the first- and second-order temporal aberration coefficients in the high-potential region F >> en (see Monastyrskiy, 1989) easily follow from Eq. (8.100). This relation is advantageous because it also contains information on the asymptotic properties of the temporal aberration coefficients in the low-potential region. For sim1=2 plicity, let us put et and r0 equal to zero and study the low-potential 1=2 asymptotics of Dt with respect to en in detail. After simple algebraic transformations, Eq. (8.100) takes the form 

Dt ¼

rffiffiffiffiffiffiffi 1=2 2m en  0 ð1  B Þ e F0

where B is the boundary-layer function

ð8:101Þ

Static and Time-Analyzing Image Tubes With Axial Symmetry

m ðzÞ ffi pffiffiffi ; B ¼ B ðB; zÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B þ L ðzÞ þ B

B¼

F ; en

275

ð8:102Þ

with the functions l ¼ l(z), m ¼ m(z) defined as l ðzÞ ¼

F

0

02

F0



0

F

 1 02  F 0 z 22 ; 2

m ðzÞ ¼

 1 0 F 0 F0

 1 02  F 0 z 22 : 2

ð8:103Þ

The second-order tau-variation z22 is the regular solution of the corresponding differential equation with zero initial condition at z ¼ 0, and zþ 22 is the continuously differentiable continuation of z22 along the principal trajectory through the turning point zP. Therefore, the direct and inverse branches l(z), m(z) are also continuously differentiable on the principal trajectory at the turning point zP. The relation (8.101) represents the asymptotic aberration expansion of the temporal dispersion Dt, being uniformly accurate everywhere on the principal trajectory up to the turning point zP. In the high-potential region B >> 1, the boundary-layer function B is uniformly small, so that expanding B with respect to B 1, we can obtain from Eq. (8.101) the regular temporal aberration expansion mentioned previously. It should be emphasized that the first term of those expansions - both on the direct and inverse branches of the principal trajectory - is given by the ZavoiskyFanchenko formula (8.46). In the low-potential region F  en adjoining the cathode, we have z22 ! 0, l ! 1, m ! 1, B ! 1 at B ! 0, and therefore the temporal dispersion Dt on the direct branch of the principal trajectory vanishes at the starting point z ¼ 0. When the direct branch of the charged particle trajectory enters the low-potential region F  en adjoining the turning point z ¼ zP, the temporal dispersion Dt tends to the limiting value  Dt zp ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 rffiffiffiffiffiffiffi 1=2 0 02 2m en @ 1F0 A; 1þ 1 0 ðz22 ÞP 0 e F0 2F P

ð8:104Þ

whereas on the inverse branch of the principal trajectory, the temporal dispersion Dtþ at the turning point z ¼ zP is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 rffiffiffiffiffiffiffi 1=2 0 02  2m en @ 1F0 ð8:105Þ Þ A: 1 1 þ Dtþ zp ¼ 0 ðz 0 e F0 2 F P 22 P The upper index ‘‘ ‘‘ for ðz22 ÞP in Eqs. (8.104) and (8.105) is unessential because ðz22 ÞP ¼ ðzþ 22 ÞP . We leave it to the reader to show that at the turning point z ¼ zP

276

Static and Time-Analyzing Image Tubes With Axial Symmetry

2 1 ðz22 ÞP ¼ 0 jF P j

02

FP 02

F0

!

ð8:106Þ

;

so that we immediately obtain from Eqs. (8.104) and (8.105) ! rffiffiffiffiffiffiffi 1=2 rffiffiffiffiffiffiffi 1=2 0   2m e F 2m en n 0 þ Dt zP ¼ ; Dt ¼ 1 þ 1 0 0 0 zP e F0 e F0 jF P j

0

F0 0 jF P j

!

ð8:107Þ

We can see that at the turning point on the direct branch of principal trajectory the temporal dispersion Dt jzP exceeds the value given by the Zavoisky-Fanchenko formula (8.46) (exactly by two times if the absolute 0 0 values of the field intensities F 0 and F P at the starting and turning points are equal, which, according to Eq. (8.106) implies (z22)P ¼ 0). On the contrary, the temporal dispersion DtþjzP on the inverse branch at the turning point can be negative, positive, or zero, depending on which of 0 0 0 0 0 0 the correlations jF P j > F 0 , jF P j < F 0 , or jF P j ¼ F 0 holds true. In the 0 0 approximation considered, with F 0 , en fixed and F P varied, the turning points of all charged particle trajectories are located at the z-coordinate 0 ~zP ¼ zP þ en =F P in the same temporal plane t ¼ ~tP (Figure 131) ! rffiffiffiffiffiffiffi 1=2   1 2m en þ  ~tP ¼ tP þ Dt zP þ Dt zP ¼ tP ð8:108Þ 0 2 e F0 z

3 2 4

zP

1

O

∼t P

tP

t

FIGURE 131 Temporal dispersion Dt in the electron mirror (1, principal trajectory; 2, (z22)P ¼ 0; 3, (z22)P > 0; 4, (z22)P < 0). The low-potential regions adjoining the starting and turning points are shaded gray.

Static and Time-Analyzing Image Tubes With Axial Symmetry

277

It follows from Eq. (8.107) that the temporal dispersion at the turning point z ¼ zP undergoes a jump rffiffiffiffiffiffiffi 1=2   þ  ¼ 2 2m en0 ¼ Dt Dt ð8:109Þ ½DtŠ P zP zP e FP equal to the doubled value of the Zavoisky-Fanchenko aberration coefficient in Eq. (8.46). Thus, we have shown that the first-order temporal chromatic aberration can be compensated using a smooth electrostatic mirror but only within the boundary layer located in the turning point vicinity, where 0 F  en. Obviously, the width of this boundary layer is  en =jF P j.

CHAPTER

9 Spatial and Temporal Focusing of Photoelectron Bunches in Time-Dependent Electric Fields

Contents

9.1. Two Different Jobs That Ultrashort Electron Bunches Can Do 9.2. The Master Equation of First-Order Temporal Focusing 9.3. Moving Potential Well as a Simple Example of Temporal Focusing 9.4. Thin Temporal Lens Approximation 9.5. Second-Order Aberrations and Quantum-Mechanical Limitations 9.6. Approximate Estimation of the Space Charge Effects Contribution 9.7. Simulation of a Photoelectron Gun with Time-Dependent Electric Field and Some Experimental Results

280 282 287 288 291 292

294

In this chapter, we consider the peculiarities of spatial and temporal focusing of electron bunches in time-dependent fields as applied to the problem of generating the ultrashort electron probes for time-resolved electron diffraction (TRED) experiments. As Siwick et al. (2004) emphasize, ‘‘. . .those experiments, being in fact alternative to time-resolved experimentation with ultrashort optical pulses, provide much higher spatial resolution while maintaining femtosecond temporal resolution, thus offering the possibility of investigating the most intricate and sophisticated regularities of atomic/molecular dynamics in matter. . .’’ TRED experiments allow the researcher to observe the breaking and making of chemical bonds in chemical reactions on a real time scale, this is why Advances in Imaging and Electron Physics, Volume 155 ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)00809-4

2009 Elsevier Inc. All rights reserved.

#

279

280

Spatial and Temporal Focusing of Photoelectron Bunches

Irvine and Ellezabi (2006) refer to TRED as making the ‘‘molecular movie’’ (see also Lobastov, Srinivasan, and Zewqil, 2005; and Williamson et al.,1992). In Section 9.1, we consider the principal difference in requirements that electron bunch should obey in conventional streak image technique and TRED, and thus reach the necessity of the time-dependent electric fields for temporal focusing. In Section 9.2, we consider some theoretical grounds of first-order temporal focusing on the basis of the aberration approach and the tau-variation technique. A ‘‘master equation’’ derived here shows that the first-order (‘‘ideal’’) temporal focusing is conditioned by joint evolution of the first-order temporal chromatic aberration and the first variation of the full energy of electrons during the bunch’s traveling through the time-dependent field region. Two simple model problems analyzed in Sections 9.3 and 9.4 illustrate the possibility of first-order temporal focusing in time-dependent electric fields. The first of the model problems represents 1D motion of electrons in a parabolic potential well moving along the direction of electron bunch propagation; the second one introduces the thin temporal lens approximation. Contribution of the second-order temporal aberrations and quantum-mechanical limitations on electron bunch compression are the subjects of Section 9.5. Section 9.6 considers a simplified model for evaluation of the space-charge effects in electron bunches being compressed in time-dependent electric fields. Finally, in Section 9.7, we discuss some aspects of numerical simulation of a photoelectron gun with timedependent electric field and present some experimental results on temporal focusing, which have been recently obtained with the use of the photoelectron gun in question.

9.1. TWO DIFFERENT JOBS THAT ULTRASHORT ELECTRON BUNCHES CAN DO As seen in the previous chapter, the main role of the electron bunch in the conventional streak image tube is to preserve and bring to the image receiver the information on the temporal structure of incident optical radiation. This requirement makes only the stationary fields applicable for bunch focusing and acceleration. The main characteristic of an image streak tube is temporal resolution. According to the Zavoisky-Fanchenko formula (8.46), the limiting duration of an electron bunch emitted by a photocathode in any static electromagnetic field is restricted mainly by the first-order temporal chromatic aberration being directly proportional to the spread of photoelectrons in initial axial velocities and inversely proportional to the electric field intensity nearby the photocathode. For almost 60 years, since the Courtney-Pratt (1949) time-analyzing image

Spatial and Temporal Focusing of Photoelectron Bunches

281

converter tube (in modern terminology – the streak image tube) was invented, the field intensity increase near the photocathode has been always used as the main possibility of diminishing the first-order temporal chromatic aberration and thus improving the temporal resolution. Basically, the pressing requirement of high field intensity in the near-cathode region has always served as a stimulating goal for many sophisticated engineering solutions related to the streak image tube design, including, for example, placing a fine-structure grid nearby the photocathode, development of a special, ‘‘grid-oriented’’ technology for photocathode evaporation, or design of high-voltage electronic circuitry to control the streak tube operation modes. As a result, temporal resolution of the most advanced streak cameras available on the market constitutes 200 fs, with the near-cathode field intensity raised, in the pulse mode, as high as 20 kV/mm and even more. The electrical breakdown limitations make a somewhat significant advance in temporal resolution rather problematic. Obviously, the true physical reasons of those difficulties arise from the need to pursue two different purposes simultaneously in conventional streak image tube technique: first, to essentially shorten the electron bunch duration, and second, to maintain in the same bunch the temporal structure of the incident optical radiation. A very different situation presents itself if we consider application of the ultrashort electron bunches to TRED experiments. The principal scheme of TRED experiments is schematically shown in Figure 132. The laser (denoted as 1 in Figure 132) generates the original femtosecond laser pulse, which is divided into two fractions by the beam splitter (2). Having then passed through the optical delay line (3), one of the fractions 1

7

6

5

4 2

3

FIGURE 132 Principal scheme of TRED experiments. 1, femtosecond laser; 2, laser beam splitter; 3, optical delay line; 4, sample under investigation; 5, photocathode; 6, system of electrodes; 7, – image receiver. See text for details.

282

Spatial and Temporal Focusing of Photoelectron Bunches

excites the sample (4) under investigation while the other one illuminates the photocathode (5), thus giving ‘‘birth’’ to the bunch of photoelectrons. The photoelectron bunch is accelerated by the electric field and focused onto the sample. The time interval between the time moments when the laser pulse and the probing electron bunch hit the sample is controlled by the optical delay line (3). Experiencing interaction with atoms and molecules of the sample, the electrons form a diffraction pattern on the image receiver (7). This diffraction pattern, being ‘‘attached’’ to a particular time moment of the laser-initiated transition process in the sample, provides direct information regarding the atomic/molecular dynamics under investigation. It is important that, in contrast to the streak image technique, the electron bunch in TRED experiments is considered not as a carrier of information on the temporal profile of the incident laser pulse but as an experimental tool to probe the matter. Therefore, we should no longer take care in preserving the initial temporal structure within the electron bunch. Our main task in this case is to make the electron bunch on the sample as short as possible. In turn, this allows the use of the timedependent electric fields to focus the electron bunches temporally, in full analogy to spatial focusing of electron beams by means of spatially nonhomogeneous electric fields, as is the case in a variety of static electron-optical devices. It is most remarkable that a properly chosen time-dependent electric field may ensure the first-order (ideal) temporal focusing of the bunch, which, as shown below, implies simultaneous elimination, at some points of the main optical axis of the device, of both the ZavoiskyFanchenko first-order temporal chromatic aberration and the temporal aberration caused by the finiteness of the incident laser pulse duration. At those points, which are further called the points of temporal focusing, the electron bunch duration can be made substantially shorter than the incident laser pulse. Moreover, in contrast to the streak image tubes, temporal focusing does not impose any ultrahigh constraints on the electric field strength in the near-cathode region.

9.2. THE MASTER EQUATION OF FIRST-ORDER TEMPORAL FOCUSING Consider a bunch of emitted photoelectrons traveling toward the sample in an axially symmetric electron-optical system with the time-dependent potential distribution ’(z, r, r*,t). As before, the photocathode center is positioned at z ¼ 0, and the z-axis is considered as the axis of symmetry. Due to axial symmetry, the potential ’ depends on the product rr* ¼ r2, where r* is complex number conjugated to r ¼ x þ iy. Let t0, en, et, and r0 be, respectively, the start time moment, the axial and radial energy components of an individual particle leaving the photocathode, and the particle’s initial distance from the photocathode center.

Spatial and Temporal Focusing of Photoelectron Bunches

283

The motion equations in complex form appear as r¨ ¼

2e @’ðz; r; r∗ ; tÞ m @r∗

z¨ ¼

e @’ðz; r; r ; tÞ : m @z 1=2

ð9:1Þ

1=2

1=2

1=2

The general solution rðt; t0 ; en ; et ; r0 Þ; zðt; t0 ; en ; et ; r0 Þ to Eq (9.1) obeys the initial conditions on the cathode 1=2

1=2

1=2

1=2

sffiffiffiffiffi 2e 1=2 io e e m t sffiffiffiffiffi 2e 1=2 _ 0 ; t0 ; en1=2 ; et1=2 ; r0 Þ ¼ en zðt m

1=2 1=2 r_ ðt0 ; t0 ; en ; et ; r0 Þ ¼

rðt0 ; t0 ; en ; et ; r0 Þ ¼ r0 eib; zðt0 ; t0 ; en ; et ; r0 Þ ¼ z0 ðr0 Þ;

1=2

ð9:2Þ

1=2

valid for any values of the initial parameters t0 ; en ; et ; r0 . Our main concern in this section is the evolution in time of the axial 1=2 1=2 1=2 1=2 coordinate zðt; t0 ; en ; et ; r0 Þ and arrival time Tðt; t0 ; en ; et ; r0 Þ, which can be described in terms of the aberration expansions 1=2

zðt; t0 ; en1=2 ; et ; r0 Þ ¼ z0 ðtÞ þ ðzj t0 Þt0 þ ðz j en1=2 Þen1=2 þ ðz j t20 Þt20 þ ðz j t0 en1=2 Þt0 e1=2 n þ ðz j en Þen þ ðz j et Þet 1=2

1=2

þ ðz j et r0 Þet r0 þ ðz j r20 Þr20 þ . . .

ð9:3Þ

1=2

Tðt; t0 ; en1=2 ; et ; r0 Þ ¼ t þ ðTj t0 Þt0 þ ðT j en1=2 Þen1=2 þ ðT j t20 Þt20

þ ðT j t0 en1=2 Þt0 en1=2 þ ðT j en Þen þ ðT j et Þet 1=2

1=2

þ ðT j et r0 Þet r0 þ ðT j r20 Þr20 þ . . .

ð9:4Þ

constructed in the vicinity of the principal trajectory 1=2

z0 ðtÞ ¼ zðt; t0 ; en1=2 ; et ; r0 Þ jt0 ¼ 0;en ¼ 0;et ¼ 0;r0 ¼ 0 ;

 

r0 ðtÞ ¼ 0

ð9:5Þ

The symbols ( j ) denote the tau-variations introduced in Chapter 5 and considered as functions of time t. 1=2 1=2 It is necessary to clarify that the function Tðt; t0 ; en ; et ; r0 Þ represents 0 the time of particles’ arrival at the plane z ¼ z (t) ¼ z(t, 0, 0) at t fixed, being considered as a function of the initial small parameters 1=2 1=2 t0 ; en ; et ; r0 . In other words, the argument t in Eqs. (9.3) and (9.4) represents the time being counted along the principal trajectory z0(t).

284

Spatial and Temporal Focusing of Photoelectron Bunches





The function T( ) is by definition inverse to the function z( ) at any t fixed, which means that the equality   1=2 1=2 ð9:6Þ z Tðt; t0 ; en1=2 ; et ; r0 Þ; t0 ; en1=2 ; et ; r0 ¼ z0 ðtÞ holds identically true. As soon as, due to axial symmetry, the aberration terms connected 1=2 with the parameters et ; r0 do not enter the linear part of the expansions (9.3) and (9.5), in this and the subsequent three sections we will study the 1=2 dependence of the arrival time T on the parameters t0 ; en . In Section 9.6, we briefly touch on the contribution of the aberration terms connected 1=2 with the parameters et ; r0 and the possibility of its partial elimination. For now, we omit those parameters as the arguments of the corresponding functions. 1=2 The differential equations for the tau-variations (zjt0) and (z j en ) can be easily obtained, along with corresponding initial conditions, by varying Eqs. (9.1) and (9.2) on the principal trajectory (9.5):   dðzjt0 Þ e dðzjt0 Þ e ¼ F00 z0 ðtÞ; t ðzjt0 Þ; ð0Þ ¼ E0 ðzjt0 Þð0Þ ¼ 0; dt m dt m sffiffiffiffiffi 1=2 1=2  dðzjen Þ e 00  0 dðzjen Þ 2e 1=2 1=2 ¼ F z ðtÞ; t ðzjen Þ; ðzjen Þð0Þ ¼ 0; ð0Þ ¼ ; dt m dt m ð9:7Þ where E0 ¼ F0 (0,0) is the on-cathode electric field intensity at the initial 1=2 time moment t ¼ 0. Differentiating Eq. (9.6) with respect to t 0 and en gives the relations ðTjt0 Þ ¼

ðzjt0 Þ ; z_ 0

1=2

ðTjen1=2 Þ ¼

ðzjen Þ ; z_ 0

ð9:8Þ

which are valid at any point on the principal trajectory with nonzero velocity z
0(t) 6¼ 0. The relations for the higher-order aberration coefficients in Eqs. (9.3) and (9.4) can be derived quite similarly. 1=2 It is important to note that the tau-variations (zjt0)(t), ðz j en ÞðtÞ obey identical linear homogeneous differential equations, with the initial conditions being proportional. This immediately implies that the tau1=2 variations (zjt0), (zjen ) remain strictly proportional for all t > 0. According to Eq. (9.8), the same statement holds true for the first-order temporal 1=2 aberration coefficients (Tjt0), (T jen ): rffiffiffiffiffiffiffi 1 2m 1=2 : ð9:9Þ ðTjen ÞðtÞ ¼ lðTjt0 ÞðtÞ; l ¼ E0 e

Spatial and Temporal Focusing of Photoelectron Bunches

285

(We leave it to the reader to show that such proportionality, with the coefficient l inversely proportional to the field intensity E0 at the cathode, takes place regardless the assumption of the axial symmetry.) Denote by 1=2

W ¼ Wðt; t0 ; en Þ ¼

m 2 z_ ðt; t0 ; en1=2 Þ 2e

1=2

1=2

E ¼ Eðt; t0 ; en Þ ¼ Wðt; t0 ; en Þ

1=2

Fðzðt; t0 ; en Þ; tÞ;

ð9:10Þ

respectively, the kinetic and full particle energy (referred to the charge of electron) on the individual trajectory. The motion equation (9.1) implies the full energy evolution equation 1=2

dEðt; t0 ; en Þ ¼ dt

1=2

@Fðzðt; t0 ; en Þ; tÞ @t

ð9:11Þ

valid on any charged particle trajectory. Varying Eq. (9.10) on the principal trajectory by the initial parameter t0 and using the permutability of differentiating by time t and varying by t0 immediately gives rffiffiffiffiffi dðEjt0 Þ @2F @2F 2epffiffiffiffiffi @ 2 F _ ð9:12Þ ¼ ðzjt0 Þ ¼ ðTjt0 Þ; z0 ðTjt0 Þ ¼ W dt @z@t @z@t m @z@t where (Ejt0) ¼ @E/@t0 is the first-order aberration coefficient in the corresponding aberrational expansion for the function E, being similar to the expansions (9.3) and (9.4). Quite similarly, differentiating by t the first of the relations (9.8) gives dðTjt0 Þ ¼ dt

ð_zjt0 Þ ´z 0 þ 2 ðzjt0 Þ ¼ z_ 0 z_ 0

ðWjt0 Þ ðFjt0 Þ ¼ 2W

ðEjt0 Þ : 2W

ð9:13Þ

Combining Eqs. (9.12) and (9.13), we come to the Hamiltonian system of two linear homogeneous differential equations 8 dðTjt0 Þ ðEjt0 Þ > > ¼ > > < dt 2W sffiffiffiffiffi dðEjt0 Þ 2epffiffiffiffiffi @ 2 F > > > ¼ ðTjt0 Þ W ð9:14Þ > : dt m @z@t

with obvious initial conditions (Tjt0)(0) ¼ 1, (Ejt0)(0) ¼ 0. This linear system describes the joint evolution in time of the first-order temporal aberration coefficient (Tjt0) and the first variation (Ejt0) of the particle’s full energy inside the bunch with respect to the start time moment t0.

286

Spatial and Temporal Focusing of Photoelectron Bunches

The coefficients of this system represent some functions of the time t along the principal trajectory z0(t). At times, it is more convenient to analyze or numerically solve the system (9.14) with respect to the spatial change of variables coordinate z ¼ z0(t). Having made the corresponding pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in Eq. (9.14) with regard to the correlation z_ 0 ¼ ð2e=mÞW , we obtain 8 dðTjt0 Þ > > ¼ > > < dz > dðEjt0 Þ > > > : dz

sffiffiffiffi m ðEjt0 Þ e ð2WÞ3=2

@2F ¼ ðTjt0 Þ @z@t

: ð9:15Þ

If the electric field does not depend on time, we obtain the trivial solution (Tjt0) ¼ 1, (Ejt0) ¼ 0, which, according to the proportionality condition (9.9), returns us to the Zavoisky-Fanchenko formula (8.46). Thus, it may be concluded that the system of differential equations (9.14) and (9.15) represents direct generalization of the Zavoisky-Fanchenko formula to the case of time-dependent electric fields. As we can see, the first-order temporal chromatic aberration coefficient 1=2 ðTjen Þ is principally unavoidable in stationary fields. A different situation arises if the electric potential F depends on time. In this case, one might expect (we will show further that such expectation is a reality) that in a properly chosen nonstationary electric field the aberration coefficient 1=2 (Tjt0) would vanish, simultaneously with ðTjen Þ, at a point z ¼ z*, which we call the first-order temporal focus of the bunch. Accordingly, Eq. (9.14) may be called the master equation of the first-order temporal focusing. The bunch duration at the point of temporal focusing is determined by the second- and higher-order terms of the aberration expansion (9.4). A historical analogy is appropriate here. In the very first electronoptical image intensifier developed in the mid-1930s, the famous ‘‘Holst glass’’ tube (Holst et al., 1934), the electron image was created by the electrons moving along the parabolic trajectories in a static homogeneous electric field generated by two parallel plates, one of them representing the cathode, and the other one the phosphor screen. Very low spatial resolution of this tube (a few lines per millimeter) was conditioned by the transversal spread of photoelectrons in initial energies, which could not be compensated due to lack of spatial focusing. Only the discovery and then practical implementation of the remarkable fact that a spatially inhomogeneous electrostatic field is capable of focusing the electrons emitted from the same point of the cathode in different directions, has allowed essential advance of spatial resolution in imaging electron optics. Returning to our consideration, we may thus conclude that the principal capability of temporally inhomogeneous (e.g., time-dependent) electric

Spatial and Temporal Focusing of Photoelectron Bunches

287

fields to focus electrons in time is very similar to that of spatially inhomogeneous stationary electric fields to focus electrons in space. It is important to emphasize that the temporal focusing in question may essentially improve the temporal resolution of TRED experiments, in which the electron probe should be made as short as possible. Indeed, the ‘‘temporal magnification’’ (Tjt0) vanishes at the point of temporal focusing together with the first-order temporal chromatic aberration coefficient 1=2 ðTjen Þ, thus making the bunch duration minimal. As to the information on initial temporal structure of the bunch, it is completely lost at that point. This is a direct manifestation of the principal difference in the application of ultrashort electron bunches to streak image tube technique and TRED experiments, mentioned above.

9.3. MOVING POTENTIAL WELL AS A SIMPLE EXAMPLE OF TEMPORAL FOCUSING One may ask whether any time-dependent electric fields exist, in which the condition of the first-order temporal focusing (Tjt0) ¼ 0 is met. The model of a 1D parabolic potential well moving with the velocity V0 along the direction of the bunch propagation gives the positive answer to this question. The corresponding time-dependent potential distribution appears as i2 mo2 h e z ð9:16Þ E0 V 0 t : Fðz; tÞ ¼ 2 2e mo Here, as above, E0 ¼ F0 (0,0) is field intensity at the cathode z ¼ 0 at the time moment t ¼ 0. We also suppose that the electrons are emitted with rather small energies een  mV02 =2 in the positive direction of the z-axis within the initial time interval [-dt0/2, dt0/2], where dt0  2p/o. In the coordinate system moving together with the potential well, the motion of electrons represents isochronous harmonic oscillation with the frequency o. In the basic (stationary) coordinate system, such motion is described as   eE0 eE0 1=2 cos½oðt t0 ފ V 0 t0 þ zðt; t0 ; en Þ ¼ V0 t þ mo2 mo2 ! rffiffiffiffiffi 2e 1=2 sin½oðt t0 ފ þ e : V0 ð9:17Þ m n o At the time moment equal to the half-period of oscillation t* ¼ p/o, all the particles emitted at the same time moment t0 ¼ 0 are focused at the point z* ¼ pV0/o þ 2eE0/m o2 independent of their initial energy en.

288

Spatial and Temporal Focusing of Photoelectron Bunches

1=2

Accordingly, @ k z=@ðen Þkjt0 ¼ 0;en ¼ 0 ¼ 0 at t ¼ t* for any natural number k, which means that the temporal focusing at the point z* in the case under consideration has not the first but infinite order with respect to the particles’ velocity. According to the proportionality condition (9.9), the particles emitted at various time moments t0 are also focused at the point z* at t ¼ t*, but only within the first-order approximation with respect to t0. This fact can be easily verified by means of direct differentiating of Eq. (9.17) and is shown in Figure 133. Thus, the time moment t* ¼ p/o and, correspondingly, the point z* ¼ pV0/o þ 2eE0/m o2 represent the point of first-order temporal 1=2 focusing with respect to the parameters ft0 ; en g. It is noteworthy that the bunch velocity at the point of temporal focusing, with an accuracy to the initial energy spread, is the velocity on the principal trajectory z˙ 0(t*) ¼ 2V0 6¼ 0. In other words, the temporal focusing is achieved at the nonzero velocity of the particles comprising the bunch, which, as we have seen in Section 8.5, is impossible in the stationary limit V0 ¼ 0. This example confirms the principal possibility of the first-order focusing in time-dependent electric fields but is yet too removed from practice. The next section considers the so-called thin temporal lens model which is more close to practical implementation.

9.4. THIN TEMPORAL LENS APPROXIMATION Consider a lens comprising three electrodes positioned in a manner shown in Figure 134. The voltages of the left and right diaphragms are the constant values U1 and U2, accordingly. The middle diaphragm (a)

(b) Z*

Coordinate

Coordinate

Z*

t*

Time

t*

Time

FIGURE 133 Motion of electrons in the moving potential well. (a) trajectories of the particles emitted with different initial energies; (b) trajectories of the particles emitted at different starting time moments. The principal trajectories are shown with the bold lines.

Spatial and Temporal Focusing of Photoelectron Bunches

U1

289

U2

Ud(t)

Φ(z,t f) Φ(z,t i) 1 Φd(z)

Δz1 z1

Δz2 zd

z2

FIGURE 134 Thin temporal lens (qualitative view of the unit potential function Fd (z) of the middle diaphragm along with the time-dependent axial potential distribution F(z, t) at the time moments ti and tf is shown).

voltage Ud (t) is changing in time while the bunch is moving along the main optical axis from the left diaphragm toward the right one. According to the superposition principle, the axial potential distribution in the area of interest appears in the form Fðz; tÞ ¼ F1 ðzÞU1 þ Fd ðzÞUd ðtÞ þ F2 ðzÞU2 ;

ð9:18Þ

with F1(z), Fd(z), F2(z) representing the unit potential function of corresponding electrodes. If apertures of the diaphragms are small enough, the electric field between the electrodes may be regarded as homogeneous, and the unit function Fd (z) may be approximately considered as piecewise linear. Within the thin temporal lens approximation, which is quite analogous to the thin lens approximation in static charged particle optics, we may neglect the variation of the first-order aberration coefficient (Tjt0) inside the lens under consideration compared with its initial value, so we can put (Tjt0) ¼ 1 in the right part of the master equation (9.15). As soon as       0 2 0 2 0 @F z ðtÞ; t @ F z ðtÞ; t @ F z ðtÞ; t d z_ 0 ðtÞ þ ¼ ð9:19Þ @t @z@t @t2 dt on the principal trajectory z ¼ z0(t), we can easily calculate from Eq. (9.12) the full energy variation

290

Spatial and Temporal Focusing of Photoelectron Bunches

t

ðEjt0 Þ jtfi ¼ ðEjt0 Þðtf Þ ðEjt0 Þðti Þ ¼ ðEjt0 Þðtf Þ  tf rffiffiffiffiffi ðtf ðtf 2 2e @ 2 F pffiffiffiffiffi @ F 0 @F  z_ dt ¼ W dt ¼ ¼  m @z@t @z@t @t  ti

ti

ti

ðtf

ti

@2F dt @2t

ð9:20Þ

between the endpoints ti, tf of the time interval during which the middle diaphragm voltage Ud(t) is changing in time. If Ud(t) depends on time _ d(t) ¼ const), the second integral in Eq. (9.20) vanishes, and, linearly (U according to Eq. (9.18), the full energy variation within the temporal lens takes the very simple form     _ d ½Fd z0 ðtf Þ; tf Fd z0 ðti Þ; ti Š: ð9:21Þ ðEjt0 Þðtf Þ ¼ U _ d > 0). In Let us assume for distinctness that Ud is increasing in time (U this case, as can be easily seen from Eq. (9.21), the full energy variation _ d > 0 if the electric field starts to change attains its maximum (E jt0)(tf) ¼ U in time just at the time moment when the electron bunch is entering the lens area through the left diaphragm (Fd ¼ 0), and stops changing at the time moment when the bunch is passing through the middle diaphragm (Fd ¼ 1). This means that the full energy of the particles traveling at the bunch tail exceeds that of the particles traveling at the leading edge of the bunch at the time moment t ¼ tf. After a definite time interval, the rear particles overtake the leading ones, and the bunch duration becomes minimal as shown in Figure 135.

U2

U1 T t0

Ud(t )

FIGURE 135 Temporal focusing of electron bunch in thin temporal lens (the arrows indicate the correlation between the full energies of electrons at the tail and leading edge of the bunch).

Spatial and Temporal Focusing of Photoelectron Bunches

Introducing the full energy increment   ðtf @F z0 ðtÞ; t dt DE ¼ @t

291

ð9:22Þ

ti

the bunch acquires within the temporal lens, and assuming that, having left the lens, the bunch enters the free-of-field region, we can easily calculate from Eq. (9.15) the distance f between the thin temporal lens and the temporal focus z* at which the temporal aberration coefficient (Tjt0) vanishes rffiffiffiffiffi 8e ðU2 þ DEÞ3=2 : ð9:23Þ f ¼ _d m U By analogy with the case of spatial focusing, this value may be regarded as the focus length of the thin temporal lens under consideration. If Ud decreases in time, a similar effect of temporal focusing takes place in the field area located between the middle and right diaphragms. Thus, we have shown that the thin temporal lens allows making zero the firstorder terms of the expansion Eqs. (9.3) and (9.4) in a certain plane of the main optical axis, where a sample for TRED should be placed. The electron bunch duration is mainly determined in this case by the secondorder terms (which are briefly analyzed in the following section).

9.5. SECOND-ORDER ABERRATIONS AND QUANTUM-MECHANICAL LIMITATIONS The equations of charged particle motion in time-dependent field remain Hamiltonian. According to the Liouville theorem, the bunch volume in the phase space (E,T), where E,T are, correspondingly, full energy of the individual particle in any plane perpendicular to the z-axis and the time of arrival of the particle at that plane, remains invariable. This immediately implies that an additional spread in energies should be attached to the particles comprising the bunch to compress the bunch temporally. This fact becomes most important if we suppose that not only the first1=2 order coefficients (Tjt0) and ðTjen Þ are eliminated at the temporal focus, but the second-order temporal aberration coefficients in Eq. (9.4) also should be diminished in order to achieve better temporal compression. Let us analyze the contribution of the second-order terms of the expansions (9.3), (9.4) at the point of temporal focusing. The phase volume conservation theorem implies that the functional determinant @(T, E)/@(t0, en) is identically equal to unit for any values of the initial

292

Spatial and Temporal Focusing of Photoelectron Bunches

1=2

1=2

parameters t0, en . By differentiating this equation with respect to en we obtain ðTjt0 Þ½2ðEjen Þ þ lðEjt0 en1=2 ފ þ ðEjt0 Þ½2ðTjen Þ þ lðTjt0 en1=2 ފ ¼

2:

ð9:24Þ

We here used the proportionality condition (9.9), as well as the similar 1=2 proportionality condition ðEjen Þ ¼ lðEjt0 Þ for the first variation of the full energy E, which easily follows from Eq. (9.10). The first-order aberration coefficient (Tjt0) vanishes at the point of temporal focusing, and Eq. (9.24) takes the simpler form ðEjt0 Þ½2ðTjen Þ þ lðTjt0 en1=2 ފ ¼

2;

ð9:25Þ

which shows, first, that the full energy variation (Ejt0) cannot be made zero, and, second, that the second-order temporal aberration coefficients 1=2 (Tjen), ðTjt0 en Þ cannot be made zero simultaneously at the point of temporal focusing. These two conclusions are closely connected with quantum limitation on temporal compression of the bunch. Indeed, suppose we have designed a photoelectron device with the second-order temporal chromatic aberration also eliminated: (Tjen) ¼ 0. In this case 1=2 1=2 Eq. (9.24) and (9.25) give ðEjen ÞðTjt0 en Þ ¼ 2, and, using the Heisenberg uncertainty relation dt0de  2ph  for the initial spread, we obtain approximate estimation for the product of energy and temporal spreads of electron bunch at the point of temporal focusing  dEdT  ½ðEjen1=2 Þden1=2 Š½ðTjt0 en1=2 Þdt0 den1=2 Š ¼ 2dt0 den  4ph

ð9:26Þ

(the additional coefficient 2 appears because the temporal and energy distributions at the temporal focus are not independent). The relation (9.26) clearly shows that temporal compression can be achieved only at the expense of increasing the energy spread in the electron bunch, in full accordance with the principles of quantum mechanics.

9.6. APPROXIMATE ESTIMATION OF THE SPACE CHARGE EFFECTS CONTRIBUTION Let us now turn to the approximate estimation of the contribution of space charge effects to the bunch duration. Accurate numerical evaluation of this effect is given in the following section on the basis of the tree-type algorithm described in Chapter 6. Our main goal here is to outline the principal qualitative difference in Coulomb dynamics of electron bunches in stationary and time-dependent fields. Let us assume that the bunch length in the axial direction is much less than its radial dimensions, and the space charge is uniformly distributed within the bunch volume.

Spatial and Temporal Focusing of Photoelectron Bunches

2R

FC

293

FC z

d

FIGURE 136 The estimation of space charge effect contribution: the model shape of an electron bunch.

For instance, we may conceive of the bunch as a flat round ‘‘pancake’’ with d  2R (Figure 136). Using these assumptions and denoting J0 the photoemission current and S(t) the area of the local radial cross-section of the bunch, we can introduce a term responsible for space charge repulsion into the second of Eqs. (9.15) 8 sffiffiffiffi dðTjt0 Þ m ðEjt0 Þ > > ¼ > > dz < e ð2WÞ3=2 > dðEjt0 Þ > > > : dz

¼

@2F ðTjt0 Þ @z@t

4pJ0 : SðtÞ

ð9:27Þ

In the stationary field, the mixed derivative @ 2F/@z@t is zero and space charge repulsion makes the value (Ejt0) negative. In turn, the first of Eqs. (9.27) says that (Tjt0) increases compared with its initial value, which is a direct manifestation of the irreversible Coulomb broadening of the bunch in this case. On the contrary, in the case of time-dependent field, the mixed derivative @ 2F/@z@t may be intentionally chosen to compensate for the energy spread induced by the space charge interaction. This fact points to the principal possibility of first–order temporal focusing even in the presence of space charge effects. This situation is similar to the contribution of space charge effects to spatial focusing in the stationary case, when adjustment of the spatially focusing electric field may help to significantly compensate for Coulomb broadening of the bunch in the radial direction. The counteraction of these two factors in the stationary case mainly results in a shift of the image plane position compared with that of the bunch with a space charge that can be assumed negligibly small. Coincidentally, as seen above, there are no forces compressing the bunch in the axial direction and preventing from the space charge-induced axial broadening of the bunch in stationary fields. In the time-dependent field, inversely, the extra energy spread the

294

Spatial and Temporal Focusing of Photoelectron Bunches

time-dependent field introduces into the bunch is able to compensate, at least partially, for the space charge effects. By analogy with stationary charged particle optics, in which the second spatial derivative d2F/dz2 determines the optical force and, correspondingly, the focal length of static lenses, in charged particle optics of time-dependent fields the mixed derivative @ 2F/@z@t determines the optical force and, correspondingly, the focal length of temporal lenses. With regard to the said above, it can be expected that the estimation for dynamic range of the bunch duration in time-dependent electric fields would appear more optimistic than in stationary ones. The results of numerical modeling presented in the next paragraph confirm this expectation.

9.7. SIMULATION OF A PHOTOELECTRON GUN WITH TIME-DEPENDENT ELECTRIC FIELD AND SOME EXPERIMENTAL RESULTS To confirm the principal possibility of temporal focusing of photoelectron bunch in time-dependent electric fields, an experimental prototype of a photoelectron gun was simulated, designed, manufactured, and tested at the Photoelectronics Department of A.M. Prokhorov General Physics Institute, RAS (Monastyrskiy et al., 2003, 2007). The authors of this monograph are grateful to their colleagues for the opportunity to present here the experimental results of this long-term work. All calculations provided below were made with ELIM/DYNAMICS (Monastyrskiy et al., 1999) and MASIM 3D (Monastyrskiy, Greenfield, and Tarasov, 2006) software packages. The external view of the photoelectron gun in question and its structural scheme are shown in Figures 137 and 138, respectively.

FIGURE 137 External view of the photoelectron gun for temporal focusing of electron bunches.

Spatial and Temporal Focusing of Photoelectron Bunches

295

(a)

0.5 mm

n

T t0 w

2

1

3

4

5

6

7

8

(b) F F+(z)

F-(z) zg

z

FIGURE 138 (a) Structural scheme of the photoelectron gun. 1, photocathode; 2, finestructure grid; 3, temporally-focusing electrode; 4, spatially-focusing electrode; 5, anode; 6, crossover point; 7, temporal focus point; 8, image receiver. The bunch shape at different time moments is essentially scaled with respect to the electrode dimensions. (b) The axial potential distributions before triggering the high-voltage electric ramp (F (z)) and after the electric ramp has ceased (Fþ(z)) are displayed (zg is the grid position). See the text for details.

The photoelectron gun operates as follows. A picosecond laser illuminates a 1mm0.05 mm slit-shaped area of the photocathode 1 (see Figure 138) and produces a photoelectron bunch that is then accelerated by the 3-kV/mm electric field in the gap between the photocathode and the fine-structure grid 2. Having passed through the grid, the electrons enter the time-dependent field area governed by the electrode 3, which is supplied with electric voltage ramping by 1-2 kV during 250-400 ps (Figure 139). The ramp generator is synchronized with the laser pulse to ensure a controllable delay between the time moments of emitting the bunch from the photocathode and triggering the high-voltage electric ramp. While the bunch is traveling in the time-dependent field, the energies of the front and rear electrons become different, and, as shown in Figure 138, the first-order aberration coefficient (Tjt0) starts to decrease.

Spatial and Temporal Focusing of Photoelectron Bunches

Voltage

296

Amplitude ∼1 kV

Rise time ∼400 ps

Time

FIGURE 139

High-voltage electric ramp on the temporally-focusing electrode.

Eventually, this aberration coefficient vanishes simultaneously with the 1=2 coefficient ðTjen Þ at the point of temporal focusing 7 located in the fieldfree region downstream from the anode 5. The electrode 4 adjusts spatial focusing of the electron bunch in such a way that the limiting image plane coincides with the position of the image receiver 8, whereas the crossover 6 turns to be located rather close to the point of temporal focusing. In the diffraction mode, the bunch hits a sample located at the point of temporal focusing (not shown in Figure 138), and a diffraction pattern is formed on the image receiver 8. In the testing mode intended for measuring the bunch duration, the sample is lifted, and a dynamic deflector located close to the sample (also not shown in Figure 138) sweeps the bunch along the image receiver. The time profile of the swept image of the bunch provides direct information as to the bunch duration on the sample. From the viewpoint of temporal focusing, it is critical to strictly obey certain matching conditions between the time moment of electron bunch emission and the time moment of high-voltage electric ramp triggering. In the stationary case, the time T(t0) of the particle arrival at the sample, being considered versus the initial time moment t0 at zero initial energy of photoelectrons (e ¼ 0) and the sample plane fixed, is the linear function T(t0) ¼ t0 þ DT, with the constant DT being the traveling time between the photocathode and the sample. The situation differs if the electric field

Spatial and Temporal Focusing of Photoelectron Bunches

297

8 12 ps

6

4

T, ns

2 1 4

2 1,7 ps

3 0 −6

−4

−2

0 t0, ns

2

4

FIGURE 140 The arrival time T(t0) versus the start time t0 in the photoelectron gun designed. The regions to the left from point 3 and to the right from point 4 correspond to the bunch motion in different stationary fields. The histograms show the arrival time distributions at each of the two temporal focuses (points 1 and 2) for the initial laser pulse duration dt0 ¼ 100 ps.

depends on time. In this case, the traveling time DT depends itself upon t0. Figure 140 displays the arrival time T(t0) as a function of t0, calculated for the photoelectron gun. If the bunch is emitted ‘‘too early’’ with respect to the time moment of the high-voltage electric pulse triggering (point 3 in Figure 140), it travels in the stationary field with axial potential distribution F (z) shown in Figure 138. In this limiting case, the traveling time DT1 is unambiguously determined by the potential distribution F (z). Quite similarly, if the bunch is emitted ‘‘too late’’ (point 4 in Figure 140), it travels in the final, also stationary, electric field with axial potential distribution Fþ(z), with another traveling time constant DT2 < DT1 determined by Fþ(z). In both cases, the electron bunch does not ‘‘feel’’ the time-dependent voltage applied. The entire information on temporal compressing of the bunch is concentrated in the nonlinear part of the curve T(t0) located between the points 3 and 4. This part contains two temporal focuses (points 1 and 2), at which the first-order terms of the expansions (9.3), (9.4) vanish. The shape of the time dispersion histograms calculated at the temporal focuses are in full agreement with the local parabolic behavior of the

298

Spatial and Temporal Focusing of Photoelectron Bunches

1.2

(E|t 0), V/ps

1.0

3

2

0.8 0.6

4

5

0.4 0.2 1

0 −0.2

0

0.2

0.4 0.6 (T|t 0)

0.8

1.0

1.2

FIGURE 141 The transition of the electron bunch from the photocathode to the point of temporal focusing in the phase plane {(Tjt0), (Ejt0)}. 1, initial state of the bunch on the photocathode; 2, the bunch as it enters the temporally focusing electrode; 3, the bunch as it leaves the temporally focusing electrode; 4, the point of temporal focusing.

arrival time in the vicinity of those points. The existence of these temporal focuses has been experimentally observed in the course of the photoelectron gun testing. It can be clearly seen that the first temporal focus (point 1) provides better temporal compressing of the bunch than the second one. Figure 141 displays the transition of the electron bunch from the photocathode to the temporal focus 1 in the phase plane {(Tjt0), (Ejt0)}. Originally, the aberration coefficient (Tjt0) is unit and the full energy variation (Ejt0) is zero (point 1). Then, due to the extra energy spread acquired by the particles in the time-dependent field located in the left vicinity of the temporally focusing electrode 3 (see Figure 138), the aberration coefficient (Tjt0) starts to decrease (point 2). The mixed derivative @ 2F/@z@t is close to zero while the bunch is traveling inside the temporally focusing electrode, and the extra energy spread remains practically constant up to the point 3. The time-dependent field located in the right vicinity of the temporally focusing electrode results in a slight decrease of the extra energy spread, and, correspondingly, the (Tjt0) decrease becomes slower (point 4). Then the bunch enters the static field region, the extra energy spread becomes constant again, the aberration coefficient (Tjt0) continues to decrease, and, finally, the bunch reaches the point of temporal focusing at which (Tjt0) vanishes (point 5). Four series of experiments on temporal focusing of electron bunches with time-dependent electric fields were conducted with the photoelectron gun designed. We started from the comparatively long electron bunches to make the time-compressing effect more definitive. In the first series, an electron bunch of 150-ps initial duration was compressed

Spatial and Temporal Focusing of Photoelectron Bunches

299

Compressed bunch duration, ps

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0

50 100 150 Initial bunch duration, ps

200

FIGURE 142 The results of temporal compressing the photoelectrons bunches obtained in the first three series of experiments (circles denote experimental results, the solid line shows numerical results).

to 3 ps using the 1.5-kV/ns high-voltage electric ramp. In the second and third series of experiments, the electron pulses of 70-ps and 10-ps duration were compressed to 2 ps and 0.4 ps, respectively. The results of those experiments are shown in Figure 142 with the results of numerical calculation. Two extreme experimental points (initial bunch duration 150 ps and 10 ps) are in good agreement with the theoretically predicted points, while the middle experimental point (initial bunch duration 70 ps) is somewhat apart. With regard to the possible shape of such dependence, this discrepancy may be seemingly attributed to some experimental inaccuracy. The limiting capabilities of temporal compressing of electron bunches are essentially restricted by the rise time of the high-voltage electric ramp applied to the temporally focusing electrode. In the fourth series of experiments, the original photoelectron bunch of 7-ps duration was compressed to 280 fs using the rise time approximately twofold increased (Figure 143). The calculations have shown (Figure 144) that the result obtained in the fourth series of experiments is rather close to the limiting one, being determined by the electrical and electrodynamic parameters of the tube. Let us now discuss the contribution of the space charge effect to the temporal broadening of the bunch in these experiments. We have seen that the space charge term in Eq. (9.27) describes this effect only qualitatively.

300

Spatial and Temporal Focusing of Photoelectron Bunches

(a)

7 ps (b) 280 fs

FIGURE 143 (a) Swept image of the original 7-ps electron bunch (no temporal focusing); (b) Swept image of the same electron bunch compressed to 280 fs with the use of the high-voltage electric ramp. The streak speed is 2.8c (c is the speed of light).

(a)

(b) 100 80 200 fs

60

200 fs

40 20 0 4.8875

4.8880

4.8885 Time, ns

4.8890

FIGURE 144 Limiting spatiotemporal figure of diffusion (a) and temporal spread function (b) resulting from a point source on the photocathode.

However, more accurate computational analysis is needed in practice because of two factors not described by this approximate estimation: the non-homogeneity of the space charge distribution over the bunch crosssection and the stochastic Bo¨rsch effect responsible for particle-to-particle interaction. To take those effects into account, we used the tree-type structure algorithm described in Chapter 6. The calculation conducted for different numbers of electrons constituting the bunch have shown that it is possible to partially compensate for the contribution of the space-charge repulsion by adjusting the high-voltage ramp amplitude. For instance, as it follows from Figure 145, the bunch of 1,000 electrons is broadened up to 1 ps unless the ramp amplitude adjustment is applied.

Spatial and Temporal Focusing of Photoelectron Bunches

301

1600 1400 3

4

Bunch duration, fs

1200 1000 2 800 600 1 400 200 0 800

1000

1200

1400

1600

1800

Electric ramp amplitude, V

FIGURE 145 Temporal spread of the bunch versus the high-voltage ramp amplitude for different numbers N of electrons in the bunch. 1, space charge is not taken into account; 2, N ¼ 1,000; 3, N ¼2,000; 4, N ¼ 3,000.

With the ramp amplitude increased by 300 V, the bunch duration rises only by 150 fs. Now let us discuss some perspectives of the temporal focusing of electron bunches with the use of time-dependent electric fields. First, it should be noticed that modern pulse lasers are capable of generating exceedingly short pulses of a few femtoseconds duration, which makes the contribution of the terms connected with the parameter t0 in the aberration expansion (9.4) rather small at the point of temporal focusing. To essentially shorten the electron bunch duration on the sample, it is desirable to make zero at the point of temporal focusing the most influential aberration coefficients (Tjen), (Tjet) of the expansion (9.4). We re1=2 iterate that, according to Eq. (9.25), the term ðTjt0 en Þ cannot be eliminated simultaneously with the term (Tjen). The example in Figure 146 illustrates the principal possibility of simultaneous elimination of the two second-order temporal aberration coefficients (Tjen), (Tjet) at the point of first-order temporal focusing. The system in question comprises not one (as before) but two electrodes responsible for temporal focusing of the bunch, with one voltage linearly increasing and the other linearly decreasing in time. Spatial focusing of the bunch in the radial direction is ensured by a magnetic lens positioned downstream from the diaphragm that separates the timedependent electric and static magnetic field areas. Our calculations have shown that, with the voltage change rate not higher than 10 kV/ns (which

302

Spatial and Temporal Focusing of Photoelectron Bunches

2

3

4

5 6

1 7

170 mm

FIGURE 146 Principal scheme of the electron-optical system with two ‘‘time-dependent’’ diaphragm and static magnetic focusing. 1, photocathode; 2, 3, ‘‘time-dependent’’ diaphragms; 4, diaphragm; 5, 6, magnetic focusing system; 7, sample. The arrows indicate the direction of the voltage change in time (down arrow indicates a decrease, and up arrow indicates an increase).

3 1 2 2 3

1

0 0

50

100 z, mm

150

FIGURE 147 The first-order aberration coefficients. 1, 2, solutions v(z), w(z) of the limiting paraxial equation; 3, aberration coefficient (Tjt0).

is quite attainable for modern high-voltage electric pulse generators) and maximum magnetic field  250 Gauss, it is possible to place the first-order temporal focus at the crossover point of the tube, with the

Spatial and Temporal Focusing of Photoelectron Bunches

303

fs/V

150 100 2 1

50

0 0

FIGURE 148 2, (Tjet).

50

100 z, mm

150

Second-order temporal chromatic aberration coefficients: 1, (Tjen) ;

aberration coefficients (Tjen), (Tjet) eliminated at the same point as shown in Figures 147 and 148. With the laser pulse duration not exceeding 50 fs, it is principally possible to compress the photoelectron bunch down to some tenths of a femtosecond.

APPENDICES APPENDIX 1: SOME GAUSS QUADRATURE FORMULAS The general idea of the Gauss quadrature method is as follows. Suppose we need to integrate the function FðxÞ ¼ Fðx1 . . . xM Þ with a weight function GðxÞ over the domain O  RM. In the particular case, when the integrand is represented by a K-order polynomial 
ðKÞ ð2Þ ð1Þ ðA1:1Þ F ¼ Fð0Þ þ Fi xi þ Fij xi xj þ . . . þ Fijk... xi xj xk . . . ;

we get

ð ð2Þ ð1Þ I ¼ FðxÞGðxÞdx ¼ SFð0Þ þ Si Fi þ Sij Fij þ . . . ;

ðA1:2Þ

O

where ð S ¼ GðxÞdx;

dx ¼ dx1 . . . dxM

ðA1:3Þ

O

and ð Sijk... ¼ ðxi xj xk . . .ÞGðxÞdx:

ðA1:4Þ

O

The total number of coefficients in Eqs. (A1.3) and (A1.4) is given by the binomial expression N ¼ (M þ K)!/(M!K!). Let us represent the integral (A1.2) by means of the finite sum I0 ¼

P X p¼1


 op F x ðpÞ ;

ðA1:5Þ

with P ¼ N/(M þ 1) nodal points  xðpÞ 2 O and the corresponding weights op. Here we assume that P is an integer, which means that N is a multiple of M þ 1. Obviously, this holds true for 1D integrals (M ¼ 1) if K is any odd number. In the 2D case (M ¼ 2), the Gauss quadrature formulas may be constructed for K ¼ 1, 2, 4, 5, 7, 8,. . ., or, in other words, for all positive integers except for the multiples of three. The number of the unknown values to be determined in Eq. (A1.5), namely, the number of components x ðpÞ and weights op, is equal to the number of coefficients in Eq. (A1.2), which allows, in principle, making

305

306

Appendices

the sums (A1.2) and (A1.5) strictly coincide for any polynomial (A1.1). Indeed, by equating the similar terms in (A1.2) and (A1.5), we arrive at the system of N nonlinear equations P X p¼1

ðpÞ ðpÞ

op xi xj . . . ¼ Sij... ;

ðA1:6Þ

which can be solved, either analytically or numerically, with respect to the components of x ðpÞ and the weights op. In that way we can obtain the Gauss quadrature formulas approximating the integrals (A1.2) with the accuracy up to the K-th order with respect to the size DO ¼ max j x y j of the integration domain O for any function F being x;y2O

smooth enough. We consider below two types of the integration domains: 1D segment (bearing in mind that the integration over the M-dimensional parallelepiped may be reduced to the iterated 1D integral) and 2D domain being triangle-shaped.

A1.1 One-Dimensional Integrals Consider the case O ¼ ½ 1=2; þ1=2Š;

GðxÞ ¼ ð0:5 þ xÞ

g

ðA1:7Þ

and the corresponding Gauss quadrature formula þ1=2 ð

FðxÞð0:5 þ xÞ

1=2

g

dx 

X p

op Fðxp Þ

ðA1:8Þ

If g ¼ 0, the quadrature formulas are referred to as regular ones, whereas for 0 < g < 1 we have to deal with the singular quadrature formulas. The coordinates of the third- and fifth-order Gauss quadrature nodes, along with the corresponding weights, are listed up to the six-digit accuracy in Tables A1.1 and A1.2.

A1.2 Standard Triangle Integrals Let us define two-dimensional integration domain as the triangle O ¼ fu  0; v  0; u þ v  1g and consider the integrals

ðA1:9Þ

307

Appendices

TABLE A1.1

Third-Order Gauss Quadrature Coefficients Gauss nodes

Weights

g

t1

t2

o1

o2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-0.28868 -0.30633 -0.32471 -0.34383 -0.36372 -0.38441 -0.40592 -0.42825 -0.45140 -0.47533

0.28868 0.28070 0.27208 0.26275 0.25261 0.24156 0.22945 0.21613 0.20140 0.18501

0.50000 0.58111 0.68624 0.82661 1.02112 1.30429 1.74598 2.51074 4.09571 8.99685

0.50000 0.53000 0.56376 0.60196 0.64554 0.69571 0.75402 0.82259 0.90429 1.00315

TABLE A1.2

Fifth-Order Gauss Quadrature Coefficients Gauss nodes

Weights

g

t1

t2

t3

o1

o2

o3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-0.38730 -0.39816 -0.40915 -0.42040 -0.43169 -0.44306 -0.45450 -0.46597 -0.47741 -0.48878

0.00000 -0.01132 -0.02317 -0.03580 -0.04896 -0.0628 -0.07740 -0.09283 -0.10917 -0.12650

0.38730 0.38415 0.3808 0.37726 0.37350 0.36950 0.36524 0.36068 0.35581 0.35058

0.27778 0.33824 0.41945 0.53140 0.69255 0.93583 1.32926 2.03486 3.54616 8.35390

0.44444 0.48415 0.52999 0.58372 0.64671 0.72152 0.81141 0.92080 1.05592 1.22570

0.27778 0.28872 0.30056 0.31345 0.32741 0.34265 0.35933 0.37767 0.39792 0.42040

ð1 0

du

1ð u

Fðu; vÞGðu; vÞdv ¼

0

7 X p¼1

op Fðup ; vp Þ

ðA1:10Þ

with the weight functions G ¼ G0  1 (the regular quadrature formulas), G ¼ G1 ¼ (1 u v) g (the edge singularity), and G ¼ G2 ¼ (u þ v) g (the corner singularity). We presume that 0 < g < 1 for both of the singular integrals in question. The fifth-order Gauss quadrature scheme operates with the nodes located symmetrically with respect to the line u ¼ v, as shown in Figure A1.1. The u and v coordinates of the nodal points, along with the corresponding weights o, are given in Tables A1.3 – A.1.5.

308

Appendices

v=1

6

2

4 1

7

5 3

u=v=0

FIGURE A1.1 triangle.

u=1

Location of the fifth-order Gauss quadrature nodal points in the standard

TABLE A1.3 The Regular Gauss Quadrature Coefficients for Integrating Over the Standard Triangle Nodal index 1

u v o

2

3

4

5

6

7

0.333333 0.059716 0.470142 0.470142 0.797427 0.101287 0.101287 0.333333 0.470142 0.059716 0.470142 0.101287 0.797427 0.101287 0.112500 0.066197 0.066197 0.066197 0.062970 0.062970 0.062970

Appendices

TABLE A1.4

309

The Edge Singularity Gauss Quadrature Coefficients Nodal index

g

0.1 u v o 0.2 u v o 0.3 u v o 0.4 u v o 0.5 u v o 0.6 u v o 0.7 u v o 0.8 u v o 0.9 u v o

1

2

3

4

5

6

7

0.327869 0.327869 0.099254 0.322581 0.322581 0.088072 0.317461 0.317461 0.078563 0.312500 0.312500 0.070421 0.307692 0.307692 0.063406 0.303031 0.303031 0.057326 0.298508 0.298508 0.052028 0.294119 0.294119 0.047388 0.289855 0.289855 0.043306

0.058542 0.460897 0.059602 0.057416 0.452031 0.053911 0.056334 0.443519 0.048970 0.055295 0.435339 0.044654 0.054296 0.427471 0.040865 0.053334 0.419895 0.037521 0.052407 0.412596 0.034558 0.051513 0.405556 0.031920 0.050649 0.398764 0.029563

0.460897 0.058542 0.059602 0.452031 0.057416 0.053911 0.443519 0.056334 0.048970 0.435339 0.055295 0.044654 0.427471 0.054296 0.040865 0.419895 0.053334 0.037521 0.412596 0.052407 0.034558 0.405556 0.051513 0.031920 0.398764 0.050649 0.029563

0.466753 0.466753 0.052606 0.463341 0.463341 0.042428 0.459912 0.459912 0.034663 0.456473 0.456473 0.028644 0.453032 0.453032 0.023913 0.449592 0.449592 0.020145 0.446158 0.446158 0.017112 0.442735 0.442735 0.014645 0.439324 0.439324 0.012621

0.788819 0.100193 0.051477 0.780329 0.099115 0.042643 0.771959 0.098052 0.035733 0.763713 0.097004 0.030247 0.755594 0.095973 0.025834 0.747603 0.094958 0.022243 0.739740 0.093959 0.019290 0.732005 0.092977 0.016840 0.724399 0.092011 0.014789

0.100193 0.788819 0.051477 0.099115 0.780329 0.042643 0.098052 0.771959 0.035733 0.097004 0.763713 0.030247 0.095973 0.755594 0.025834 0.094958 0.747603 0.022243 0.093959 0.739740 0.019290 0.092977 0.732005 0.016840 0.092011 0.724399 0.014789

0.098936 0.098936 0.058882 0.096695 0.096695 0.055180 0.094557 0.094557 0.051817 0.092513 0.092513 0.048752 0.090558 0.090558 0.045951 0.088686 0.088686 0.043385 0.086891 0.086891 0.041028 0.085170 0.085170 0.038858 0.083516 0.083516 0.036856

310

Appendices

TABLE A1.5 The Corner Singularity Gauss Quadrature Coefficients Nodal index g

0.1 u v o 0.2 u v o 0.3 u v o 0.4 u v o 0.5 u v o 0.6 u v o 0.7 u v o 0.8 u v o 0.9 u v o

1

2

3

4

5

6

7

0.337122 0.337122 0.108175 0.340721 0.340721 0.104171 0.344145 0.344145 0.100453 0.347407 0.347407 0.096991 0.350517 0.350517 0.093756 0.353487 0.353487 0.090726 0.356327 0.356327 0.087882 0.359047 0.359047 0.085207 0.361654 0.361654 0.082686

0.060429 0.475753 0.062443 0.061123 0.481217 0.059009 0.061799 0.486540 0.055859 0.062457 0.491726 0.052963 0.063100 0.496782 0.050296 0.063726 0.501711 0.047833 0.064336 0.506519 0.045555 0.064932 0.511210 0.043443 0.065514 0.515787 0.041482

0.475753 0.060429 0.062443 0.481217 0.061123 0.059009 0.486540 0.061799 0.055859 0.491726 0.062457 0.052963 0.496782 0.063100 0.050296 0.501711 0.063726 0.047833 0.506519 0.064336 0.045555 0.511210 0.064932 0.043443 0.515787 0.065514 0.041482

0.471313 0.471313 0.063827 0.472416 0.472416 0.061618 0.473455 0.473455 0.059555 0.474435 0.474435 0.057624 0.475360 0.475360 0.055816 0.476233 0.476233 0.054118 0.477059 0.477059 0.052523 0.477840 0.477840 0.051019 0.478580 0.478580 0.049601

0.798957 0.101481 0.061420 0.800436 0.101669 0.059943 0.801866 0.101850 0.058533 0.803249 0.102026 0.057186 0.804589 0.102196 0.055898 0.805886 0.102361 0.054664 0.807143 0.102521 0.053483 0.808362 0.102675 0.052350 0.809545 0.102826 0.051263

0.101481 0.798957 0.061420 0.101669 0.800436 0.059943 0.101850 0.801866 0.058533 0.102026 0.803249 0.057186 0.102196 0.804589 0.055898 0.102361 0.805886 0.054664 0.102521 0.807143 0.053483 0.102675 0.808362 0.052350 0.102826 0.809545 0.051263

0.105447 0.105447 0.056462 0.109528 0.109528 0.050853 0.113531 0.113531 0.045991 0.117457 0.117457 0.041753 0.121309 0.121309 0.038042 0.125089 0.125089 0.034776 0.128798 0.128798 0.031890 0.132438 0.132438 0.029331 0.136012 0.136012 0.027051

311

Appendices

APPENDIX 2: NUMERICAL INTEGRATION OF THE GREEN FUNCTIONS WITH COULOMB SINGULARITIES IN THE COINCIDENCE LIMIT Consider 1D element S, maybe curved, given in the R2 space by the formulas S:

x ¼ xðtÞ;

y ¼ yðtÞ;

1=2  t  1=2:

ðA2:1Þ

The points P ¼ {x ( 1/2), y( 1/2)} and Q ¼ {x (1/2), y(1/2)} are the element ends, as shown in Figure A2.1. Our task is to calculate the Coulomb potential ’ðRÞ ¼

1=2 ð

1=2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x0 2 þ y0 2 sðtÞG xR

 yðtÞ dt;

xðtÞ; yR

R ¼ fxR ; yR g ðA2:2Þ

(a)

R

S* P* d1

h

P

C

t = −1/2

t = tc

d2 Q*

S t = 1/2

Q

(b) P* S*

P R t = tc

t = −1/2 S

h Q = C = Q* t = 1/2 G

FIGURE A2.1 On numerical integration of the Green function with Coulomb singularity over a curved 1D element. (a) The case that the point nearest to the point R is strictly inside the element; (b) the case that this point coincides with one of the element ends.

312

Appendices

generated by the surface charge distribution s(t) defined on S. Here we restrict ourselves to the case of planar symmetry, when the Green function appears as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA2:3Þ Gðx; yÞ ¼ 2ln x2 þ y2

The difficulty of numerical integration in Eq. (A2.2) consists in the presence of the logarithmic singularity in the integrand in the coincidence limit when the point R is approaching S. This makes the standard Gauss quadrature method inapplicable for calculating the potential not only upon the element itself but also in the element vicinity. The appropriate method to calculate the integrals like that presented by Eq. (A2.2) consists in representation of the singular integrand as a sum of two terms, one of them being regular and the other one allowing exact analytical integration. Here we consider the method of tangential elements, geometrical essence of which is displayed in Figure A2.1. Let the point C ¼ {x(tc), y(tc)} 2 S be the nearest to the point R over all points of the element S. It may be located strictly inside the element (Figure A2.1a) or coincide with one of its end (Figure A2.1b). We construct an auxiliary straight element S* parameterized as 0

x ðtÞ ¼ xðtc Þ þ x ðtc Þðt 0 y ðtÞ ¼ yðtc Þ þ y ðtc Þðt

tc Þ tc Þ

ðA2:4Þ

The auxiliary element S* is referred to as tangential element because (1) it has the common point C with the curved element at t ¼ tc: x*(tc) ¼ x(tc), y*(tc) ¼ y(tc), and (2) it is colinear with the tangential vector of the curved element: x*(1/2) x*( 1/2) ¼ x0 (tc), y*(1/2) y*( 1/2) ¼ y0 (tc). We ascribe the constant charge density s* ¼ s(tc) to the tangential element and divide the integral (A2.2) into two parts ’ðxR; yR Þ ¼ ’1 þ ’2 ; ’1 ¼

1=2 ð

½


1=2 L s G xR  

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x0 2 þ y0 2 sðtÞG xR

’2 ¼ L s

1=2 ð

yðtÞ



 y ðtÞ Š dt

x ðtÞ; yR  

xðtÞ; yR

ðA2:5Þ


G xR

1=2

x ðtÞ; yR

ðA2:6Þ  y ðtÞ dt;

ðA2:7Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where L ¼ x0 ðtc Þ2 þ y0 ðtc Þ2 is the tangential element’ length. With the charge density function s(t) being regular, the integrand in Eq. (A2.6)

Appendices

313

contains no singular points and, therefore, the integral ’1 may be calculated with the use of the standard Gauss quadrature formulas. More sophisticated procedure is required if the charge density has a singularity at one of the element’s ends, say sðtÞ ¼ ðt þ 1=2Þ g s0 ðtÞ; g > 0; j s0 ðtÞ j< 1. In this case, we can extract in Eq. (A2.6) the singular term as common multiplier 1=2 ð "qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 x0 2 þ y0 2 s0 ðtÞG xR xðtÞ; yR yðtÞ ’1 ¼ 1=2

1g
1 @t þ A L s G xR 2 0

x ðtÞ; yR

1 g #0  1 y ðtÞ @t þ A dt 2

ðA2:8Þ

and then apply the singular Gauss quadrature scheme according to Appendix 1. To improve the integration accuracy, one can also use additional subdivision in the vicinity of the left integration limit. The second integral in Eq. (A2.7) may be calculated analytically     d1 d2 2 2 2 2  þ 2L s d1 ln½d1 þ h Š d2 ln½d2 þ h Š 2h arctan þ arctan ’2 ¼ h h ðA2:9Þ h¼

½xR

0

xðtc ފy ðtc Þ

L

½yR

0

yðtc ފx ðtc Þ

:

ðA2:10Þ

The values d1 and d2 are the distances between the projection of the point R onto the tangential element and the points P*, Q* correspondingly d1

¼

d2

¼

0

yR Šy ðtc Þ

0

yR Šy ðtc Þ

½xð0Þ

xR Šx ðtc Þ þ ½yð0Þ L

½xð1Þ

xR Šx ðtc Þ þ ½yð1Þ L

0

0

ðA2:11Þ

In the particular case when the point C is internal point of the element, 2tc)/2. If C then d1 ¼ jP*Cj ¼ L* (1 þ 2tc)/2 and d2 ¼ jQ*Cj ¼ L*(1 coincides with P or Q, one of these values may be negative. For example, d1 ¼ jP*Gj and d2 ¼ jQGj in the case C ¼ Q shown in Figure A2.1b. The right-hand side in Eq. (A2.9) is finite everywhere on the tangential element. In the strictly inner points (d1 > 0, d2 > 0) its normal derivative @’2 / @h experiences the 4ps* discontinuity, but still stays finite. The only points where the derivatives of ’2 become singular are the ends of the tangential element (tc ¼ 1/2). However, as can be easily seen, the singularity of this type disappears if we sum the contributions of the adjacent elements constituting a smooth line with smooth charge density.

314

Appendices

Let us now consider numerical calculation of the integral ð1

’ðRÞ ¼ du 0

1ð u 0

jR

sðu; vÞ Jðu; vÞdudv Pðu; vÞ j

ðA2:12Þ

taken over a triangular element parameterized as P(u,v) in the domain O¼{u, v: 0  u  1, 0  v  1, u þ v  1}. Here s(u,v) is the surface charge density, and @P @P dudv Jðu; vÞdudv ¼ ðA2:13Þ  @u @v

is the elementary area on the surface S. Suppose that the inner point C ¼ P (uc,vc) of the element is the nearest to the point R, and the vector H ¼ R C, whose length we denote by h, is orthogonal to the element surface. We construct the auxiliary flat triangular element being tangential to S, with the corners P1 ¼ C

uc U

vc V;

U¼

P2 ¼ P1 þ U;

@P ; @u u ¼ uc v ¼ vc

V¼

@P @v u ¼ uc v ¼ vc

and parameterize this element as P ðu; vÞ ¼ ð1 doing so, we obtain the conditions P ðuc ; vc Þ ¼ Pðuc ; vc Þ ¼ C;

P3 ¼ P1 þ V

@P ¼ U; @u

u

ðA2:14Þ ðA2:15Þ

vÞP1 þ uP2 þ vP3 . In

@P ¼ V: @v

ðA2:16Þ

Similar to the case of a 1D element, we ascribe the constant charge s* ¼ s(uc,vc) to the tangential element and divide Eq. (A2.12) into two parts ’ ¼ ’1 þ ’2, where  ð

sðu; vÞJðu; vÞ s J  du dv; ðA2:17Þ ’1 ¼ j R Pðu; vÞ j j R P ðu; vÞ j O

’2 ¼ s  J 

ð

O

du dv ; j R P ðu; vÞ j

ðA2:18Þ

and J* ¼ jUVj is doubled area of the auxiliary flat element. The integrand in Eq. (A2.17) allows use of the Gauss quadrature scheme (A1.10), either regular or singular. The second integral in Eq. (A2.18) may be calculated analytically. For this purpose, we subdivide the auxiliary element into six right triangles, as shown in Figure A2.2, and integrate

315

Appendices

R

h

P*3

Ω1

a3,1

a2,2

b3 S*

b2

C

a3,2

a2,1

b1 P*1

P*2 a1,1

a1,2

S

FIGURE A2.2 On numerical integration of the Green function with the Coulomb singularity over a curved triangular element.

over each of those triangles separately. For instance, the integral taken over the hatched triangle O1 appears as ð du dv a ¼ Pða1;1 ; b1 ; hÞ j h j atan J ðA2:19Þ  j R P ðu; vÞ j b O1

a ah Pða; b; hÞ ¼ b asinh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ h asin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 þ h2 a2 þ b2 b 2 þ h2

ðA2:20Þ

The meaning of the parameters a, b is clear from Figure A2.2. The last term in Eq. (A2.19), after having been summed over the six triangles, gives 2pjhj. Thus, we obtain the representation  3  X  pðas;1 ; bs ; hÞ þ pðas;2 ; bs ; hÞ 2 ps j h j; ðA2:21Þ ’2 ¼ s s¼1

with the parameters as1, as2, bs defined according to Figure A2.2.

316

Appendices

APPENDIX 3: FIRST VARIATION OF A FUNCTIONAL UPON THE EQUALITY-TYPE OPERATOR CONSTRAINTS (FEDORENKO’S VARIATIONAL SCHEME) Consider the functional F[X, U] and the operator R[X, U], both defined and differentiable on the product of the linear normalized spaces {X}{U}. Following the control theory terminology, we will call {X} the phase space and {U} the space of controls. Consider the operator equation R½X; UŠ ¼ 0

ðA3:1Þ

~ and suppose this equation to possess a single solution XðUÞ 2 fXg for any U 2 {U}. Let us define the functional ~ UŠ F0 ½UŠ ¼ F½XðUÞ;

ðA3:2Þ

and calculate its variation dF0[dU] in the vicinity of a fixed control U ¼ U0. By varying Eq. (A3.2) with respect to U at U ¼ U0 we have dF0 ¼ hhFX ; dXii þ hhFU ; dUii:

ðA3:3Þ

Here FX, FU denote the corresponding functional derivatives at the ~ 0). By varying the operator equation (A3.1) in the point U ¼ U0, X0 ¼ X(U similar way, we obtain dR½dX; dUŠ ¼ RX ½dXŠ þ RU ½dUŠ ¼ 0:

ðA3:4Þ

Using the definition of conjugate operator (which is marked by the upper index *), we get the chain of equalities hhC; RX ½dXŠii ¼ hhR∗ X ½CŠ; dXii ¼

hhC; RU ½dUŠii ¼

hhR∗ U ½CŠ; dUii

ðA3:5Þ

If we now specify vector C as the solution of the conjugate equation R∗ X ½CŠ ¼ FX ;

ðA3:6Þ

the full variation of the functional F0[U] will take the form dF0 ½dUŠ ¼ hh R∗ U ½CŠ þ FU ; dUii:

ðA3:7Þ

In fact, the introduction of the conjugate function C is a direct functional generalization of the Lagrange multiplier method to project the functional gradient {FX, FU} onto the plane dR[dX, dU] ¼ 0 being tangent to the nonlinear operator equality constraint (A3.1) at the point ~ 0 Þ. This scheme, being simultaneously most simple U ¼ U0 ; X0 ¼ XðU and general, is often used in many variational and control theory problems.

317

Appendices

APPENDIX 4: JUMP CONDITION FOR VARIATIONS OF THE ORDINARY DIFFERENTIAL EQUATIONS WITH NONSMOOTH RIGHT-HAND SIDE Consider the Cauchy problem for the nonautonomous system of differential equations x_ ¼ f ðt; x; xÞ;

xð0; xÞ ¼ x0 ðxÞ

ðA4:1Þ

in the n-dimensional space Rn. Here x ¼ (x1,. . ., xn), f ¼ (f1,. . ., fn) are n-vectors, t is independent argument, which is further called time, and x ¼ (x1,. . ., xm) is vectorial parameter. We suppose that the solution x(t, x) of the problem (A4.1) exists and is unique for any x belonging to the vicinity  of the fixed parameter value x0. With no loss of generality, we may assume x0 ¼ 0. Let the phase velocity f undergo a finite discontinuity (jump) on the smooth surface S given by the equation gðx; xÞ ¼ 0;

ðA4:2Þ

which also depends on the parameter x. For any x 2 , let us define the domains in Rn Dx ¼ fx : gðx; xÞ < 0g;

Dþ x ¼ fx : gðx; xÞ > 0g

and represent the phase velocity in the form

x 2 Dx f ðt; x; xÞ; f ðt; x; xÞ ¼ f þ ðt; x; xÞ; x 2 Dþ x

ðA4:3Þ

ðA4:4Þ

with f , f þ being smooth enough functions of their arguments in Dx , Dþ x, respectively. For the sake of definiteness, let us assume the initial point x0(x), x 2  to be located in the domain Dx . Now we must define the solution of the differential equation (A4.1) with the discontinuous right-hand side (A4.4). We follow a standard definition, which seems to be the most natural from the physical standpoint. Denote tS(x) the time moment at which the solution x (t,x) of the Cauchy problem x_ ¼ f ðt; x; xÞ;

x ð0; xÞ ¼ x0 ðxÞ

ðA4:5Þ

defined in the smoothness domain Dx reaches the surface S at the point xS(x) 2 S (we assume that such time moment does exist). We have at t ¼ tS(x)   


ðA4:6Þ x tS ðxÞ; x ¼ xS ðxÞ; g x tS ðxÞ; x ; x ¼ 0:

318

Appendices

To simplify our consideration, we will consider the so-called rough case, when the principal trajectory x0 ðtÞ ¼ x ðt; 0Þ arrives at the surface S at a nonzero angle, so that 
ðA4:7Þ < gx ; f > 6¼ 0 at x ¼ 0; t ¼ tS ð0Þ; x ¼ x0 tS ð0Þ : With regard to the continuity reasons, we may assume this condition to be satisfied not only for x ¼ 0 but for all x 2 . Let us define the continuation xþ(t, x) of the trajectory x (t, x) through the surface S into the smoothness domain Dþ x as the solution of the Cauchy problem 
x_ ¼ f þ ðt; x; xÞ; xþ tS ðxÞ; x ¼ xS ðxÞ; t  tS ðxÞ: ðA4:8Þ This implies the continuity condition  

x tS ðxÞ; x ¼ xþ tS ðxÞ; x ;

ðA4:9Þ

which is obeyed on the surface S for all x 2 . Thus, we have defined the solution

x ðt; xÞ; 0  t  tS ðxÞ ðA4:10Þ xðt; xÞ ¼ xþ ðt; xÞ; t  tS ðxÞ of the differential equation (A4.1) with the right-hand side given by Eq. (A4.4), being continuous on the discontinuity surface S for all x 2 . Let us denote xþ 0 ðtÞ the continuation of the principal trajectory x0 ðtÞ through the surface S into the domain Dþ x . In each of the smoothness we can define the tau-variations domains Dx , Dþ x x i ðtÞ ¼

@x ðt; xÞ @ 2 x ðt; xÞ @ 3 x ðt; xÞ ; x ; x ;... ij ðtÞ ¼ ijk ðtÞ ¼ x¼0 x¼0 @xi @xi @xj @xi @xj @xk x¼0

ðA4:11Þ

of the perturbed trajectory x(t, x) on the principal trajectory x 0 ðtÞ, and expand the smooth branches x(t, x) of the trajectory x(t, x) into the series X  1X  1X  xij ðtÞxi xj þ x ðtÞxi xj xk þ . . . xi ðtÞxi þ x ðt; xÞ ¼ x 0 ðtÞ þ 2 i;j 6 i;j;k ijk i ðA4:12Þ which, according to the well-known Poincare theorem, is uniformly convergent on the time interval being not ‘‘too’’ big (the reader can find the exact formulation of this remarkable theorem, which constitutes the basis of aberration theory, in the corresponding textbooks on analytical theory of ordinal differential equations).

Appendices

319

The tau-variations being the coefficients of the expansion (A4.12) satisfy the differential equations in variations (see Chapter 5) with the only addition that the upper index ‘‘ ‘‘ or ‘‘þ’’ should be chosen in each case. The question to be answered here is what are the initial conditions  þ for the tau-variations xþ i , xij , xijk ,. . . on the discontinuity surface S. Let us introduce the limiting values of the phase velocity
 ðA4:13Þ fS ¼ lim f t; x ðtÞ; 0 t!tS ð0Þ0

and the limiting values of the tau-variations   ðx i ÞS ; ðxij ÞS ; ðxijk ÞS ; . . . ¼

lim

t!tS ð0Þ0

  x i ðtÞ; xij ðtÞ; xijk ðtÞ; . . .

ðA4:14Þ

along the principal trajectory on both sides of the discontinuity surface S. The relationship between the tau-variations upon S directly follows from the continuity condition (A4.9). Indeed, first differentiation of Eq. (A4.9) by xi at x ¼ 0 gives ðtS Þi fS þ ðx i ÞS ¼ ðtS Þi fSþ þ ðxþ i ÞS ;

ðA4:15Þ

whence, denoting the jump Fþ FS of any function F on the surface S as S [F]S, we obtain the jump condition for the first-order tau-variation xi ½xi ŠS ¼

ðtS Þi ½ f ŠS :

ðA4:16Þ

The first derivative (tS)i (which from the viewpoint of charged particle optics represent the first-order temporal aberration coefficient on the surface S) can be easily connected with the tau-variation ðxi ÞS and differential properties of the surface S Indeed, differentiating of the second of the relations (A4.6) gives < gx ; fS > ðtS Þi þ < gx ; ðxi ÞS > þgi ¼ 0

ðA4:17Þ

whence, inasmuch as < gx ; fS > 6¼ 0, we obtain ðtS Þi ¼

< gx ; ðxi ÞS > þgi ; < gx ; f S >

ðA4:18Þ

and the jump condition (A4.15) takes the final form ½xi ŠS ¼

< gx ; ðxi ÞS > þgi ½ f ŠS : < gx ; fS >

ðA4:19Þ

We can see that the first-order tau-variation xi appears continuous on the surface S if there is no jump of the phase velocity f on S ([ f ]S ¼ 0). In the same manner, the jump conditions for the high-order tau-variations xij, xijk ,. . . on the discontinuity surface S can be obtained.

320

Appendices

APPENDIX 5: SOME GENERAL PROPERTIES OF LINEAR SYSTEMS Let us consider a system S, the internal properties of which can be described by means of the operator A in the sense that the input signal X and the output signal Y are interrelated by the operator equation Y ¼ A½XŠ:

ðA5:1Þ

We will assume the system S and, as a consequence, the operator A, to be linear. According to the definition, this implies that for any input signals X1, X2 and any complex l, the two following conditions are simultaneously satisfied: 1Þ 2Þ

A½X1 þ X2 Š ¼ A½X1 Š þ A½X2 Š A½lX1 Š ¼ lA½X1 Š:

ðA5:2Þ

For definiteness sake, we will assume that both the input signal X ¼ X(x) and the output signal Y ¼ Y(y) represent the complex-valued functions of the vectorial arguments x 2 Rn, y 2 Rm, respectively. Depending on the particular nature of the system S, physical variables such as time, coordinates, velocities, voltages, currents, and so on, may be regarded as components of the vectors x, y. Let us fix an arbitrary point x0 2 Rn and consider a function of two variables Iðy; x0 Þ, which represents a ‘‘response’’ of the system S to the input pulse signal dD(x x0) located at the point x0 Iðy; x0 Þ ¼ A½dD ðx

x0 ފ:

ðA5:3Þ

It is noteworthy that the response function Iðy; x0 Þ has different names in different areas of natural and engineering sciences. In particular, in mathematical physics, it is commonly called the Green function, in applied mechanics and control theory – pulse transition function, in optics and laser physics – either instrumental, response, or spread function. With regard to the optical orientation of this monograph, we will further preferably use the two latter terms. By representing any input signal X(x) as a linear combination of dDpulses, ð ðA5:4Þ XðxÞ ¼ Xðx0 Þ dD ðx x0 Þ dx0 ; and applying the linear operator A to both the left and right parts of Eq. (A5.4), with regard to Eq. (A2.2), we obtain: ð ð YðyÞ ¼ A½ Xðx0 Þ dD ðx x0 Þ dx0 Š ¼ Xðx0 Þ A½dD ðx x0 ފ dx ð (A5.5) ¼ Iðy; x0 Þ Xðx0 Þ dx0 :

321

Appendices

This result indicates that the response function Iðy; x0 Þ contains full information as to internal properties of the linear system S because, as seen from Eq. (A5.5), the response of the linear system S to any input signal X(x) can be easily derived by means of simple integration, being the corresponding response function Iðy; x0 Þ previously known. Now we need to introduce one more, most important, assumption as to the properties of the linear system S in question. Let us assume that the function Iðy; x0 Þ obeys the generalized invariance condition Iðy; x0 Þ ¼ Iðy

ðA5:6Þ

Mx0 ; 0Þ;

with M being a constant n  m-matrix. Equation (A5.6) states that the response of the linear system S to the input dD-pulse located at any point x0 can be expressed as a shift (in the sense of the variables change y ! y Mx0) of the system’s response to the input dD-pulse located at the point x0 ¼ 0. Physically, this condition, being especially transparent in the case that M is identity matrix, guarantees the invariance of the response function with respect to the shifts of the pulse source dD in the {x} space (Figure A5.1). For brevity sake, we will denote I ðy Mx0 Þ the function in the right part of Eq. (A5.6). Using optical terminology, we will call the linear system isoplanatic in a broad sense, or simply, isoplanatic, if it obeys the isoplanatic condition (A5.6) for any x0 2 Rn. In the case that the isoplanatic condition (A5.6) is satisfied with sufficient accuracy in a vicinity of the point x0 ¼ 0, we will use the term locally isoplanatic system. It is evident that the simple change of variables x ! x0 x∗ 0 allows extension of the concept of local isoplanatism to a vicinity of any fixed point x∗ 0 in the {x} space.

x2

0

x1

A

y2 = Mx2

0

y1 = Mx1

FIGURE A5.1 The definition of the isoplanatism condition (thin arrows denote the input dD-pulses in the {x}-space).

322

Appendices

It immediately follows from Eq. (A5.5) that the input and output signals in the isoplanatic system are interrelated by means of the convolution equation ð M ðA5:7Þ YðyÞ ¼ Iðy MxÞXðxÞdx ¼ IðyÞ ∗ XðxÞ (zero index indicating the location of the dD-pulse source is here omitted as non-essential). The set of the linear systems possessing invariant structure in the sense of Eq. (A5.6) is rather wide and plays an important role in theory and applications. In particular, this set involves the systems that can be described by linear differential equations (both ordinary and partial) with constant coefficients. Many of the well-known principles and facts in mathematical physics, optics, radiophysics, acoustics, and electric circuit theory represent direct consequences of the convolution equation (A5.7). As an example, let us consider the system S described by the linear nonhomogeneous ordinary differential equation in vectorial form dy ¼ AðtÞy þ f ðtÞ dt

ðA5:8Þ

with zero initial condition y(0) ¼ 0. Here, y ¼ {y1,. . ., yn}, f(t) ¼ {f1(t),. . ., fn(t)} are n-vectors, and A(t) is n  n - matrix continuously dependent on time. Obviously, S represents a linear system if f(t) and y(t) are considered as the input and output signals, respectively. Our aim here is to construct a response function for this system using the general definition given above. Let us fix any

time moment t  0and consider the vectorial pulse 0; . . . ; dD ðt tÞ; . . . 0 ðkÞ , in which the k-th coordinate signal fd ðt; tÞ ¼ k represents the dD-function located at t ¼ t. It can be easily seen that the ðkÞ response of the system S to the signal fd ðt; tÞ is the solution IðkÞ ðt; tÞ of the homogeneous differential equation dIðkÞ ¼ AðtÞIðkÞ dt

ðA5:9Þ

with the initial condition IðkÞ ðt; tÞ ¼ eðkÞ ¼ columnf0; . . .1; . . .0g. The k matrix Iðt; tÞ ¼ kIðkÞ ðt; tÞk, the columns of which are the partial solutions IðkÞ ðt; tÞ extended as zero into the domain t < t, is the sought response function of the system S. According to Eq. (A5.5), for any input signal f(t), the corresponding output signal takes the form

Appendices

yðtÞ ¼

1 ð

1

Iðt; tÞf ðtÞ dt:

323

ðA5:10Þ

Denoting I0 ðtÞ ¼ Iðt; 0Þ, we have the equality Iðt; tÞ ¼ I0 ðtÞ I0 1 ðtÞ, which trivially follows from the uniqueness theorem for the ordinary differential equations, because both the left and right-hand sides of this equality, being considered as functions of t, represent the matrix solutions of Eq. (A5.9) coinciding at t ¼ t. Equation (A5.10) may be written as ðt yðtÞ ¼ I0 ðtÞ I0 1 ðtÞf ðtÞ dt

ðA5:11Þ

0

which brings us to the well-known Cauchy formula for the partial solution of the linear nonhomogeneous differential equation (A5.8). If the matrix A in Eq. (A5.8) does not depend on time, we may state even more, namely that Iðt; tÞ ¼ I0 ðtÞI0 1 ðtÞ ¼ I0 ðt

tÞ;

ðA5:12Þ

which indicates the invariance of the response function Iðt; tÞ with respect to the shifts of the ‘‘delta-source’’ dD(t) in time (Figure A5.2) and ensures the isoplanatism condition (with M ¼ 1) for the system in question. We now return to the general case and consider the question of how the linear isoplanatic system S transforms harmonic signals. Let the input signal be the harmonic oscillation X(x) ¼ exp [i < ox, x >] with the angular frequency vector ox. Assuming the matrix M to be a nondegenerate n  nmatrix and substituting X(x) into Eq.(A5.7), we have ð YðyÞ ¼ Iðy MxÞ exp ½i < ox ; x >Š dx ðA5:13Þ 1 ∗ 1 ∗ 1 ¼ IðM ox Þexp½i < M ox ; y >Š; j detM j

(t,0) E (t,t)= (t−t) t

t

FIGURE A5.2 Geometrical essence of the isoplanatism condition in Eq. (A5.8), with the matrix A independent of time.

324

Appendices

which shows that the response Y(y) to the harmonic signal X(x) is also 1 harmonic, with the frequency oy ¼ M∗ ox and the complex amplitude 1 j det M j 1 I ðM∗ ox Þ both dependent on the input frequency ox. The complex-valued function I ðoy Þ, being the Fourier transform of the response function IðyÞ, is commonly called the point transfer function (in optics it is called the optical transfer function [OTF]) and plays an important role in linear systems theory. We can attach a more transparent form to (A5.13) if we use the linear transformation y ¼ Mx, which interconnects the arguments of the input and output signals. Along with any output signal Y(y), let us consider the referred output signal Y(r)(x) ¼ jdet MjY(Mx) as a function of x. Obviously, both of the signals are equally normalized with respect to the corresponding variables x, y ð ð ð ðA5:14Þ YðyÞ dy ¼j det M j YðMxÞdx ¼ YðrÞ ðxÞdx Introducing the referred response function IðrÞ ðxÞ ¼j det M jIðMxÞ, we can bring Eq. (A5.13) to the form YðrÞ ðxÞ ¼ I ðrÞ ðoÞexp½i < o; x >Š ¼j I ðrÞ ðoÞ j exp½i < o; x > þiargI ðrÞ ðoފ:

ðA5:15Þ

It is natural to call the function I ðrÞ ðoÞ, representing the Fourier transform of the referred response function IðrÞ ðxÞ, the referred optical transfer function of the linear isoplanatic system S. In this Appendix we further consider the referred characteristics only; therefore, hereafter there is no need in using the term ‘‘referred’’ and attaching the index x to the frequency vector o. The module j I ðrÞ ðoÞ j and argument arg I ðrÞ ðoÞ of the transfer function I ðrÞ ðoÞ are called the modulation transfer function (MTF) and phase ðrÞ transfer function (PTF), respectively (arg I ðoÞ denotes here the singleðrÞ valued branch of Arg I ðoÞ, vanishing at o ¼ 0). Before we discuss how the MTF and PTF reflect the linear system properties, let us additionally assume the response function IðrÞ ðxÞ to be a real nonnegative function satisfying the normalization condition ð ðA5:16Þ IðrÞ ðxÞ dx ¼ 1: Consider the harmonic input signal X(x) ¼ A þ B exp{i < o,x >}, with A, B being any complex constants (0Š . For any harmonic signal, let us define the contrast w(o) as a ratio of the modules of its oscillating and constant components. Thus, we have

Appendices

wX ðoÞ ¼

jBj ; jAj

wY ðoÞ ¼

jBj jAj

j I ðrÞ ðoÞ j :

The ratio of the output and input contrasts wY ðoÞ  ðrÞ wðoÞ ¼ ¼ I ðoÞ ; w ðoÞ X

325

ðA5:17Þ

ðA5:18Þ

represents the MTF of the linear isoplanatic system S and determines the distortion that the input signal acquires after having been transformed by the system S. We can see that MTF depends only on the input signal frequency o and does not depend on the coefficients A, B, which might have been taken equal to unit. With regard to the inequality þ1 þ1 ð ð ðrÞ ðrÞ  ðoÞ ¼  I I ðxÞexp½ i < o; x >Šdx IðrÞ ðxÞdx ¼ 1 1

1

ðA5:19Þ

valid for all o, we may conclude that MTF is ranged between unit and zero at all frequencies. It can also be easily seen that MTF ! 1 when o ! 0, which means that the input signal of extremely low frequency is transformed with no contrast change. If a linear isoplanatic system is ideal in the sense that IðrÞ ðxÞ ¼ dD ðxÞ, the corresponding MTF is identically equal to unit, which means that such an ideal system does not distort the contrast of the input harmonic signals independently of their frequency. If a linear isoplanatic system S is not ideal, the normalization condition (A5.16) implies that MTF ! 0 when o tends to infinity. According to Eq. (A5.15), the PTF  ðrÞ ðoÞ fðoÞ ¼ argI

ðA5:20Þ

determines the phase shift between the input and output signals versus the input signal frequency o. In optical applications, we often come across the input signals, which represent 1D harmonic test patterns being harmonically distributed along a fixed direction k in the {x}-space (Figure A5.3). Let us represent such input signal in the form Xk ðxÞ ¼ 1 þ cos < Ok; x >;

ðA5:21Þ

with k, x being n-vectors and frequency O being scalar. According to Eq. (A5.17), the contrast of this signal is equal to unit. The referred output signal appears as Yk ðxÞ ¼ 1 þ wðOkÞcos½< Ok; x > þfðOkފ

ðA5:22Þ

326

Appendices

x2

x2

x1

k x1

FIGURE A5.3

One-dimensional input harmonic test pattern in the 2D space (x1, x2).

which shows that the MTF and PTF of the test pattern (A5.21) are, in fact, 1D ‘‘cross-sections’’ o ¼ Ok of the corresponding n-dimensional MTF and PTF w(o), f(o) in the frequency space {o}. By considering various orientations of the test pattern (A5.21) in the {x}-space, or, in other words, by rotating the vector k in the frequency space {o}, we can extract full information as to the properties of the system S. We have introduced the response function as the linear system reaction to the input signal dD(x x0) localized at a certain point x ¼ x0 of the {x}-space. It is natural to call such function the point response function, or, using optical terminology, the point spread function (PSF). We have seen that PSF and OTF considered together, contain the entire information as to the isoplanatic linear system properties. Very often the point sources like dD(x x0) appear to be continually distributed over some geometrical sets (lines, surfaces, etc) in Rn. In this case, by integrating the PSF over the corresponding sets, we come to the conception of the line (or surface) spread function. The classic examples of such spread functions are the ordinary and double layer potentials in mathematical physics, and the line spread function (LSF) in optics. Let us refer to strict definitions. Let L be any set in the space {Rn}, which can be presented in the parametric form L ¼ fx 2 Rn : x ¼ xL ðuÞ; u 2 Ug:

ðA5:23Þ

Here xL(u) is a smooth enough function of the vectorial argument u belonging to the measurable set U, the dimension of which is less than n. Consider the input signal

327

Appendices

ðdÞ

XL ðxÞ ¼

ð

pðuÞ dD ½x

xL ðuފ du;

ðA5:24Þ

U

which represents a sum of the point sources being continually distributed ðdÞ on L with the non-negative density p(u). The signal XL ðxÞ is located on L ðdÞ = L. Obviously, in the sense that XL ðxÞ ¼ 0 for x 2 ð ð ð ð ð ð ðdÞ XL ðxÞdx ¼ dx pðuÞdD ½x xðuފdu ¼ pðuÞdu dD ½x xðuފdx ¼ pðuÞdu: U

U

U

ðA5:25Þ It should be noted that in some cases the integrals in the left-hand and right-hand sides of Eq. (A5.25) may be simultaneously equal to infinity (see the example below). In accordance with (A5.5), the response IL ðyÞ of ðdÞ the linear system S to the distributed input signal XL ðxÞ takes the form ð ð ð ðdÞ IL ðyÞ ¼ Iðy; xÞXL ðxÞdx ¼ Iðy; xÞ pðuÞdD ½x xL ðuފdudx U

¼

ð

U




 pðuÞI y; xL ðuÞ du:

(A5.26)

The output signal IL ðyÞ is called the spread function of the set L. In the case that L is a straight line in Rn, IL ðyÞ is called the line spread function. Obviously, the case of the distribution density p(u) being a point source
 on L: p(u) ¼ dD(u u0), u0 2 U returns us to the PSF I y; xðu0 Þ .

If the linear system S is isoplanatic in the sense of Eq. (A5.6), then ð
 ðA5:27Þ IL ðyÞ ¼ pðuÞI y MxL ðuÞ du; U

and, as can be easily understood, the invariance of the PSF Iðy; xÞ with respect to the shifts of the point source in Rn entails the invariance of the LSF IL ðyÞ (A5.27) with respect to the shifts of the set L in Rn. Indeed, if we define the shift of the set L by the constant vector a 2 Rn as the set L(a)Rn with the parametrical representation LðaÞ ¼ fx 2 Rn : x ¼ xL ðuÞ þ a; u 2 Ug;

ðA5:28Þ

n

then, for any vector a 2 R , we have ð

 ILðaÞ ðyÞ ¼ pðuÞI y M xL ðuÞ þ a du ¼ IL ðy U

MaÞ:

ðA5:29Þ

328

Appendices

Quite similar to the definition of the referred PSF, the referred spread function of the set L is defined as ð 
ðrÞ ðrÞ ðA5:30Þ IL ðxÞ ¼j det M j IL ðMxÞ ¼ pðuÞIL x xL ðuÞ du: U

Consider the example when the set L is the coordinate plane x1 ¼ 0 with parametrical representation xL ðuÞ ¼ fx1 ¼ 0; x2 ¼ u1 ; . . . ; xn ¼ un 1 ;

1 < u1 ; . . .; un

ðdÞ XL ðxÞ

1

< 1g:

ðA5:31Þ

The input signal of (A5.24) takes the form ð ðdÞ XL ðxÞ ¼ pðuÞdD ðx1 ; x2 u1 ; . . . ; xn un 1 Þ du ¼ dD ðx1 Þpðx2 ; . . . ; xn Þ: U

ðA5:32Þ and represents a 1D delta-source with the distributed density p. According to Eq. (A5.30), the spread function of the plane L appears as ð ðrÞ pðuÞIðrÞ ðx1 ; x2 u1 ; . . . ; xn un Þdu1 . . . dun 1 : ðA5:33Þ IL ðxÞ ¼ Rn

1

In the case of p(u)  1, the spread function IL ðxÞ of the plane x1 ¼ 0 represents the integral of the PSF IðrÞ ðxÞ over the variables x2,. . .,xn, and, consequently, depends on the variable x1 only ð ðrÞ IL ðx1 Þ ¼ IðrÞ ðx1 ; x2 ; . . . ; xn Þdx2 . . . dxn : ðA5:34Þ Rn

1

ðrÞ

Obviously, the spread functions IðrÞ ðxÞ and IL ðx1 Þ are equally normalized. The correlation (A5.34) allows us to express the transfer function  ðrÞ ðoÞ:  ðrÞ ðo1 Þ of the plane x1 ¼ 0 in terms of the point transfer function I I L ð ð  ðrÞ ðo1 Þ ¼ expð io1 x1 Þdx1 IðrÞ ðx1 ; x2 ; . . . ; xn Þdx2 . . . dxn I L ð Rn 1 ðA5:35Þ  ðrÞ ðo1 ; 0; . . . 0Þ: ¼ IðrÞ ðxÞexpð io1 x1 Þdx ¼ I Rn

Following to O’Neil (1963), let us consider, as another example, one interesting optical effect that may occur in the case that the image formed by a linear optical system is strongly defocused. We shall assume that the arguments of the input and output signals are, accordingly, x ¼ (x1,x2) and y ¼ (y1, y2), and the PSF has the form of finite cylinder

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=pR2 ; r0  R ðr0 ¼ x21 þ x22 Þ: ðA5:36Þ IðrÞ ðr0 Þ ¼ 0; r0  R

Appendices

329

ℑ(w)

1

2

3.83 R

7.01 R

w

FIGURE A5.4 The ‘‘false contrast’’ effect illustrated by O’Neil (1963). 1, the convergent test pattern image; 2, optical transfer function.

Simple calculation of the Fourier transform gives the optical transfer function in the form  ðrÞ ðoÞ ¼ 2J1 ðoRÞ ; I oR

ðA5:37Þ

with J1(oR) being the Bessel function of the index 1. We can see that the  ðrÞ ðoÞ changes its sign at the points real-valued optical transfer function I (k) (k) o ¼ x /R (k ¼ 1,. . .), where the values x(k) represent the consecutive zeros of the function J1(x). Accordingly, the MTF and PTF in the case in question appear as wðoÞ ¼

2 j J1 ðoRÞ j oR

fðoÞ ¼ pðk

1Þ;

ðk 1Þ

o

oÞdy  ð ð
¼ fx ðxÞdx exp i < o; y > dD ½ y Gðxފdy ð
 (A6.14) ¼ fx ðxÞexp i < o; GðxÞ > dx:

332

Appendices

The so-called phase integral in the right-hand side of (A6.14) may be calculated either numerically or by means of the stationary phase method (see Appendix 7).

APPENDIX 7: THE MULTIDIMENSIONAL STATIONARY PHASE METHOD In this Appendix, we follow, with necessary simplifications, the monographs by Weinberg (1982) and Fedoryuk (1987). Let h(x), S(x) be real infinitely smooth functions in Rn, and h(x) be zero outside the closed restricted domain O  Rn. Consider the asymptotic behavior of the phase integral ð ðA7:1Þ IðlÞ ¼ hðxÞexp½ilSðxފdx O

at large l. Theorem 1. Let f(x), S(x) obey the conditions above, and ▽xS 6¼ 0 for x 2 O. Then IðlÞ ¼ Oðl

1

Þ;

l ! 1;

ðA7:2Þ

which means that the phase integral IðlÞ at l ! 1 tends to zero faster than any positive power of l. The point x0 2 Rn is called the stationary phase point if ▽xS(x0) ¼ 0. Let us 00 00 denote Sxx the matrix k@ 2 S=@xi @xj k, sgnSxx the difference between the number of its positive and negative eigenvalues. The stationary phase 00 point is called nondegenerate if detkSxx ðx0 Þk 6¼ 0. Theorem 2. Let us assume that there exists a single nondegenerate stationary point x0 located strictly inside the domain O. In this case, the main term of the asymptotics of IðlÞ at l ! 1 appears as IðlÞ  Cl

n=2

;

ðA7:3Þ

where h i ð2pÞn=2 hðx0 Þ p 00 ffi sgnS Þ þ i ðx Þ exp ilSðx C ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 xx 4 j detkS00xx ðx0 Þk j

ðA7:4Þ

It can be shown that generally the high-frequency asymptotics of the integral (A7.1) represents a sum of partial asymptotics connected with the contribution of the internal stationary phase points and the stationary phase points located on the boundary @O. The asymptotics also includes the contribution of the singularities that the preexponential factor h(x) may have in the closure of the domain O.

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CONTENTS OF VOLUMES 151–154

VOLUME 1511 C. Bontus and T. Ko¨hler, Reconstruction algorithms for computed tomography L. Busin, N. Vandenbroucke and L. Macaire, Color spaces and image segmentation G. R. Easley and F. Colonna, Generalized discrete Radon transforms and applications to image processing T. Radlicˇka, Lie agebraic methods in charged particle optics V. Randle, Recent developments in electron backscatter diffraction

VOLUME 152 N. S. T. Hirata, Stack filters: from definition to design algorithms S. A. Khan, The Foldy–Wouthuysen transformation technique in optics S. Morfu, P. Marquie´, B. Nofie´le´ and D. Ginhac, Nonlinear systems for image processing T. Nitta, Complex-valued neural network and complex-valued backpropagation learning algorithm J. Bobin, J.-L. Starck, Y. Moudden and M. J. Fadili, Blind source separation: the sparsity revoloution R. L. Withers, ‘‘Disorder’’: structured diffuse scattering and local crystal chemistry

VOLUME 153 Aberration-corrected Electron Microscopy H. Rose, History of direct aberration correction M. Haider, H. Mu¨ller and S. Uhlemann, Present and future hexapole aberration correctors for high-resolution electron microscopy O. L. Krivanek, N. Dellby, R. J. Kyse, M. F. Murfitt, C. S. Own and Z. S. Szilagyi, Advances in aberration-corrected scanning transmission electron microscopy and electron energy-loss spectroscopy

1

Lists of the contents of volumes 100–149 are to be found in volume 150; the entire series can be searched on ScienceDirect.com

341

342

Contents of Volumes

P. E. Batson, First results using the Nion third-order scanning transmission electron microscope corrector A. L. Bleloch, Scanning transmission electron microscopy and electron energy loss spectroscopy: mapping materials atom by atom F. Houdellier, M. Hy¨tch, F. Hu¨e and E. Snoeck, Aberration correction with the SACTEM-Toulouse: from imaging to diffraction B. Kabius and H. Rose, Novel aberration correction concepts A. I. Kirkland, P. D. Nellist, L.-y. Chang and S. J. Haigh, Aberration-corrected imaging in conventional transmission electron microscopy and scanning transmission electron microscopy S. J. Pennycook, M. F. Chisholm, A. R. Lupini, M. Varela, K. van Benthem, A. Y. Borisevich, M. P. Oxley, W. Luo and S. T. Pantelides, Materials applications of aberration-corrected scanning transmission electron microscopy N. Tanaka, Spherical aberration-corrected transmission electron microscopy for nanomaterials K. Urban, L. Houben, C.-l. Jia, M. Lentzen, S.-b. Mi, A. Thust and K. Tillmann, Atomic-resolution aberration-corrected transmission electron microscopy Y. Zhu and J. Wall, Aberration-corrected electron microscopes at Brookhaven National Laboratory

VOLUME 154 H. F. Harmuth and B. Meffert, Dirac’s difference equation and the physics of finite differences

INDEX

A Aberration chromatic 147, 168, 254–256, 271, 280 coma 168, 254–255, 260 corrector 114, 115 of deflection 231, 237 dynamic 258 off-axis 91, 253 spatial 177, 245, 257 spherical 147, 168, 252, 256 spherochromatic 251–252, 260 temporal 150, 177, 208, 255, 271, 274, 280 temporal chromatic 286, 303 tolerance 167 Zavoisky-Fanchenko 277, 282 Alternating directions sweep method 29 Anti-paraxial expansions 188 Arrival time 297 Assembly operation 7 Astigmatism 102, 253 B Beltrami-Laplace operator 55–59, 64–67 spectral problem 57–58 Binomial distribution 244 Boundary potential variation 76, 101 Boundary value problem 146–147 Boundary-layer function 272 Bruns-Bertein method 70–71 C Capacitor cylindrical 98–100 spherical 90–91 Cauchy problem 27, 188, 214, 317 Celestial mechanics 153, 197 Coincidence-limit singularity 24 Cold-cathode approximation 187 Collocation method 14 Conductive disk 85 Conjugate equation 71 Conjugate function 76–78 Conjugate gradients method 66 Contact transformation 150, 154–155, 158, 160–161, 270

Coordinate system Cartesian 27, 29, 62, 87, 151, 198, 212, 220 conical 63 curvilinear 27–28, 79, 117, 188–190 cylindrical 65, 81 local 6, 28, 65, 124, 129, 221, 230, 242, 254 polar 41 spherical 55, 59, 62, 243 Cosine-power angular distributions 244 Coulomb integral 4, 15, 181 Crossover plane 250 Current contours 114–117 Curved triangular element 315 D Defocusing parameter 259, 263 Delta-source 323 Density transformation 212, 329–330 Dielectric permittivity 50 Diffusion figure 227 Diode 183 Direct ray-tracing 150, 164 Dirichlet problem 71 Domain decomposition 31, 138, 180 Dynamic defocusing 258 E Effective (averaged) potential 37 Eigenfunction 50, 62, 65, 104 Eigenvalue 43, 50, 53, 65, 105, 332 Einzel lens 171 Electrode fringes 103 Electron flood 189 Electron-optical system 69, 90, 146, 148, 282 Elliptic integral 25, 81, 133 Emission current density 195 Emitter surface 148, 152, 170, 180, 245 virtual 155, 170 Energy conservation law 189–190 Energy functional, integral 53, 119–120, 132, 139–141, 192 Equation Boltzmann 173 conjugate 71

343

344

Index

Equation (cont.) Fredholm first-kind 4, 14, 23, 36, 71, 141, 181 Fredholm second-kind 5, 19, 38, 117, 136, 141 Lorenz 148, 150, 155, 159, 178, 188, 201, 218 Equiphase lines 231 Extrusion and intersection operations 7–8 F False contrast 269, 329 Fine-structure grid 26, 32–33, 90, 162, 173, 258, 281 Finite element mesh generation 122 quadrilateral 123 triangular 123–124 Finite element method 106, 112, 133 Finite volumes method 3 Finite-difference operator 29–30 Fourier expansion, series 81–83, 98, 117 integral 98 transform 229, 234, 242, 324, 329 Fringe field penetration 109 Full energy evolution equation 285 G Gauss plane 249 quadrature formulas 305 Generalized invariance condition 321 Golden section 182 Gradient search method 128 Green function 2D 22 3D 4 conjugate 78 harmonics 83 magnetic 133 periodic 33–37 H Hybrid BEM-FDM method 138

I Image conformal 40 curvature 165, 253–254 defocusing 251 electron 211–212, 219–220 formation 146, 148 input 211, 222 intensifier 164–165, 8 ion 207, 211, 220 output 222, 224 plane 249–250, 253, 15, 18 spatiotemporal 222 streaked 234, 236, 258 sweeping 231, 239, 255, 257, 286 tube 91–92, 143, 148, 235, 251, 255, 280 virtual 232 Initial parameters variation method 204 Intensity profiles 169 Interface boundary conditions 12 Ion injector 20–21 Isoplanatism condition 222 J Jump condition 162–163 L Lagrange variation 75 Lagrange–Helmholtz invariant 247, 251 Lagrangian derivative 72 Local isoplanatism approximation 227 Local magnifications matrix 226 M Magnification electron-optical 91, 171, 249 local 226, 231–232 temporal 287 complex 248 average 249 Magnetic permeability 112–113, 117, 131 superelement 112 Mean curvature surface 253 Metric tensor 126, 189 Modulation transfer function 225, 228, 324 Monte-Carlo method 173–174

Index

N Narrow and wide beams 147, 150, 161 Near-axis approximation 137 Nodal points 9, 14, 26, 129, 137, 164, 192, 305 Node numbering 131 Normalization condition 53, 66, 106, 244, 324, 330 Numerical minimization 66, 133, 142, 192 O Objective lens 167 Optical transfer function 324 Optimal quick-action problem 50–51 Ordinary and double layer potentials 326 Orthogonality condition 54 P Paraxial trajectories 247, 271 Periodic structure 33–34, 171 Perturbation axis misalignment 81, 89, 92 ellipticity 81, 95 locally strong 100–103 off-axis shift 87–89 radius variation 85 tilt 89, 92, 95 Perturbed surface charge density 82 Phase transfer function 225, 228, 324 Phase velocity 317 Photoelectron gun 91–92 Pierce electron gun 186 Point transfer function 324 Pontryagin maximum principle 51 Principal trajectory 159–160, 318 Probability density function 181, 213–215, 330 Pulse duration 213, 282, 297 transition function 320 Q Quadrupole ion trap 107–108 Quasi-stationary approximation 223 R Resolution pure temporal 234 spatial 166, 229

345

temporal 149–150 Response function 322 S Scattering effect 172–174, 204, 241, 256 event 173, 206 function 33, 37 Self-consistent equations 106 Shape functions definition 8–9, 121 singular 10 second-order 10 Singularity index 11 Space charge effects 292–294 Specific capacitance 36 Spread function definition 320 line 260, 268 local 225 point 326 spatial 225 spatiotemporal 224, 230 Standard triangle integral 306–307 Stationary current 213, 220 Stationary phase method 258, 332 Streak speed vector 232 Stress-strain transformation 248 Surface approximation 6–7 charge density 33, 314 redefined 74 perturbation harmonics 97 Symmetrical and antisymmetrical basis solutions 37 T Tangential elements method 312–313 Tau-variations definition 160 tensor form 155 Taylor expansion, series 28, 128, 153, 158, 172, 273 Temporal lens 288–291 Test pattern 229, 325, 329 Theorem Bertein–Bruns 100 Fichera 68 Gauss 140 Liouville 291

346

Index

Theorem (cont.) Poincare 153, 318 Scherzer 147 uniqueness 23, 106, 188, 323 Voronin-Tsetsoho 6, 19, 40 Three-channel electrostatic lens 21 Time-of-flight mass-spectrometer 207–209 Tolerance analysis 166 Total charge 24 Tracking technique 169 Transit time 218, 271 TRED experiments 281–282

U Ultrafast events 149 Unit potential distribution 77 Unit sphere 59, 65

V Vector potential 111 Velocity field circulation 189 Virtual current density 135 Virtual sweeping 232